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+arXiv:2301.13239v1  [math.QA]  30 Jan 2023
+Periodic Y-systems and Nahm sums: the rank 2 case
+Yuma Mizuno
+Abstract
+We classify periodic Y-systems of rank 2 satisfying the symplectic property. We find
+that there are six such Y-systems. In all cases, the periodicity follows from the existence
+of two reddening sequences associated with the time evolution of the Y-systems in positive
+and negative directions, which gives rise to quantum dilogarithm identities associated with
+Donaldson-Thomas invariants. We also consider q-series called the Nahm sums associated
+with these Y-systems. We see that they are included in Zagier’s list of rank 2 Nahm sums
+that are likely to be modular functions. It was recently shown by Wang that they are indeed
+modular functions.
+1
+Introduction
+1.1
+Background
+The Y-system is a system of algebraic relations satisfied by coefficients of a cluster algebra,
+which has the following form:
+Yi(u)Yi(u − ri) =
+�
+j∈I
+ri−1
+�
+p=1
+Yj(u − p)[nij;p]+�
+1 + Yj(u − p)
+�−nij;p
+(1.1)
+where I is a finite index set, Yi(u) for i ∈ I, u ∈ Z are commuting variables, ri ∈ Z≥1, and
+nij;p ∈ Z. We also use the notation [n]+ := max(0, n).
+Such equations are first discovered
+by Zamolodchikov in the study of thermodynamic Bethe ansats [36], prior to the discovery of
+cluster algebras by Fomin and Zelevinsky [7]. The most striking feature of Zamolodchikov’s Y-
+systems, as well as their generalizations [22, 30] defined shortly after the Zamolodchikov’s work,
+is that they are periodic, which was fully proved by applying the theory of cluster algebras
+[9, 10, 15, 16, 21].
+A systematic treatment of the Y-systems in the general setting of cluster algebras, including
+the Y-systems arising from the thermodynamic Bethe ansatz as spacial cases, was given by
+Nakanishi [27]. This approach was further developed in [24], and it was shown that the algebraic
+relation (1.1) arises from a cluster algebra if and only if the data ri, nij;p have a certain symplectic
+property. This allows the “axiomatic” study of Y-systems without explicitly referring to cluster
+algebras. In this general setting, however, the Y-system is typically not periodic, and so the
+study of periodic Y-systems as a generalization of Zamolodchikov’s Y-systems would be further
+developed. In particular, the classification problem for periodic Y-systems is a challenging open
+problem (see the last comments in [27, Section 3]).
+There are several classification results in the literature. Fomin and Zelevinsky [10] showed
+that the classification when ri = 2, nij;p ≤ 0, and nii;p = 0 for any i, j, p coincides with the
+Cartan-Killing classification. Galashin and Pylyavskyy [12] generalized this result to show that
+the classification when ri = 2 and nii;p = 0 for any i, p coincides with the classification of ADE
+bigraphs of Stembridge [31]. On the other hand, the situation is more complicated when ri > 2
+for some i, and so far there has been no comprehensive classification results except when |I| = 1
+1
+
+where it is not difficult to give a complete classification thanks to the work by Fordy and Marsh
+[11] (e.g. see [24, Example 5.6]).
+In this paper, we make a first attempt to give a classification result involving the case ri > 2
+for some i. Precisely, we classify the periodic Y-systems of the form (1.1) with |I| = 2 satisfying
+the symplectic property. We would like to emphasize that we consider general ri, nij;p in the
+classification. The result is given in the next section.
+We also discuss the relation to Nahm’s conjecture on q-series [26, 35] in Section 1.3.
+1.2
+Main result
+Let I be a finite set.
+We denote by Y0 the set of pairs (r, n) where r = (ri)i∈I and n =
+(nij;p)i,j∈I,p∈N are families of integers satisfying ri ≥ 1 for any i and
+n±
+ij;p = 0 unless 0 < p < ri
+(1.2)
+for any i, j, p.
+Definition 1.1. Let (r, n) ∈ Y0. Let P be a semifield, and (Yi(u))i∈I,u∈Z be a family of elements
+in P. We say that (Yi(u)) satisfies the Y-system associated with the pair (r, n) if the relation
+(1.1) holds for any i, u. The equation (1.1) itself is called the Y-system associated with (r, n).
+We also say that (Yi(u)) is a solution of the Y-system if it satisfies the Y-system.
+It is useful to think a pair (r, n) ∈ Y0 as a triple of matrices with polynomial entries by the
+map (r, n) �→ (N0(z), N+(z), N−(z)) : Y0 → (MatI×I N[z])3 defined by
+N0(z) := diag(1 + zri)i∈I,
+N±(z) :=
+� �
+p∈N
+n±
+ij;pzp
+�
+i,j∈I
+(1.3)
+where we set n±
+ij;p := [±nij;p]+. We also define the map (r, n) �→ A±(z) : Y0 → (MatI×I Z[z])2
+by A±(z) := N0(z) − N±(z). Since this map is injective by the condition (1.2), we will identify
+Y0 with the image of this map. For example, we will use the term “the Y-system associated
+with A±(z) ∈ Y0”.
+Definition 1.2. We say that A±(z) ∈ Y0 satisfies the symplectic property if
+A+(z)A−(z−1)T = A−(z)A+(z−1)T,
+(1.4)
+where T is the transpose of a matrix. We denote by Y the subset of Y0 consisting of pairs
+satisfying the symplectic property.
+The pair A±(z) ∈ Y0 satisfies the simplectic property if and only if the Y-system associated
+with A±(z) ∈ Y0 is realized as the exchange relations of coefficients in a cluster algebra [24].
+We review this fact in Section 2.1.
+Definition 1.3. We say that a solution of a Y-system is periodic if there is a positive integer
+Ω > 0 such that Yi(u + Ω) = Yi(u) for any i, u.
+Definition 1.4. We say that a pair A±(z) ∈ Y is of finite type if any solution (in any semifield)
+of the Y-system associated with this pair is periodic. In this case, we also say that Y-system
+itself is periodic.
+The purpose of this paper is to classify periodic Y-systems of rank 2. Before stating the
+result, we give a few remarks. We say that A±(z) ∈ YI is decomposable if it is a direct sum
+of some A′
+±(z) ∈ YI′ and A′′
+±(z) ∈ YI′′ with nonempty I′ and I′′. We say that A±(z) ∈ YI
+is indecomposable if it is not decomposable. It is enough to consider indecomposable pairs in
+the classification. We also note that A±(z) is of finite type if and only if A±(z)op := A∓(z) is
+of finite type by the correspondence between solutions Yi(u) �→ Yi(u)−1. The main results are
+summarized as follows:
+2
+
+A+(z)
+A−(z)
+h+
+h−
+�
+1 + z2
+−z
+−z
+1 + z2
+�
+�
+1 + z2
+0
+0
+1 + z2
+�
+3
+2
+(1)
+�
+1 + z2
+−z
+−z − z5
+1 + z6
+�
+�
+1 + z2
+0
+−z3
+1 + z6
+�
+8
+6
+(2)
+�
+1 + z2
+−z
+−z − z5 − z9
+1 + z10
+�
+�
+1 + z2
+0
+−z3 − z7
+1 + z10
+�
+18
+10
+(3)
+�
+1 + z2
+−z
+−z
+1 + z2
+�
+�
+1 + z2 − z
+0
+0
+1 + z2 − z
+�
+3
+3
+(4)
+�
+1 + z2
+−z
+−z − z2
+1 + z3
+�
+�
+1 + z2 − z
+0
+0
+1 + z3
+�
+5
+3
+(5)
+�
+1 + z2
+−z
+−z
+1 + z2 − z
+�
+�
+1 + z2
+0
+0
+1 + z2
+�
+5
+2
+(6)
+Table 1: Finite type classification for Y-systems of rank 2. The numbers h± are the length of
+reddening sequences in positive and negative directions, respectively.
+Theorem 1.5. Suppose that I = {1, 2}.
+(1) Any pair A±(z) ∈ Y in Table 1 is of finite type.
+(2) Any indecomposable pair A±(z) ∈ Y of finite type is reduced to exactly one pair in Table 1
+by permuting the indices, changing sign, and changing slices (see Section 3.1), if necessary.
+The claim (1) can be proved by concrete calculation in a suitable universal algebra since
+A±(z) in Table 1 is concrete. We, however, give another proof involving cluster algebras. We
+give a quiver and a sequence of mutations for each A±(z) in Table 1 that yields the Y-system as
+the exchange relation of coefficients in the cluster algebra. See Table 3 for quivers and mutations.
+We can verify that some iteration of this sequence of mutations, as well as its inverse, is a
+reddening sequence (Theorem 2.8). Thanks to the deep results in the theory of cluster algebras,
+this property is enough to imply the periodicity (Proposition 2.7). The number h± in Table
+1 are the length of reddening sequences in positive and negative directions, respectively. This
+verification of the periodicity is interesting not only because it is computationally more efficient,
+but also because it leads to nontrivial dilogarithm identities associated with Donaldson-Thomas
+invariants (Corollary 2.10).
+The claim (2) is proved in Section 3.2 by the following steps:
+Step 1. We recall the result in [24] that asserts that A±(1) satisfies a certain positivity, which
+in particular implies that tr A±(1) and det A±(1) are positive. This allows us to significantly
+reduce the candidates for finite type A±(z).
+Step 2. For a fixed A+(1) in the candidates obtained in Step 1, we search for A−(1) satisfying
+the symplectic property (1.4) at z = 1.
+Step 3. During the search in Step 2, we discard the pair A±(1) that cannot be endowed with
+the parameter z (Lemma 3.3 and 3.4).
+Step 4. At this point, we have six candidates up to a permutation of the indices and a change of
+sign. For each A±(1) in the six candidates, we try to endow with the parameter z. It turns out
+that this is possible for all the six candidates. We give all possible A±(z) in Lemma 3.5–3.8.
+3
+
+Step 5. We finally check that each remaining candidate reduces to one of A±(z) in Table 1 by
+change of slices.
+Remark 1.6. Most of the Y-systems obtained from Table 1 are already known in the litera-
+ture. (1)op and (6)op are Zamolodchikov’s Y-system of type A2 [36] and T2 (“tadpole”) [30],
+respectively. (2)op is the reduced sine-Gordon Y-system associated with the continued fraction
+3/4 = [1, 3] = 1/(1+ 1/3), and (5)op with z replaced by z2 is the reduced sine-Gordon Y-system
+associated with 3/5 = [1, 1, 2] = 1/(1 + 1/(1 + 1/2)) [32] . (4) is the “half” of the Y-system
+associated with the pair (A2, A2) [30]. (3) appears to be new:
+Y1(u)Y1(u − 2) =
+1
+1 + Y2(u − 1)−1
+Y2(u)Y2(u − 10) =
+�
+1 + Y1(u − 3)
+��
+1 + Y1(u − 7)
+�
+�
+1 + Y1(u − 1)−1��
+1 + Y1(u − 5)−1��
+1 + Y1(u − 9)−1�
+although it is implicitly given in the author’s previous work [24, Table 2].
+Remark 1.7. The pair A±(z) ∈ Y is called the T-datum in [24] since it describes the T-systems,
+which is a companion to the Y-systems. We do not use this term since we only consider the
+Y-systems in this paper. Moreover, the definition of the T-datum in [24] allows to have a non-
+diagonal N0 and have a nontrivial symmetrizer D, which is more general than the definition in
+this paper. See also Section 1.4 for the Y-systems involving nontrivial symmetrizers.
+Remark 1.8. There is another expression of the Y-system using a pair of matrices A±(z)
+directly. Let A±(z) ∈ Y0, and define aij;p ∈ Z by
+A±(z) =
+��
+p∈N
+a±
+ij;pzp
+�
+i,j∈I
+.
+Let (P ±
+i (u))i∈I,u∈Z be a family of elements in a multiplicative abelian group P. We say that
+(P ±
+i (u)) satisfies the multiplicative Y-system associated with A±(z) if
+�
+j∈I
+�
+p∈N
+P +
+j (u − p)a+
+ij;p =
+�
+j∈I
+�
+p∈N
+P −
+j (u − p)a−
+ij;p
+(1.5)
+for any i, u (schematically, “A+(z) · log P + = A−(z) · log P −” under the action z : u �→ u − 1).
+The solution (P ±
+i (u)) is called normalized if P is endowed with a semifield structure, and
+P +
+i (u) + P −
+i (u) = 1
+for any i, u. We have a one-to-one correspondence between solutions of the Y-system (1.1) and
+normalized solutions of the multiplicative Y-system (1.5). The correspondence is given by
+Yi(u) �→ P +
+i (u)
+P −
+i (u),
+P +
+i (u) �→
+Yi(u)
+1 + Yi(u),
+P −
+i (u) �→
+1
+1 + Yi(u).
+In the setting of cluster algebras, this correspondence is nothing but the normalization of the
+coefficients described by Fomin and Zelevinsky [7, Section 5].
+1.3
+Relation to Nahm sums
+Consider the q-series defined by
+G(q) =
+∞
+�
+n=0
+qn2
+(q)n
+,
+H(q) =
+∞
+�
+n=0
+qn2+n
+(q)n
+,
+(1.6)
+4
+
+where (q)n := (1 − q)(1 − q2) · · · (1 − qn) is the q-Pochhammer symbol. The famous Rogers-
+Ramanujan identities express these q-series as the following infinite products:
+G(q) =
+�
+n≡±1 mod 5
+1
+1 − qn,
+H(q) =
+�
+n≡±2 mod 5
+1
+1 − qn.
+These expressions in particular implies that q−1/60G(q) and q11/60H(q) are modular functions
+on some finite index subgroup of SL(2, Z). In fact, it is a rare case that an infinite sum of the
+form (1.6) is modular. It is known that the q-series
+∞
+�
+n=0
+q
+1
+2 an2+bn+c
+(q)n
+(1.7)
+with a, b, c ∈ Q is modular only if a = 1/2, 1, or 2 [35].
+Nahm [26] considered higher rank generalization of (1.7), which we call the Nahm sum. Let
+I be a finite set, and suppose that A ∈ QI×I is a symmetric positive definite matrix, B ∈ QI is
+a vector, and C ∈ Q is a scalar. The Nahm sum is the q-series defined by
+fA,B,C(q) :=
+�
+n∈NI
+q
+1
+2 nTAn+nTB+C
+�
+a(q)ni
+.
+When |I| ≥ 2, it is not well understood when fA,B,C(q) is modular. Nahm gave a conjecture
+providing a criterion on the modularity of fA,B,C(q) in terms of torsion elements in the Bloch
+group [26, 35]. See [3, 33] for the development of this conjecture.
+Nahm used Zamolodchikov’s periodicity to provide an evidence of the conjecture. In fact,
+there is a natural way to give a candidate of modular Nahm sums from finite type A±(z) ∈ Y
+in general. Precisely, the matrix K := A+(1)−1A−(1) is always symmetric and positive definite
+for finite type A±(z) ∈ Y, and it is conjectured that it gives a modular Nahm sum fK,0,C(q) for
+some C [24]. (This construction is essentially the same as that in [18], except that they did not
+prove that K is symmetric and positive definite. A special case can also be found in [23].) We
+note that the symplectic property (1.4) at z = 1 plays an important role here since it implies
+that K is symmetric. On the other hand, the positive definiteness is related to the periodicity
+of the Y-system.
+Based on our classification, we find that:
+Theorem 1.9. Suppose that I = {1, 2}. The Nahm sum fK,0,C(q) is modular for any finite type
+A±(z) ∈ Y, where C is given in Table 2.
+In fact, every K from finite type A±(z) is included in the Zagier’s list [35, Table 2] for rank
+2 candidates of modular Nahm sums. There are Rogers-Ramanujan type identities that enable
+us to write each Nahm sum in the list in terms of theta functions. The proof of the desired
+identities was partially given in [1, 4, 6, 33, 35], and was recently completed by Wang [34] except
+for one candidate that does not appears in our construction from Y-systems. See Table 2.
+Remark 1.10. We can define the refinement f (s)
+A±(z)(q) of the Nahm sum fK,0,0(q), which is
+parametrized by s ∈ H for an abelian group H of order det A+(1) such that it reduces to the
+original one by taking summation [24, Definition 5.12]:
+fK,0,0(q) =
+�
+s∈H
+f (s)
+A±(z)(q).
+It is conjectured that each f (s)
+A±(z)(q) is already modular after multiplying qC for some C. We
+note that the symplectic property (1.4) at z = 1 again plays an important role in the definition
+of the refinement. We will discuss this refinement for rank 2 case in more detail elsewhere. We
+remark that similar refinement also appears in the context of 3-dimensional quantum topology
+[13, Section 6.3].
+5
+
+A±(z)
+K
+−24C
+RR
+A±(z)
+K
+−24C
+RR
+(1)
+�
+4/3
+2/3
+2/3
+4/3
+�
+4
+5
+[6]
+(1)op
+�
+1
+−1/2
+−1/2
+1
+�
+6
+5
+[33]
+(2)
+�3/2
+1
+1
+2
+�
+5
+7
+[34]
+(2)op
+�
+1
+−1/2
+−1/2
+3/4
+�
+9
+7
+[34]
+(3), (6)
+�2
+2
+2
+4
+�
+4
+7
+[1]
+(3)op, (6)op
+�
+1
+−1/2
+−1/2
+1/2
+�
+10
+7
+[34]
+(4)
+�
+2/3
+1/3
+1/3
+2/3
+�
+1
+[35]
+(4)op
+� 2
+−1
+−1
+2
+�
+1
+[35]
+(5)
+�1
+1
+1
+2
+�
+3
+4
+[34]
+(5)op
+� 2
+−1
+−1
+1
+�
+5
+4
+[4]
+Table 2: The list of the matrix K = A+(1)−1A−(1). The Nahm sum fK,0,C(q) is modular, which
+can be proved by using Rogers-Ramanujan type identities (RR for short) given in the references
+in the table.
+1.4
+Remarks on higher rank and skew-symmetrizable case
+We have seen that the following properties hold for rank 2 case:
+(P1) We have reddening sequences in both positive and negative directions.
+(P2) The map A±(z) �→ A+(1)−1A−(1) gives modular Nahm sums.
+We expect that the properties (P1) and (P2) also hold for any finite type A±(z) ∈ Y of general
+rank. The followings are some known examples:
+• For the Y-system associated with the untwisted quantum affine algebras Uq(X(1)
+r ) with
+level ℓ restriction [22], (P1) holds with h+ = tℓ and h− = t · (dual Coxeter number of Xr)
+where t = 1, 2, or 3 is the multiplicity in the Dynkin diagram of Xr [15, 16], and (P2)
+holds under the assumption [14, Conjecture 5.3] by the result of Kac and Peterson [17].
+• For the Y-system associated with a pair of finite type simply laced Dynkin type (Xr, X′
+r′)
+[30], (P1) holds with h+ = (Coxeter number of Xr) and h− = (Coxeter number of X′
+r′)
+[19, 21].
+• For the (reduced) sine-Gordon Y-system associated with the continued fraction p/q =
+[nF, . . . , n1] = 1/(nF + 1/(· · · + 1/n1)) [29, 32], (P1) appears to hold with h+ = 2p and
+h− = 2q.
+• For the Y-system associated with an admissible ADE bigraph (Γ, ∆) [12], (P1) appears to
+hold with h+ = (Coxeter number of Γ) and h− = (Coxeter number of ∆).
+Moreover, we can consider Y-systems associated with skew-symmetrizable cluster algebras
+rather than skew-symmetric ones discussed in this paper. In this case, the symplectic property
+(1.4) becomes
+A+(z)DA−(z−1)T = A−(z)DA+(z−1)T,
+where D is a diagonal matrix called symmetrizer [24]. We also expect that the properties (P1)
+and (P2) also hold for skew-symmetrizable case. See [24, Definition 5.12] for the definition of
+the Nahm sum in skew-symmetrizable case.
+Acknowledgment. This work is supported by JSPS KAKENHI Grant Number JP21J00050.
+6
+
+2
+Y-systems and cluster algebras
+2.1
+Preliminaries on cluster algebras
+In this paper, a semifield is a multiplicative abelian group equipped with an addition that is
+commutative, associative, and distributive with respect to the multiplication.
+Definition 2.1. Let I be a set.
+The set of all nonzero rational functions in the variables
+y = (yi)i∈I with natural number coefficients is a semifield with respect to the usual addition
+and multiplication. This semifield is called the universal semifield, and denoted by Q>0(y). We
+have a canonical bijection Homsemifield(Q>0(y), P) ∼= Homset(I, P) for any set I and semifield P.
+Definition 2.2. Let I be a set. The tropical semifield Trop(y) is the multiplicative free abelian
+group generated by the variables y = (yi)i∈I equipped with the addition defined by
+�
+i
+yai
+i +
+�
+i
+ybi
+i =
+�
+i
+ymin(ai,bi)
+i
+.
+Let I be a finite set and P be a semifield. A Y-seed is a pair (B, y) where B = (Bij)i,j∈I
+is a skew-symmetric integer matrix and y = (yi)i∈I is a tuple of elements in P. We sometimes
+represent B as the quiver whose signed adjacency matrix is B. For a Y-seed (B, y) and k ∈ I,
+the mutation in direction k transforms (B, y) into the new Y-seed µk(B, y) = (B′, y′) given by
+B′
+ij :=
+�
+−Bij
+if i = k or j = k,
+Bij + [−Bik]+Bkj + Bik[Bkj]+
+otherwise,
+(2.1)
+y′
+i :=
+�
+yk
+if i = k,
+yiy[Bki]+
+k
+(1 + yk)−Bki
+otherwise.
+(2.2)
+A mutation is involutive, that is, µk(B, y) = (B′, y′) implies (B, y) = µk(B′, y′). We have the
+commutativity
+µiµj = µjµi
+if Bij = 0,
+(2.3)
+which allows us to write µi for a set i ⊆ I such that Bij = 0 for any i, j ∈ i to mean the
+successive mutations along arbitrarily chosen order on i.
+For a Y-seed (B, y) and a bijection ν : I → I, we define a new Y-seed ν(B, y) = (B′, y′) by
+B′
+ν(i)ν(j) := Bij and y′
+ν(i) := yi.
+2.2
+Solving Y-systems by cluster algebras
+Let A±(z) ∈ Y. We will construct a solution of the Y-system associated with A±(z) based on
+[24, Section 3.3]. We first define a subset R ⊆ I × Z by
+R := {(i, u) ∈ I × Z | 0 ≤ u < ri},
+(2.4)
+and define a skew-symmetric R × R integer matrix B by
+B(i,p)(j,q) = −nij;p−q + nji;q−p +
+�
+k∈I
+min(p,q)
+�
+v=0
+�
+n+
+ik;p−vn−
+jk;q−v − n−
+ik;p−vn+
+jk;q−v
+�
+,
+(2.5)
+where we understand nij;p = 0 if p < 0. We then define i := {(i, u) | u = 0} ⊆ R. We also define
+a bijection ν : R → R by
+ν(i, p) =
+�
+(i, p − 1)
+if p > 0,
+(i, ri)
+if p = 0.
+(2.6)
+7
+
+Then the symplectic property (1.4) ensures that ν(µi(B)) = B [24, Lemma 3.16]. We finally
+define a sequence of Y-seeds
+· · · → (B, y(−1)) → (B, y(0)) → (B, y(1)) → · · ·
+(2.7)
+in Q>0(y) by y(0) := y and (B, y(u + 1)) = ν(µi(B, y(u))). The sequence (2.7) gives a solution
+of the Y-system:
+Lemma 2.3. [24, Theorem 3.13] (yi,0(u))i∈I,u∈Z satisfies the Y-system associated with A±(z).
+This solution is universal in the following sense.
+Lemma 2.4. [24, Theorem 3.19] Suppose that a family (Yi(u))i∈I,u∈Z satisfies the Y-system
+associated with A±(z). Define a semifield homomorphism f : Q>0(y) → P by
+f(yi,p) := Yi(p)
+�
+j∈I
+p
+�
+q=0
+Yj(p − q)−[nij;q]+�
+1 + Yj(p − q)
+�nij;q.
+(2.8)
+Then f(yi,0(u)) = Yi(u) for any i, u.
+Corollary 2.5. A±(z) ∈ Y is of finite type if and only if there are different integers u, v such
+that y(u) = y(v) in (2.7).
+2.3
+Periodicity and reddening sequences
+Similarly to (2.7), we define a sequence of Y-seeds
+· · · → (B, y(−1)) → (B, y(0)) → (B, y(1)) → · · ·
+(2.9)
+by the same formulas but now in Trop(y) rather than Q(y).
+Definition 2.6. We say that the Y-system associated with A±(z) ∈ Y is positive (resp. negative)
+reddening if there is a positive integer u such that all the exponents in yi(u) (resp. yi(−u)) in
+(2.9) are nonpositive for any i. We denote by h+ (resp. h−) the least such positive integer u.
+Equivalently, the Y-system is positive (resp. negative) reddening if and only if all the en-
+tries in the C-matrix associated with the sequence of mutations (B, y(0)) → (B, y(u)) (resp.
+(B, y(0)) → (B, y(−u))) are nonpositive for some u > 0.
+Proposition 2.7. Suppose that the Y-system associated with A±(z) is positive and negative
+reddening. Then A±(z) is of finite type.
+Proof. We verify the equivalent condition in Corollary 2.5. By [2, Proposition 2.10], there are
+bijections σ, σ′ : R → R such that yi(h+) = y−1
+σ(i) and yi(−h−) = y−1
+σ′(i) for any i (in other
+words, the C-matrices associated with them are the minus of permutation matrices). Now the
+claim follows from the separation formula for y-variables [10, Proposition 3.13] and the result
+on C-matrices shown by Cao, Huang, and Li [5, Theorem 2.5]. See also [28, Theorem 5.2] for
+the corresponding statement dealing with permutations that is actually suitable here.
+Theorem 2.8. The Y-system associated with each A±(z) in Table 1 is positive and negative
+reddening.
+Proof. The quiver B associated with A±(z) is given in Table 3. We can verify the assertion by
+concrete calculation on the quiver. The numbers h± are given in Table 1.
+8
+
+Quiver B
+A±(z)
+(1, 0)
+(2, 1)
+(1, 1)
+(2, 0)
+(1)
+(1, 0)
+(2, 1)
+(2, 3)
+(2, 5)
+(1, 1)
+(2, 0)
+(2, 2)
+(2, 4)
+(2)
+(1, 0)
+(2, 1)
+(2, 3)
+(2, 5)
+(2, 7)
+(2, 9)
+(1, 1)
+(2, 0)
+(2, 2)
+(2, 4)
+(2, 6)
+(2, 8)
+(3)
+(1, 0)
+(2, 1)
+(2, 0)
+(1, 1)
+(4)
+(2, 0)
+(1, 1)
+(1, 0)
+(2, 2)
+(2, 1)
+(5)
+(1, 0)
+(2, 1)
+(2, 0)
+(1, 1)
+(6)
+Table 3: Quivers associated with A±(z) in Table 1. Each quiver is preserved by the mutation at
+(∗, 0) followed by the permutation (i, p) �→ (i, p − 1) (the second argument is considered modulo
+ri), which yields Y-system. For (1)–(3), this operation interchanges the connected components
+(see Section 3.1).
+Theorem 1.5 (1) now follows from Proposition 2.7 and Theorem 2.8.
+Remark 2.9. A connected component of each quiver in Table 3 has the following cluster type:
+(1) A2
+(2) A4
+(3) E6
+(4) D4
+(5) A5
+(6) A4
+These are of finite type in the sense of [8], which also implies Theorem 1.5 (1). We remark,
+however, that this observation is somewhat misleading since the quiver associated with a periodic
+Y-system of general rank is typically of infinite type.
+It might be better to think that the
+appearance of only finite type quivers happens “by chance” due to the smallness of 2, the rank
+of Y-systems considered in this paper.
+Theorem 2.8 also gives quantum dilogarithm identifies associated with Donaldson-Thomas
+invariants. For any reddening sequence i starting from a quiver B, we can define a quantity
+E(i) by using the quantum dilogarithm. We refer to [20, Remark 6.6] as the definition. This
+quantity coincides with Kontsevich-Soibelman’s refined Donaldson-Thomas invariant associated
+with B [20, 25]. In particular, E(i) does not depend on i, which gives the quantum dilogarithm
+identifies. In our case, we have:
+9
+
+Corollary 2.10. For each A±(z) in Table 1, we have
+E(µh+) = E(µ−h−),
+where µ := ν ◦ µi is the sequence of mutations (together with the permutation) (B, y(0)) →
+(B, y(1)) in (2.7).
+For example, the pair (1) in Table 1 yields the famous pentagon identity of the quantum
+dilogarithm.
+3
+Classification
+3.1
+Change of slices
+We need to introduce an appropriate equivalence relation on the set Y, which identifies essentially
+the same Y-systems. Before we get into the definition, we will see a typical example. Consider
+the following Y-system:
+Y1(u)Y1(u − 2) = (1 + Y2(u − 1)−1)−1
+Y2(u)Y2(u − 2) = (1 + Y1(u − 1)−1)−1
+(3.1)
+which corresponds to A±(z) ∈ Y given by (1) in Table 1. This system of equations are defined
+on the set [1, 2] × Z, but actually can be defined on each component of the following disjoint
+union:
+[1, 2] × Z =
+1�
+k=0
+{(i, u) | i − u ≡ k mod 2}.
+We informally call the algebraic relation defined on each subset the slice of the whole Y-system.
+If (Yi(u)) is a solution of the Y-system for i − u ≡ 0 mod 2, then (Yi(u + 1)) is a solution of the
+Y-system for i − u ≡ 1 mod 2. Thus it is enough to consider only one slice when considering
+solutions. Now we consider another Y-system:
+Y ′
+1(u)Y ′
+1(u − 3) = (1 + Y ′
+2(u − 2)−1)−1
+Y ′
+2(u)Y ′
+2(u − 3) = (1 + Y ′
+1(u − 1)−1)−1.
+(3.2)
+which corresponds to A′
+±(z) ∈ Y given by
+A′
++(z) :=
+�
+1 + z3
+−z2
+−z
+1 + z3
+�
+,
+A′
+−(z) :=
+�
+1 + z3
+0
+0
+1 + z3
+�
+.
+The Y-system (3.2) is decomposed into three slices:
+[1, 2] × Z =
+2�
+k=0
+{(i, u) | i − u ≡ k mod 3}.
+We see that for any solution of (3.1) for i − u ≡ 0 mod 2,
+Y ′
+1(u) := Y1
+�2
+3u − 1
+3
+�
+,
+Y ′
+2(u) := Y2
+�2
+3u
+�
+is a solution of (3.2) for i − u ≡ 2 mod 3. We also obtain solutions for the other two slices by
+shifting u. Conversely, any solution of (3.1) is obtained from a solution of (3.2). Therefore, it
+10
+
+is enough to consider one of the Y-systems (3.1) and (3.2). In particular, A±(z) is of finite type
+if and only if A′
+±(z) is.
+Now we work in the general setting. The idea is that each slice corresponds to each connected
+component of the quiver associated with the matrix B defined by (2.5). Let A±(z) ∈ Y, and
+assume that it is indecomposable. By [24, Proposition 3.24], we have a decomposition of the
+matrix B and its index set R:
+B =
+t−1
+�
+u=0
+B(u),
+R =
+t−1
+�
+u=0
+R(u)
+such that each B(u) is indecomposable and we have a cyclic sequence of mutations
+B(0)
+ν|R(0)◦µi(0)
+−−−−−−−→ B(1) −→ · · · −→ B(t − 1)
+ν|R(t−1)◦µi(t−1)
+−−−−−−−−−−→ B(0)
+(3.3)
+where i(u) := i ∩ R(u). We say that two pairs A±(z) and A′
+±(z) are related by change of slices
+if they yield the same cyclic sequence (3.3) up to a change of indices and the commutativity of
+mutations (2.3). (This commutativity is already implicitly used to justify the notation µi(u) as
+stated below (2.3).)
+Example 3.1. The pairs A±(z) and A′
+±(z) associated with (3.1) and (3.2), respectively, are
+related by change of slices. Indeed, we see that the sequence (3.3) for (3.1) is
+(1, 0)
+(2, 1)
+ν◦µ(0,0)
+−−−−−→ (1, 1)
+(2, 0)
+ν◦µ(1,0)
+−−−−−→ (1, 0)
+(2, 1) ,
+whereas the sequence (3.3) for (3.2) is
+(1, 0)
+(2, 1)
+ν′◦µ(0,0)
+−−−−−→ (1, 2)
+(2, 0)
+ν′◦µ(1,0)
+−−−−−→ (1, 1)
+(2, 2)
+ν′
+−→ (1, 0)
+(2, 1) .
+These are the same sequence up to a change of indices.
+3.2
+Proof of the classification
+In this section, we will prove Theorem 1.5 (2). We first recall the following result.
+Lemma 3.2 ([24, Theorem 5.5]). Let A±(z) ∈ Y. Assume that A±(z) is of finite type. Then
+there is a vector v ∈ RI such that v > 0, vA+(1) > 0, and vA−(1) > 0.
+In particular,
+tr A±(1) > 0 and det A±(1) > 0.
+By Lemma 3.2, A+(1) and A−(1) are equal to one of the following matrices:
+� 2
+−1
+−1
+2
+�
+,
+� 2
+−1
+−2
+2
+�
+,
+� 2
+−1
+−3
+2
+�
+,
+� 2
+−1
+−1
+1
+�
+,
+� 2
+0
+−n
+2
+�
+,
+� 2
+0
+−n
+1
+�
+,
+� 1
+0
+−n
+1
+�
+up to a permutation of the indices. We give several lemmas about impossible pairs. Before
+giving lemmas, we note that
+n+
+ij;p = 0
+or
+n−
+ij;p = 0
+(3.4)
+for any i, j, p.
+Lemma 3.3. It is impossible that A±(z) ∈ Y has the following forms:
+(1) A+(1) =
+�2
+−a
+∗
+∗
+�
+, A−(1) =
+�2
+−b
+∗
+∗
+�
+for odd a, b.
+11
+
+(2) A+(1) =
+�2
+−a
+∗
+∗
+�
+, A−(1) =
+�1
+−b
+∗
+∗
+�
+for odd a, b.
+(3) A+(1) =
+�1
+−1
+∗
+∗
+�
+, A−(1) =
+�1
+−1
+∗
+∗
+�
+.
+(4) A+(1) =
+�1
+0
+∗
+∗
+�
+, A−(1) =
+�1
+∗
+∗
+∗
+�
+.
+Proof. For (2), we can set
+A+(z) =
+�1 + zr
+−f(z)
+∗
+∗
+�
+,
+A−(z) =
+�1 + zr − za
+−g(z)
+∗
+∗
+�
+.
+By the symplectic property (1.4), we have
+za + za−r + f(z)g(z−1) = z−a + zr−a + g(z)f(z−1).
+(3.5)
+Since 0 < a and a − r < 0 by (1.2), the sum of the coefficients of the terms in f(z)g(z−1)
+with positive exponents is equal to that with negative exponents. Since f(1)g(1) (=ab) is odd,
+f(z)g(z−1) should contain the constant term z0, which contradicts (3.4). The proof for (1) is
+similar.
+For (3), we can set
+A+(z) =
+���
+1 + zr − za
+−zb
+∗
+∗
+�
+,
+A−(z) =
+�
+1 + zr − zc
+−zd
+∗
+∗
+�
+with 0 < a, b, c, d < r. Without loss of generality, we can assume a < c. By (1.4), we have
+z−c + zr−c + za + za−r + zc−a + zd−b = zc + zc−r + z−a + zr−a + za−c + zb−d
+Since c − a > 0, we see that c − a is equal to c, r − a, or b − d. However, the first two cases are
+impossible by (1.2). Thus c − a = b − d, which implies that
+z−c + zr−c + za + za−r = zc + zc−r + z−a + zr−a.
+Since a > 0, we see that a is equal to c or r − a. However, a = c is impossible by (3.4). Thus
+a = r − a, which implies that
+z−c + zr−c = zc + zc−r.
+Since c > 0, we see that c = r − c. However, this implies that a = r/2 = c, which is impossible
+by (3.4).
+For (4), we can set
+A+(z) =
+�1 + zr − za
+0
+∗
+∗
+�
+,
+A−(z) =
+�
+1 + zr − zb
+∗
+∗
+∗
+�
+.
+By (1.4), we have
+za + za−r + z−b + zr−b + zb−a = z−a + zr−a + zb + zb−r + za−b.
+Comparing the number of the terms with positive and negative exponents, we should have a = b.
+This is impossible by (3.4).
+12
+
+Lemma 3.4. It is impossible that indecomposable A±(z) ∈ Y has the form
+A+(1) =
+�∗
+0
+∗
+∗
+�
+,
+A−(1) =
+�∗
+0
+∗
+∗
+�
+.
+Proof. We can set
+A±(z) =
+�
+1 + zr1 − f±(z)
+0
+−g±(z)
+1 + zr2 − h±(z)
+�
+.
+Since g+(z) ̸= 0 or g−(z) ̸= 0, we can pick the least integer c among the exponents in g+(z) and
+g−(z). Without loss of generality, we can assume g+(1) contains the term zc. By (1.4), we have
+f+(z)g−(z−1) + (1 + zr1)g+(z−1) = f−(z)g+(z−1) + (1 + zr1)g−(z−1).
+(3.6)
+The left-hand side in (3.6) contains the term zr1−c, but any exponent in the right-hand side is
+strictly smaller that r1 − c by (1.2) and (3.4), which is a contradiction.
+We now search for possible pairs A±(1) case by case using the symplectic property (1.4) at
+z = 1 together with Lemma 3.3 and 3.4:
+• Case: A+(1) =
+� 2
+−1
+−1
+2
+�
+. The possibilities for A−(1) are:
+�2
+0
+0
+2
+�
+,
+�1
+0
+0
+1
+�
+.
+• Case: A+(1) =
+� 2
+−1
+−1
+2
+�
+. The possibilities for A−(1) are:
+�2
+0
+0
+2
+�
+,
+�1
+0
+0
+1
+�
+.
+• Case: A+(1) =
+� 2
+−1
+−2
+2
+�
+. The possibilities for A−(1) are:
+� 2
+0
+−1
+2
+�
+,
+�1
+0
+0
+2
+�
+.
+• Case: A+(1) =
+� 2
+−1
+−3
+2
+�
+. The possibilities for A−(1) are:
+� 2
+0
+−2
+2
+�
+.
+• Case: A+(1) =
+� 2
+−1
+−1
+1
+�
+. The possibilities for A−(1) are:
+�2
+0
+0
+2
+�
+,
+�1
+0
+0
+1
+�
+.
+13
+
+• Case: A+(1) =
+� 2
+0
+−n
+2
+�
+. The possibilities for A±(1) are:
+��2
+0
+0
+2
+�
+,
+� 2
+−1
+−1
+2
+��
+,
+�� 2
+0
+−1
+2
+�
+,
+� 2
+−1
+−2
+2
+��
+,
+��2
+0
+0
+2
+�
+,
+� 1
+−1
+−1
+2
+��
+.
+• Case: A+(1) =
+� 2
+0
+−n
+1
+�
+. The possibilities for A±(1) are:
+��2
+0
+0
+1
+�
+,
+� 2
+−2
+−1
+2
+��
+.
+• Case: A+(1) =
+� 1
+0
+−n
+1
+�
+. The possibilities for A±(1) are:
+��2
+0
+0
+1
+�
+,
+� 2
+−2
+−1
+2
+��
+.
+In summary, the remaining possible pairs, up to a permutation of the indices and an change
+of sign, are given in the following table:
+A+(1)
+A−(1)
+� 2
+−1
+−1
+2
+�
+�2
+0
+0
+2
+�
+� 2
+−1
+−2
+2
+�
+� 2
+0
+−1
+2
+�
+� 2
+−1
+−3
+2
+�
+� 2
+0
+−2
+2
+�
+A+(1)
+A−(1)
+� 2
+−1
+−1
+2
+�
+�2
+0
+0
+2
+�
+� 2
+−1
+−2
+2
+�
+� 2
+0
+−1
+2
+�
+� 2
+−1
+−3
+2
+�
+� 2
+0
+−2
+2
+�
+(3.7)
+We now start searching for possible A±(z).
+Lemma 3.5. Let n ≥ 1. Suppose that
+A+(1) =
+� 2
+−1
+−n
+2
+�
+,
+A−(1) =
+�
+2
+0
+−(n − 1)
+2
+�
+.
+Then
+A+(z) =
+�
+[2]r
+−z−a
+−zr−a[n]2r
+[2](2n−1)r
+�
+,
+A−(z) =
+�
+[2]r
+0
+−z2r−a[n − 1]2r
+[2](2n−1)r
+�
+for some r, a, where [n]r is the z-integer defined by
+[n]r := 1 − zrn
+1 − zr .
+(3.8)
+Proof. We can set
+A+(z) =
+�
+[2]r1
+−za
+− �n
+i=1 zbi
+[2]r2
+�
+,
+A−(z) =
+�
+[2]r1
+0
+− �n−1
+i=1 zci
+[2]r2
+�
+.
+14
+
+Without loss of generality, we can assume that
+b1 ≤ b2 ≤ · · · ≤ bn,
+c1 ≤ c2 ≤ · · · ≤ cn−1.
+By the symplectic property (1.4), we have
+n−1
+�
+i=1
+(z−ci + zr1−ci) + za + za−r2 =
+n
+�
+i=1
+(z−bi + zr1−bi).
+Comparing the degree by using the conditions (1.2) and (3.4), we obtain the system of linear
+equations
+a = r1 − b1,
+a − r2 = −bn,
+r1 = ci − bi = bi+1 − ci
+(i = 1, . . . , n − 1),
+which implies that
+r2 = (2n − 1)r1,
+bi = (2i − 1)r1 − a,
+ci = 2ir1 − a.
+Lemma 3.6. Suppose that
+A+(1) =
+� 2
+−1
+−1
+2
+�
+,
+A−(1) =
+�1
+0
+0
+1
+�
+.
+Then
+A+(z) =
+�
+1 + z2r
+−za
+−z2r−a
+1 + z2r
+�
+,
+A−(z) =
+�
+1 + z2r − zr
+0
+0
+1 + z2r − zr
+�
+for some r, a.
+Proof. We can set
+A+(z) =
+�
+1 + zr1
+−za
+−zb
+1 + zr2
+�
+,
+A−(z) =
+�
+1 + zr1 − zc
+0
+0
+1 + zr2 − zd
+�
+.
+By (1.4), we have r1 = r2 = a + b = 2c = 2d.
+Lemma 3.7. Suppose that
+A+(1) =
+� 2
+−1
+−2
+2
+�
+,
+A−(1) =
+�1
+0
+0
+2
+�
+.
+Then
+A+(z) =
+�
+1 + z2r
+−za
+−z2r−a − z3r−a
+1 + z3r
+�
+,
+A−(z) =
+�
+1 + z2r − zr
+0
+0
+1 + z2r
+�
+for some r, a.
+Proof. We can set
+A+(z) =
+�
+1 + zr1
+−za
+−zb1 − zb2
+1 + zr2
+�
+,
+A−(z) =
+�1 + zr1 − zc
+0
+0
+1 + zr2
+�
+.
+Without loss of generality, we can assume b1 ≤ b2. By (1.4), we have r1 = 2c, r2 = 3c, b1 = 2c−a,
+and b2 = 3c − a.
+15
+
+Lemma 3.8. Suppose that
+A+(1) =
+� 2
+−1
+−1
+1
+�
+,
+A−(1) =
+�2
+0
+0
+2
+�
+.
+Then
+A+(z) =
+�
+1 + z2r
+−za
+−z2r−a
+1 + z2r
+�
+,
+A−(z) =
+�
+1 + z2r
+0
+0
+1 + z2r
+�
+for some r, a.
+Proof. We can set
+A+(z) =
+�
+1 + zr1
+−za
+−zb
+1 + zr2
+�
+,
+A−(z) =
+�1 + zr1
+0
+0
+1 + zr2
+�
+.
+By (1.4), we have r1 = r2 = a + b = 2c.
+Proof of Theorem 1.5 (2). The remaining possibilities for finite type A±(z) ∈ Y, up to a per-
+mutation of indices and change of sign, are the six families of the pairs given in Lemma 3.5–3.8,
+which contain the parameters r, a. We can verify that these six families belong to Y, and they
+can be reduced to the pairs in Table 1 by change of slices.
+References
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+17
+

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@@ -0,0 +1,2864 @@
+arXiv:2301.02381v1  [math.NT]  6 Jan 2023
+Existence of primitive pairs with two
+prescribed traces over finite fields
+Aakash Choudhary∗and R. K. Sharma †
+Department of Mathematics,
+Indian Institute of Technology Delhi-110016, India
+Abstract
+Given F = Fpt, a field with pt elements, where p is a prime power,
+t ≥ 7, n are positive integers and f = f1/f2 is a rational func-
+tion, where f1, f2 are relatively prime, irreducible polynomials with
+deg(f1) + deg(f2) = n in F[x]. We construct a sufficient condition on
+(p, t) which guarantees primitive pairing (ǫ, f(ǫ)) exists in F such that
+TrFpt/Fp(ǫ) = a and TrFpt/Fp(f(ǫ)) = b for any prescribed a, b ∈ Fp.
+Further, we demonstrate for any positive integer n, such a pair defi-
+nitely exists for large t. The scenario when n = 2 is handled separately
+and we verified that such a pair exists for all (p, t) except from possible
+71 values of p. A result for the case n = 3 is given as well.
+Keywords: Character, Finite fields, Primitive elements.
+2020 Mathematics Subject Classification: 12E20, 11T23
+1
+Introduction
+Let Fp represent a field of finite order p, where p = qr for some prime q and
+r, a positive integer. The multiplicative group of Fp is cyclic, it is denoted
+by F∗
+p and a generator of F∗
+p is referred to as a primitive element in Fp.
+∗email: [email protected]
+†email: [email protected]
+1
+
+The field Fp has φ(p − 1) primitive elements, where φ is the Euler’s totient
+function. Let Fpt denote an extension of Fp of degree t for some positive
+integer t. A necessary and sufficient condition for an element ǫ ∈ F∗
+pt to be
+primitive is that it is a root of an irreducible polynomial of degree t over Fp
+and such an irreducible polynomial is referred to as primitive polynomial.
+For ǫ ∈ Fpt, the trace of ǫ over Fp denoted by TrFpt/Fp(ǫ), is defined as
+TrFpt/Fp(ǫ) = ǫ + ǫp + ǫp2 + · · · + ǫpt−1.
+In Cryptographic schemes such as Elgamel encryption scheme and the
+Diffie-Hellman key exchange, primitive elements serve as the fundamental
+building blocks. Numerous applications of primitive elements can be found
+in Coding theory and Cryptography [10], making the study of primitive el-
+ements and primitive polynomials an active research field. Please refer to
+[9] for more information about the existence of primitive elements in finite
+fields. For any rational function f(x) ∈ Fp(x) and ǫ ∈ Fp we call the pair
+(ǫ, f(ǫ)), a primitive pair if both ǫ and f(ǫ) are primitive elements in Fp.
+In general, if ǫ is primitive, f(ǫ) need not be primitive. For instance, take
+x2 + 3x + 2 ∈ F7[x], then 3, 5 are primitive elements in F7 but none of f(3)
+and f(5) are.
+In 1985, Cohen [3] introduced the term ”primitive pair” and he verified
+the existence of primitive pairs (ǫ, f(ǫ)) in Fp for linear polynomials f(x) =
+x + k ∈ Fp[x]. Since then many researchers have conducted studies in this
+area [12, 13, 7, 14].
+Most recently, Cohen, Sharma and Sharma [4] have
+supplied a condition that ensures the occurrence of primitive pair (ǫ, f(ǫ)) in
+Fp for non-exceptional rational function f, i.e., f is not of the form cxjgk(x),
+where j ∈ Z, k > 1 that divides p − 1 and c ∈ F∗
+p, for any g(x) ∈ Fp(x).
+Jungnickel, Vanstone [8] identified a sufficient condition for the occurrence
+of primitive elements ǫ ∈ Fpt with a prescribed trace of ǫ. Later Cohen [5]
+extended the result with some exceptions.
+Chou and Cohen [2], in 2014,
+addressed the issue of the existence of primitive element ǫ ∈ Fpt such that
+TrFpt/Fp(ǫ) = TrFpt/Fp(ǫ−1) = 0. Cao and Wang [1], for t ≥ 29, established
+a condition for the existence of primitive pair (ǫ, f(ǫ)) with f(x) = x2+1
+x
+∈
+Fpt(x) such that for prescribed a, b ∈ F∗
+p, TrFpt/Fp(ǫ) = a and TrFpt/Fp(ǫ−1) =
+b. In 2018, Gupta, Sharma and Cohen [7], for the same rational function
+and prescribed a ∈ Fp, presented a condition that ensures the existence
+of primitive pair (ǫ, f(ǫ)) in Fpt with TrFpt/Fp(ǫ) = a for t ≥ 5. Then in
+2019, Gupta and Sharma [14] extended the result to the rational function
+2
+
+ΓM(x) =
+a11x2+a12x+a13
+a22x+a23
+, where M =
+�
+a11
+a12
+a13
+0
+a22
+a23
+�
+∈ M2×3(Fpt) is any
+matrix of rank 2, and if ΓM(x) = λx or λx2 for some λ ∈ Fpt, then λ = 1.
+In 2021, Sharma and Sharma [11] examined the rational function f = f1/f2
+in Fpt(x), where f1 and f2 are relatively prime, irreducible polynomials and
+proved that for prescribed a, b ∈ Fp, the existence of primitive pair (ǫ, f(ǫ))
+in Fpt such that TrFpt/Fp(ǫ) = a and TrFpt/Fp(ǫ−1) = b for t ≥ 7.
+Prior to this article, for primitive pairs, traces were considered for ǫ and
+ǫ−1. In this article, we will consider the trace onto the element ǫ and its
+image under f, i.e, f(ǫ). Some terminology and conventions are introduced
+for explanation. We say that a non-zero polynomial f over Fp[x] has degree
+k ≥ 0, if f(x) = akxk + ak−1xk−1 + · · ·+ a1x + a0, where ak ̸= 0 and we write
+the degree of f as deg(f) = k. Next, we suppose that, for a rational function
+f(x) = f1(x)
+f2(x) ∈ Fp(x), f1 and f2 are relatively prime, irreducible polynomials
+and define the degree-sum as degsum(f) = deg(f1) + deg(f2). We will now
+define various sets that will play a crucial role in this article.
+1. We define Rp,t(n1, n2) to represent the set of all rational function f(x) =
+f1(x)
+f2(x) ∈ Fpt(x) such that f1 and f2 are relatively prime, irreducible
+polynomials over Fpt with deg(f1) = n1 and deg(f2) = n2.
+2. Denote An1,n2 as the set consisting of pairs (p, t) ∈ N × N such that
+for any f ∈ Rp,t(n1, n2) and prescribed a, b ∈ Fp, Fpt contains an ele-
+ment ǫ such that (ǫ, f(ǫ)) is a primitive pair with TrFpt/Fp(ǫ) = a and
+TrFpt/Fp(f(ǫ)) = b.
+3. Define, Rp,t(n) = �
+n1+n2=n Rp,t(n1, n2) and An = �
+n1+n2=n An1,n2.
+First, in this paper, for n ∈ N, we consider f(x) ∈ Rp,t(n) and a, b ∈ Fp,
+and then verify that there exists an element ǫ ∈ Fpt such that (ǫ, f(ǫ)) is
+a primitive pair in Fpt with TrFpt/Fp(ǫ) = a and TrFpt/Fp(f(ǫ)) = b, i.e., we
+provide a sufficient condition on pt such that (p, t) ∈ An. Furthermore, using
+a sieve variation of this sufficient condition, we prove the following result:
+Theorem 1.1. Let t, q, r, p ∈ N be such that q is a prime number, t ≥ 7
+and p = qr. Suppose p and t assumes none of the following values:
+1. 2 ≤ p ≤ 16 or p = 19, 23, 25, 27, 31, 37, 43, 49, 61, 67, 79 and t = 7;
+3
+
+2. 2 ≤ p ≤ 31 or p = 32, 37, 41, 43, 47, 83 and t = 8;
+3. 2 ≤ p ≤ 8 or p = 11, 16 and t = 9;
+4. p = 2, 3, 4, 5, 7 and t = 10, 12;
+5. p = 2, 3, 4 and t = 11;
+6. p = 2 and t = 14, 15, 16, 18, 20, 24.
+Then (p, t) ∈ A2.
+Note:- The exceptions in above theorem need not be true exceptions, they
+are possible exceptions.
+SageMath [16] is used to perform all nontrivial calculations required
+throughout this article.
+2
+Preliminaries
+In this section, we present some basic concepts, notations, and results that
+will be used in forthcoming sections of this article. Throughout the article,
+t is a positive integer, p is an arbitrary prime power and Fp is a finite field
+of order p.
+2.1
+Definitions
+1. A character of a finite abelian group G is a homomorphism χ from the
+set G into Z1, where Z1 is the set of all elements of complex field C with
+absolute value 1. The trivial character of G denoted by χ0, is defined
+as χ0(g) = 1 for all g ∈ G. In addition, the set of all characters of G,
+denoted by ˆG, forms a group under multiplication, which is isomorphic
+to G. The order of a character χ is the least positive integer d such
+that χd = χ0. For a finite field Fpt, a character of the additive group
+Fpt is called an additive character and that of the multiplicative group
+F∗
+pt is called a multiplicative character.
+2. For u, a divisor of pt−1, an element ζ ∈ F∗
+pt is called u-free, if whenever
+ζ = ξs, where ξ ∈ Fpt and s|u implies s = 1. We see that an element
+ζ ∈ F∗
+pt is (pt − 1)-free if and only if it is a primitive element of Fpt.
+4
+
+For more information on characters, primitive elements and finite fields, we
+refer the reader to [9].
+The following conclusion holds as a particular case of [15, Lemma 10]:
+Lemma 2.1. Let u be a divisor of pt − 1, ζ ∈ F∗
+pt, then we have:
+�
+s|u
+µ(s)
+φ(s)
+�
+χs
+χs(ζ) =
+�
+u
+φ(u)
+if ζ is u − free,
+0
+otherwise
+where µ(.) is the Mobius function and φ(.) is the Euler function, χs runs
+through all the φ(s) multiplicative characters over F∗
+pt with order s.
+Therefore for u, a divisor of pt − 1
+ρu : ǫ �→ θ(u)
+�
+s|u
+µ(s)
+φ(s)
+�
+χs
+χs(ǫ)
+(1)
+gives a characteristic function for the subset of u-free elements of F∗
+pt, where
+θ(u) = φ(u)/u.
+Also for a ∈ Fp,
+τa : ǫ �→ 1
+p
+�
+ψ∈ ˆ
+Fp
+ψ(TrFpt/Fp(ǫ) − a)
+(2)
+is a characteristic function for the subset of Fpt whose elements satisfy
+TrFpt/Fp(ǫ) = a. From [9, Theorem 5.7], any additive character ψ of Fp can
+be derived by ψ(a) = ψ0(ua), where ψ0 is the canonical additive character of
+Fp and u is an element of Fp corresponding to ψ. Thus
+τa = 1
+p
+�
+ψ∈ ˆ
+Fp
+ψ0(TrFpt/Fp(uǫ) − ua)
+= 1
+p
+�
+u∈Fp
+ˆψ0(uǫ)ψ0(−ua)
+(3)
+where ˆψ0 is the additive character of Fpt defined by ˆψ0(ǫ) = ψ0(TrFpt/Fp(ǫ)).
+In next theorem, we will make major use of the results given below by Wang
+and Fu [6] in 2014.
+5
+
+Lemma 2.2. [6, Theorem 5.5] Let F(x) ∈ Fpd(x) be a rational function.
+Write F(x) = �k
+j=1 fj(x)rj, where fj(x) ∈ Fpd[x] are irreducible polynomials
+and rj are non zero integers.
+Let χ be a multiplicative character of Fpd.
+Suppose that the rational function �d−1
+i=1 f(xpi) is not of the form h(x)ord(χ) ∈
+Fpd(x), where ord(χ) is the order of χ, Then we have
+����
+�
+ǫ∈Fp,f(ǫ)̸=0,∞
+χ(F(ǫ))
+���� ≤
+�
+d
+k
+�
+j=1
+deg(fj) − 1
+�
+p
+1
+2.
+Lemma 2.3. [6, Theorem 5.6] Let f(x), g(x) ∈ Fpt(x) be rational functions.
+Write f(x) = �k
+j=1 fj(x)rj, where fj(x) ∈ Fpt[x] are irreducible polynomials
+and rj are non-zero integers. Let D1 = �k
+j=1 deg(fj), D2 = max{deg(g), 0},
+D3 is the degree of denominator of g(x) and D4 is the sum of degrees of those
+irreducible polynomials dividing denominator of g but distinct from fj(x)( j=
+1,2,...,k). Let χ be a multiplicative character of Fpt, and let ψ be a nontrivial
+additive character of Fpt. Suppose g(x) is not of the form v(x)pt − v(x) in
+Fpt(x). Then we have
+����
+�
+ǫ∈Fpt,f(ǫ)̸=0,∞,g(ǫ)̸=∞
+χ(f(ǫ))ψ(g(ǫ))
+���� ≤ (D1 + D2 + D3 + D4 − 1)p
+t
+2.
+Evidently, both the sufficient condition (Theorem 3.1) and its sieving
+variation (Theorem 3.4) are entirely dependent on pt and the degrees of the
+numerator and denominator polynomials of the rational function. It is easy
+to see that the Trace part of the main result in [11] is a special case of our
+finding for f(x) = 1
+x.
+For every κ ∈ N, we will use ω(κ) to represent the number of distinct
+prime divisors of κ, and W(κ) to represent the number of square free divisors
+of κ. Clearly, W(κ) = 2ω(κ).
+3
+Sufficient Condition
+Let k1, k2, p, t ∈ N be such that p is a prime power and k1, k2 are positive
+integers which divide pt − 1. Let a, b ∈ Fp, f(x) ∈ Rp,t(n). Let Af,a,b(k1, k2)
+represents the set consisting of all those elements ǫ ∈ Fpt such that ǫ is k1-free,
+f(ǫ) is k2-free, TrFpt/Fp(ǫ) = a, and TrFpt/Ft(f(ǫ)) = b.
+We now verify the sufficient condition as follows:
+6
+
+Theorem 3.1. Suppose t, n, p ∈ N and p is a prime power. Suppose that
+p
+t
+2−2 > (2n + 1)W(pt − 1)2.
+Then (p, t) ∈ An.
+Proof. In order to prove this result it suffices to demonstrate that Af,a,b(k1, k2) >
+0 for every f(x) ∈ Rp,t(n) and for every prescribed a, b ∈ Fp . Suppose that
+f(x) ∈ Rp,t(n) be a rational function and that a, b ∈ Fp. Let P represent the
+collection of zeroes and poles of f(x) ∈ Fpt and P
+′ = P ∪ {0}. Let k1, k2 be
+divisors of pt − 1. Then by definition, Af,a,b(k1, k2) will be given by
+Af,a,b(k1, k2) =
+�
+ǫ∈Fpt−P ′
+ρk1(ǫ)ρk2(f(ǫ))τa(ǫ)τb(f(ǫ)).
+Using the characteristic functions (1) and (3) defined in the previous section,
+we obtain
+Af,a,b(k1, k2) = θ(k1)θ(k2)
+p2
+�
+s1|k1,s2|k2
+µ(s1)µ(s2)
+φ(s1)φ(s2)
+�
+s1,s2
+χf,a,b(s1, s2)
+where θ(ki) = φ(ki)
+ki
+; i = 1, 2 and
+χf,a,b(s1, s2) =
+�
+u,v∈Fp
+ψ0(−au − bv)
+�
+ǫ∈Fpt−P ′
+χs1(ǫ)χs2(ǫ0) ˆψ0(uǫ + vǫ0)
+where ǫ0 = f(ǫ). It follows from [9, Example 5.1] that, for any divisors s1, s2
+of pt − 1, there exist integers m1, m2 with 0 < m1, m2 < pt − 1 such that
+χs1(x) = χpt−1(xm1) and χs2(x) = χpt−1(xm2). Thus
+χf,a,b(s1, s2) =
+�
+u,v∈Fp
+ψ0(−au − bv)
+�
+ǫ∈Fpt−P ′
+χpt−1(ǫm1f(ǫ)m2) ˆψ0(uǫ + vǫ0)
+(4)
+=
+�
+u,v∈Fp
+ψ0(−au − bv)
+�
+ǫ∈Fpt−P ′
+χpt−1(F1(ǫ)) ˆψ0(F2(ǫ)),
+(5)
+where F1(x) = xm1f(x)m2 ∈ Fpt(x) and F2(x) = ux + vf(x) ∈ Fpt(x).
+7
+
+First we consider the situation when F2(x) = l(x)pt −l(x) for some l(x) ∈
+Fpt(x), where l(x) = l1(x)
+l2(x) with (l1, l2) = 1. We have, ux+vf1(x)
+f2(x) = l1(x)pt
+l2(x)pt −
+l1(x)
+l2(x), that is,
+f2(x)(l1(x)pt − l1(x)l2(x)pt−1) = l2(x)pt(uxf2(x) + vf1(x)).
+Since (l1(x)pt − l1(x)l2(x)pt−1, l2(x)pt) = 1, it implies that, l2(x)pt divides
+f2(x), which can only happen if l2(x) is constant. That is, we have
+c−(pt)f2(x)(l1(x)pt − l1(x)cpt−1) = uxf2(x) + vf1(x)
+where c = l2. Now, the above equation only applies if v = 0. Substituting
+it to the equation above yields, c−(pt)(l1(x)pt − l1(x)cpt−1) = ux, which can
+happen only if l1 is constant and u = 0. Moreover, if F1(x) ̸= r(x)pt−1 for
+any r(x) ∈ Fpt(x), then it follows form Lemma 2.2 that
+|χf,a,b(s1, s2)| ≤ np
+t
+2 +2.
+(6)
+And, when F1(x) = r(x)pt−1 for some r(x) ∈ Fpt(x), where r(x) = r1(x)
+r2(x) is
+such that (r1, r2) = 1. Following [11], it happens only if m1 = m2 = 0, a
+contradiction.
+If F2(x) ̸= d(x)pt − d(x) for any d(x) ∈ Fpt(x) then,
+Case 1 : When n1 ≤ n2. Then in accordance with Lemma 2.3 we have D2
+= 1, and
+|χf,a,b(s1, s2)| ≤ (2n + 1)p
+t
+2 +2.
+(7)
+Case 2 : When n1 > n2. We have D2 = n1 − n2 and
+|χf,a,b(s1, s2)| ≤ 2np
+t
+2+2.
+(8)
+Thus, if (χs1, χs2, u, v) ̸= (χ1, χ1, 0, 0) then based on the discussion above,
+and using (6), (7) and (8), we get, |χf,a,b(s1, s2)| ≤ (2n + 1)p
+t
+2+2. From this
+and the definition of Af,a,b(k1, k2), we get
+Af,a,b(k1, k2) ≥ θ(k1)θ(k2)
+p2
+((pt − |P
+′|) − (2n + 1)p
+t
+2+2(W(k1)W(k2) − 1))
+(9)
+≥ θ(k1)θ(k2)
+p2
+((pt − (n + 1)) − (2n + 1)p
+t
+2+2(W(k1)W(k2) − 1))
+(10)
+8
+
+Therefore, if p
+t
+2−2 > (2n + 1)W(k1)W(k2), then Af,a,b(k1, k2) > 0 for every
+f(x) ∈ Rp,t(n) and prescribed a, b ∈ Fp. Considering k1 = k2 = pt − 1, result
+follows.
+Now, we provide the bounds for the absolute values for Af,a,b(mk, k) −
+θ(m)Af,a,b(k, k) and Af,a,b(k, mk) − θ(m)Af,a,b(k, k). Proofs are omitted as
+they follow from the idea of [7].
+Lemma 3.2. Let k be a positive integer that divides pt −1 and m is a prime
+dividing pt − 1 but not k. Then
+|Af,a,b(mk, k) − θ(m)Af,a,b(k, k)| ≤ θ(k)2θ(m)
+p2
+(2n + 1)W(k)2p
+t
+2 +2
+and
+|Af,a,b(k, mk) − θ(m)Af,a,b(k, k)| ≤ θ(k)2θ(m)
+p2
+(2n + 1)W(k)2p
+t
+2+2.
+Lemma 3.3. Let k be a positive integer that divides pt−1 and {q1, q2, . . . , qm}
+be the collection of all primes dividing pt − 1 but not k. Then
+Af,a,b(pt−1, pt−1) ≥
+m
+�
+i=1
+Af,a,b(k, qik)+
+m
+�
+i=1
+Af,a,b(qik, k)−(2m−1)Af,a,b,(k, k).
+Sieve variation of sufficient condition (Theorem 3.1) is given below, proof
+of which is not given as it follows from Lemmas 3.2, 3.3 and ideas in [7].
+Theorem 3.4. Let t, n, p, k ∈ N be such that k divides pt − 1, where p is
+a prime power. Assume {q1, q2, . . . , qm} is the collection of all those primes
+that divide pt −1 but not k. Suppose δ = 1 −2 �m
+i=1
+1
+qi
+and ∆ = 2m − 1
+δ
++ 2.
+If δ > 0 and
+p
+t
+2 −2 > (2n + 1)∆W(k)2
+then (p, t) ∈ An.
+Lemma 3.5. Suppose that κ ∈ N is such that ω(κ) ≥ 1547, then W(κ) ≤
+κ1/12.
+9
+
+Proof. Let V = {2, 3, 5, . . ., 12983} is the set of first 1547 primes. We see
+that the product of all elements of V exceeds K = 6.57 × 105588. Let κ =
+κ1κ2, where κ1 and κ2 are co-prime integers such that all prime divisors
+of κ1 come from the least 1547 prime divisors of κ and remaining prime
+divisors are divisors of κ2. Hence, κ1/12
+1
+> K1/12 > 5.42 × 10465, whereas
+W(κ1) < 4.93 × 10465. The conclusion follows, since ρ1/12 > 2 for all primes
+ρ > 12983.
+We shall need Theorem 3.4 and Lemma 3.5 for calculation work in the
+next section.
+4
+Proof of Theorem 1.1
+Proof will be carried out for the situation t ≥ 7, since according to [2]
+there is no primitive element ǫ, for t ≤ 4, such that TrFpt/Fp(ǫ) = 0 and
+TrFpt/Fp(ǫ−1) = 0. The cases t = 5 and 6 necessitate substantial computation
+and appear to demand a different technique. As a result, we postpone further
+examination of these situations.
+We assume initially that, ω(pt − 1) ≥ 1547.
+Using Theorem 3.1 and
+Lemma 3.5, if p
+t
+2 −2 > 5p
+t
+6, that is, if pt > 5
+3t
+t−6 then (p, t) ∈ A2.
+But
+t ≥ 7 gives
+3t
+t−6 ≤ 21. Hence, if pt > 521 then (p, t) ∈ A2, and this holds
+true for ω(pt − 1) ≥ 1547. Therefore, we may suppose ω(pt − 1) ≤ 1546.
+We shall use sieve variation in order to carry forward computational work.
+Let 62 ≤ ω(pt − 1) ≤ 1546.
+To use Theorem 3.4 assume k to be the
+product of least 62 primes that divide pt − 1, that is, W(k) = 262, then
+m ≤ 1485 and δ assumes its least positive value when {q1, q2, . . . , q1485} =
+{307, 311, 313, . . ., 12979}. This yields δ > 0.004174 and ∆ < 710770.7395.
+Hence 5∆W(k)2 < 7.558211 × 1043. Let Z = 7.558211 × 1043. By sieve
+variation, (p, t) ∈ A2 if q
+t
+2 −2 > Z i.e., if pt > Z
+2t
+t−4. Since t ≥ 7, it gives
+2t
+t−4 ≤ 14
+3 . Therefore, (p, t) ∈ A2 under the condition that pt > 5.834 × 10204.
+Hence, ω(pt − 1) ≥ 95 implies (p, t) ∈ A2. In a similar manner (p, t) ∈ A3,
+A4 if ω(pt − 1) ≥ 95, and (p, t) ∈ A5 if ω(pt − 1) ≥ 96.
+10
+
+Table 1.
+Sr.No.
+a ≤ ω(pt − 1) ≤ b
+W(k)
+δ >
+∆ <
+5∆W(k)2 <
+1
+a = 13, b = 94
+213
+0.04481712
+3594.3767988
+1,206,072,718,756
+2
+a = 7, b = 34
+27
+0.04609692
+1151.7513186
+94,351,469
+3
+a = 6, b = 25
+26
+0.08241088
+450.9698124
+9,235,862
+4
+a = 6, b = 23
+26
+0.12550135
+264.9453729
+5,426,082
+5
+a = 6, b = 22
+26
+0.14959773
+209.2223842
+4,284,875
+6
+a = 5, b = 19
+25
+0.07663431
+354.3225878
+1,814,132
+7
+a = 5, b = 17
+25
+0.13927194
+167.1445296
+855,780
+8
+a = 5, b = 16
+25
+0.17317025
+123.2679422
+631,132
+9
+a = 5, b = 15
+25
+0.21090610
+92.0874844
+471,488
+Using the values in the Table 1 above and repeating the process of sieve
+variation, we determine that (p, t) ∈ A2 if pt > (4284875)
+14
+3 or pt > 8.8929 ×
+1030 for t ≥ 7 and since t ≥ 8 implies
+2t
+t−4 ≤ 4, so (p, t) ∈ A2 if pt > 3.371×1026
+for t ≥ 8.
+Therefore, for t ≥ 8 it is sufficient that ω(pt − 1) ≥ 20, We deduce,
+utilising sieve variation repeatedly for values in the second section of the
+preceding table that, (p, t) ∈ A2 if pt > 1.084 × 1025.
+Similarly, ω(pt − 1) ≥ 18 is sufficient for inclusion of (p, t) in A2, and based
+on the table above (p, t) ∈ A2 if pt > 2.2725 × 1021 for t ≥ 9, and (p, t) ∈ A2
+if pt > 8.158 × 1018 for t ≥ 10.
+Hence (p, t) ∈ A2 unless t = 7 and p < 26382, t = 8 and p < 1347, t = 9
+and p < 237, t = 10 and p < 78, t = 11 and p < 53, t = 12 and p < 38,
+t = 13 and p < 29, t = 14 and p < 23, t = 15 and p < 19, t = 16 and p < 16,
+t = 17 and p < 13, t = 18 and p < 12, t = 19 and p < 10, t = 20 and p < 9,
+t = 21, 22 and p < 8, t = 23, 24 and p < 7, t = 25, 26, 27 and p = 2, 3, 4, 5.
+28 ≤ t ≤ 31 and p = 2, 3, 4. 32 ≤ t ≤ 39 and for p = 2, 3. 40 ≤ t ≤ 62 and
+p = 2.
+From the preceding discussion for every (p, t), we validated Theorem 3.1
+and compiled a list of 570 potential exceptions (listed in the Appendix).
+11
+
+Then, for these potential exceptions, we discover that sieve variation is true
+for the large majority of prime powers, with the exception of those mentioned
+in Theorem 1.1.
+(see Appendix).
+Theorem 1.1 derives from this.
+Using
+similar reasoning, it is possible to find a subset of An for any n ∈ N. In
+particular, for A3, we have the following result.
+Theorem 4.1. Suppose t q, r, p ∈ N be such that q is a prime number, t
+≥ 7 and p = qr. Let p and t assumes none of the following values:
+1. 2 ≤ p ≤ 31 or p = 37, 41, 43, 49, 61, 67, 71, 79, 103, 121 and t = 7;
+2. 2 ≤ p ≤ 47 or p = 53, 83 and t = 8;
+3. 2 ≤ p ≤ 7 or p = 9, 11, 16 and t = 9;
+4. 2 ≤ p ≤ 8 and t = 10;
+5. p = 2, 3, 4 and t = 11;
+6. 2 ≤ p ≤ 7 and t = 12;
+7. p = 2 and t = 14, 15, 16, 18, 20, 24.
+Then (p, t) ∈ A3.
+References
+[1] Cao, X., Wang, P. (2014). Primitive elements with prescribed trace.
+AAECC 25(5):339–345.
+[2] Chou, W. S., Cohen, S. D. (2001). Primitive elements with zero traces.
+Finite Fields Appl. 7(1):125–141.
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+Amer. Math. Soc. 93(2):189–197.
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+tional functions at primitive elements of a finite field. J. Number Theory
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+Cambridge University Press.
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+prescribed traces over finite fields. Commun. Algebra 49(4):1773-1780.
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+scribed traces over finite fields. Commun. Algebra 47(3):1278–1286.
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+(version 9.0), https:// www.sagemath.org, 2020.
+13
+
+Appendix
+List of 570 values of (p, t) for which the condition of
+Theorem 3.1 of the this article fails:
+For t=7:
+p= 2, 4, 8, 16, 32, 64, 256, 512, 1024, 4096, 3, 9, 27, 81, 243, 729, 6561, 5,
+25, 125, 625, 3125, 15625, 7, 49, 343, 2401, 11, 121, 1331, 14641, 13, 169,
+2197, 17, 19, 361, 23, 529, 29, 31, 37, 41, 1681, 43, 1849, 47, 53, 59, 3481, 61,
+67, 4489, 71, 79, 6241, 83, 6889, 97, 9409, 101, 103, 10609, 107, 109, 11881,
+113, 127, 131, 17161, 139, 19321, 151, 22801, 157, 181, 191, 197, 199, 211,
+223, 227, 229, 233, 239, 241, 263, 269, 271, 277, 281, 283, 293, 311, 331, 359,
+367, 389, 397, 401, 409, 431, 439, 463, 491, 499, 509, 547, 571, 593, 601, 607,
+613, 619, 631, 643, 659, 661, 683, 691, 709, 727, 733, 739, 743, 877, 919, 953,
+967, 1021, 1051, 1063, 1093, 1123, 1151, 1171, 1181, 1231, 1283, 1301, 1303,
+1321, 1381, 1399, 1453, 1481, 1483, 1499, 1523, 1531, 1597, 1607, 1693, 1723,
+1741, 1759, 1789, 1801, 1823, 1877, 1879, 1951, 2003, 2141, 2161, 2281, 2311,
+2381, 2591, 2713, 2731, 2791, 2887, 2971, 3041, 3083, 3191, 3221, 3229, 3271,
+3301, 3307, 3313, 3499, 3547, 3571, 3739, 3851, 3911, 4013, 4159, 4219, 4241,
+4243, 4261, 4327, 4421, 4423, 4567, 4621, 4663, 4691, 4751, 4957, 5419, 5923,
+5981, 6067, 6211, 6491, 6577, 7159, 7759, 8009, 8053, 8191, 8807, 9103, 9403,
+9421, 9463, 9719, 9767, 9871, 9901, 9967, 10427, 10949, 10957, 10979, 11311,
+11593, 11621, 12959, 14323, 15313, 15511, 16381, 17431, 17491, 19483, 19687,
+19891, 20011, 20441, 21391, 22543, 23143, 23671, 24181, 24683, 25171, 25411.
+For t=8:
+p = 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 3, 9, 27, 81, 243, 729, 5, 25, 125,
+7, 49, 343, 11, 121, 1331, 13, 169, 17, 19, 361, 23, 529, 29, 841, 31, 961, 37,
+41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127,
+131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 211, 223,
+227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311,
+313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409,
+419, 421, 433, 439, 443, 457, 461, 463, 467, 491, 499, 509, 521, 523, 541, 547,
+557, 563, 569, 571, 587, 593, 599, 601, 617, 619, 631, 647, 653, 659, 661, 683,
+691, 701, 709, 727, 733, 739, 743, 757, 773, 787, 797, 809, 811, 823, 827, 829,
+839, 853, 857, 859, 863, 881, 887, 911, 919, 929, 937, 941, 947, 953, 967, 971,
+977, 983, 991, 1009, 1013, 1021, 1033, 1039, 1061, 1063, 1069, 1087, 1091,
+14
+
+1093, 1097, 1103, 1109, 1117, 1123, 1201, 1213, 1217, 1223, 1231, 1277, 1279,
+1283, 1289, 1291, 1301, 1303, 1319, 1321.
+For t=9:
+p = 2, 4, 8, 16, 32, 3, 9, 27, 81, 5, 25, 125, 7, 49, 11, 121, 13, 169, 17, 19, 23,
+29, 31, 37, 43, 47, 53, 61, 79, 83, 137, 139, 149, 157, 211.
+For t=10:
+p = 2, 4, 8, 16, 32, 64, 3, 9, 27, 5, 25, 7, 49, 11, 13, 17, 19, 23, 29, 31, 37, 41,
+53, 59, 61, 67.
+For t=11:
+p = 2, 4, 16, 3, 9, 7, 13.
+For t=12:
+p = 2, 4, 8, 16, 32, 3, 9, 27, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37.
+For t=14:
+p = 2, 4, 3, 5, 7.
+For t=15:
+p = 2, 4, 16, 3, 9, 5.
+For t=16:
+p = 2, 4, 8, 3, 5.
+For t=18:
+p = 2, 3, 4, 5.
+For t=20:
+p = 2, 4, 8.
+15
+
+For t=24:
+p = 2, 3, 4.
+For t=22, 28, 30, 36:
+p = 2.
+16
+
+List consisting values of k, m corresponding to (p, t) for
+which the condition of theorem 3.1 fails but sieve vari-
+ation is satisfied for some choice of k in this article.
+for t=7:
+Sr.No.
+p
+k
+m
+1
+32
+1
+4
+2
+64
+3
+5
+3
+256
+3
+7
+4
+512
+1
+6
+5
+1024
+3
+8
+6
+4096
+3
+11
+7
+81
+2
+5
+8
+243
+2
+4
+9
+729
+2
+7
+10
+6561
+2
+8
+11
+125
+2
+4
+12
+625
+6
+5
+13
+3125
+2
+7
+14
+15625
+6
+10
+15
+343
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+4
+16
+2401
+6
+6
+17
+121
+6
+4
+18
+1331
+2
+8
+19
+14641
+6
+8
+20
+169
+6
+4
+21
+2197
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+6
+22
+17
+2
+1
+23
+361
+6
+5
+24
+529
+6
+6
+25
+29
+2
+2
+26
+41
+2
+3
+27
+1681
+6
+6
+28
+1849
+6
+5
+29
+47
+2
+3
+30
+53
+2
+3
+31
+59
+2
+5
+32
+3481
+6
+8
+33
+4489
+6
+8
+Sr.No.
+p
+k
+m
+34
+71
+2
+4
+35
+6241
+6
+8
+36
+83
+2
+3
+37
+6889
+6
+7
+38
+97
+6
+3
+39
+9409
+6
+7
+40
+101
+2
+3
+41
+103
+6
+4
+42
+10609
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+43
+107
+2
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+44
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+6
+3
+45
+11881
+6
+7
+46
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+2
+3
+47
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+4
+48
+131
+2
+5
+49
+17161
+6
+9
+50
+139
+6
+4
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+19321
+6
+9
+52
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+3
+53
+22801
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+9
+54
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+6
+3
+55
+181
+6
+4
+56
+191
+2
+6
+57
+197
+2
+4
+58
+199
+6
+5
+59
+211
+6
+4
+60
+223
+6
+5
+61
+227
+2
+4
+62
+229
+6
+3
+63
+233
+2
+4
+64
+239
+2
+5
+65
+241
+6
+3
+66
+263
+2
+4
+17
+
+Sr.No.
+q
+l
+s
+67
+269
+2
+6
+68
+271
+6
+3
+69
+277
+6
+4
+70
+281
+2
+5
+71
+283
+6
+3
+72
+293
+2
+4
+73
+311
+2
+5
+74
+331
+6
+4
+75
+359
+2
+6
+76
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+6
+4
+77
+389
+2
+6
+78
+397
+6
+5
+79
+401
+2
+5
+80
+409
+6
+4
+81
+431
+2
+7
+82
+439
+6
+4
+83
+463
+6
+4
+84
+491
+2
+5
+85
+499
+6
+4
+86
+509
+2
+5
+87
+547
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+4
+88
+571
+6
+4
+89
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+90
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+607
+6
+4
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+4
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+631
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+4
+95
+643
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+4
+96
+659
+2
+5
+97
+661
+6
+5
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+683
+2
+5
+99
+691
+6
+6
+100
+709
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+4
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+727
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+5
+102
+733
+6
+4
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+739
+6
+4
+104
+743
+2
+5
+105
+877
+6
+5
+106
+919
+6
+6
+Sr.No.
+q
+l
+s
+107
+953
+2
+8
+108
+967
+6
+5
+109
+1021
+6
+5
+110
+1051
+6
+5
+111
+1063
+6
+5
+112
+1093
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+5
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+115
+1171
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+5
+116
+1181
+2
+6
+117
+1231
+6
+6
+118
+1283
+2
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+119
+1301
+2
+6
+120
+1303
+6
+5
+121
+1321
+6
+5
+122
+1381
+6
+5
+123
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+1481
+2
+6
+126
+1483
+6
+6
+127
+1499
+2
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+2
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+1531
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+130
+1597
+6
+5
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+1693
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+6
+133
+1723
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+1789
+6
+5
+137
+1801
+6
+6
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+6
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+140
+1879
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+5
+141
+1951
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+2003
+2
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+2141
+2
+7
+144
+2161
+6
+7
+145
+2281
+6
+6
+146
+2311
+6
+6
+18
+
+Sr.No.
+p
+k
+m
+147
+2381
+2
+8
+148
+2591
+2
+7
+149
+2713
+6
+6
+150
+2731
+6
+7
+151
+2791
+6
+7
+152
+2887
+6
+6
+153
+2971
+6
+6
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+2
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+2
+7
+156
+3191
+2
+7
+157
+3221
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+3229
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+6
+159
+3271
+6
+6
+160
+3301
+6
+7
+161
+3307
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+3571
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+6
+166
+3739
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+6
+167
+3851
+2
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+168
+3911
+2
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+169
+4013
+27
+7
+170
+4159
+6
+6
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+4219
+6
+7
+172
+4241
+2
+7
+173
+4243
+6
+7
+174
+4261
+6
+6
+175
+4327
+6
+7
+176
+4421
+2
+7
+177
+4423
+6
+6
+178
+4567
+6
+6
+179
+4621
+6
+6
+180
+4663
+6
+6
+181
+4691
+2
+7
+182
+4751
+2
+7
+183
+4957
+6
+7
+184
+5419
+6
+7
+185
+5923
+6
+7
+186
+5981
+2
+8
+187
+6067
+6
+7
+188
+6211
+6
+7
+189
+6491
+2
+8
+Sr.No.
+p
+k
+m
+190
+6577
+6
+7
+191
+7159
+6
+7
+192
+7759
+6
+8
+193
+8009
+2
+9
+194
+8053
+6
+9
+195
+8191
+6
+7
+196
+8807
+2
+8
+197
+9103
+6
+7
+198
+9403
+6
+7
+199
+9421
+6
+7
+200
+9463
+6
+7
+201
+9719
+2
+8
+202
+9767
+2
+8
+203
+9871
+6
+7
+204
+9901
+6
+7
+205
+9967
+6
+7
+206
+10427
+2
+8
+207
+10949
+2
+10
+208
+10957
+6
+8
+209
+10979
+2
+8
+210
+11311
+6
+7
+211
+11593
+6
+7
+212
+11621
+2
+8
+213
+12959
+2
+9
+214
+14232
+6
+8
+215
+15313
+6
+8
+216
+15511
+6
+8
+217
+16381
+6
+8
+218
+17431
+6
+9
+219
+17491
+6
+9
+220
+19483
+6
+8
+221
+19687
+6
+8
+222
+19891
+6
+8
+223
+20011
+6
+8
+224
+20441
+2
+10
+225
+21391
+6
+8
+226
+22543
+6
+8
+227
+23143
+6
+8
+228
+23671
+6
+9
+229
+24181
+6
+8
+230
+24683
+2
+9
+231
+25171
+6
+9
+232
+25411
+6
+8
+19
+
+for t=8:
+Sr.No.
+p
+k
+m
+1
+64
+6
+7
+2
+128
+3
+6
+3
+256
+3
+6
+4
+512
+6
+10
+5
+1024
+3
+8
+6
+81
+2
+5
+7
+243
+2
+6
+8
+125
+6
+6
+9
+49
+6
+4
+10
+343
+6
+9
+11
+121
+6
+5
+12
+169
+6
+5
+13
+361
+6
+6
+14
+529
+6
+7
+15
+841
+6
+7
+16
+961
+6
+7
+17
+53
+6
+5
+18
+59
+6
+6
+19
+61
+6
+4
+20
+67
+6
+6
+21
+71
+6
+4
+22
+73
+6
+5
+23
+79
+6
+6
+24
+89
+6
+6
+25
+97
+6
+5
+26
+101
+6
+5
+27
+103
+6
+5
+28
+107
+6
+5
+29
+109
+6
+6
+30
+113
+6
+5
+31
+127
+6
+6
+32
+131
+6
+6
+33
+137
+6
+7
+34
+139
+6
+6
+35
+149
+6
+6
+36
+151
+6
+6
+37
+157
+6
+7
+38
+163
+6
+5
+Sr.No.
+p
+k
+m
+39
+167
+6
+6
+40
+173
+6
+7
+41
+179
+6
+6
+42
+181
+6
+6
+43
+191
+6
+7
+44
+197
+6
+6
+45
+211
+6
+7
+46
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+6
+7
+47
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+7
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+6
+8
+49
+233
+6
+8
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+6
+7
+51
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+7
+52
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+257
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+5
+54
+263
+6
+7
+55
+269
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+6
+56
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+6
+57
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+6
+58
+283
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+6
+59
+293
+6
+7
+60
+311
+6
+7
+61
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+6
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+65
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+6
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+6
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+6
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+70
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+7
+71
+379
+6
+6
+72
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+6
+7
+73
+389
+6
+8
+74
+397
+6
+6
+75
+401
+6
+8
+76
+409
+6
+6
+20
+
+Sr.No.
+p
+k
+m
+77
+433
+6
+7
+78
+439
+6
+7
+79
+443
+6
+7
+80
+457
+6
+7
+81
+467
+6
+7
+82
+491
+6
+7
+83
+499
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+8
+84
+509
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+85
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+7
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+90
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+569
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+97
+617
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+102
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+683
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+7
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+107
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+757
+6
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+111
+773
+6
+8
+112
+787
+6
+8
+113
+797
+6
+8
+114
+809
+6
+8
+Sr.No.
+p
+k
+m
+115
+811
+6
+7
+116
+823
+6
+8
+117
+827
+6
+8
+118
+829
+6
+7
+119
+839
+6
+7
+120
+857
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+7
+121
+859
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+863
+6
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+123
+881
+6
+7
+124
+887
+6
+7
+125
+919
+76
+7
+126
+929
+6
+7
+127
+941
+6
+7
+128
+947
+6
+8
+129
+953
+6
+7
+130
+971
+6
+8
+131
+977
+6
+8
+132
+983
+6
+7
+133
+991
+6
+7
+134
+1009
+6
+7
+135
+1013
+6
+7
+136
+1021
+6
+8
+137
+1033
+6
+8
+138
+1039
+6
+8
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+1061
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+9
+140
+1063
+6
+8
+141
+1069
+6
+7
+142
+1087
+6
+7
+143
+1091
+6
+8
+144
+1093
+6
+7
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+1097
+6
+8
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+1103
+6
+8
+147
+1109
+6
+8
+148
+1117
+6
+8
+149
+1123
+6
+8
+150
+1201
+6
+9
+151
+1213
+6
+8
+152
+1223
+6
+9
+21
+
+Sr.No.
+p
+k
+m
+153
+1231
+6
+8
+154
+1277
+6
+9
+155
+1279
+6
+8
+156
+1283
+6
+9
+157
+1289
+6
+9
+158
+1291
+6
+9
+159
+1303
+6
+8
+160
+1319
+6
+8
+161
+1321
+6
+8
+162
+191
+6
+7
+163
+911
+30
+9
+164
+659
+30
+8
+165
+1301
+30
+10
+Sr.No.
+p
+k
+m
+166
+281
+30
+6
+167
+410
+30
+6
+168
+421
+30
+8
+169
+937
+30
+7
+170
+1331
+30
+9
+171
+307
+30
+7
+172
+1217
+30
+7
+173
+967
+30
+8
+174
+461
+30
+7
+175
+463
+30
+7
+176
+853
+30
+9
+177
+727
+30
+7
+178
+729
+30
+9
+for t=9:
+Sr.No.
+p
+k
+m
+1
+8
+7
+2
+2
+32
+7
+5
+3
+27
+2
+5
+4
+81
+14
+7
+5
+25
+6
+5
+6
+125
+2
+7
+7
+49
+6
+5
+8
+121
+6
+6
+9
+13
+6
+2
+Sr.No.
+p
+k
+m
+10
+169
+6
+7
+11
+17
+2
+3
+12
+19
+6
+3
+13
+23
+2
+5
+14
+29
+2
+5
+15
+31
+6
+4
+16
+37
+6
+5
+17
+43
+6
+6
+18
+47
+2
+5
+Sr.No.
+p
+k
+m
+19
+53
+2
+7
+20
+61
+6
+5
+21
+79
+6
+5
+22
+83
+2
+6
+23
+137
+2
+7
+24
+139
+6
+6
+25
+149
+2
+7
+26
+157
+6
+6
+27
+211
+6
+7
+for t=10:
+Sr.No.
+p
+k
+m
+1
+8
+3
+5
+2
+16
+3
+6
+3
+32
+3
+6
+4
+64
+6
+9
+5
+9
+2
+4
+6
+27
+2
+7
+7
+25
+6
+6
+Sr.No.
+p
+k
+m
+8
+49
+6
+6
+9
+11
+6
+3
+10
+13
+6
+4
+11
+17
+6
+4
+12
+19
+6
+5
+13
+23
+6
+5
+14
+29
+6
+6
+Sr.No.
+p
+k
+m
+15
+31
+6
+5
+16
+37
+6
+5
+17
+41
+6
+6
+18
+53
+6
+6
+19
+59
+6
+7
+20
+61
+6
+6
+21
+67
+6
+6
+22
+
+for t=11:
+Sr.No.
+p
+k
+m
+1
+16
+3
+6
+2
+9
+2
+4
+3
+7
+2
+3
+4
+13
+2
+5
+for t=12:
+Sr.No.
+p
+k
+m
+1
+8
+39
+6
+2
+16
+39
+7
+3
+32
+39
+9
+4
+9
+2
+6
+5
+13
+6
+6
+Sr.No.
+p
+k
+m
+6
+17
+6
+6
+7
+19
+6
+6
+8
+29
+6
+9
+9
+27
+30
+6
+Sr.No.
+p
+k
+m
+10
+11
+30
+6
+11
+23
+30
+7
+12
+31
+30
+6
+13
+37
+30
+8
+for t=14:
+Sr.No.
+p
+k
+m
+1
+4
+15
+4
+2
+5
+6
+3
+3
+7
+6
+4
+4
+3
+2
+2
+for t=15:
+Sr.No.
+p
+k
+m
+1
+4
+3
+5
+2
+3
+2
+3
+3
+9
+2
+7
+4
+5
+2
+5
+5
+16
+15
+9
+for t=16:
+Sr.No.
+p
+k
+m
+1
+4
+3
+4
+2
+8
+3
+8
+3
+3
+2
+4
+4
+5
+2
+5
+23
+
+for t=18:
+Sr.No.
+p
+k
+m
+1
+3
+2
+5
+2
+4
+15
+6
+3
+5
+6
+5
+for t=20:
+Sr.No.
+p
+k
+m
+1
+4
+3
+6
+2
+8
+15
+9
+for t=22:
+Sr.No.
+p
+k
+m
+1
+2
+15
+2
+for t=24:
+Sr.No.
+p
+k
+m
+1
+4
+3
+8
+2
+3
+2
+6
+for t=28:
+Sr.No.
+p
+k
+m
+1
+2
+15
+4
+for t=30:
+Sr.No.
+p
+k
+m
+1
+2
+3
+5
+for t=36:
+Sr.No.
+p
+k
+m
+1
+2
+15
+6
+24
+

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+ 
+1 
+Giant	resonant	enhancement	for	photo-induced		
+superconductivity	in	K3C60	
+	
+E.	Rowe1,*,	B.	Yuan1,	M.	Buzzi1,	G.	Jotzu1,	Y.	Zhu1,	M.	Fechner1,	M.	Först1,	B.	Liu1,2	
+D.	Pontiroli3,	M.	Riccò3,	A.	Cavalleri1,4,*	
+	
+1	Max	Planck	Institute	for	the	Structure	and	Dynamics	of	Matter,	Hamburg,	Germany	
+2	Paul	Scherrer	Institute,	Villigen,	Switzerland	
+3	Dipartimento	di	Scienze	Matematiche,	Fisiche	e	Informatiche,	Università	degli	Studi	di	Parma,	Italy	
+4	Department	of	Physics,	Clarendon	Laboratory,	University	of	Oxford,	United	Kingdom	
+*	e-mail:	[email protected],	[email protected]		
+	
+Photo-excitation	at	terahertz	and	mid-infrared	frequencies	has	emerged	as	a	new	
+way	 to	manipulate	functionalities	in	quantum	materials,	in	some	cases	creating	
+non-equilibrium	phases	that	have	no	equilibrium	analogue.	In	K3C60,	a	metastable	
+zero-resistance	phase	was	documented	with	optical	properties	and	pressure	de-
+pendences	compatible	with	non-equilibrium	high	temperature	superconductivity.	
+Here,	we	report	the	discovery	of	a	dominant	energy	scale	for	this	phenomenon,	
+along	 with	 the	 demonstration	 of	 a	 giant	 increase	 in	 photo-susceptibility	 near	
+10	THz	 excitation	 frequency.	 At	 these	 drive	 frequencies	 a	 metastable	 supercon-
+ducting-like	 phase	 is	 observed	 up	 to	 room	 temperature	 for	 fluences	 as	 low	 as	
+~400	µJ/cm2.	These	findings	shed	light	on	the	microscopic	mechanism	underlying	
+photo-induced	superconductivity.	They	also	trace	a	path	towards	steady	state	op-
+eration,	currently	limited	by	the	availability	of	a	suitable	high-repetition	rate	opti-
+cal	source	at	these	frequencies.	
+	
+	
+
+ 
+2 
+The	search	for	new	non-equilibrium	functional	phases	in	quantum	materials,	such	as	op-
+tically	induced	ferroelectricity1,2,	magnetism3-5,	charge	density	wave	order6,7,	non-trivial	
+topology8,9	and	superconductivity10-18,	has	become	a	central	research	theme	in	condensed	
+matter	physics.	In	the	case	of	K3C60	(Fig.	1a),	mid	infrared	optical	pulses	have	been	exten-
+sively	documented	to	yield	an	unconventional	non-equilibrium	phase	which	exhibits	met-
+astable	zero-resistance14,	an	extraordinarily	high	mobility	and	a	superconducting-like	gap	
+in	the	optical	conductivity12,14	that	reduce	with	applied	pressure13,	and	nonlinear	I-V	char-
+acteristics19.	All	these	observations	are	indicative	of	non-equilibrium	high	temperature	
+superconductivity,	observed	at	base	temperatures	far	exceeding	the	highest	equilibrium	
+superconducting	critical	temperature	of	any	alkali-doped	fulleride	(Fig.	1b).		
+Typical	experimental	results	reported	to	date	are	displayed	in	Fig.	2c.	K3C60	powders	were	
+held	at	a	base	temperature	T = 100	K	 ≫ 	T! = 	20	K	and	irradiated	with	1	ps-long	pulses	
+with	170	meV	photon	energy	(l	~ 7.3	µm,	)	~	41	THz)	at	a	fluence	of	18	mJ/cm².	This	
+strong	excitation	regime	yielded	a	long-lived	transient	state	with	dramatic	changes	in	
+both	 the	 real	 and	 imaginary	 parts	 of	 the	 optical	 conductivity,	 measured	 using	 phase	
+	
+Figure	1.	Crystal	structure	and	phase	diagram	K3C60.	(a)	Crystal	structure	of	the	organic	molec-
+ular	solid	K3C60.	C60	molecules	are	situated	at	the	vertices	of	a	face-centered-cubic	lattice.	Potassium	
+atoms	(red)	occupy	the	interstitial	voids.	(b)	Pressure-temperature	phase	diagram	of	the	fcc-A3C60	
+alkali-doped	fulleride	family	of	compounds.	Physical	pressure	tunes	the	spacing	between	the	C60	
+molecules.	The	grey	line	indicates	the	boundary	between	the	insulating	and	metallic/superconduct-
+ing	compounds.	The	blue	shaded	area	indicates	where	superconductivity	is	observed	at	equilibrium.	
+The	star	indicates	the	K3C60	compound	investigated	in	this	work,	which	superconducts	at	tempera-
+tures	! ≤ !! = 20K.	
+
+Insulator
+Superconductor 
+3 
+sensitive	terahertz	time-domain	spectroscopy.	The	transient	optical	properties	displayed	
+in	Fig.	2c	are	reminiscent	of	those	of	the	equilibrium	superconducting	state	measured	in	
+the	same	material	at	T	 ≪ 	T!	 = 	20	K	(cf.	Fig.	2b),	and	are	suggestive	of	transient	high	
+temperature	superconductivity.	These	signatures	consist	of	a	saturated	reflectivity,	a	gap	
+in	the	real	part	of	the	optical	conductivity	+"(-),	and	an	imaginary	conductivity	+#(-)	
+which	diverges	towards	low	frequencies	as	~1/-.	The	divergent	+#(-)	implies	(through	
+Kramers-Kronig	relations)	the	presence	of	a	peak	in	+"	centred	at	zero	frequency,	with	a	
+width	limited	by	the	lifetime	of	the	state	which	also	determines	the	carrier	mobility.	
+These	data	were	obtained	by	accounting	for	the	inhomogeneous	excitation	of	the	probed	
+volume	using	a	multilayer	model.	Here	we	show	the	results	of	this	reconstruction	under	
+the	assumption	of	a	linear	(open	symbols)	and	sublinear	(filled	symbols)20	dependence	of	
+the	photo-induced	changes	in	the	terahertz	refractive	index	on	the	mid-infrared	pump	
+fluence,	as	detailed	in	supplementary	section	S6.	Allowing	for	a	finite	temperature	super-
+conductor,	in	which	a	varying	density	of	uncondensed	quasi-particles	also	contributes	to	
+the	terahertz	response,	the	superconducting-like	nature	of	the	transient	state	is	inde-
+pendent	of	the	specific	choice	of	assumption.	Only	quantitative	differences,	associated	
+with	the	relative	densities	of	induced	superfluid	and	heated	quasi-particles,	which	can	be	
+extracted	by	fitting	with	a	two-fluid	model,	emerge.	
+Note	that	the	enhancement	of	conductivity	observed	in	these	experiments	is	not	con-
+nected	to	an	increase	in	the	carrier	density,	but	is	solely	caused	by	a	transfer	of	spectral	
+weight	from	the	real	part	(resistive)	to	the	imaginary	part	(inductive)	of	the	conductivity,	
+and	hence	reflects	a	colossal	increase	in	the	carrier	mobility	at	constant	density.		
+Three	spectrally-integrated	figures	of	merit	are	extracted	from	the	snapshots	of	R(-, 2),	
++"(-, 2)	and	+#(-, 2),	and	plotted	as	a	function	of	pump-probe	time	delay	2	in	Fig.	2e,	
+showing	the	time	evolution	of	the	system.		
+
+ 
+4 
+	
+Figure	2:	Photo-induced	metastable	superconductivity	in	K3C60	generated	with	intense	170	
+meV	excitation	pulses.	(a)	Schematic	of	the	experimental	set-up.	Pump	pulses	with	170	meV	pho-
+ton	energy	were	generated	in	an	optical	parametric	amplifier	(OPA)	and	subsequent	difference	fre-
+quency	generation	(DFG)	of	the	signal	and	idler	beams.	These	pulses	were	stretched	to	a	duration	
+of	~1	ps	by	linear	propagation	in	a	highly	dispersive	CaF2	rod.	The	photoinduced	changes	in	the	far-
+infrared	optical	properties	of	K3C60	were	detected	with	phase-sensitive	transient	THz	time-domain	
+spectroscopy.	(b)	Reflectivity	(sample–diamond	interface),	real	and	imaginary	part	of	the	optical	
+conductivity	of	K3C60	measured	upon	cooling	across	the	equilibrium	superconducting	transition.	
+The	blue	shading	indicates	the	change	of	spectral	weight	in	these	quantities	across	the	thermally	
+driven	superconducting	transition.	(c)	Same	quantities	measured	at	equilibrium	(red	lines)	and	
+10	ps	after	excitation	(filled	and	open	symbols).	The	data	in	filled	(open)	symbols	are	obtained	ac-
+counting	for	the	inhomogeneous	excitation	of	the	probed	volume	under	the	assumption	of	a	square	
+root	(linear)	fluence	dependence	of	the	photo-induced	changes	in	the	complex	refractive	index	of	
+the	material	(Supplementary	Section	S6).	The	blue	shading	indicates	the	change	of	spectral	weight	
+in	these	quantities	after	photo-excitation.	The	blue	solid	lines	are	fits	to	the	transient	optical	data	
+with	a	Drude-Lorentz	model	(Supplementary	Section	S7).		These	data	were	acquired	at	a	base	tem-
+perature	T	=	100	K	with	an	excitation	fluence	of	18	mJ	cm-2.	(d)	Same	quantities	as	in	(c)	but	meas-
+ured	at	a	base	temperature	T	=	295	K	with	an	excitation	fluence	of	18	mJ	cm-2.	(e)	Time	dependence	
+of	the	average	reflectivity,	average	real	part	of	the	optical	conductivity	&"((),	and	light-induced	“su-
+perfluid	density”	extracted	from	a	two-fluid	model	fit	and	expressed	as	a	fraction	of	the	total	charge	
+carrier	density.	All	quantities	are	evaluated	in	the	region	of	the	photo-induced	gap	(5–10	meV).	
+Filled	and	open	symbols	indicate	the	results	of	two	different	reconstructions	as	in	(c).	The	red	dotted	
+lines	indicate	the	value	of	the	corresponding	quantity	at	equilibrium.	These	data	were	acquired	at	a	
+base	temperature	T	=	100	K	with	a	fluence	of	18	mJ	cm−2	and	a	pump-pulse	duration	of	~1	ps.	
+
+a
+3-Stage OPA
+WLG
+DFG
+Ti:SaOscillator
+Ti:SaAmplifierx2
+THz gen.
+b
+d
+e
+1.0
+Reflectivity*
+Reflectivity
+1.0
+0.5
+0.5
+25K > Tc
+Equilibrium
+Equilibrium
+18 mJ cm-2
+5K< Tc
+Photoexcited
+Photoexcited
+0.0
+0.0
+900
+Equil.
+T = 100 K
+T = 295 K
+T= i0 ps
+= iops
+cm
+0%
+Gapping
+cm
+600
+150
+300
+*
+6
+6
+100%
+0
+900
+1.0
+cm
+600
+0.5
+300
+(S
+02
+102040
+102040
+04
+102040
+C
+0 5 10
+50
+100
+Energy (meV) Energy (meV) Energy (meV)
+Time (ps) 
+5 
+The	first	two	quantities	are	the	frequency-averaged	values	of	the	reflectivity	and	of	+"(-)	
+below	the	energy	gap,	for	which	a	zero-temperature	superconductor	with	infinite	lifetime	
+would	give	values	of	1	and	0	respectively.	The	third	figure	of	merit	is	the	fractional	super-
+fluid	density	which	is	proportional		
+to	the	divergence	of	+#(-).	This	is	determined	by	fitting	the	photoexcited	optical	proper-
+ties	with	a	two-fluid	model	where	one	fluid	represents	the	remaining	normal	carriers	with	
+a	finite	scattering	rate	and	the	other	has	zero	scattering	rate,	giving	a	superconducting-
+like	contribution.	Details	of	this	fitting	procedure	are	given	in	supplementary	section	S7.		
+For	low	excitation	fluences	the	system	becomes	superconducting-like	after	photoexcita-
+tion,	and	relaxes	on	a	time	scale	of	a	few	picoseconds.	As	already	seen	in	the	spectrally	
+resolved	measurements,	for	high	excitation	fluences	the	system	enters	a	metastable	re-
+gime	in	which	the	superconducting-like	optical	properties	persist	for	much	longer	times,	
+up	to	several	nanoseconds.		
+We	note	that	the	temperature	dependence	reported	in	Ref.	12	shows	transient	supercon-
+ducting-like	optical	properties	up	to	a	temperature	of	150-200	K.	For	higher	tempera-
+tures	the	gapping	and	extracted	superfluid	density	are	severely	reduced.	Examples	of	such	
+spectra	measured	at	room	temperature	are	shown	in	Fig.	2d.	Nevertheless,	the	pressure	
+scaling	reported	in	Ref.	13	suggests	that	traces	of	non-equilibrium	superconductivity	may	
+survive	up	to	higher	temperatures,	raising	the	prospect	that	with	more	effective	driving	a	
+full	manifestation	of	the	metastable	superconducting-like	state	may	be	possible	at	300	K.	
+To	date,	these	experiments	have	been	limited	to	excitation	photon	energies	between	80	
+and	165	meV	(20-40	THz),	such	that	a	more	comprehensive	search	for	a	dominant	excita-
+tion	frequency	scale	has	remained	out	of	reach.	Many	potentially	important	resonances	at	
+lower	frequencies	(ℎ4 <	80	meV)	have	remained	unexplored,	primarily	due	to	the	lack	of	
+a	suitable	high-intensity	pump	source	that	operates	in	this	range.	In	the	present	work,	we	
+explore	excitation	at	energies	between	24	and	80	meV	(6-20	THz).		
+
+ 
+6 
+	
+Figure	3:	Photo-induced	metastable	superconductivity	in	K3C60	generated	with	41	meV	exci-
+tation	pulses.	(a)	Schematic	of	the	experimental	set-up.	Pump	pulses	with	41	meV	(10	THz)	photon	
+energy	are	generated	in	a	twin	optical	parametric	amplifier	(OPA)	and	subsequent	chirped-pulse	
+difference	 frequency	 generation	 (DFG)	 of	 the	 two	 stretched	 signal	 beams.	 The	 photoinduced	
+changes	in	the	far-infrared	optical	properties	of	K3C60	are	detected	with	phase-sensitive	transient	
+THz	time-domain	spectroscopy.	(b)	Reflectivity	(sample–diamond	interface),	real	and	imaginary	
+part	of	the	optical	conductivity	of	K3C60	measured	at	equilibrium	(red	lines)	and	50	ps	after	excita-
+tion	(filled	and	open	symbols).	The	data	in	filled	(open)	symbols	are	obtained	accounting	for	the	
+inhomogeneous	excitation	of	the	probed	volume	under	the	assumption	of	a	square	root	(linear)	flu-
+ence	dependence	of	the	photo-induced	changes	in	the	complex	refractive	index	of	the	material.	The	
+blue	shading	indicates	the	change	of	spectral	weight	in	these	quantities	after	photo-excitation.	These	
+data	were	acquired	at	a	base	temperature	T	=	100	K	with	pump	pulses	tuned	to	41	meV	(10	THz)	
+center	frequency	and	excitation	fluence	of	0.4	mJ	cm-2.	(c)	Same	quantities	as	in	(b)	but	measured	
+10	ps	after	photoexcitation	at	a	base	temperature	T	=	295	K.	(d)	Same	quantities	as	in	(c)	but	meas-
+ured	50	ps	after	photoexcitation.	(e)	Time	dependence	of	the	average	reflectivity,	average	real	part	
+of	the	optical	conductivity	&"((),	and	light-induced	“superfluid	density”	extracted	from	a	two-fluid	
+model	fit	and	expressed	as	a	fraction	of	the	total	charge	carrier	density.	All	quantities	are	evaluated	
+in	the	region	of	the	photo-induced	gap	(5–10	meV).	Filled	and	open	symbols	indicate	the	results	of	
+two	different	reconstruction	as	in	(b).	The	inset	in	the	top	panel	highlights	the	early	time	delays	
+region	where	light	amplification	(* > 1)	is	observed	(red	shading).	The	red	dotted	lines	indicate	the	
+value	of	the	corresponding	quantity	at	equilibrium.	These	data	were	acquired	at	a	base	temperature	
+T	=	100	K	with	pump	pulses	tuned	to	45	meV	(11	THz)	photon	energy	and	excitation	fluence	of	
+0.5	mJ	cm-2	
+
+a
+2x3-stageOPAs
+Pulsestretchers
+WLG
+DFG
+THz gen.
+Ti:SaAmplifier
+b
+e
+Reflectivity
+Reflectivity*
+.0
+0.5
+0.5
+Equilibrium
+Equilibrium
+Equilibrium
+Photoexcited
+Photoexcited
+Photoexcited
+0.0
+0.0
+900
+T=100K
+T= 295K
+T= 295K
+0%
+cm
+T = 50 ps
+T = 10 ps
+T = 50 ps
+600
+cm
+Gapping
+150
+0.5 mJ cm-2
+300
+*
+100%
+0
+6
+900
+1.0
+600
+ntot
+0.5
+nsf
+300
+6
+0
+4
+102040
+4102040
+41020
+0 5 10
+50
+100
+Energy (meV)
+Energy (meV)
+E
+Energy (meV)
+Time (ps) 
+7 
+This	energy	range	hosts	a	number	of	excitations,	both	vibrational	(phonons)	and	elec-
+tronic	in	nature,	including	a	broad	polaronic	peak	seen	in	+"	centered	at	approximately	
+60	meV	(15	THz).	The	possible	relevance	of	this	excitation	has	been	highlighted	in	Ref.	21,	
+although	this	prediction	could	not	be	tested	to	date.	
+To	achieve	wide	tuneability,	we	made	use	of	a	terahertz	source	based	on	chirped	pulse	
+difference	frequency	generation,	mixing	the	near-infrared	signal	beams	of	two	phase-
+locked	optical	parametric	amplifiers22.	This	source,	illustrated	schematically	in	Fig.	3a	and	
+described	in	detail	in	supplementary	section	S4,	was	used	to	generate	narrow-bandwidth	
+pulses	with	photon	energies	spanning	the	range	from	24	to	145	meV	(6-35	THz).	All	meas-
+urements	 reported	 here	 were	 carried	 out	 with	 an	 excitation	 bandwidth	 of	 ~4	meV	
+(1	THz)	and	~600	fs	pulse	duration.	The	same	probing	protocol	as	that	reported	in	Fig.	2	
+was	utilized	here	to	detect	changes	in	the	complex	optical	properties	for	probe	energies	
+spanning	4-18	meV	(1-4.5	THz).	
+Figures	3b-d	show	reflectivity	and	complex	conductivity	spectra	measured	after	photoex-
+citation	with	pulses	tuned	to	41	meV	photon	energy	(l	~ 30	µm,	)	~	10	THz)	at	base	tem-
+peratures	of	100	K	and	room	temperature,	respectively.	Figure	3e	displays	the	time-evo-
+lution	of	the	optical	properties.	The	response	is	very	similar	to	the	case	reported	in	Fig.	2	
+for	170	meV	(41	THz)	excitation,	manifested	on	metastable	timescales	but	persisting	here	
+up	to	room	temperature	–	despite	an	almost	two	orders	of	magnitude	weaker	excitation	
+fluence.	
+Figure	4a	shows	the	scaling	with	fluence	of	the	below-gap	averaged	values	of	8(-)	and	
++"(-),	 as	 well	 as	 the	 fractional	 superfluid	 density	 in	 response	 to	 photoexcitation	 at	
+170	meV	 (41	THz)	 and	 41	meV	 (10	THz).	 These	 measurements	 were	 carried	 out	 at	 a	
+pump-probe	time	delay	of	10	ps,	and	thus	refer	to	the	metastable	phase.	The	figure	shows	
+how	all	figures	of	merit	approach	their	equilibrium	superconducting-state	values	as	the		
+
+ 
+8 
+fluence	increases,	with	the	fluence	required	being	approximately	50	times	less	for	41	meV	
+(10	THz)	compared	to	170	meV	(41	THz)	excitation.	
+Similar	fluence	dependence	measurements	were	carried	out	by	varying	the	photon	en-
+ergy	of	the	pump	and	maintaining	a	constant	4	meV	(1	THz)	bandwidth	with	600	fs	pulse	
+duration.	 For	 all	 excitation	 photon	 energies	 between	 24	meV	 (6	THz)	 and	 145	meV	
+(35	THz)	the	photoinduced	changes	in	the	optical	properties	were	qualitatively	similar	to	
+those	shown	in	Figs.	2,	3	with	only	the	size	of	the	response	for	a	given	fluence	differing.	
+From	each	fluence	dependence	we	extracted	a	figure	of	merit	for	the	photo-susceptibility,	
+	
+Figure	4:	Scaling	of	the	out-of-equilibrium	features	of	photo-induced	metastable	supercon-
+ductivity	in	K3C60.	(a)	Fluence	dependence	of	the	average	reflectivity,	average	real	part	of	the	op-
+tical	conductivity	&"((),	and	light-induced	“superfluid	density”	extracted	from	a	two-fluid	model	fit	
+and	expressed	as	a	fraction	of	the	total	charge	carrier	density.	All	quantities	are	evaluated	in	the	
+region	of	the	photo-induced	gap	(5–10	meV).	Red	and	blue	symbols	indicate	measurements	with	
+excitation	pulses	tuned	to	41	meV	(10	THz)	and	170	meV	(41	THz)	central	frequency.	The	red	dotted	
+lines	indicate	the	value	of	the	corresponding	quantity	at	equilibrium.	These	data	were	acquired	at	a	
+base	temperature	T	=	100	K,	at	a	time-delay	∆t	=	10ps,	and	with	a	pump	pulse	duration	of	~600	fs.	
+(b)	Frequency	dependence	of	the	photo-susceptibility	of	K3C60	defined	as	the	gradient	of	the	lost	
+spectral	weight	in	&"	in	the	low-fluence	limit	(Supplementary	Section	S8)	measured	10	ps	and	50	ps	
+after	 photo-excitation.	 These	 measurements	 were	 carried	 out	 at	 a	 base	 temperature	 T	=	100	K.	
+These	data	are	obtained	by	accounting	for	the	inhomogeneous	excitation	of	the	probed	volume	un-
+der	the	assumption	of	a	square	root	fluence	dependence	of	the	photo-induced	changes	in	the	com-
+plex	refractive	index	of	the	material. 
+
+41 me
+170 meV 
+9 
+defined	as	the	rate	of	growth	of	the	+"	gap	with	excitation	fluence	in	the	limit	of	low	flu-
+ence	(see	supplementary	section	S8).	Plots	of	the	pump-frequency-dependent	photo-sus-
+ceptibility	are	shown	in	Fig.	4b	for	both	10	ps	and	50	ps	pump-probe	time	delay.	A	peak	
+centered	at	41	meV	(10	THz)	with	approximately	16	meV	FWHM	bandwidth	is	observed	
+in	these	measurements.	In	the	next	section	we	will	discuss	three	distinct	energy	scales	
+which	coincide	with	this	resonance,	sequentially	these	relate	to	“on-ball”	orbital	excita-
+tions,	phonons	and	finally	excitons.	
+Superconductivity	in	alkali-doped	fullerides	is	believed	to	be	mediated	by	a	dynamical	
+Jahn-Teller	distortion,	which	leads	to	an	effective	negative	Hund’s	coupling	for	the	orbit-
+als	of	a	single	buckyball23	and	to	a	low	spin	S=1/2	state.	A	theoretical	model	based	on	this	
+assumption	has	been	successful	at	providing	a	quantitatively	correct	phase	diagram	for	
+fulleride	superconductors,	based	on	ab-initio	calculations24,25.	Within	this	model,	the	local	
+ground	state	of	the	system	is	a	six-fold	degenerate	low-spin	state,	which	features	intra-
+orbital	pairs	that	de-localize	over	two	molecular	orbitals.	As	detailed	in	supplementary	
+section	S10,	a	first	set	of	local	excited	states	also	features	such	pairs,	albeit	with	a	different	
+angular	momentum	(i.e.	a	different	inter-orbital	phase	for	the	delocalized	pair).	Ab-initio	
+calculations	predict	an	energy	splitting	of	37	meV	between	these	two	sets	of	states24,25.	
+The	observed	resonance	may	therefore	be	related	to	the	creation	of	interorbital	pairs	with	
+local	angular	momentum,	which	may	also	contribute	to	superconductivity,	as	suggested	
+in	the	Suhl-Kondo	mechanism26,27.	However,	it	is	not	yet	clear	how	exactly	this	excitation	
+is	transformed	in	the	presence	of	tunneling	between	neighboring	C60	molecules,	and	why	
+the	creation	of	such	pairs	may	support	metastable	superconductivity	at	such	high	tem-
+peratures.	Furthermore,	as	the	local	parity	of	this	excited	state	would	be	different	from	
+that	of	the	ground	state,	condensation	in	this	configuration	may	give	rise	to	a	supercon-
+ductor	with	different	symmetry.	This	possibility,	whilst	tantalizing,	remains	speculative	
+and	should	be	tested	with	more	comprehensive	ultrafast	probing	methods.		
+
+ 
+10 
+Turning	to	phonon	excitations,	we	also	note	that	the	41	meV	resonance	frequency	identi-
+fied	here	coincides	with	an	infrared-active	T1u	phonon	which	predominantly	consists	of	
+intramolecular	motion	of	the	C	atoms.	While	the	atomic	motions	of	the	170	meV	molecular	
+mode	discussed	previously	in	Ref.	12	are	directed	along	the	tangential	directions	of	the	
+C603-	molecule,	those	of	the	41	meV	mode	are	predominantly	along	the	radial	directions	
+(see	supplementary	section	S9).	By	performing	frozen	phonon	calculations	using	density	
+functional	theory	(DFT)	we	evaluated	the	different	impact	of	these	distortions	on	the	
+three	t1u	molecular	levels	at	the	Fermi	energy,	which	we	map	out	from	DFT	wave	functions	
+as	maximally-localized	Wannier	functions	(supplementary	section	S9).	In	the	undistorted	
+C603-	structure	these	molecular	levels	are	degenerate.	Applying	a	distortion	along	a	T1u	
+coordinate	lifts	this	degeneracy	leaving	a	doubly	degenerate	t1u	orbital	lowered	in	energy.	
+This	electronic	configuration	is	prone	to	developing	a	Jahn-Teller	distortion	that	may	lead	
+to	an	enhanced	negative	Hund’s	coupling,	possibly	facilitating	the	onset	of	superconduc-
+tivity	at	higher	temperatures.		
+The	strength	of	the	induced	splitting	is	quadratic	in	the	phonon	coordinate	and	is	more	
+significant	for	when	driving	the	41	meV	mode	compared	to	the	170	meV	one,	suggesting	
+that	the	observed	resonance	may	arise	from	a	more	efficient	manipulation	of	the	elec-
+tronic	degrees	of	freedom	when	driving	the	41	meV	T1u	mode.	
+Finally,	we	address	the	electronic	excitations	discussed	in	Ref.	21,	in	which	the	existence	
+of	a	polaronic	mode	was	predicted	at	the	same	energy	scales	as	the	resonance	reported	
+here.	However,	we	also	note	that	the	proposed	mechanism	for	the	formation	of	the	non-
+equilibrium	superconducting-like	state	was	one	in	which	the	quasi-particles	are	cooled	
+incoherently	via	coupling	to	the	polaronic	bath.	As	already	reported	in	Ref.	28	and	shown	
+here	in	the	inset	to	Fig.	3e,	the	response	of	the	sample	in	the	first	few	picoseconds	after	
+photoexcitation	yields	amplification	of	the	terahertz	probe	light,	which	is	likely	to	reflect	
+coherent	 dynamics	 of	 the	 driven	 degrees	 of	 freedom.	 Assuming	 that	 the	 mechanism	
+
+ 
+11 
+proposed	in	Ref.	21	were	to	be	valid,	the	early-time	dynamics	of	that	model	would	require	
+further	investigation	to	understand	how	such	coherences	would	arise	at	early	times.	The	
+amplification	observed	here	and	in	Ref.	28	has	so	far	been	attributed	to	the	existence	of	a	
+parametric	resonance	that	couples	amplitude	(Higgs)	modes	to	phase	(Goldstone)	modes,	
+an	effect	possible	at	the	sample	surface	because	of	reduced	screening.	
+We	expect	the	significance	of	this	discovery	to	be	capitalized	upon	in	future	work.	The	
+extreme	efficiency	improvement	due	to	resonant	enhancement,	nearing	two	orders	of	
+magnitude,	is	expected	to	also	dramatically	reduce	unwanted	dissipation.	This,	taken	in	
+conjunction	with	the	observed	nanosecond-long	lifetime	suggests	that	excitation	of	the	
+sample	with	a	train	of	pulses	of	only	400	µJ/cm2	delivered	at	100	MHz	repetition	rate	–	as	
+determined	by	the	inverse	lifetime	of	this	state	-	may	yield	continuous	wave	operation.	
+Because	this	effect	is	documented	here	to	persist	up	to	room	temperature,	continuous	
+wave	operation	would	likely	have	important	practical	implications.	To	make	this	regime	
+experimentally	accessible,	single	order-of-magnitude	improvements	in	the	efficiency	of	
+the	process,	or	in	the	light	matter	coupling	strength,	combined	with	suitable	develop-
+ments	in	high	repetition	rate	THz	sources	would	be	required.	
+	
+Acknowledgments	
+The	research	leading	to	these	results	received	funding	from	the	European	Research	Council	under	
+the	European	Union’s	Seventh	Framework	Programme	(FP7/2007-2013)/ERC	Grant	Agreement	
+No.	319286	(QMAC).	We	acknowledge	support	from	the	Deutsche	Forschungsgemeinschaft	(DFG)	
+via	the	Cluster	of	Excellence	‘The	Hamburg	Centre	for	Ultrafast	Imaging’	(EXC	1074	–	project	ID	
+194651731).	We	thank	Michael	Volkmann	and	Peter	Licht	for	their	technical	assistance.	We	are	
+also	grateful	to	Boris	Fiedler	and	Birger	Höhling	for	their	support	in	the	fabrication	of	the	elec-
+tronic	devices	used	on	the	measurement	setup,	and	to	Jörg	Harms	for	assistance	with	graphics.	
+	
+	
+
+ 
+12 
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+	
+
+ 
+ 
+14 
+Giant	resonant	enhancement	for	photo-induced	
+superconductivity	in	K3C60	
+E.	Rowe1,*,	B.	Yuan1,	M.	Buzzi1,	G.	Jotzu1,	Y.	Zhu1,	M.	Fechner1,	M.	Först1,	B.	Liu1,2	
+D.	Pontiroli3,	M.	Riccò3,	A.	Cavalleri1,4,*	
+	
+1	Max	Planck	Institute	for	the	Structure	and	Dynamics	of	Matter,	Hamburg,	Germany	
+2	Paul	Scherrer	Institute,	Villigen,	Switzerland	
+3	Dipartimento	di	Scienze	Matematiche,	Fisiche	e	Informatiche,	Università	degli	Studi	di	Parma,	Italy	
+4	Department	of	Physics,	Clarendon	Laboratory,	University	of	Oxford,	United	Kingdom	
+*	e-mail:	[email protected],	[email protected]		
+	
+ 
+Supplemental	Material	
+S1.	Sample	growth	and	characterization	
+S2.	Determination	of	the	equilibrium	optical	properties	
+S3.	High	fluence	mid-infrared	source	
+S4.	Frequency-tunable	narrowband	terahertz	and	mid-infrared	source	
+S5.	Measurements	of	the	transient	THz	reflectivity	
+S6.	Determination	of	the	transient	optical	properties	
+S7.	Fitting	the	transient	optical	spectra	
+S8.	Extracting	the	frequency-dependent	photosusceptibility	
+S9.	Density	functional	theory	calculations	
+S10.	Local	electronic	hamiltonian	calculations	
+	
+	
+	
+	
+
+ 
+ 
+15 
+S1.	Sample	growth	and	characterization	
+	
+The	K3C60	powder	pellets	used	in	this	work	were	prepared	and	characterized	as	reported	
+previously1-3.	Stoichiometric	amounts	of	ground	C60	powder	and	potassium	were	placed	
+in	a	sealed	pyrex	vial,	which	was	evacuated	to	a	pressure	of	10-6	mbar.	Whilst	keeping	the	
+C60	powder	and	solid	potassium	separated,	the	vial	was	kept	at	523	K	for	72	h	and	then	
+at	623	K	for	28	h	such	that	the	C60	powder	was	exposed	to	pure	potassium	vapor.	The	vial	
+was	then	opened	inside	an	Ar	glovebox	(<0.1	ppm	O2	and	H2O),	where	the	powder	was	
+reground	 and	 pelletized	 before	 annealing	 at	 623K	 for	 5	 days.	 X-ray	 diffraction	
+measurements	were	then	carried	out	on	the	resulting	K3C60	powder,	which	confirmed	
+that	it	was	phase	pure,	with	an	average	grain	size	ranging	between	100	and	400	nm.	The	
+static	superconducting	transition	temperature	was	measured	to	be	19.8	K	(in	agreement	
+with	literature	values)	via	magnetic	susceptibility	measurements	upon	zero	field	cooling	
+and	cooling	in	field	with	a	field	strength	of	400	A/m.	
+	
+	
+	
+Figure	S1.1:	a.	X-ray	diffraction	data	and	single	f.c.c.	phase	Rietveld	refinement	for	the	K3C60	
+powder	used	in	this	work.	b.	Temperature	dependence	of	the	sample	magnetic	susceptibility	
+measured	by	SQUID	magnetometry	upon	cooling	without	(ZFC:	zero	field	cooling)	and	with	a	
+magnetic	field	applied	(FCC:	field	cooled	cooling).	 
+	
+	
+
+observed
+(10emu/(g0e)
+calculated
+residual
+reflections
+20
+ZFC
+FCC
+30
+40
+50
+10
+20
+30
+40
+50
+60
+5
+10
+15
+20
+2A
+tdearees
+Temperature 
+ 
+16 
+S2.	Determination	of	the	equilibrium	optical	properties	
+	
+The	equilibrium	reflectivity	was	measured	for	photon	energies	between	5	meV	and	500	
+meV	using	a	commercial	Fourier-transform	infrared	spectrometer	(FTIR)	equipped	with	
+a	microscope	at	the	SISSI	beamline	in	the	Elettra	Synchrotron	Facility	(Trieste,	Italy),	as	
+reported	 previously1-3.	 The	 sample	 was	 pressed	 by	 a	 diamond	 window	 into	 a	 sealed	
+holder	in	order	to	obtain	an	optically	flat	interface	and	prevent	exposure	to	air.	This	
+procedure	was	carried	out	inside	an	Ar	filled	glove	box	(<0.1	ppm	O2	and	H2O)	before	the	
+sealed	sample	was	removed	and	mounted	on	a	He	cooled	cryostat	to	enable	temperature	
+dependent	measurements.	The	K3C60	reflectivity	spectra	were	referenced	against	a	gold	
+mirror	placed	at	the	sample	position.	
+In	 order	 to	 extract	 the	 complex	 optical	 conductivity	 a	 Kramers-Kronig	 algorithm	 for	
+samples	 in	 contact	 with	 a	 transparent	 window4	 was	 used.	 This	 requires	 data	 at	 all	
+frequencies,	which	were	obtained,	at	low	energies	(<5	meV)	using	an	extrapolation	based	
+on	a	Drude-Lorentz	fit,	and	at	high	energies	(>500	meV)	using	data	measured	on	single	
+crystal	samples	reported	in	Refs.	5,6.	
+The	equilibrium	properties	are	shown	in	figure	S2.1	for	temperatures	of	100	K	and	300	
+K.		This	and	further	data	measured	at	different	temperatures	and	pressures	were	already	
+reported	in	Refs.	1,2	and	discussed	also	in	comparison	with	data	obtained	from	single	
+crystals.		
+These	 data	 were	 fitted	 with	 a	 Drude-Lorentz	 model,	 which	 is	 given	 by	 the	 following	
+equation:	
+𝜎!(𝜔) + 𝑖𝜎"(𝜔) =	𝜔#"
+4𝜋
+1
+𝛾$ − 𝑖𝜔 + 𝜔#,&'(
+"
+4𝜋
+𝜔
+𝑖.𝜔),&'(
+"
+− 𝜔"/ + 𝛾&'(𝜔	
+Here	the	first	term	represents	the	Drude	response	of	the	free	carriers	with	𝜔#	and	𝛾$	
+representing	the	plasma	frequency	and	scattering	rate	respectively,	whereas	the	second	
+term	captures	the	mid	infrared	absorption	in	the	form	of	a	Lorentz	oscillator	centered	at	
+frequency	𝜔),&'(	with	plasma	frequency	𝜔#,&'(	and	damping	rate	𝛾&'(.	The	equilibrium	
+data	reported	here	was	used	to	normalize	the	transient	optical	spectra	of	K3C60	measured	
+upon	photoexcitation,	as	discussed	in	detail	in	section	S6.	
+	
+	
+
+ 
+ 
+17 
+Figure	S2.1:	Equilibrium	optical	properties	(reflectivity,	real,	and	imaginary	part	of	the	optical	
+conductivity)	of	K3C60	measured	at	a	temperature	of	100	K	(blue)	and	300	K	(green).	The	black	
+dashed	curve	is	a	Drude-Lorentz	fit	to	the	optical	conductivity	at	100	K	in	the	range	from	3	meV	
+to	60	meV	as	described	in	the	text.		
+	
+S3.	High	fluence	mid-infrared	source	
+ 
+For	the	data	reported	in	figure	2	and	in	figure	4(a)	at	170	meV	(41	THz)	excitation,	the	
+pump	pulses	were	generated	via	difference	frequency	mixing	(DFG)	of	the	signal	and	idler	
+output	 of	 a	 three-stage	 home-built	 optical	 parametric	 amplifier	 (OPA).	 A	 commercial	
+Ti:Al2O3	amplifier	delivering	60	fs	duration	pulses	at	800	nm	central	wavelength	was	used	
+to	drive	the	OPA,	and	the	DFG	process	was	performed	using	a	0.5	mm	thick	GaSe	crystal,	
+resulting	in	~100	fs	long	pulses.	The	170	meV	pulses	were	then	propagated	through	a	
+highly	dispersive	16	mm	long	CaF2	rod,	stretching	their	duration	to	~1	ps.	The	spectrum	
+of	 the	 pump	 pulses	 was	 characterized	 using	 a	 home	 built	 FTIR	 spectrometer.	 Their	
+duration	was	measured	by	cross-correlation	with	a	synchronized,	35	fs	long,	800	nm	
+wavelength	pulse	in	a	50	μm	thick	GaSe	crystal.	While	a	certain	degree	of	tunability	is	also	
+given	by	this	source,	its	useful	operation	range	spans	between	80	and	320	meV,	hence	it	
+was	only	used	for	the	high-intensity	experiments	at	170	meV	excitation.	
+	
+	
+	
+	
+
+900
+E
+900㎡
+1.0
+300 K
+100K
+Reflectivity
+T-
+T-
+600
+600
+Fit 100 K
+0.5
+300
+300
+02
+0.04
+0
+10
+30
+100
+410
+30
+100
+4
+10
+30
+100
+Energy (meV)
+Energy (meV)
+Energy (meV) 
+ 
+18 
+S4.	Frequency-tunable	narrowband	terahertz	and	mid-infrared	source	
+	
+For	the	experiments	that	required	tunability	of	the	excitation	pulses	down	to	the	THz	gap,	
+a	 different	 source	 was	 used.	 This	 source	 is	 based	 on	 the	 principle	 of	 chirped-pulse	
+difference	frequency	generation	(CP-DFG)	in	organic	non-linear	optical	crystals,	namely	
+DAST	and	DSTMS	of	approximately	600	μm	thickness.	The	principle	of	operation	of	this	
+new	source	is	described	in	detail	in	Ref.	7.	A	commercial	Ti:Al2O3	amplifier	is	used	to	drive	
+two	identical	three-stage	OPAs	which	are	seeded	by	the	same	white-light,	such	that	the	
+signal	 beams	 have	 the	 same	 phase-fluctuations.	 The	 ~100	 fs	 signal	 pulses	 are	 then	
+chirped	using	a	pair	of	transmission-grating-based	stretchers	as	depicted	in	figure	3(a).	
+This	 arrangement	 enables	 continuous	 tuning	 of	 the	 pulse	 durations	 by	 varying	 the	
+distance	between	the	gratings	in	each	pair,	effectively	enabling	continuous	tuning	of	the	
+pump-pulse	 bandwidth.	 For	 this	 experiment	 the	 pump	 pulse	 bandwidth	 was	 kept	
+constant	at	4	meV	by	maintaining	a	signal	pulse	duration	of	~600	fs,	as	measured	using	a	
+home-built	second	harmonic-based	Frequency-Resolved-Optical-Gating	(FROG)	device.	
+Frequency	tuning	of	the	generated	excitation	pulses	was	carried	out	both	by	varying	the	
+central	wavelengths	of	the	two	OPA	signal	beams,	and	by	varying	the	time	delay	between	
+the	chirped	signal	pulses	in	the	DFG	crystal	(for	fine	tuning).	For	each	measurement	the	
+pump	 frequency	 spectrum	 was	 measured	 via	 FTIR	 (Fourier	 Transform	 Infrared	
+Spectroscopy).	
+	
+S5.	Measurements	of	the	transient	THz	reflectivity	
+	
+The	experiments	presented	in	Figures	2,	3,	and	4	were	performed	on	compacted	K3C60	
+powder	pellets	pressed	against	a	diamond	window	to	ensure	an	optically	flat	interface.	
+As	K3C60	is	water	and	oxygen	sensitive,	the	pellets	were	sealed	in	an	air-tight	holder	and	
+all	sample	handling	operations	were	performed	in	an	Argon	filled	glove	box	with	<0.1	
+ppm	O2	and	H2O.	The	sample	holder	was	then	installed	at	the	end	of	a	commercial	Helium	
+cold-finger	(base	temperature	5K),	to	cool	the	pellets	down	to	a	temperature	of	100	K.		
+The	changes	in	the	properties	of	the	sample	following	photoexcitation	were	measured	
+using	time-domain	THz-spectroscopy.		
+
+ 
+ 
+19 
+The	mid-infrared	pump	induced	changes	in	the	low	frequency	optical	properties,	were	
+retrieved	using	transient	THz	time	domain	spectroscopy	in	two	different	experimental	
+setups.	The	THz	probe	pulses	were	generated	via	optical	rectification	in	a	0.2	mm	thick	
+(110)-cut	GaP	crystal	starting	from	800	nm	pulses	with	a	duration	of	~80	fs	and	35	fs,	
+respectively.	Whilst	in	one	setup	these	800	nm	were	derived	from	the	same	laser	used	
+for	pumping	the	source	described	in	section	S4,	the	35	fs,	800	nm	pulses	were	generated	
+by	a	second	Ti:Al2O3	amplifier	optically	synchronized	to	that	used	to	pump	the	high-
+intensity	mid-infrared	source	described	in	section	S3.	The	THz	probe	pulses	were	then	
+focused	onto	the	sample	with	incidence	angles	of	30	and	0	degrees,	respectively.	After	
+reflection	from	the	sample,	the	electric	field	profile	of	the	THz	pulses	was	reconstructed	
+in	 a	 standard	 electro-optic	 sampling	 setup,	 using	 a	 (110)-cut	 0.2	mm	 GaP	 crystal	
+supported	on	a	1	mm	thick	(100)-cut	GaP	substrate	to	delay	internal	reflections.	The	
+setup	 combined	 with	 the	 frequency	 tunable	 narrowband	 source	 had	 a	 measurement	
+bandwidth	 that	 extended	 between	 4	 and	 18	 meV,	 while	 the	 other	 spanned	 between	
+4	meV	to	29	meV.	The	time	resolution	of	both	setups	is	determined	by	the	measurement	
+bandwidth	and	is	~250	fs	and	~150	fs	respectively.  
+To	minimize	the	effects	on	the	pump-probe	time	resolution	due	to	the	finite	duration	of	
+the	THz	probe	pulse,	the	experiments	were	performed	as	described	in	Refs.	8,	9.	The	
+pump-probe	time	delay	was	controlled	by	fixing	the	delay	between	the	800	nm	gating	
+pulse	and	the	mid-infrared	pump	pulse	𝜏.	The	transient	THz	field	was	then	obtained	by	
+scanning	the	delay	𝑡	relative	to	both.		
+In	 order	 to	 simultaneously	 retrieve	 both	 the	 ‘pump	 on’	 (𝐸*+,
+&- (𝑡, 𝜏))	 and	 ‘pump	 off’	
+(𝐸*+,
+&..(𝑡))	probe	fields,	a	differential	chopping	scheme	was	deployed.	The	scheme	was	
+different	for	the	two	above	mentioned	setup.	For	the	narrowband,	frequency	tunable	
+setup	which	operated	at	a	repetition	rate	of	1	kHz,	the	THz	probe	pulse	was	chopped	at	a	
+frequency	of	500	Hz	and	the	mid-infrared	pump	pulse	was	chopped	at	~	357	Hz.	The	
+electro-optic	sampling	signal	was	then	fed	to	two	lock-in	amplifiers	reading	out	𝑉/01!	at	
+500	Hz	and	𝑉/01"	at	143	Hz	respectively.	For	the	high-intensity	setup,	operating	at	2	kHz	
+repetition	rate,	the	THz	probe	pulse	was	chopped	at	a	frequency	of	1	kHz	and	the	mid-
+infrared	pump	was	chopped	at	500	Hz.	In	this	case,	the	electro-optic	sampling	signal	was	
+filtered	by	two	lock-in	amplifiers	operating	at	1	kHz	and	500	Hz	respectively.	𝐸*+,
+&..(𝑡)	and	
+Δ𝐸*+,(𝑡, 𝜏)	were	then	extracted	from	the	signals	in	the	two	lock-ins	using	the	following	
+formulas:	
+
+ 
+ 
+20 
+𝐸*+,
+&..(𝑡) = 𝑉𝐿𝐼𝐴1(𝑡, 𝜏) − 	𝛼𝑉𝐿𝐼𝐴2(𝑡, 𝜏)	
+Δ𝐸*+,(𝑡, 𝜏) = 𝐸*+,
+&- (𝑡, 𝜏) − 𝐸*+,
+&..(𝑡) = 𝛼𝑉𝐿𝐼𝐴2(𝑡, 𝜏)	
+	
+where	 𝛼	 is	 a	 calibration	 constant	 determined	 experimentally	 on	 an	 InSb	 reference	
+sample.	 This	 is	 done	 by	 extracting	 Δ𝐸*+,(𝑡, 𝜏)	 as	 the	 difference	 of	 two	 separate	
+measurements	of	𝐸*+,
+&- (𝑡, 0)	and	𝐸*+,
+&..(𝑡)	performed	with	the	first	lock-in	amplifier	and	by	
+chopping	only	the	THz	probe	pulse	while	leaving	the	mid-infrared	pump	pulse	either	
+always	on	or	always	off.	Equating	the	value	of	Δ𝐸*+,(𝑡, 𝜏)	determined	in	this	way	to	the	
+one	with	differential	chopping	yields	the	calibration	constant.	
+	
+S6.	Determination	of	the	transient	optical	properties	
+	
+From	the	measured	changes	in	the	reflected	probe	field	(see	section	S5),	the	transient	
+complex	 reflection	 coefficient	 of	 the	 sample	 𝑟̃(𝜔, 𝜏)	 can	 be	 determined	 by	 taking	 the	
+Fourier	 transform	 along	 t	 of	 both	 𝐸*+,
+&..(𝑡)	 and	 Δ𝐸*+,(𝑡, 𝜏)	 and	 using	 the	 following	
+equation:	
+	
+Δ𝐸:*+,(𝜔, 𝜏)
+𝐸:*+,
+&..(𝜔)
+= 𝑟̃(𝜔, 𝜏) − 𝑟̃)(𝜔)
+𝑟̃)(𝜔)
+	
+	
+where	𝑟̃)(𝜔)	is	the	equilibrium	complex	reflection	coefficient,	obtained	as	described	in	
+section	S2.		
+In	the	cases	where	the	pump	light	penetrates	in	the	sample	several	times	deeper	than	the	
+probe	light,	one	can	assume	that	the	probe	pulse	samples	a	volume	in	the	material	that	
+has	been	homogeneously	excited	by	the	pump.	In	this	case,	it	is	possible	to	directly	extract	
+the	complex-valued	optical	response	functions	by	inverting	the	Fresnel	equations.	
+However,	 in	 K3C60	 the	 penetration	 depth	 of	 the	 probe	 electric	 field	 (~600-900	nm)	
+exceeds	that	of	the	pump	(~500	nm	at	10	THz,	~200	nm	at	41	THz),	such	that	the	probe	
+interrogates	an	inhomogeneously	excited	volume	(Figure	S6.1(a)).		
+
+ 
+ 
+21 
+As	 the	 pump	 penetrates	 into	 the	 material,	 its	 intensity	 is	 reduced,	 and	 it	 will	 induce	
+progressively	 weaker	 changes	 in	 the	 refractive	 index	 of	 the	 sample.	 This	 situation	 is	
+modeled	by	“slicing”	the	probed	thickness	of	the	material	into	thin	layers	(figure	S6.1(b)),	
+where	 we	 assume	 that	 the	 pump-induced	 changes	 in	 the	 refractive	 index	 ∆𝑛=	scale	
+according	to	the	pump	intensity	in	the	layer,	i.e.	𝑛=(𝜔, 𝑧, 𝜏) = 𝑛=)(𝜔) + ∆𝑛=(𝜔, 𝜏, 𝐼(𝑧)).	The	
+pump	 intensity	 𝐼(𝑧)	 is	 assumed	 to	 follow	 the	 dependence	 𝐼(𝑧) = 𝐼)𝑒2,/4!"#!,	 where	
+𝑑#56# = 𝜆#56# 4𝜋𝐼𝑚 D𝑛).𝜔#56#/E
+F
+.	 Here,	 the	 refractive	 index	 of	 the	 material	 at	 the	
+pump	 frequency,	 𝑛).𝜔#56#/	 is	 taken	 to	 be	 the	 one	 at	 equilibrium.	 Additionally,	 an	
+assumption	 is	 made	 on	 the	 functional	 form	 for	 the	 dependence	 of	 ∆𝑛=	 on	 the	 pump	
+intensity.	Here,	we	consider	two	different	forms	given	by:	
+(1) ∆𝑛=(𝜔, 𝜏, 𝑧) ∝ 𝐼(𝑧)	
+(2) ∆𝑛=(𝜔, 𝜏, 𝑧) ∝	H𝐼(𝑧)	
+Respectively,	these	equations	result	in	the	following	depth-dependent	functional	forms	
+for	the	spatial	profile	of	the	refractive	index:	
+(1) 𝑛=(𝑧, 𝜔, 𝜏) = 𝑛=)(𝜔) + Δ𝑛=(𝜔, 𝜏)𝑒2,/4!"#!	
+(2) 𝑛=(𝑧, 𝜔, 𝜏) = 𝑛=)(𝜔) + ∆𝑛=(𝜔, 𝜏)𝑒2,/"4!"#!	
+where	 Δ𝑛=(𝜔, 𝜏)	 represents	 the	 pump-induced	 change	 in	 the	 refractive	 index	 of	 the	
+material	at	the	sample	surface.	
+	
+Figure	S6.1:	a.	Schematics	of	pump-probe	penetration	depth	mismatch.	b.	Multi-layer	model	
+with	exponential	decay	used	to	calculate	the	pump-induced	changes	in	the	complex	refractive	
+index	 𝑛#(𝜔, 𝜏)	 for	 each	 pump-probe	 delay	 𝜏.	 The	 transition	 from	 red	 to	 background	 (grey)	
+represents	the	decaying	pump-induced	changes	in	𝑛#(𝜔, 𝑧).	
+
+Sample
+Sample
+Probe
+Pump 
+ 
+22 
+For	each	time	delay	𝜏	and	probe	frequency	𝜔7,	the	complex	reflection	coefficient	𝑟̃(∆𝑛=)	of	
+the	multilayer	stack	described	above	is	calculated	using	the	transfer	matrix	method10,	
+keeping	∆𝑛=	as	a	free	parameter.	To	numerically	extract	the	value	of	∆𝑛=(𝜔, 𝜏)	we	minimize	
+the	following	function:	
+	
+IΔ𝐸:*+,(𝜔7)
+𝐸:*+,
+&..(𝜔7)
+− 𝑟̃(𝜔7, Δn) − 𝑟̃)(𝜔7)
+𝑟̃)(𝜔7)
+I	
+	
+By	 then	 taking	 𝑛=(𝜔, 𝜏) = 𝑛=)(𝜔) + Δ𝑛=(𝜔, 𝜏),	 one	 obtains	 the	 refractive	 index	 of	 the	
+material	as	if	it	had	been	homogeneously	excited.	From	𝑛=(𝜔, 𝜏)	we	then	calculate	𝑅(𝜔, 𝜏),	
+𝜎!(𝜔, 𝜏)	and	𝜎"(𝜔, 𝜏)	as	plotted	in	the	main	text.	
+Figures	S6.2	and	S6.3	display	extended	data	sets	measured	at	increasing	pump-probe	
+delays	 with	 pump	 photon	 energies	 of	 170	 meV	 (41	 THz)	 and	 45	 meV	 (11	 THz)	
+respectively.	 Therein	 we	 report	 reflectivity	 (sample-diamond	 interface),	 real	 and	
+imaginary	part	of	the	optical	conductivity	after	reconstruction	under	the	assumptions	of	
+models	(1)	and	(2),	identified	with	hollow	and	filled	circles	respectively.		
+At	 early	 delays,	 for	 both	 excitation	 mechanisms	 and	 reconstruction	 assumptions,	 the	
+reconstructed	reflectivity	is	higher	than	one,	and	the	real	part	of	the	optical	conductivity	
+is	negative,	indicative	of	amplification	of	the	incoming	THz	probe	radiation,	as	discussed	
+previously	in	Ref.	11.	In	all	cases,	this	non-equilibrium	driven	state	then	relaxes	into	a	
+superconducting-like	 state	 with	 a	 fully	 gapped	 𝜎!(𝜔)	 and	 a	 divergence	 ∝ 1 𝜔
+⁄ 	 in	 the	
+𝜎"(𝜔)	spectrum.	At	even	later	delays	the	optical	spectra	are	those	of	a	finite	temperature	
+superconductor.	These	optical	properties	can	be	interpreted	in	the	context	of	a	two	fluid	
+model,	in	which	a	varying	density	of	uncondensed	quasi-particles	also	contributes	to	the	
+terahertz	response.	
+Importantly	the	time-evolution	of	K3C60	following	photo-excitation	is	independent	of	the	
+used	 reconstruction,	 and	 only	 the	 specific	 values	 of	 pump-probe	 delay	 up	 to	 which	
+amplification,	 fully	 gapped	 superconductor,	 and	 finite	 temperature	 superconductor	
+appear	are	affected	by	this	choice.	
+
+ 
+ 
+23 
+	
+Figure	S6.2:	Comparison	of	linear	and	sub-linear	reconstruction	in	the	transient	optical	spectra	
+at	 170	 meV	 (41	 THz)	 pump-photon	 energy.	 Reflectivity	 (sample-diamond	 interface),	 real,	 and	
+imaginary	 parts	 of	 the	 optical	 conductivity	 measured	 at	 equilibrium	 (red	 lines)	 and	 after	
+photoexcitation	(blue	symbols)	at	increasing	pump-probe	time	delays	indicated	in	the	figure.	The	data	
+in	 filled	 (open)	 symbols	 reconstructed	 under	 the	 assumption	 of	 a	 square-root	 (linear)	 fluence	
+dependence	of	the	changes	in	complex	refractive	index	of	the	material.	These	data	were	measured	at	
+18	mJ	cm-2	excitation	fluence,	and	at	a	base	temperature	of	100	K.	
+
+0 ps
+1 ps
+2 ps
+5 ps
+10 ps
+50 ps 
+ 
+24 
+	
+Figure	S6.3:	Comparison	of	linear	and	sub-linear	reconstruction	in	the	transient	optical	spectra	
+at	 45	 meV	 (11	 THz)	 pump-photon	 energy.	 Reflectivity	 (sample-diamond	 interface),	 real,	 and	
+imaginary	 parts	 of	 the	 optical	 conductivity	 measured	 at	 equilibrium	 (red	 lines)	 and	 after	
+photoexcitation	(blue	symbols)	at	increasing	pump-probe	time	delays	indicated	in	the	figure.	The	data	
+in	 filled	 (open)	 symbols	 reconstructed	 under	 the	 assumption	 of	 a	 square-root	 (linear)	 fluence	
+dependence	of	the	changes	in	complex	refractive	index	of	the	material.	These	data	were	measured	at	
+0.5	mJ	cm-2	excitation	fluence,	and	at	a	base	temperature	of	100	K.	
+
+1.5 ps
+3.5 ps
+5.5 ps
+11.5 ps
+50.5 ps 
+ 
+25 
+S7.	Fitting	of	the	transient	optical	spectra	
+	
+The	transient	optical	conductivity	spectra	presented	in	figures	2-3	as	well	as	for	each	
+fluence	in	figure	4	were	fitted	with	a	two-fluid	model	according	to	the	following	equation:	
+𝜎=(𝜔, 𝜏) = 𝜋
+2
+Λ'(𝜏)	𝑒"
+𝑚
+𝛿[𝜔 = 0] + 𝑖 Λ'(𝜏)	𝑒"
+𝑚
+1
+𝜔	
++ Λ-(𝜏)	𝑒"
+𝑚
+1
+𝛾$ − 𝑖𝜔	
++ R
+𝐵-𝜔
+𝑖(Ω-" − 𝜔") + 𝛾-𝜔
+"
+-8!
+	
+	
+Here	 the	 first	 term	 captures	 the	 frequency	 dependent	 contribution	 from	 the	
+supercarriers	with	density	Λ',	the	second	term	captures	the	Drude	contribution	of	the	
+normal	carriers	with	density	Λ-	and	scattering	rate	𝛾$.	Finally,	we	include	a	sum	over	
+	
+Figure	S7.1:	Two-fluid	fit	to	the	transient	spectrum.	Reflectivity,	real	(𝜎!)	and	imaginary	(𝜎")	
+parts	 of	 the	 optical	 conductivity	 measured	 in	 equilibrium	 at	 100	 K	 (red)	 and	 50	 ps	 after	
+photoexcitation	with	a	fluence	of	0.5	mJ	cm-2	at	45	meV	(11	THz)	photon	energy.	The	fit	to	the	
+equilibrium	data	using	the	procedure	described	in	this	section	is	shown	as	a	dashed	black	line	and	
+gives	 zero	 superfluid	 density.	 The	 two-fluid	 fit	 to	 the	 transient	 data	 generated	 using	 the	 same	
+procedure	is	shown	as	a	solid	blue	line	and	returns	a	superfluid	fraction	Λ#	 (Λ$ + Λ#)
+⁄
+= 73%.	The	
+data	in	this	figure	was	reconstructed	under	the	assumption	of	a	square	root	dependence	of	the	
+change	in	refractive	index	on	excitation	fluence	(see	supplementary	section	S6).	
+
+ 
+ 
+26 
+two	 Lorentz	 oscillators	 in	 order	 to	 capture	 the	 broad	 midinfrared	 absorption	 peak	
+centered	at	around	60	meV.	
+The	transient	data	are	fitted	at	each	delay	𝜏	using	the	parameter-set	that	captures	the	
+equilibrium	optical	conductivity	spectra	as	a	starting	condition,	and	leaving	only	Λ'	and	
+Λ-	free	to	vary,	as	though	the	effect	of	the	pump	is	to	simply	convert	carriers	from	the	
+normal	to	the	superconducting	fluid.	
+Figure	 S7.1	 shows	 representative	 fits	 to	 transient	 data	 measured	 at	 100	 K	 base	
+temperature	 and	 at	 50	ps	 time	 delay,	 as	 well	 as	 to	 the	 100	K	 equilibrium	 spectra.	
+Importantly,	 while	 the	 fit	 of	 the	 equilibrium	 data	 converges	 to	 a	 superfluid	 fraction	
+Λ'	 (Λ- + Λ')
+⁄
+	which	is	equal	to	zero,	the	fit	to	the	transient	data	yields	Λ'	 (Λ-
+⁄
++ Λ')	=	
+0.73.	 The	 transient	 optical	 data	 was	 fitted	 at	 each	 time	 delay	 and	 driving	 frequency,	
+yielding	the	time	and	frequency	dependence	of	the	superfluid	fractions	shown	in	figures	
+2(e),	3(e),	and	4(a).		
+	
+S8.	Extracting	the	frequency-dependent	photosusceptibility	
+	
+In	 figure	 4(b)	 we	 introduce	 a	 figure	 of	 merit,	 referred	 to	 as	 the	 ‘photosusceptibility’,	
+which	can	be	used	to	quantitatively	compare	the	efficiency	with	which	the	metastable	
+light-induced	 superconducting	 state	 is	 generated	 in	 K3C60	 for	 different	 excitation	
+frequencies.	
+For	each	excitation	photon	energy,	transient	optical	spectra	were	measured	at	different	
+excitation	 fluences	 ℱ.	 From	 these	 fluence	 dependent	 spectra	 we	 extract	 the	 loss	 in	
+spectral	weight	of	𝜎!(𝜔)	after	photoexcitation	in	the	5-10	meV	spectral	range,	calculated	
+as:	
+𝑆𝑊𝐿(ℱ) = Z
+𝜎!
+9:(𝜔) − 𝜎!
+#;&<&(𝜔, ℱ)	𝑑𝜔
+!)	meV/ℏ
+B	meV/ℏ
+	
+where	𝜎!
+9:(𝜔)	and	𝜎!
+#;&<&(𝜔, ℱ)	are	the	𝜎!(𝜔)	spectra	measured	in	equilibrium	and	upon	
+photoexcitation	 respectively.	 The	 𝑆𝑊𝐿(ℱ)	 data	 is	 then	 fitted	 with	 the	 following	
+phenomenological	function:	
+𝐴 \
+1
+1 + 𝐵𝑒2CDℱ
+1
+− 1
+2]	
+
+ 
+ 
+27 
+where	 ℱ	 represents	 the	 excitation	 fluence	 and	 𝐴,	 𝐵	 are	 free	 parameters.	 The	
+‘photosusceptibility’	plotted	in	figure	4(b)	is	equal	to	𝐵,	which	is	the	gradient	of	this	
+function	evaluated	at	zero	fluence.	Figure	S8.1	shows	the	fluence-dependent	data	and	
+corresponding	fit	for	one	exemplary	dataset.	
+	
+Figure	 S8.1:	 Extracting	 photosusceptibility	 from	 the	 fluence-dependent	 data.	 Lost	 spectral	
+weight	in	the	real	part	of	the	optical	conductivity	between	5	and	10	meV	as	a	function	of	fluence	(red	
+circles),	measured	10	ps	after	photoexcitation	at	100	K	with	a	pump	spectrum	centered	at	41	meV	(10	
+THz).	The	fit	is	shown	as	a	solid	green	line,	with	the	gradient	at	zero	fluence	(which	we	define	as	the	
+photosusceptibility)	shown	as	a	dashed	blue	line.	The	data	in	this	figure	was	reconstructed	under	the	
+assumption	of	a	square	root	dependence	of	the	change	in	refractive	index	on	excitation	fluence	(see	
+supplementary	section	S6).	
+	
+S9.	Density	functional	theory	calculations	
+	
+In	this	section,	we	address	how	the	displacement	of	phonon	modes	affects	the	electronic	
+properties	of	K3C60.	Specifically,	we	consider	the	molecular	orbitals	and	their	response	to	
+the	 change	 in	 the	 crystal	 structure.	 To	 carry	 out	 this	 investigation,	 a	 first-principles	
+approach	based	on	density	functional	theory	(DFT)	was	used.	The	starting	point	is	the	
+unit	cell	of	K3C60	containing	sixty	carbon	and	three	potassium	atoms.	Before	computing	
+
+ 
+ 
+28 
+the	 phonon	 spectrum,	 this	 unit	 cell	 is	 structurally	 relaxed,	 and	 the	 resulting	 lattice	
+constants	and	atomic	coordinates	are	listed	in	table	S9.1.	
+Next,	the	phonon	spectrum	of	K3C60	is	computed	from	the	force	constant	matrix	utilizing	
+a	finite	displacement	approach12.	In	total,	there	are	186	non-translational	phonon	modes	
+covering	the	symmetries	of	point	group	m-3.	Specifically,	there	are	24	Tu,	7	Eu,	23	Tg,	8	Eg,	
+and	8	Ag	modes.	Note	that	only	the	modes	of	Tu	character	are	infrared	active,	and	we	list	
+their	computed	frequencies	in	the	table	S9.2.	
+We	utilized	a	frozen	phonon	approach	to	estimate	the	impact	of	these	distortions	on	the	
+molecular	levels.	Therefore,	we	modulated	our	equilibrium	crystal	structure	with	the	
+eigen-displacements	of	these	modes.	We	then	created	a	low	energy	Hamiltonian	for	these	
+structures	by	computing	the	maximally	localized	Wannier	functions	for	the	valence	band	
+electrons.	Note	that	since	the	three	valence	bands	are	well	separated	in	energy	from	other	
+orbital-like	bands	our	method	does	not	require	a	disentanglement	procedure.		
+Our	calculations	focused	on	the	three	degenerate	t1u	molecular	levels	at	the	Fermi	energy,	
+which	 we	 mapped	 out	 from	 DFT	 wave	 functions	 as	 maximally-localized	 Wannier	
+functions.	In	the	equilibrium	structure,	the	onsite	energy	of	these	molecular	levels	is	
+degenerate;	however,	deforming	the	crystal	by	applying	a	T1u	polar	distortion	lifts	this	
+degeneracy.	Thereby,	similar	to	a	Jahn-Teller	distortion,	the	symmetry	breaking	of	the	
+crystal	structure	splits	the	level	into	a	double	and	a	single	degenerate	orbital.	For	the	43.2	
+meV	and	173.4	meV	phonon	modes,	this	splitting	manifests	as	a	lowering	in	the	energy	
+of	the	double	degenerate	orbital.	A	schematic	visualization	of	this	is	depicted	in	the	inset	
+to	figure	S9.1(a).	Diagrams	illustrating	the	distortion	of	the	C60	molecule	for	the	43.2	meV	
+Lattice	vectors		
+a		
+14.175	Å	
+Alpha	
+90˚	
+90˚	
+90˚	
+b	
+14.175	Å	
+Beta	
+c	
+14.175	Å	
+Gamma	
+Atomic	positions	according	to	Space	Group	202	(Fm-3)	
+Element	
+Wykoff	label	
+X	
+y	
+c	
+C	
+H	
+0.00000	
+0.54991	
+0.24682	
+C	
+I	
+0.58242	
+0.10057	
+0.21408	
+C	
+I	
+0.66275	
+0.05092	
+0.18294	
+K	
+C	
+0.25000	
+0.25000	
+0.25000	
+K	
+A	
+0.00000	
+0.00000	
+0.00000	
+	
+Table	S9.1:	Structural	parameters	of	K3C60	from	first-principles	computations	
+
+ 
+ 
+29 
+and	173.4	meV	modes	(labelled	‘A’	and	‘B’	and	corresponding	to	mode	numbers	4	and	21	
+in	table	S9.2	respectively)	are	shown	in	figure	S9.1(b).	
+Besides	this	qualitative	difference	of	the	phonon-mode	distortion	on	the	molecular	levels,	
+we	also	examined	the	strength	of	the	induced	splitting.	From	group-theory,	the	size	of	the	
+splitting	scales	with	the	square	of	the	distortion.	Figure	S9.1(a)	displays	how	the	splitting		
+develops	 as	 a	 function	 of	 the	 fluence	 of	 the	 incoming	 THz	 pulse.	 Each	 phonon	 mode	
+distortion	was	weighted	according	to	its	eigenfrequency	and	mode	effective	charge	in	this		
+	plot.	For	the	same	strength	of	the	driving	electric	field,	the	splitting	induced	by		phonon	
+A	produces	a	more	significant	separation	of	the	t1u	levels	compared	to	phonon	B.	Due	to	
+the	square	scaling	of	the	splitting	with	the	electric	field,	this	effect	is	further	enhanced	at	
+higher	field	strengths.		
+	The	 computations	 were	 performed	 with	 the	 Vienna	 ab-initio	 simulation	 package	
+VASP.6.213-15.	For	the	phonon	calculations,	we	used	the	Phonopy	software	package16	and	
+the	 Wannier90	 package	 for	 wannierization12.	 The	 computations	 further	 utilized	
+Number:	
+ℎ𝜈#56#	(meV)	
+1	
+2.2	
+2	
+14.1	
+3	
+42.4	
+4	(A)	
+43.2	
+5	
+48.3	
+6	
+60.5	
+7	
+62.6	
+8	
+71.5	
+9	
+80.6	
+10	
+83.9	
+11	
+85.9	
+12	
+91.1	
+13	
+92.0	
+14	
+118.6	
+15	
+122.8	
+16	
+147.5	
+17	
+148.3	
+18	
+149.8	
+19	
+163.7	
+20	
+165.4	
+21	(B)	
+173.4	
+22	
+176.9	
+23	
+184.8	
+24	
+185.3	
+	
+Table	S9.2:	List	of	the	IR	active	phonon	modes	of	Tu	symmetry.	
+
+ 
+ 
+30 
+pseudopotentials	 generated	 within	 the	 Projected	 Augmented	 Wave	 (PAW)	 method16.	
+Specifically,	the	following	default	potentials	were	used:	C	2s22p2	and	K	3s23p64s1.	The	
+Generalized	 Gradient	 Approximation	 (GGA17)	 approximation	 for	 the	 exchange-
+correlation	 potential	 was	 used.	 For	 the	 final	 numerical	 setting,	 a	 4x4x4	 Monkhorst18	
+generated	k-point-mesh	sampling	of	the	Brillouin	zone	and	a	plane-wave	energy	cutoff	of	
+600	eV	were	chosen.	The	calculations	were	re-iterated	self-consistently	until	the	change	
+in	total	energy	converged	within	10-8	eV.		
+	
+	
+Figure	S9.1:	Effect	of	vibrational	distortions	on	the	t1u	molecular	levels	from	first-principle	
+computations.		(a)	shows	the	induced	splitting	of	the	molecular	orbital	of	t1u	symmetry	at	the	Fermi	
+energy	(as	illustrated	by	the	inset)	as	a	function	of	drive	fluence.	The	two	curves	represent	the	effect	
+of	the	two	distinct	T1u	IR-phonon	modes	with	eigenfrequencies	of	43.2	(red)	and	173.4	(blue)	meV.	
+The	eigen	displacement	of	these	modes	are	shown	in	(b).	Note,	that	due	to	the	symmetry	character	of	
+the	phonon	modes	the	t1u	level	split	into	a	single	and	double	degenerate	orbital.	Lastly,	in	(c)	we	show	
+the	 induced	 splitting	 as	 a	 function	 of	 frequency	 for	 a	 fixed	 fluence.	 Here	 we	 consider	 the	 whole	
+spectrum	of	T1u	IR	modes	of	K3C60,	as	listed	in	table	S9.2.	
+	
+S10.	Local	electronic	hamiltonian	calculations	
+ 
+The Hamiltonian proposed in Ref. 19 in order to model superconductivity in alkali-doped 
+fullerides is based on an effective negative Hund’s coupling J. It arises from a combination of 
+the usual Hund’s coupling with a dynamical Jahn-Teller distortion. This causes states featuring 
+intra-orbital pairing on a buckyball to be energetically favourable. Using ab-initio calculations, 
+
+B 
+ 
+31 
+values of the intra-orbital interaction U = 0.826 eV and of J = −18.5meV were predicted for 
+K3C6020. The phase diagram for the A3C60 family of compounds was computed using DMFT 
+starting from this Hamiltonian and was found to be in quantitative agreement with experimental 
+data21. 
+The Hamiltonian can be written as: 
+ 
+𝐻 = 𝐻Intra + 𝐻Inter + 𝐻Pairhop + 𝐻Spinswap 
+ 
+with an intra-orbital interaction with magnitude U given by: 
+ 
+𝐻Intra = 𝑈 R 	𝑛7,↑𝑛7,↓
+U
+7
+ 
+ 
+where 𝑛7,V = 𝑎7,V
+W 𝑎7,V is the number operator for a spin down electron on orbital i with spin 𝜎 ∈
+{↑, ↓}. 𝑎7,V
+W  and 𝑎7,V are fermion creation and annihilation operators, respectively. The inter-
+orbital interaction appears as: 
+ 
+𝐻Inter = (𝑈 − 2𝐽) R R .1 − δ7X/𝑛7,↑𝑛X,↓
+U
+X
+U
+7
++ (𝑈 − 3𝐽) R R R
+𝑛7,V𝑛X,V
+72!
+X
+U
+7
+V
+ 
+ 
+with δ7X denoting the Kronecker delta. which, given that J is negative, makes these terms higher 
+in energy. In addition, there is a pair hopping term, which corresponds to a transfer of a pair of 
+electrons from one orbital to another. It is given by: 
+ 
+𝐻Pairhop = 𝐽 R R .1 − δ7X/𝑎7,↑
+W 𝑎7,↓
+W 𝑎X,↓𝑎X,↑
+U
+X
+U
+7
+ 
+ 
+This term was found to be crucial for the appearance of superconductivity21. Finally, there is a 
+“spin 
+swapping” 
+term, 
+where 
+two 
+opposite 
+spins 
+exchange 
+orbitals: 
+ 
+−𝐽 R R .1 − δ7X/𝑎7,↑
+W 𝑎7,↓𝑎X,↓
+W 𝑎X,↑
+U
+X
+U
+7
+ 
+ 
+
+ 
+ 
+32 
+When restricting ourselves to a Hilbert space where the three degenerate orbitals are populated 
+by three electrons (as is appropriate for A3C60 in the atomic limit), we can use a basis given by 
+the different possible arrangements in which the orbitals can be populated: 
+ 
+{|↑, ↑↓ ,0⟩,|↑ ,0, ↑↓⟩,|↑↓, ↑ ,0⟩,|0, ↑, ↑↓⟩,|↑↓ ,0, ↑⟩,|0, ↑↓, ↑⟩, |↓, ↑, ↑⟩,|↑, ↓, ↑⟩,|↑, ↑, ↓⟩,|↑, ↑, ↑⟩} 
+ 
+as well as a second set of states created by flipping all spins in the set above. 
+In this basis, the Hamiltonian takes on the form: 
+ 
+𝐻m − (3𝑈 + 5𝐽)𝐼o = −𝐽
+⎝
+⎜
+⎜
+⎜
+⎜
+⎜
+⎜
+⎛
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
++1
+0
+0
+0
+0
+0
+0
+0
+0
++1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
++2
+−1
++1
+0
+0
+0
+0
+0
+0
+0
+−1
++2
+−1
+0
+0
+0
+0
+0
+0
+0
++1
+−1
++2
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
++4⎠
+⎟
+⎟
+⎟
+⎟
+⎟
+⎟
+⎞
+ 
+ 
+where 𝐼o is the identity matrix, which encodes an overall energy offset. This matrix is block-
+diagonal, meaning that different sectors of the Hilbert space are not coupled to each other: For 
+example, there is no term that destroys or creates pairs. Because of the inverted Hund’s 
+coupling, i.e. because J is negative, the stretched state |↑, ↑, ↑⟩ as well as its global spin-flip 
+partner |↓, ↓, ↓⟩ are now the most energetic local eigenstates.  
+The local ground state is 6-fold degenerate, with an exemplary instance given by: 
+|𝑔!⟩ = (|↑, ↑↓ ,0⟩ +|↑ ,0, ↑↓⟩)/√2, i.e. it is a state where one singlet pair of electrons has de-
+localized over two orbitals. 
+The first excited manifold is 10-fold degenerate. Six of those states are of the type 
+|𝑒!⟩ = ((|↑, ↑↓ ,0⟩ −|↑ ,0, ↑↓⟩)/√2 i.e. identical to the ground state except for the phase of the 
+de-localized singlet pair (and hence corresponding to a different local angular momentum) – 
+as illustrated in Figure S10.1.  
+The energy difference between these two manifolds is given by 2J=37meV, remarkably close 
+to the observed resonance in the experiment. However, several questions remain in order to 
+determine whether an excitation of this transition is responsible for the experimental 
+observation:  
+
+ 
+ 
+33 
+Firstly, how does the light field of the laser couple to 
+this excitation? As the size of a buckyball is 
+comparable to the distance between buckyballs both 
+inter-site and intra-site driving terms may be 
+comparable in terms of the associated energy. 
+Understanding possible inter-site driving terms 
+(arising from the oscillating energy difference 
+between neighbouring sites, given by the electric 
+field multiplied with the charge and the lattice 
+spacing) will require a calculation featuring multiple 
+buckyballs. Locally, because the dynamical Jahn-
+Teller distortion causes the populated orbitals to be 
+superpositions of several undistorted orbitals, we 
+may expect the electric field to lift the orbital 
+degeneracy, for example through an orbital offset 
+term of the type 𝐻offset = Δ(𝑛U,↑ + 𝑛U,↓), where Δ 
+encodes the amplitude of the drive and is oscillating 
+in time. Such a term would in fact cause an excitation from |𝑒!⟩ to |𝑔!⟩, but it would not 
+populate any un-paired state (which are not affected by this driving term, as all orbitals are 
+equally occupied).  
+Secondly, K3C60 has an electronic bandwidth of about 0.5eV21, meaning that the system is far 
+away from the atomic limit (i.e. zero inter-site tunneling). Nevertheless, because the excitation 
+here does not require inter-site tunneling (unlike e.g., double occupancy creation in a regular 
+one-band Hubbard model), it may remain sufficiently separable. 
+Finally, how does this excitation generate superconductivity? Indeed, the Suhl-Kondo 
+mechanism suggests that in a multi-band system, pairs in any local superposition can contribute 
+to superconductivity, but how the generation of excited-state pairs can lead to superconducting 
+properties starting from a normal state remains to be investigated. 
+	
+	
+Figure	 S10.1:	 Ground	 state	 and	 first	
+excited	state	of	the	local	Hamiltonian.	
+The	yellow	lines	indicate	the	phase	of	the	
+pair	 which	 is	 de-localized	 over	 two	
+orbitals.	 The	 energy	 spacing	 between	
+these	two	states	is	given	by	-2J	
+
+-2J 
+ 
+34 
+References	
+1	
+Mitrano,	M.	et	al.	Possible	light-induced	superconductivity	in	K3C60	at	high	temperature.	
+Nature	530,	461-464,	(2016).	
+2	
+Cantaluppi,	A.	et	al.	Pressure	tuning	of	light-induced	superconductivity	in	K3C60.	Nature	
+Physics	14,	837-841,	(2018).	
+3	
+Budden,	 M.	 et	 al.	 Evidence	 for	 metastable	 photo-induced	 superconductivity	 in	 K3C60.	
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+

7tAzT4oBgHgl3EQfgfwd/content/tmp_files/2301.01468v1.pdf.txt

@@ -0,0 +1,1582 @@
+Black hole interiors
+in holographic topological semimetals
+Ling-Long Gao a,b1, Yan Liu a,b2 and Hong-Da Lyu a,b3
+aCenter for Gravitational Physics, Department of Space Science
+and International Research Institute of Multidisciplinary Science,
+Beihang University, Beijing 100191, China
+bPeng Huanwu Collaborative Center for Research and Education,
+Beihang University, Beijing 100191, China
+Abstract
+We study the black hole interiors in holographic Weyl semimetals and holo-
+graphic nodal line semimetals.
+We find that the black hole singularities are of
+Kasner form. In the topologically nontrivial phase at low temperature, both the
+Kasner exponents of the metric fields and the proper time from the horizon to the
+singularity are almost constant, likely reflecting the topological nature of the topo-
+logical semimetals. We also find some specific behaviors inside the horizon in each
+holographic semimetal model.
+1Email: [email protected]
+2Email: [email protected]
+3Email: [email protected]
+arXiv:2301.01468v1  [hep-th]  4 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Inside holographic Weyl semimetal
+3
+2.1
+Inner structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.2
+Behaviors of Kasner exponents . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.3
+Proper time of timelike geodesics
+. . . . . . . . . . . . . . . . . . . . . .
+9
+3
+Inside holographic nodal line semimetal
+10
+3.1
+Kasner exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+3.2
+Proper time of timelike geodesics
+. . . . . . . . . . . . . . . . . . . . . .
+15
+4
+Conclusion and discussion
+15
+A Equations in holographic WSM
+16
+B Equations in holographic NLSM
+18
+1
+Introduction
+The conventional classifications on the phases of matter are rooted in the Landau paradigm
+of symmetry breaking theory [1]. Over the past thirty years, new states of matter have
+been found which are beyond the concept of Landau paradigm.
+One example is the
+topological states of matter, including the quantum Hall states, topological insulators,
+topological semimetals and so on [2]. Different from the conventional Landau paradigm,
+there is no symmetry breaking during the topological phase transition and it attracts lots
+of research attention.
+In recent years, the strongly interacting topological Weyl semimetals (WSM) [3,4] and
+nodal line semimetals (NLSM) [5,6] have been explicitly constructed from the holographic
+duality. Both holographic WSM and NLSM are shown to possess nontrivial topological
+invariants [7]. Remarkably, holographic WSM exhibits interesting effects inherited from
+the boundary states [8]. These features suggest that the physical properties associated to
+topology from the weakly coupled field theories persist in the strongly coupled topologi-
+cal systems from the holography. Moreover, the systems could go through a topological
+phase transition to a topologically trivial semimetal phase, see [9] for a review on the
+1
+
+developments.4
+In the holographic WSM, during the topological phase transition the
+anomalous Hall conductivity could be served as an order parameter, while in the holo-
+graphic NLSM it is not clear about the order parameters. Whether possible universal
+“order parameter” exist for the topological phase transitions? What is the topological
+nature in the topological phase from holography? These are elusive problems we aim to
+explore from the holographic duality.
+In holography, the thermal states are dual to black hole geometries in the bulk. The
+black hole interior is expected to encode important information of the dual field the-
+ory [28–30]. In the case that the thermal states are described by the black holes with
+simple Kasner singularities, it has been shown recently in [31] that the order of the ther-
+mal phase transition in the dual field theory is connected to the behavior of the Kasner
+exponents of the black hole singularity.5 For the topological phase transitions in holo-
+graphic topological semimetals at finite temperature, the systems experience a smooth
+crossover from a topological phase, a critical phase to a trivial phase. Although the phase
+crossover is different from thermal phase transitions, it is still interesting to explore the
+interior geometries in holographic topological semimetals, in order to uncover possible
+universal behavior during the topological phase transitions.
+It turns out that there exist both universal and special behaviors of the singularities
+in holographic topological semimetals. The universal behavior is similar to the topolog-
+ical nature of topological phase and might give hints to the problems we raised for the
+topological semimetals, while the special features can be understood from the fact that
+the holographic WSM and the holographic NLSM share similarities and also differences
+in the constructions as emphasized in [5, 7]. More precisely, in both cases, two matters
+fields are added which play same role from the point of view of the boundary field theory,
+while they play different roles in the bulk geometry. In the boundary field theory, one of
+the two matter fields is to deform the Dirac point into two Weyl nodes or a nodal line,
+while the other matter field is to gap the system. In the bulk, in the topological phase
+of holographic WSM the IR geometry of Schwarzschild black hole is not deformed by the
+matter fields, while the backreaction of the matter fields on the gravitational geometry is
+quite strong in IR in the topological phase of holographic NLSM. We will see that these
+two different situations lead to different properties of the black hole singularities in the
+topological phases.
+It is known that the information of the interior geometry can be probed from the
+geodesics which correspond to certain correlators in the dual field theory. For example,
+the proper time from the horizon to the singularity can be extracted from the thermal
+one point function of certain heavy operator [30]. We will compute this quantity in the
+4Other interesting developments can be found in e.g. [10–27].
+5Other studies on the geometric aspects of black hole singularities can be found in e.g. [32–53].
+2
+
+bulk and study its behavior in the topological phases and trivial phases.
+This paper is organized as follows. In Sec. 2, we will first review the holographic WSM
+and then study its interior geometry as well as the proper time of the timelike geodesics.
+In Sec. 3, we will review the holographic NLSM and then also study its interior geometry
+and the proper time of the timelike geodesics. Sec. 4 is devoted to the conclusions and
+open questions. The details of calculations are in the appendices.
+2
+Inside holographic Weyl semimetal
+In this section we first briefly review the holographic WSM which describes a topological
+phase transition from topological WSM phase to a trivial semimetal phase. Then we
+study the interior geometry of the black hole solutions and discuss the possible universal
+behavior of the black hole singularities as well as the interior geometry. We also comment
+on the possible observable as the role of “order parameter” during the topological phase
+transition.
+The action of the holographic WSM [3,4] is
+S =
+�
+d5x√−g
+� 1
+2κ2
+�
+R + 12
+L2
+�
+− 1
+4F2 − 1
+4F 2 + α
+3 ϵabcdeAa
+�
+FbcFde + 3FbcFde
+�
+− (DaΦ)∗(DaΦ) − V (Φ)
+�
+,
+(2.1)
+where two gauge fields are dual to vector and axial currents respectively. A special Chern-
+Simons structure is introduced to match the Wald identity for these currents. An axially
+charged scalar field Φ is also introduced in the model with the source interpreted as the
+mass term. Note that DaΦ = ∂aΦ − iqAaΦ where Aa is the axial U(1) gauge potential,
+and V (Φ) = m2|Φ|2 + λ
+2|Φ|4. We set 2κ2 = L = 1.
+We focus on the finite temperature and use the following ansatz
+ds2 = −udt2 + dr2
+u + f(dx2 + dy2) + hdz2 ,
+A = Azdz ,
+Φ = φ .
+(2.2)
+The equations of motion for the fields can be found in the appendix A. In the following we
+consider m2 = −3, q = 1, λ = 1/10. Generalization to the other values of the parameters
+is straightforward.
+We use the following boundary conditions for the matter fields
+lim
+r→∞ Az = b ,
+lim
+r→∞ rφ = M ,
+(2.3)
+3
+
+where b is the time reversal symmetry breaking parameter which play the role of split-
+ting a Dirac point into two Weyl points, and M is the mass parameter which gaps the
+Dirac point. The competing between these two effects leads to interesting topological
+phase transitions. The system is completely determined by the dimensionless parameters
+T/b, M/b.
+In the weakly coupled WSM, the quantum topological phase transition could be man-
+ifest from both the band structure and equivalently the behavior of the anomalous Hall
+conductivity. In the strongly coupled model from holography, the anomalous Hall con-
+ductivity behaves similarly to the weakly coupled case, indicating that there is a topo-
+logical phase transition, as shown in Fig. 1. The lines in red, blue and purple are for
+T/b = 0.05, 0.02, 0.01 respectively. The transition becomes sharp at zero temperature
+and the dashed gray line is the critical value of the transition at zero temperature.
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+M
+b
+σAHE
+8 α b
+Figure 1:
+Plot of anomalous Hall conductivity as a function of M/b at the temperatures
+T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). The gray dashed line is the critical value of M/b
+of the quantum phase transition at zero temperature.
+2.1
+Inner structures
+The phase transitions could be parameterized by the anomalous Hall conductivity which
+is completely determined by the horizon value of the axial gauge field Az. Given the
+possible connection between the physics inside and outside the horizon, it is interesting
+to study the black hole inner structures during the topological phase transitions.
+From the black hole solutions we have obtained, we could integrate the system further
+to the singularity since the geometry is smooth at the horizon. We find that at low
+temperature, the matter field φ oscillates inside the horizon only in the topological phase
+4
+
+(i.e. M/b < 0.744). The typical behavior is shown in Fig. 2, where the oscillation regime
+of the scalar field φ (which has been rescaled according to φ/φh) as a function of r/rh
+at fixed T/b (left) or M/b (right) are plotted respectively. We find that when we fix
+the temperature T/b, the times of oscilation become less when we increase M/b from 0
+to (M/b)c. Furthermore, when we fix M/b < (M/b)c, the lower temperature, the more
+times that φ oscillates. Note that the other fields do not show any oscillation from the
+horizon to the singularity.
+Different from the holographic superconductor cases, the oscillation here is not related
+to the collapse of Einstein-Rosen bridge [34], since there is no inner horizon any more
+for holographic WSM. Similar oscillation behavior has been found previously in neutral
+helical black holes [35].
+���
+���
+���
+���
+���
+���
+-���
+-���
+-���
+���
+���
+���
+r
+rh
+ϕ
+ϕh
+���
+���
+���
+���
+���
+���
+-���
+-���
+���
+���
+���
+r
+rh
+ϕ
+ϕh
+Figure 2: The plots of φ/φh along radial direction in the oscillation region at fixed T/b = 0.02
+(left) while M/b = 0.1 (purple), 0.4 (blue), 0.6 (orange), 0.74 (red), as well as at fixed M/b = 0.1
+(right) while T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). Here φh is the horizon value of φ.
+2.2
+Behaviors of Kasner exponents
+The interior solution can be further integrated to the singularity. Near the singularity
+rs, we assume that at the leading order the fields behave as
+u ∼ −u0(r − rs)nu ,
+f ∼ f0(r − rs)nf ,
+h ∼ h0(r − rs)nh ,
+φ ∼ nφ ln(r − rs) , (2.4)
+where u0, f0, h0 and nu, nf, nh, nφ are all constants. Here u0, f0, h0 depend on the scaling
+symmetry in (A.3),(A.4),(A.5) while nu, nf, nh, nφ are not. Also note that here rs is not
+necessarily to be zero since there is a shift symmetry of the system r → r + α along the
+radial direction which was used to set the boundary behavior (A.8). Moreover, as we
+shall see later, the axial gauge field Az is determined by the ansatz (2.4).
+5
+
+Near the singularity the equations of motion (A.6) can be simplified under the assump-
+tion that the ignored terms are subleading which will be numerically checked afterward,
+u′′ + h′
+2hu′ −
+�
+f ′′ + f ′h′
+2h
+� u
+f = 0 ,
+f ′′
+f + u′′
+2u − f ′2
+4f 2 + f ′u′
+fu + 1
+2φ′2 = 0 ,
+1
+2φ′2 − u′
+2u
+�f ′
+f + h′
+2h
+�
+− f ′h′
+2fh − f ′2
+4f 2 = 0 ,
+A′′
+z +
+�f ′
+f − h′
+2h + u′
+u
+�
+A′
+z = 0 ,
+φ′′ +
+�f ′
+f + h′
+2h + u′
+u
+�
+φ′ = 0 .
+(2.5)
+Substituting (2.4) into (2.5), we obtain
+nh = 2 (1 − nu − nf) ,
+nφ = ±
+�
+(2nf + nu)(1 − nu) − 3n2
+f
+2 .
+(2.6)
+We can also solve the fourth equation in (2.5) to obtain at leading order Az
+Az ≃ Azs0 + Azs1(r − rs)nh .
+(2.7)
+Note that the leading term Azs0 can be rescaled to be 1, while Azs1 could be determined
+from the radial conserved quantities as will be discussed later. Thus there are only two
+independent parameters in (2.4) and (2.7).
+Note that in the above equations (2.5), we have assumed that the terms ignored are
+subleading. More explicitly, we have assumed
+nu < 2 ,
+nf + nu < 1 ,
+2nf + nu > 0 .
+(2.8)
+Numerically we have checked that all the above relations are satisfied for the parameters
+we have considered, which indicates that the singularities are stable and of form (2.4)
+and (2.7).
+There are two radical conserved charge associated to the scaling symmetries of the
+system,
+Q1 =
+√
+h(u′f − uf ′) ,
+(2.9)
+Q2 = u′√
+hf − h′
+√
+h
+uf − AzA′
+z
+uf
+√
+h
+.
+(2.10)
+6
+
+We have used them to check the accuracy of the numerics. Moreover, evaluate them at
+the horizon and at the singularity we obtain
+4πTf1
+�
+h1 = Ts = u0f0
+�
+h0(nf − nu)
+(2.11)
+= u0f0
+√h0
+(nhAzs0Azs1 − h0(2nf + 3nu − 2) )
+(2.12)
+where s is the density of entropy. From (2.11), we have nf > nu in addition to the con-
+straints (2.8). Moreover, the above two conserved quantities give the relations nhAzs0Azs1 =
+h0(3nf +2nu −2) which turns out to be zero in the topological phase at low temperature
+where Azs1 = 3nf + 2nu − 2 = 0.
+Starting from (2.2, 2.4) and performing the coordinate transformation
+τ = −
+2
+√n0(nu − 2)(r − rs)(2−nu)/2 ,
+(2.13)
+we obtain the Kasner form for the fields
+ds2 = −dτ 2 + ctτ 2ptdt2 + cxτ 2px(dx2 + dy2) + czτ 2pzdz2 ,
+φ = pφ log τ + cφ ,
+(2.14)
+where
+pt =
+nu
+2 − nu
+,
+px =
+nf
+2 − nu
+,
+pz =
+nh
+2 − nu
+,
+pφ =
+2nφ
+2 − nu
+.
+(2.15)
+Note that Az is a constant at the leading order. Using the relations (2.6), the above
+Kasner exponents can be expressed in terms of nu and nf,
+pt =
+nu
+2 − nu
+,
+px =
+nf
+2 − nu
+,
+pz = 2(1 − nu − nf)
+2 − nu
+,
+pφ = ±
+�
+4(2nf + nu)(1 − nu) − 6n2
+f
+2 − nu
+.
+(2.16)
+Note that the sign of pφ in (2.16) can only be determined from numerics. They satisfy
+the following Kasner relations
+pt + 2px + pz = 1 ,
+p2
+t + 2p2
+x + p2
+z + p2
+φ = 1 .
+(2.17)
+It indicates that only two of the four Kasner exponents are independent.
+In Fig. 3, we show the Kasner exponents as a function of M/b at different tempera-
+tures T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). We find that at low temperature, the
+Kasner exponents in the Weyl semimetal phase take the same value of the Schwarzschild
+black hole (e.g. within the difference of order less than 10−9 between M/b = 0.5 and
+M/b = 0 at T/b = 0.01). This reminds us the topological feature in terms of the black
+7
+
+hole singularity. It is related to the fact that the matter fields do not backreact relevantly
+to the Schwarzschild solution in the topological phase, i.e. the probe limit of system in
+terms of matter fields in the Schwarzschild black hole background works well. We have
+also checked that inside the black holes, in the topological phase the matter fields ob-
+tained from the backreacted case match well with the solutions obtained from the probe
+limit. In the quantum critical regime, the Kasner exponents oscillate. While in the trivial
+phase, the Kasner exponent does not have any oscillate behavior.
+���
+���
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+���
+���
+���
+���
+-����
+-����
+-����
+-����
+-����
+-����
+M
+b
+pt
+���
+���
+���
+���
+���
+���
+���
+-���
+-���
+-���
+���
+���
+���
+M
+b
+pϕ
+���
+���
+���
+���
+���
+���
+���
+����
+����
+����
+����
+����
+����
+M
+b
+px
+���
+���
+���
+���
+���
+���
+���
+����
+����
+����
+����
+M
+b
+pz
+Figure 3: Plots of Kasner exponents as a function of M/b. For all cases we have T/b = 0.05
+(red), 0.02 (blue), 0.01 (purple). The dashed gray vertical lines are the Kasner exponents of
+five dimension Schwarzschild black hole.
+Note that in [5], a paradigm for constructing the topological phase was proposed
+and the holographic Weyl semimetal belongs to the first type, where the matter fields are
+irrelevant in the IR of the Schwarzschild black hole. It seems likely that in any topological
+phase of this kind, the singularities are of Kasner form taking values of Schwarzschild
+black hole.
+8
+
+2.3
+Proper time of timelike geodesics
+One of interesting connection between the interior geometry and the boundary observable
+is given in [30] that the proper time of radial timelike geodesic can be encoded in the
+thermal one point functions of heavy operators. It is thus interesting to study the proper
+time of radial timelike geodesics to see if it has specific behavior during the topological
+phase transitions.
+We consider radial timelike geodesic for which gtt ˙t2 + grr ˙r2 = −1, where the dot
+denotes the derivative with respect to the proper time τ. Along the geodesic there is a
+conserved charge E = −gtt ˙t which can be interpreted as energy. Then the equation of
+motion of the geodesic becomes
+E2
+gtt
++ grr ˙r2 = −1 ,
+(2.18)
+from which we obtain
+dτ
+dr =
+1
+√
+E2 − u .
+(2.19)
+The proper time from the horizon to the singularity of a particle with E = 0 (i.e. the
+longest time) is
+τs =
+� rh
+rs
+dr
+√−u .
+(2.20)
+The plots of τs as a function of M/b for different T/b are shown in Fig. 4.
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+M
+b
+τs
+Figure 4: Plots of the proper time τs from the horizon to the singularity as a function of M/b
+at different temperatures T/b = 0.05 (red), 0.02 (blue), 0.01 (purple).
+The proper time from the horizon to the singularity in the topological phase is equal
+to the case of Schwarzschild black hole τs = π/4 (e.g. within the difference of order less
+9
+
+than 10−4 between M/b = 0.5 and M/b = 0 at T/b = 0.01). This is expected from
+the fact that in the topological phase at low temperature the interior of the black holes
+match well with the Schwarzschild black hole. In the topologically trivial phase τs is
+monotonically decreasing. Moreover, τs shows a jump behavior and takes a maximum
+value in the critical regime. Note that τs is encoded in the thermal one point function
+of heavy operators in the form of ⟨O⟩ ∝ e−imτs where the complexified mass m has
+Im(m) < 0 [30]. One might use this thermal one point function as the “order” parameter
+for the topological phase transition. The behavior of the proper time also reminds us the
+behavior of the dimensionless information screening length in [14]. One obvious difference
+is that the information screening length is determined by the quantities at the horizon,
+while τs is determined by the geometry from the horizon to the singularity.
+3
+Inside holographic nodal line semimetal
+In the previous section, we have seen that the interior of the black hole geometries
+for the holographic WSM exhibit interesting behavior. In the topological WSM phase,
+the Kasner exponents of the dual geometries take the same value of the Schwarzschild
+black hole at low temperature, as shown in Fig. 3. Moreover, the dual operator which
+encodes the proper time from the horizon to the singularity could be served as an “order
+parameter” during the topological phase transition, as shown in Fig. 4. To check if these
+behaviors are universal for any topological phase transitions, in this section we study the
+other topological phase transition model from holography, i.e. the holographic NLSM
+model which describes a phase transition from the topological NLSM phase to a trivial
+semimetal phase [5,6].
+The action for the holographic NLSM [6] is
+S =
+�
+d5x √−g
+� 1
+2κ2
+�
+R + 12
+L2
+�
+− 1
+4F2 − 1
+4F 2 + α
+3 ϵabcdeAa
+�
+FbcFde + 3FbcFde
+�
+− (DaΦ)∗(DaΦ) − V1(Φ) − 1
+6ηϵabcde�
+iBabH∗
+cde − iB∗
+abHcde
+�
+− V2(Bab) − λ|Φ|2B∗
+abBab
+�
+,
+(3.1)
+where Fab = ∂aVb − ∂bVa is the vector gauge field strength. Fab = ∂aAb − ∂bAa is the
+axial gauge field strength. Da = ∇a −iq1Aa is the covariant derivative and q1 is the axial
+charge of scalar field. α is the Chern-Simons coupling. Bab is an antisymmetric complex
+two form field with the field strength
+Habc = ∂aBbc + ∂bBca + ∂cBab − iq2AaBbc − iq2AbBca − iq2AcBab ,
+(3.2)
+10
+
+where q2 is the axial charge of the two form field. η is the Chern-Simons coupling strength
+of the two form field. The introduction of the Chern-Simons terms while not canonical
+kinetic term for the two form field follows from the self-duality condition of the two form
+operator in the weakly coupled theory [6]. The potential terms are chosen as
+V1 = m2
+1|Φ|2 + λ1
+2 |Φ|4 ,
+V2 = m2
+2B∗
+abBab ,
+(3.3)
+where m2
+1 and m2
+2 are the mass parameters of the scalar field and the two form field. The
+λ term in the action (3.1) denotes the interaction between the scalar field and the two
+form field. We set 2κ2 = L = 1.
+Similar to the holographic WSM, we focus on the finite temperature solution and take
+the ansatz
+ds2 = −udt2 + dr2
+u + f(dx2 + dy2) + hdz2 ,
+Φ = φ ,
+Bxy = −Byx = Bxy ,
+Btz = −Bzt = iBtz .
+(3.4)
+Plugging the above ansatz into the equations of motion, we could obtain the dynamical
+equations of the fields, which can be found in the appendix B. In the following we choose
+m2
+1 = −3, m2
+2 = 1, η = 2 and q1 = q2 = 1, λ = 1, λ1 = 0.1 for simplicity.
+With the following boundary conditions,
+lim
+r→∞ rφ = M ,
+lim
+r→∞
+Bxy
+r
+= lim
+r→∞
+Btz
+r
+= b ,
+(3.5)
+we can integrate the system from the boundary to the horizon. Different from the holo-
+graphic WSM, in holographic NLSM there is no sharp “order parameter” like anomalous
+Hall conductivity. Nevertheless, it was found in [6] that at zero temperature, the dual
+fermionic spectral function shows multiple Fermi surfaces with the topology of nodal lines
+when M/b < (M/b)c while it is gapped when M/b > (M/b)c. This indicates that the
+system undergoes a topological phase transitions from topological NLSM to topologically
+trivial semimetal phase.
+With the regularity condition near the horizon, the system can be further integrated
+to the singularity. In the following we will discuss the interior geometries and singularities
+of the system.
+11
+
+3.1
+Kasner exponents
+Close to the singularity r → rs, similar to the holographic WSM case we again take the
+ansatz
+u ∼ −u0(r − rs)nu ,
+f ∼ f0(r − rs)nf ,
+h ∼ h0(r − rs)nh ,
+φ ∼ nφ ln(r − rs) , (3.6)
+where u0, f0, h0 and nu, nf, nh, nφ are all constants. The other two matter fields Btz and
+Bxy will be determined by the above ansatz.
+The equations of motion can be simplified close to the singularity under the assump-
+tion that the ignored terms are subleading
+u′′
+u − f ′′
+f + h′
+2h
+�u′
+u − f ′
+f
+�
+= 0 ,
+u′′
+2u + f ′′
+f − f ′2
+4f 2 + f ′u′
+fu + 1
+2φ′2 = 0 ,
+f ′2
+4f 2 + f ′h′
+2fh + u′
+2u
+�f ′
+f + h′
+2h
+�
+− 1
+2φ′2 = 0 ,
+φ′′ +
+�f ′
+f + h′
+2h + u′
+u
+�
+φ′ = 0 ,
+B′
+tz − η
+√
+h
+2f (λφ2)Bxy = 0 ,
+B′
+xy −
+ηf
+2
+√
+hu
+(λφ2)Btz = 0 .
+(3.7)
+The first four equations in (3.7) are the same as the ones in holographic WSM. Similarly,
+we obtain
+nh = 2 (1 − nu − nf) ,
+nφ = ±
+�
+(2nf + nu)(1 − nu) − 3n2
+f
+2 .
+(3.8)
+From the last two equations in (3.7) we have the following leading order solutions for the
+two form fields near the singularity
+Bxy ∼ Bxy0 + . . . ,
+Btz ∼ Btz0 + . . . ,
+(3.9)
+where the dots are subleading terms of form (r − rs)2−nu−2nf(log(r − rs))2 and (r −
+rs)2nf(log(r − rs))2 respectively. Here we have assumed 2 − nu − 2nf > 0 and nf > 0,
+otherwise the leading solution of the two form field might be divergent. Similar to the
+holographic WSM, these constants of the two form field depend on the scaling symmetry
+of the system.
+12
+
+Note that in (3.7) we have assumed that the ignored terms are subleading. More
+explicitly, we have assumed
+nu < 2 ,
+2nu + nh < 2 ,
+nu + 2nf < 2 .
+(3.10)
+Note that the last two inequalities of above are consistent with the assumptions used in
+obtaining (3.9). We have checked numerically that the inequalities (3.10) are satisfied for
+the parameters we have considered.
+Similar to the discussion in section 2.2, we can make a coordinate transformation
+(2.13) to write the metric (3.6) into the Kasner form as (2.14) with the parameters (2.16)
+and the Kasner relations (2.17). Here the leading order of the two form fields are constant
+close to the singularity.
+The two conserved charges of the scaling symmetries are
+Q1 = 8
+ηBtzBxy + u
+√
+h
+(f ′h − fh′) ,
+(3.11)
+Q2 =
+f
+√
+h
+(u′h − uh′) .
+(3.12)
+Evaluate them at the horizon and at the singularity we obtain
+8
+ηBxy0Btz0 = u0f0
+�
+h0(nf − nh)
+(3.13)
+and
+4πTf1
+�
+h1 = Ts = u0f0
+�
+h0(2 − 2nf − 3nu)
+(3.14)
+where s is the density of entropy. We have checked the above relations numerically.
+In Fig. 5, we show the Kasner exponents for the holographic NLSM as functions of
+M/b at different temperature T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). We find that
+at low temperature, the Kasner exponents pt, px, pz of the metric fields in the NLSM
+semimetal phase are almost constant in the topological phase (e.g. within the difference
+of order less than 1% between M/b = 0.5 and M/b = 0 at T/b = 0.01), which is quite
+similar to the holographic WSM, while pφ changes a lot in the topological phase. Note
+that this is consistent with the Kasner relations (2.17) since pφ is small. It is expected that
+at extremely low temperature, the properties of the Kasner exponents in the holographic
+NLSM might be the same as those in the holographic WSM, i.e. all the Kasner exponents
+are constant. Due to numerical difficulty we have not explored such a low temperature
+regime.
+Different from the holographic WSM where the geometry is the same as Schwarzschild
+black hole with a constant nonzero Az when M/b = 0. Here when M/b = 0, in the
+13
+
+���
+���
+���
+���
+���
+-����
+-����
+-����
+-����
+-����
+M
+b
+pt
+���
+���
+���
+���
+���
+-���
+-���
+-���
+���
+���
+���
+M
+b
+pϕ
+���
+���
+���
+���
+���
+����
+����
+����
+����
+����
+����
+����
+M
+b
+px
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+M
+b
+pz
+Figure 5: Plots of Kasner exponents for holographic NLSM as a function of M/b. For all cases
+we have T/b = 0.05 (red), 0.02 (blue), 0.01 (purple). The horizontal dashed gray lines represent
+the Kasner exponents for M/b = 0 at T/b = 0.01. The vertical dashed gray lines represent the
+quantum critical point at zero temperature.
+holographic NLSM, due to the fact that the matter fields backreact to the IR geometry
+and the Kasner exponents are no longer the constant exponents of Schwarzschild black
+hole and instead they depend on T/b, as shown in the first three pictures in Fig. 6.
+Nevertheless, at low enough temperature we see that the Kasner exponents are nearly
+constant.
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+-0.298
+-0.296
+-0.294
+-0.292
+-0.290
+-0.288
+-0.286
+T
+b
+pt
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.182
+0.184
+0.186
+0.188
+0.190
+0.192
+T
+b
+px
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.916
+0.918
+0.920
+0.922
+T
+b
+pz
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.902
+0.904
+0.906
+0.908
+0.910
+0.912
+0.914
+T
+b
+τs
+Figure 6: Plots of Kasner exponents and τs for holographic NLSM as a function of T/b when
+M/b = 0.
+14
+
+3.2
+Proper time of timelike geodesics
+Similar to the holographic WSM, we can also discuss the proper time from the horizon
+to the singularity in holographic NLSM. In Fig. 7, we show the proper time τs as a
+function of M/b at different temperatures. Again we see that at low temperature, the
+proper time is almost a constant in the topological phase (e.g. within the difference of
+order less than 5‰ between M/b = 0.5 and M/b = 0 at T/b = 0.01), which shows a
+topological behavior under the changes of the systems. Similar to the holographic WSM,
+we could take the operator which encodes the information of τs as the order parameter
+for the topological phase transition in holographic NLSM. In the trivial phase, the proper
+time τs is monotonically decreasing when we increase M/b.
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+���
+M
+b
+τs
+Figure 7: Plots of the proper time τs from the horizon to the singularity as a function of M/b
+at different temperatures T/b = 0.05 (red), 0.02 (blue), 0.01 (purple).
+4
+Conclusion and discussion
+We have studied the interior geometries of black holes in two different holographic topo-
+logical semimetals. We find that the singularities of the geometries are of simple Kasner
+form, together with a constant one form gauge potential or constant two form fields.
+In the topological WSM phase, all the Kasner exponents are constant taking values of
+Schwarzschild black hole at low temperature. In the topological NLSM phase, the Kasner
+exponents of the metric fields are also almost constant (the difference is of order less than
+1% at T/b = 0.01), while the Kasner exponent of the scalar field is small and changes
+a bit in the topological phase. Moreover, we find the proper times from the horizon to
+the singularity are nearly constant in both holographic WSM and holographic NLSM.
+These features seem to be of topological in the sense that they stay as constant during
+15
+
+the changes of physical parameters of the systems. The proper time in the trivial phases
+of the two holographic semimetal decreases when we increase M/b.
+In addition to the above universal behavior, specific behaviors inside the horizon are
+also found. In the topological phase of holographic WSM, we find the oscillations of
+the matter field φ inside the horizon at low temperature. In other phases we have not
+found any oscillations of fields. The Kasner exponents oscillate in the critical regime
+of holographic WSM. There is no oscillation of background fields in holographic NLSM.
+In the trivial phases of the two holographic semimetals, the Kasner exponents behave
+differently, where the details can be found in Fig. 3 and Fig. 5.
+It would be interesting to connect the topological features of Kasner exponents and
+the proper times in the topological phases of the two holographic semimetals to the
+topological invariants. It is known that they can be extracted from the correlators of
+heavy operators. It is very interesting to determine the precise observables associated
+to these quantities to understand the role played by topology. This would shed light
+on the universal theories describing the topological semimetals. Meanwhile, it is also
+interesting to check the behavior of these physical quantities in the topological phases
+of other holographic topological semimetals, e.g. [18, 25], to check if they are universal
+feature of topological semimetals.
+Acknowledgments
+We are grateful to Matteo Baggioli, Karl Landsteiner, Ya-Wen Sun, Xin-Meng Wu, Jun-
+Kun Zhao for useful discussions. This work is supported by the National Natural Science
+Foundation of China grant No.11875083.
+A
+Equations in holographic WSM
+In this appendix we list the useful equations for calculating the geometries in holographic
+WSM in section 2.
+16
+
+The equations of motion for the action (2.1) are
+Rab − 1
+2gab(R + 12) − Tab = 0 ,
+∇bF ba + αϵabcde(FbcFde + FbcFde) − iq (Φ∗(DaΦ) − Φ(DaΦ)∗) = 0 ,
+∇bFba + 2αϵabcdeFbcFde = 0 ,
+DaDaΦ − m2Φ − λΦ∗Φ2 = 0 ,
+(A.1)
+where
+Tab =1
+2(FacF c
+b − 1
+4gabF2) + 1
+2(FacF c
+b − 1
+4gabF 2) + 1
+2((DaΦ)∗DbΦ + (DbΦ)∗DaΦ)
+− 1
+2gab((DcΦ)∗DcΦ + V (Φ))
+(A.2)
+and DaΦ = ∂aΦ − iqAaΦ.
+There are three different scaling symmetries of the system
+(x, y) → a(x, y) , f → a−2f ;
+(A.3)
+z → az , h → a−2h , Az → a−1Az ;
+(A.4)
+r → ar , (t, x, y, z) → a−1(t, x, y, z) , (u, f, h) → a2(u, f, h) , Az → aAz .
+(A.5)
+For the ansatz (2.2), we have equations
+u′′ + h′
+2hu′ −
+�
+f ′′ + f ′h′
+2h
+�u
+f = 0 ,
+f ′′
+f + u′′
+2u − f ′2
+4f 2 + f ′u′
+fu − 6
+u + φ2
+2u
+�
+m2 + λ
+2φ2 − q2A2
+z
+h
+�
+− A′2
+z
+4h + 1
+2φ′2 = 0 ,
+1
+2φ′2 + 6
+u − u′
+2u
+�f ′
+f + h′
+2h
+�
+− f ′h′
+2fh − f ′2
+4f 2 + A′2
+z
+4h − φ2
+2u
+�
+m2 + λ
+2φ2 − q2A2
+z
+h
+�
+= 0 ,
+A′′
+z +
+�f ′
+f − h′
+2h + u′
+u
+�
+A′
+z − 2q2φ2
+u
+Az = 0 ,
+φ′′ +
+�f ′
+f + h′
+2h + u′
+u
+�
+φ′ − 1
+u
+�q2A2
+z
+h
++ m2 + λφ2�
+φ = 0 .
+(A.6)
+Near the horizon r = rh, the fields can be expanded as follows,
+u = 4πT(r − rh) + · · · ,
+f = f1 − f1Az2
+2m2φ2
+1 + λφ4
+1 − 24
+6Az1q2φ2
+1
+(r − rh) + · · · ,
+h = h1 −
+�
+Az1Az2 + h1Az2
+2m2φ2
+1 + λφ4
+1 − 24
+6Az1q2φ2
+1
+�
+(r − rh) + · · · ,
+Az = Az1 + Az2(r − rh) + · · · ,
+φ = φ1 + Az2
+A2
+z1q2 + h1(m2 + λφ2
+1)
+2Az1h1q2φ2
+1
+(r − rh) + · · · ,
+(A.7)
+17
+
+where T =
+φ2
+1q2Az1
+2πAz2 . Note that there is a shift symmetry r → r + α along the radial
+direction which can be used to fix rh to be any value and we choose rh = 1. There are
+five free parameters T, f1, h1, Az1, φ1 and we can use the scaling symmetries (A.3, A.4) to
+fix f1 = 1, h1 = 1 respectively. Then we can shoot three parameters T, Az1, φ1 to obtain
+the parameters T, M, b of boundary field theory, i.e. the two dimensionless parameters
+T/b, M/b according the scaling symmetry in (A.5) (we work in unit b = 1).
+When r → ∞, the UV expansions are
+u = r2 − M 2
+3
++ M 4(2 + 3λ)
+18
+ln r
+r2 − Mb
+r2 + · · · ,
+f = r2 − M 2
+3
++ M 4(2 + 3λ)
+18
+ln r
+r2 + f3
+r2 + · · · ,
+h = r2 − M 2
+3
++ M 4(2 + 3λ) + 9b2M 2q2
+18
+ln r
+r2 + h3
+r2 + · · · ,
+Az = b − bM 2q2ln r
+r2 + η
+r2 + · · · ,
+φ = M
+r − (3b2Mq2 + 2M 3 + 3λM 3))
+6
+ln r
+r3 + O
+r3 + · · · ,
+(A.8)
+where h3 =
+1
+72M(−72O + 9b2Mq2 + M 3(14 + 9λ)) − 2f3.
+Note that in order to match the expansion (A.8) we should use the shift symmetry of
+the system r → r + α which could change the location of the horizon/singularity.
+B
+Equations in holographic NLSM
+In this appendix, we list the calculations for the geometries in holographic NLSM in
+section 3.
+The equations of motion for the action (3.1) are
+Rab − 1
+2gab(R + 12) − Tab = 0 ,
+∇bFba + 2αϵabcdeFbcFde = 0 ,
+∇bF ba + αϵabcde(FbcFde + FbcFde) − iq1 (Φ∗DaΦ − (DaΦ)∗Φ) + q2
+η ϵabcdeBbcB∗
+de = 0 ,
+DaDaΦ − ∂Φ∗V1 − λΦB∗
+abBab = 0 ,
+i
+3ηϵabcdeHcde − m2
+2Bab − λΦ∗ΦBab = 0 ,
+(B.1)
+18
+
+where
+Tab = 1
+2(FacF c
+b − 1
+4gabF2) + 1
+2(FacF c
+b − 1
+4gabF 2) + 1
+2
+�
+(DaΦ)∗DbΦ + (DbΦ)∗DaΦ
+�
++ (m2
+2 + λ|Φ|2)(B∗
+acB c
+b + B∗
+bcB c
+a ) − 1
+2gab
+�
+(DcΦ)∗DcΦ + V1 + V2 + λ|Φ|2B∗
+cdBcd�
+.
+(B.2)
+With the ansatz (3.4), the equations are
+u′′
+u − f ′′
+f + h′
+2h
+�u′
+u − f ′
+f
+�
+− 4
+u(m2
+2 + λφ2)
+�B2
+tz
+uh + B2
+xy
+f 2
+�
+= 0 ,
+u′′
+2u + f ′′
+f − f ′2
+4f 2 + f ′u′
+fu − 6
+u + 1
+u(m2
+2 + λφ2)
+�B2
+tz
+uh + B2
+xy
+f 2
+�
++φ2
+2u
+�
+m2
+1 + λ1φ2
+2
+�
++ φ′2
+2 = 0 ,
+f ′2
+4f 2 + f ′h′
+2fh + u′
+2u
+�f ′
+f + h′
+2h
+�
+− 6
+u + 1
+u(m2
+2 + λφ2)
+�
+−B2
+tz
+uh + B2
+xy
+f 2
+�
++φ2
+2u
+�
+m2
+1 + λ1φ2
+2
+�
+− 1
+2φ′2 = 0 ,
+B′
+tz − η
+√
+h
+2f (m2
+2 + λφ2)Bxy = 0 ,
+B′
+xy −
+ηf
+2
+√
+hu
+(m2
+2 + λφ2)Btz = 0 ,
+φ′′ + φ′
+�u′
+u + f ′
+f + h′
+2h
+�
+−
+�
+m2
+1 + λ1φ2 − 2λB2
+tz
+uh
++ 2λB2
+xy
+f 2
+� φ
+u = 0 .
+(B.3)
+There are three different scaling symmetries of the system
+(x, y) → a(x, y) , f → a−2f , Bxy → a−2Bxy ;
+(B.4)
+z → az , h → a−2h , Btz → a−1Btz ;
+(B.5)
+r → ar , (t, x, y, z) → a−1(t, x, y, z) , (u, f, h, Bxy, Btz) → a2(u, f, h, Bxy, Btz) .
+(B.6)
+19
+
+Near the horizon r → rh, the fields can be expanded as follows,
+u = 4πT(r − rh) + · · · ,
+f = f1 − 4Bxy2
+�
+8B2
+xy1(m2
+2 + λφ2
+1) + f 2
+1(2m2
+1φ2
+1 + λ1φ4
+1 − 24)
+�
+3Bxy1f1η2(m2
+2 + λφ2
+1)2
+(r − rh) + · · · ,
+h = h1 − 4h1Bxy2
+�
+4B2
+xy1(m2
+2 + λφ2
+1) − f 2
+1(2m2
+1φ2
+1 + λ1φ4
+1 − 24)
+�
+3Bxy1f1η2(m2
+2 + λφ2
+1)2
+(r − rh) + · · · ,
+Bxy = Bxy1 + Bxy2(r − rh) + · · · ,
+Btz = η√h1Bxy1(m2
+2 + λφ2
+1)
+2f1
+(r − rh) + · · · ,
+φ = φ1 + 4Bxy2φ1
+�
+2λB2
+xy1 + f 2
+1(m2
+1 + λ1φ2
+1)
+�
+Bxy1f 2
+1η2(m2
+2 + λφ2
+1)2
+(r − rh) + · · · ,
+(B.7)
+where T = Bxy1η2(m2
+2+λφ2
+1)2
+16πBxy2
+. The strategy of the numerics the same as the holographic
+WSM. We first use the shift symmetry r → r + α to fix rh = 1. Then we also have five
+free parameters T, f1, h1, Bxy1, φ1 and we can use the scaling symmetries (B.4, B.5) to
+fix f1 = 1, h1 = 1 respectively. After that we have only three near horizon parameters
+T, Bxy1, φ1, from which we obtain T, M, b in the dual field theory, which are equivalently
+two dimensionless parameters T/b, M/b according the scaling symmetry (B.6).
+Near the boundary r → ∞, we have
+u = r2 − 2b2 − M 2
+3
++ 8b4 + M 4(2 + 3λ1)
+18
+ln r
+r2 − Mb
+r2 + · · · ,
+f = r2 − M 2
+3
++ 8b4 + M 4(2 + 3λ1)
+18
+ln r
+r2 + f3
+r2 + · · · ,
+h = r2 − 2b2 − M 2
+3
++ 8b4 + M 4(2 + 3λ1)
+18
+ln r
+r2 + h3
+r2 + · · · ,
+Bxy = br + 2b3 ln r
+r
++ b2
+r + · · · ,
+Btz = br − 2b3 ln r
+r
+− b (b2 + M 2(1 + λ)) + b2
+r
++ · · · ,
+φ = M
+r − M 3(2 + 3λ1)
+6
+ln r
+r3 + O
+r3 + · · · ,
+(B.8)
+where b2 =
+1
+48b (−56b4 + 72(2f3 + h3) − 8b2M 2(2 + 3λ) − M 4(14 + 9λ1) + 72MO).
+Note that to match the expansion (B.8) we should use the shift symmetry r → r + α
+which could change the location of the horizon/singularity .
+20
+
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+1 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Statistical Machine Translation for Indic 
+Languages 
+Sudhansu Bala Das, Divyajoti Panda, Tapas Kumar Mishra, 
+and Bidyut Kr. Patra 
+National Institute of Technology(NIT), Rourkela, Odisha, 
+India 
+Indian Institute of Technology (IIT), Varanasi, Uttar Pradesh, 
+India 
+ 
+ 
+Abstract 
+Machine Translation (MT) system generally aims at automatic 
+representation of source language into target language retaining the 
+originality of context using various Natural Language Processing (NLP) 
+techniques. Among various NLP methods, Statistical Machine Trans- 
+lation (SMT) is a very popular and successful architecture used for 
+both low as well as high-resource languages. SMT uses probabilis- 
+tic and statistical techniques to analyze information and conversion. 
+This paper canvasses about the development of bilingual SMT mod- 
+els for translating English to fifteen low-resource Indian Languages 
+(ILs) and vice versa.  At the outset, all 15 languages are briefed with 
+a short description related to our experimental need. Further, a de- 
+tailed analysis of Samanantar and OPUS dataset for model building, 
+along with standard benchmark dataset (Flores-200) for fine-tuning 
+and testing, is done as a part of our experiment. Different preprocess- 
+ing approaches are proposed in this paper to handle the noise of the 
+dataset. To create the system, MOSES open-source SMT toolkit is 
+explored. “Distance” reordering is utilized with the aim to understand 
+the rules of grammar and context-dependent adjustments through a 
+
+2 
+ 
+ 
+ 
+ 
+ 
+phrase reordering categorization framework. In our experiment, the 
+quality of the translation is evaluated using standard metrics such as 
+BLEU, METEOR, and RIBES. 
+ 
+1 Introduction 
+Technology reaches new heights through its journey from the origins of ideas 
+to their full-scale practical implementation. One such journey is heading to- 
+wards elimination of language barrier in order to establish a seamless social 
+communication in every domain. In this regard, advancement on relevant 
+fields such as Natural Language Processing (NLP), Machine Learning (ML) 
+and Artificial Intelligence (AI) based Language Modelling (LM) significantly 
+contributes for evolving a flawless automatic Machine Translation (MT) sys- 
+tem (Dorr et al ( 2004)). Irrespective of various heuristic approaches to 
+maintain both lexical and contextual interpretation of source language(s) 
+onto the translated target language(s), it is still challenging to cope with 
+required fluency, adequacy, accent, and overall accuracy (Chapelle et al ( 
+2010)). However, it is feasible with the advent of modern NLP (AI-based) 
+approaches wherein a high-quality and high resource (i.e. large quantity 
+of corpora available) parallel corpus (translation pairs in source and target 
+languages) is required to train a good translation system. Hence, for high- 
+resource languages having massive digital footprint across the globe, MT sys- 
+tems prove to be quite efficient with adequate training. On the other hand, 
+it becomes very complicated for low-resource languages suffering from uni- 
+versal recognition and scanty digital presence. Such imbalance often leads to 
+poor-quality translation in presence of low-resource language(s) in the form 
+of either target or source. Therefore, MT systems need to understand the 
+syntax (rules to combine words), semantics (meaning of words and combi- 
+nations), and morphology (rules to cover morphemes - smallest meaningful 
+units - into words) of such low-resource languages (Somers (2011)). 
+Based on the heuristic paradigms, MT models are classified into rule-based 
+(RBMT), example-based (EBMT), statistical (SMT), and neural (NMT) sys- 
+tems (Tripathi et al (2010)). Each has its own advantages and disadvantages. 
+RBMT models follow a set of rules to define a language and the interaction 
+between different linguistic devices (words, phrases, sentences) in the lan- 
+guage (Jussà et al (2012), Michael et al (2000)). These sets of rules and 
+systems defined for a translation in a language pair are hard-coded on the 
+
+3 
+ 
+ 
+ 
+ 
+ 
+machine. The linguistic information used in an RBMT model is mainly 
+the target and source languages collected from unilingual (one language), 
+bilingual (two languages), or multilingual (more than two languages) dic- 
+tionaries. In addition, the model also uses grammar covering the syntactic, 
+semantic, and morphological regularities of each language. However, a well- 
+built RBMT model requires highly skilled and expert human labour due to 
+its complexity making it hard to build. In addition, the ambiguous proper- 
+ties of languages make them prone to take more time and efforts to resolve, 
+especially in large and complex models. RBMT models require a lot of effort 
+to be made functional in day-to-day life. Hence, the need for more efficient 
+translation systems than RBMT still persists. EBMT methods make use of a 
+large number of translation examples (John (2005)). Notably, EBMT mod- 
+els make use of bilingual corpora manipulation, i.e. the breaking down of 
+a bilingual corpus into smaller parts, translating those parts into the target 
+language, and recompiling it to form whole translated sentences. They do 
+not account much for the syntax, semantic and morphological analysis of the 
+target and source language (like RBMT models). In contrast, SMT is better 
+when compared to RBMT and EBMT models, as it does not require human 
+intervention (Adam (2008)). It is a way of translation wherein a statistical- 
+based learning algorithm is applied to a large bilingual corpus that helps 
+the machine learn the translation. This method also enables the machine to 
+translate sentences not encountered by the machine during its training and 
+testing. The objective of SMT is to convert an input word sequence from the 
+source language into the target language. It has dominated academic MT 
+research and a portion of the commercial MT sector in less than two decades. 
+On the other hand, neural machine translation (NMT) is performed using a 
+neural network (NN) (Stasimioti (2020)). Unlike SMT, NMT does not have a 
+distinct translation model, language model, or model for reordering. Instead, 
+it has a single sequence model that determines one word at a time. The pre- 
+diction is based on the source sentence effort previously generated sequence 
+in the target language. NMT is a deep learning-based method of machine 
+learning that utilizes a large NN that relies on word vector representations. 
+Even though the NMT has achieved remarkable results in a few trans- 
+lation experiments using high-resource language, researchers are unsure if 
+the NMT could actually replace SMT and if its success would extend to 
+other tasks. Eventually, the experiment of (Michał (2016)) on the corpus 
+of the United Nations (consisting of 15 low-resource languages) brings the 
+fact. From the result of his experiment, it is evident that the performance of 
+
+4 
+ 
+ 
+ 
+ 
+ 
+SMT is better than that of NMT for the majority of cases, as measured by 
+BLEU score. Many researchers (Lohar et al (2019), Zhou et al (2017), Wang 
+et al (2017), Castilho et al (2017)) have pointed out various disadvantages 
+of NMT over SMT using low resource language, such as the fact that NMT 
+requires more corpus and resources than SMT. In comparison with SMT, 
+NMT training typically takes longer. Additionally, research has shown that 
+when there is a domain incompatibility between testing and training data, 
+SMT performance is superior to that of NMT (Xing et al (2018), Mahata et 
+al (2018)). Long sentences are another area where SMT excels. 
+English and ILs are languages with less parallel text data, which motivates us 
+to work with ILs. This research examines the effectiveness of SMT systems on 
+low-resource language pairs, of which many are rarely worked on. The dataset 
+used in our experiment for all fifteen Indian languages is tested for the first 
+time for all languages using SMT. Hence, the objective of this work is to build 
+an MT system using SMT for languages such as Assamese (AS), Malayalam 
+(ML), Bengali (BN), Marathi (MR), Gujarati (GU), Kannada (KN), Hindi 
+(HI), Oriya (OR), Punjabi (PA), Telugu (TE), Sindhi (SD), Sinhala (SI), 
+Nepali (NE), Tamil (TA), and Urdu (UR) to English (EN) and vice versa and 
+to check the effectiveness of SMT with low-resource language pairs. 
+Our main goal is to develop an MT system for low-resource languages, i.e., ILs, 
+that can serve as a baseline system. The following is a summary of our 
+work’s main contributions: 
+• To the best of our knowledge, this work is the first attempt to use SMT 
+with the Samanantar and OPUS Dataset to investigate the MT for all 
+fifteen IL-EN and EN-IL pairs (both directions), including both the 
+Dravidian and Indo-Aryan groups. 
+• To bring forth the linguistic approach of ILs in terms of translation. 
+Scripts, writing style, and grammar with proper examples are also dis- 
+cussed. 
+• Various data filtration methods are investigated in order to clean the 
+data and improve translation quality. 
+• Distance-based reordering is utilized to check the translation quality of 
+ILs. 
+
+5 
+ 
+| 
+| 
+ 
+ 
+ 
+ 
+• Better realistic assessment of translation quality is possible from the 
+presentation of results, as obtained using different automated metrics 
+like BLEU, METEOR, and RIBES. 
+This paper is arranged as follows. Subsections 1.1 and 1.2 give some 
+insight into SMT and cover the ILs used for our experiments. In Section 2, 
+some prominent works on SMT and NMT using ILs are described. The 
+experimental framework, including an overview of the dataset and method- 
+ology, is explained in Section 3. Section 4 narrates some of the prominent 
+metrics used for MT evaluation. Results are presented in Section 5 followed by 
+the conclusion and future direction in Section 6. 
+ 
+1.1 SMT 
+Statistical Machine Tramslation (SMT) is dependent on statistical methods 
+(Philipp et al (2007), Richard et al (2002), Mary et al (2011) ). It is a data-
+driven technique that makes use of parallel-aligned corpora. It utilizes 
+mathematical equations to calculate the likelihood of source-to-target lan- 
+guage translation. Probability P (Tl Si) is assigned by SMT. Here Tl is the target 
+language and Si is the source input. It utilizes Bayes’ theorem to 
+determine the maximum probability P (Tl|Si), which is as follows: 
+P (Tl | Si) ∝ P (Tl)P (Si | Tl) 
+(1) 
+SMT consists of three phases: the language model(LM) P (Tl) for target 
+language probability calculation, the translation model(TM) P (Si Tl) for 
+conditional probability estimation of the target to the source language, and the 
+decoder model (DM), which searches among possible source sentences the 
+one which maximizes probabilities (Kumawat et al (2014)). 
+To calculate the probability of a sentence, the LM utilizes the n-gram model. It 
+assigns the probability of a single word to the last n words that come before it 
+in the sentence and estimates the translation’s likelihood. The chain rule aids 
+in breaking down the sentence into conditional probability products. 
+ 
+P (s) = P (w1, w2, w3, ..., wn) 
+=  P (w1)P (w2|w1)P (w3|w1w2)P (w4|w1w2w3)...P (wn|w1w2...wn−1) 
+=  P (w1)P (w2|w1)P (w3|w1w2)P (w4|w1w2w3)...P (wn|w1w2...wn−k) 
+(2) 
+Where, P (s) is the probability of the sentence s, consisting of words w1, w2, 
+..., wn, assuming a k-gram model. It utilizes the bilingual parallel corpus 
+
+6 
+ 
+ 
+ 
+ 
+ 
+of the desired language pair. This is accomplished by calculating the like- 
+lihood of words or phrases extracted from sentences. The DM is the final 
+and most crucial phase of SMT. It assists in the selection of words with the 
+highest probability to be translated by maximizing the likelihood, i.e. 
+P (Tl)P (Si | Tl). 
+1.2 Language preference 
+India is a multilingual nation where people from various states use a variety of 
+regional tongues. Such diversity of language brings difficulty in commu- 
+nicating with one another for information exchange. Further, limitations in 
+public communication also bring inconvenience to share feelings, thoughts, 
+opinions and facts, as well as to deal with business purposes. Moreover, there 
+are many helpful resources available on the internet in English but many In- 
+dians struggle to take benefit of those due to language barriers. Hence, it is 
+crucial to have an easy translation solution for regional languages to support 
+effective communication and to help utilising global resources. To make it 
+possible, technological innovation are continuing to find out efficient methods 
+for a flawless translation using machines, because it is impractical to have hu- 
+man translators everywhere. For machine translation, an enormous amount 
+of resources is required for training with a proper knowledge-base (rules) for 
+better efficiency so as to fulfill the demand of a flawless translation solu- 
+tion. For translation, understanding the meaning of words is important, but 
+words are not enough to constitute a language as a whole. They must be 
+used in sentence construction that adheres to strict grammar rules and every 
+language is having its own writing style. In our work, 15 commonly spoken 
+languages (over various regions of India) are chosen. Table 1 describes the 
+languages used in our experiments with their linguistic features (ethnologue 
+(2022)). A short introduction about them in terms of translation is given 
+below. 
+English(EN) 
+English language is the primary language of roughly 45 countries and 
+is spoken by nearly 1,132 million people. It is written in Roman script, 
+which uses both uppercase and lowercase characters. English uses the 
+subject-verb-object structure. For example (expl1), “The poor man 
+took food”, and (expl2) “food took the poor man”. When the position 
+
+7 
+ 
+ 
+ 
+ 
+ 
+of the subject changes in the preceding sentences, the significance and 
+meaning of the English sentence change. 
+Assamese(AS) 
+Over 15 million native Assamese speakers live in the state of Assam in 
+the northeastern region of India. It is one of Assam’s official languages. 
+Additionally, it is spoken in various regions of other northeastern In- 
+dian states. It uses the Bengali-Assamese script and is written left to 
+right. It also follows the SOV format. “Gita is eating mango” is an 
+English sentence that when translated into Assamese became গীতাই আম 
+খাই আআছ which follows subject object verb format. গীতাই (Gita, 
+subject), আম(mango, object) and খাই আআছ(is eating, verb). 
+Malayalam(ML) 
+People in Kerala and a few societies in Karnataka and Tamil Nadu use 
+Malayalam for communication. This language is spoken by about 35 
+million citizens. It uses the SOV style of writing and a nominative- 
+accusative case marking sequence. It is written in Malayalam script in 
+left-to-right fashion. Sentence like സീതയ്ക്ക് ചിത്തരചന ഇഷ്ട മാണ് 
+which in English became “Sita loves drawing”. Here the word 
+സീതയ്ക്ക് 
+(Sita,Subject), ഇഷ്ടമാണ് 
+(loves,Verb) and ചിത്തര 
+ചന(drawing,Object). 
+Bengali(BN) 
+It is the primary language of Bangladesh and the second most spoken 
+language in India.   Over 265 million people use it as their primary 
+or second language. Approximately 11 million Bengali speakers exist 
+in Bangladesh. In India, states such as Assam, Tripura, and West 
+Bengal use this language. It is a member of the Indo-Aryan family. In 
+Bengali sentences, the standard word order is Subject-Object-Verb. For 
+example, in sentence আ রাি জ ভাত খায় which in English is “Rosy eats 
+rice ”. Here আ রাি জ (Rosy, Subject), ভাত (Rice, object) and খায় 
+(Eats,Verb). 
+Marathi(MR) 
+Marathi is associated with the Sanskrit-derived group of Indian lan- 
+guages and is used by 95 million people in India for communication, 
+primarily in the central and western regions. The fourth most widely 
+spoken language in India is Marathi, which has a sizable native-speaker 
+
+8 
+ 
+ 
+ 
+ 
+ 
+population. Similar to Hindi and Nepali, Marathi is written in the De- 
+vanagari script in left-to-right order. It follows the Subject-Object-Verb 
+order. For example, the sentence तो दध ि पतो, which means “He drinks 
+milk.”  in  English,  has  तो दध   
+ject), and ि पतो(drinks, 
+verb). 
+Gujarati(GU) 
+ि पतो where  तो(He,  subject),  
+दध 
+(Milk, ob- 
+Gujarati is spoken by 45 million citizens in Gujarat and is associated 
+with the Indo-Aryan group. It uses the SOV writing style and is drafted 
+from left to right in Gujarati script. For example, in the sentence તે 
+આઈસ્ક્રીમ ખાય છે. which in English is “He is eating ice cream.” where તે 
+is subject, આઈસ્ક્રીમ is an object and verb is ખાય છે. 
+Kannada (KN) 
+Karnataka’s official language is Kannada, which is also widely used 
+in other parts of India. In India, about 36 million people speak and 
+write Kannada. Despite being a Dravidian language with extensive 
+historical literature, Kannada has few computational linguistics re- 
+sources, making it challenging to study the language’s literature due 
+to its semantic and syntactic diversity. Subject-Object-Verb is the way 
+the Kannada language is structured. Kannada is a highly agglutina- 
+tive language. It uses the left-to-right Kannada script For example, 
+ರಾಮ ಶಾಲೆಗೆ ಹೋದ(SOV) is in English is “Rama went to school”. Here, 
+ರಾಮ(Rama, Subject), ಶಾಲೆ(school, object), and ಹೋದರು(went, verb). 
+Hindi(HI) 
+Hindi is one of the official and national languages of India. There are 
+more than 615 million people who use Hindi as their primary language, 
+and even more than 341 million who speak it as a second language. 
+However, the sentence structure is Subject Object Verb as shown in 
+the example: गीता स्क  ल जाती है। is in English is “Geeta goes to school”. In 
+this sentence, गीता (Gita ,Subject), स्क  ल (School, Object) and जाती 
+है(Goes, Verb). The Indian Constitution mandates that Hindi written 
+in Devanagari be used as the Union’s official language. 
+Oriya(OR) 
+The Oriya language is the primary language of Odisha, a state in east- 
+ern India. Oriya belongs to the Eastern Indo-Aryan group of languages. 
+
+9 
+ 
+Its standard format is subject-object-verb (SOV) and is written in the   
+ 
+Odia script from left to right.  
+Punjabi (PA) 
+Punjabi text is written in a subject-object-verb format and is spoken 
+in India and Pakistan, and a few small groups in the United Kingdom, 
+United Arab Emirates, Malaysia, the United States, South Africa, and 
+Canada. It is written in two scripts: the western Perso-Arabic Shah- 
+mukhi script and the eastern Gurmukhi script. Gurmukhi is drafted 
+from left to right, whereas Shahmukhi is written in the opposite direc- 
+tion. ਅਸ ੀਂ ਭਾਰਤ  ਹਾੀਂ is in English “We are Indians” where ਅਸ ੀਂ(We,Sub- 
+ject), ਭਾਰਤ  (Are,Verb) and ਹਾੀਂ (Indians,Object). 
+Telugu(TE) 
+Telugu is the official language of two Indian states in the south: Andhra 
+Pradesh and Telangana. It is also spoken by the Telugu-speaking im- 
+migrant communities in the United States, Canada, and the United 
+Kingdom. Text structure in Telugu takes the form of a subject-object- 
+verb and from left to right.ఆమె నన్ను  కొటి్టంది in English “she beat 
+me” where ఆమె(she, Subject), నన్ను (Me, Object) and కొటి్టంది(beat, 
+Verb). 
+Sindhi(SD) 
+Sindhi is a language spoken by 25 million speakers in Pakistan and 5 
+million in India. It is written in a modified Perso-Arabic script in Pak- 
+istan (right-to-left), whereas it is written in a variety of scripts in India, 
+like Devanagari, Khudabadi, and Gurmukhi (left-to-right). It follows the 
+Subject-Object-Verb format. For example, the sentence “Partha 
+loves books” is رٿﭘﺎ ﮐﻲ ﺘﺎﺑﻦڪ ﻦﺳﺎ رﭘﻴﺎ ﻫﻲآ where ٿﺎرﭘﺎ(Partha, subject), 
+ﮐﻲ ﺘﺎﺑﻦڪ (books, object) and ﻦﺳﺎ رﭘﻴﺎ ﻫﻲآ (loves, verb). 
+Nepali(NE) 
+It is official language and the lingua franca in Nepal, and also spoken 
+by some communities in India. Nepali is written in left-to-right De- 
+vanagari script. It is a language written in Subject-Object-Verb order For 
+example, “Sita ate apples” when converted to the Nepali language 
+
+10 
+ 
+ 
+ 
+ 
+ 
+becomes सीताले स्याउ खाइन्. Here, सीताले(Sita, subject), स्याउ(apples, 
+object) and खाइन्(ate, verb). 
+Sinhala(SI) 
+The majority of Sri Lankans speak Sinhala as their first language. Sin- 
+hala is an Indo-Aryan language that differs from English in terms of 
+grammatical structure, morphological variation, and subject-object- 
+verb (SOV) word order. It is written in right-to-left Sinhala script. 
+A sentence like “Pavan writes a letter” is in Sinhala is පවන් ලිපියක් 
+ලියයි where පවන්(Pavan, subject),    ලිපියක්(a letter, object) and 
+ලියයි(writes, verb). 
+Tamil(TA) 
+Tamil is a language spoken primarily in Tamil Nadu, a state in southern 
+India, as well as in countries with a large Tamil diaspora, which includes 
+Sri Lanka, Malaysia, and Singapore, to name a few. The phonological 
+differences exist within Tamil Nadu between southern, western, and 
+northern speech. Tamil is a Dravidian language of the southern branch, 
+with a rich literary tradition dating back over 2000 years. Tamil spoken 
+in India and Sri Lanka are two different dialects. It uses the Subject 
+Object Verb format. For example sentence: “I like paintings” in Tamil 
+becomes எனக்கு ஓவியங்கள் பிடிக்கும் where the Iஎனக்கு (I, 
+Subject), விலங்குகள் (Paintings, Object) and பிடிக்கும் (Like, 
+Verb). 
+Urdu(UR) 
+It is Pakistan’s national language and is also spoken widely in India. 
+In Pakistan and India, Urdu is spoken by over 170 million citizens and is 
+also spoken in some communities in the United Kingdom, the United 
+States, and the United Arab Emirates. Script for Urdu is a modified 
+and revised version of the Perso-Arabic script. Urdu writing structure is 
+Subject Object Verb. For example “she reads a book” which in Urdu is 
+وہ ﯾﮏا بﮐﺘﺎ ﭘﮍﻫﺘﯽ ۔ہﮯ where وہ(she, Subject), ﯾﮏا بﮐﺘﺎ(book, object) and 
+ﭘﮍﻫﺘﺎ ہﮯ(reads, verb). 
+
+11 
+ 
+ 
+ 
+ 
+ 
+Table 1: Linguistic Features of Languages Used in MT Experiments 
+ 
+ 
+Languages 
+Script 
+Word 
+Order 
+Family 
+Number 
+of Speakers 
+(in millions) 
+Writing 
+Direction 
+Assamese (AS) 
+Bengali 
+SOV 
+Indo-European 
+15 
+left to right 
+Malayalam (ML) 
+Malayalam 
+SOV 
+Dravidian 
+38 
+left to right 
+Bengali (BN) 
+Bengali 
+SOV 
+Indo-European 
+265 
+left to right 
+Marathi (MR) 
+Devanagari 
+SOV 
+Indo-European 
+95 
+left to right 
+Gujarati (GU) 
+Gujarati 
+SOV 
+Indo-European 
+60 
+left to right 
+Kannada (KN) 
+Kannada 
+SOV 
+Dravidian 
+36 
+left to right 
+Hindi (HI) 
+Devanagari 
+SOV 
+Indo-European 
+615 
+left to right 
+Oriya (OR) 
+Oriya 
+SOV 
+Indo-European 
+38 
+left to right 
+Punjabi (PA) 
+Perso-Arabic, 
+Gurmukhi 
+SOV 
+Indo-European 
+125 
+right to left 
+left to right 
+Telugu (TE) 
+Telugu 
+SOV 
+Dravidian 
+93 
+left-to-right 
+Sindhi (SD) 
+Devanagari 
+Perso -Arabic 
+SOV 
+Indo-European 
+25 
+left to right 
+right to left 
+Sinhala (SI) 
+Sinhala 
+SOV 
+Indo-European 
+17 
+left to right 
+Nepali (NE) 
+Devanagari 
+SOV 
+Indo-European 
+24 
+left to right 
+Tamil(TA) 
+Tamil 
+SOV 
+Dravidian 
+81 
+left to right 
+Urdu (UR) 
+Urdu 
+SOV 
+Indo-European 
+170 
+right to left 
+English (EN) 
+Roman 
+SVO 
+Indo-European 
+1,132 
+left to right 
+ 
+2 Related Work 
+A few works on SMT using some Indic Languages are discussed in this sec- 
+tion. 
+(Dasgupta et al (2004)) has discussed a technique for English (EN) to Bengali 
+(BN) MT that utilizes the syntax of EN sentences to BN while minimizing 
+the time of translation. In the process to create the target sentences, a dic- 
+tionary is used to know the object and subject, as well as other entities like 
+person and number in their work. 
+English-to-Hindi (EN-HI) SMT system has been created by (Ananthakrish- 
+nan et al (2009)) using morphological and syntactic pre-processing in SMT 
+
+12 
+ 
+ 
+ 
+ 
+ 
+model. In their work, the suffixes in HI language are segmented for mor- 
+phological processing before rearranging the EN source sentences as per HI 
+syntax. 
+In 2010, research has been conducted by (Zbib et al (2010)) at MIT, USA, 
+using the grammatical structures in statistical machine translation with the 
+Newswire corpus for Arabic to EN language to give better translation results. 
+Work on Kannada-to-English MTS with SMT, by (Kumar et al (2015)), using 
+Bible corpus on 20,000 sentences shows a remarkable feat with 14.5 BLEU 
+score which is even supported by (Papineni et al (2002)).(Kaur et al (2011)) 
+has presented a translation model based on SMT for English (EN) to Punjabi 
+(PA) with their own corpus containing 3844 names in both languages with 
+BLEU and word accuracy as 0.4123 (with range 0-1) and 50.22%, respec- 
+tively. 
+(Nalluri et al (2011)) has created “enTel,” an SMT-based EN to Telugu(TE) 
+MT system, using the Johns Hopkins University Open Source Architecture (Li 
+et al (2009)). For the purpose of training the translation system, TE par- allel 
+dataset from the Enabling Minority Language Engineering (EMILLE) is 
+used for their work. 
+In the year 2014, an SMT Framework for Sinhala(SI)-Tamil(TA) MT Sys- 
+tem has been created by (Randil et al (2014)). In their work, the result 
+of SMT-dependent translation between language pairs, including TA-SI and 
+SI-TA has been shown. Outcomes of the experiments using the SMT model 
+give more noticeable results for the SI-TA than the TA-SI language pair. For 
+languages closely related, SMT shows remarkable results. 
+In 2017, a survey has been conducted by (Khan et al (2017)) on the IL-EN 
+language MT models reveal the importance of SMT over 8 languages i.e. 
+Hindi (HI), Bengali (BN), Gujarati (GU), Urdu (UR), Telugu (TE), Pun- 
+jabi (PA), Tamil (TA), and Malayalam (ML). In their work, EMILLE corpus 
+(Nalluri et al (2011)) is used and Moses SMT model is preferred to make 
+the translation models, with out-of-vocabulary (OOV) words transliterated 
+to EN. In their work, the evaluation using BLEU, NIST and UNK counts as 
+metrics reveals the overall SMT performance as satisfactory (PA-EN and UR-
+EN models as the best and the HI-EN and GU-EN models as the worst). An 
+EN-BN SMT system has been presented by (Islam et al (2010)). In their work, 
+to handle OOV (out-of-vocabulary) words, a transliteration module is 
+presented. In order to address the systematic grammatical distinctions be- 
+tween EN and BN, a preposition handling module has been added. BLEU, 
+NIST and TER scores has been used to check the effectiveness of their sys- 
+
+13 
+ 
+ 
+ 
+ 
+ 
+tem. 
+Nowadays, NMT is widely appreciated for its advancement in the develop- 
+ment of machine translation with remarkable improvement in quality. Hence, 
+many researchers have compared both techniques for low and high-resource 
+languages. 
+(Antonio et al (2017)) has performed a thorough evaluation using statistical- 
+based and neural machine translation systems for nine language directions 
+along a variety of dimensions. In their experiment, for long sentences, SMT 
+systems perform better than the NMT. Recently, (Castilho et al (2017)) has 
+used automatic metrics and expert translators to conduct a thorough quan- 
+titative and qualitative comparison of NMT and SMT. SMT shows better 
+according to their experiments. 
+The comparison of NMT and SMT for the Nepali (NE) using the Nepali 
+National Corpus (NNC) with 6535 sentences has been shown by (Acharya et al 
+(2018)). The researchers have proved in their experiments that the SMT model 
+performs better than the NMT-based system with a small corpus with a 5.27 
+BLEU score. 
+In 2021, Long Short-Term Memory networks (LSTMs) integrated with atten- 
+tion mechanism using WAT corpus have been used in experiments by (Singh 
+et al (2003)) to achieve a 15.7 BLEU score as opposed to a baseline of 14.5 
+BLEU score. 
+(Abujar et al (2021)) has developed a BN-EN MT model on AmaderCAT cor- 
+pus using Sequence-to-Sequence (seq2seq) architecture, a special class of Re- 
+current Neural Networks to develop the translation system and has achieved 
+a BLEU score of 22.3. 
+In the year 2021, translation of English and Hindi-to-Tamil languages us- ing 
+both SMT and NMT has been presented by (Akshai et al (2021)). The 
+disadvantages of NMT have been shown in their experiments such as the 
+occurrence of numerous errors by NMT when interpreting domain terms and 
+OOV (Out of vocabulary) phrases. NMT frequently constructs inaccurate 
+lexical choices for polysemous words and occasionally counters reordering 
+mistakes while translating words and domain terms. The translations that 
+have been generated by the NMT models mostly include repetitions of pre- 
+viously transcribed words, odd translations, and many unexpected sentences 
+having no correlation with the original sentence. 
+
+14 
+ 
+ 
+ 
+ 
+3 Experimental Framework 
+3.1 Dataset 
+Samanantar and OPUS datasets for model building and standard benchmark 
+dataset i.e. Flores 200 for testing are utilized. Samanantar is the largest cor- 
+pus collection for ILs (Gowtham et al (2022)). The collection includes more 
+than 45 million sentence pairs in English and 11 ILs. The Samanantar Corpus 
+has been used for Assamese (AS), Malayalam (ML), Bengali (BN), Marathi 
+(MR), Gujarati (GU), Kannada (KN), Hindi (HI), Oriya (OR), Punjabi (PA), 
+Telugu (TE), and Tamil (TA) for the experiments. OPUS is a large resource 
+with freely available parallel corpora. The corpus includes data from many 
+domains and covers over 90 languages (Tiedemann (2012)). The OPUS cor- 
+pus is used for Sinhala (SI), Sindhi (SD), Urdu (UR), and Nepali (NE). Table 2 
+gives statistics of the dataset used in our experiments. 
+FLORES-200 (Marta et al (2022)) dataset is a multilingual parallel dataset 
+with 200 languages, that are used as human-translated benchmarks. It con- 
+sists of two corpora, labeled “dev” (997 lines) and “devtest” (1013 lines). 
+The “dev” dataset has been used for fine-tuning, and the “devtest” dataset 
+has been used for testing. 
+ 
+3.2 Methodology 
+Our proposed process comprises of following major steps: 
+1. Setting up SMT System Moses SMT Toolkit is used to build our 
+SMT systems. It is written in C++ and Perl. At the moment, this is 
+one of the best SMT tools available. First, Moses, GIZA++ (Och 
+(2003)), CMPH (for binarization) and SRILM in Ubuntu are installed. 
+For training, fine-tuning and testing processes, the system needs a par- 
+allel corpus of the language pair in addition to configurable phases 
+according to developer’s choice to follow. 
+2. Data Preprocessing A qualitative corpus plays a major role in any MT 
+task. While obtaining corpora from various sources, data qual- ity i.e. 
+critical for the effectiveness of an MT system, can never be 
+ascertained. So, removing unnecessary noise is an important task be- 
+fore using the data to train our statistical machine translation model. 
+Following processes are used to preprocess and clean it: 
+
+15 
+ 
+ 
+ 
+ 
+ 
+Table 2: Parallel corpus statistics 
+English to Indic 
+Parallel Corpus(Sentences) 
+Assamese (AS) 
+ 
+0.14M 
+Malayalam (ML) 
+5.85M 
+Bengali(BN) 
+8.52M 
+Marathi(MR) 
+3.32M 
+Gujarati(GU) 
+3.05M 
+Kannada(KN) 
+4.07M 
+Hindi(HI) 
+8.56M 
+Oriya(OR) 
+1.00M 
+Punjabi(PA) 
+2.42M 
+Telugu(TE) 
+4.82M 
+Sindhi(SD) 
+1.95M 
+Sinhala(SI) 
+8.68M 
+Nepali(NE) 
+3.35M 
+Tamil(TA) 
+5.16M 
+Urdu(UR) 
+8.95M 
+ 
+• Data Cleaning and Formatting The goal of data cleaning is 
+either to find and fix or to delete erroneous data from the corpus. 
+Here, characters those are used neither in ILs nor in English are 
+removed. Some of the punctuation in extended Unicode is con- 
+verted to its standard counterpart. Numbers in the IL corpus are 
+converted from English to IL scripts. Characters outside the stan- 
+dard alphabets of the language pair, extra spaces, and unprintable 
+characters are also removed from the corpus. The preprocessing 
+techniques used in our work have been summarized as follows: 
+– Removing unprintable characters 
+– Removing characters outside the language pair 
+– Removing extra spaces 
+– Deaccenting accented characters 
+– Changing non-standard Unicode punctuation characters in 
+both corpora to their standard counterparts 
+– Changing uncommon punctuations to more common ones 
+– Changing numbers to a uniform numbering system and script 
+
+16 
+ 
+ 
+ 
+ 
+ 
+3. Tokenization: It is the process of dividing a character sequence into 
+smaller units known as tokens based on a given character sequence and 
+a specified document unit. Words, punctuation, and numerals serve as 
+these tokens in our instance. The corpus is tokenized using a modified 
+Moses tokenizer (Koehn et al (2007)). Redundant punctuations (quo- 
+tation marks, apostrophes, and commas) are also removed from the 
+corpus. 
+4. Training Truecasing Model: This is the procedure for adding case 
+information to text that has been incorrectly cased or is not cased (Lita 
+et al (2003)). Data sparsity is lessened with the use of true casing. A 
+truecaser model (a model which changes the words at the beginning 
+of the sentence to the most common casing) is trained on the training 
+dataset. The Moses truecasing is used for the same. 
+5. Training Language and Translation Models: In MOSES, the 
+training procedure utilizes word and segment occurrences to draw con- 
+nections between the target and source languages. The language and 
+translation models are trained on the training dataset and binarized. 
+GIZA++ grow-diag-final-and alignment is used for word alignments, 
+which start with the intersection of the two alignments and then add 
+the additional alignment points. 
+The grow-diag-final-and model starts with the intersection of the align- 
+ments from source to target and target to source, then two steps are 
+used to add additional alignment points (Och (2003)): 
+grow-diag: For every neighboring point to the alignments measured, 
+if either source or target word is not aligned already but is present 
+in the union of the alignment, then the neighboring point is in- 
+cluded in the alignment. 
+final: If any phrase pairs are unaligned but present in the union, add 
+the point to the alignment. 
+• Word Alignment Model: After preprocessing the words, the 
+next step is word alignment. The proposed work employs the 
+GIZA++ (Och (2003)) incorporation of the IBM models to ac- 
+complish the word procedure. The GIZA++ model assesses the 
+likelihood of word-to-word alignment for each source and target 
+word in each sentence. To produce a good-quality word alignment, 
+
+17 
+ 
+− 
+− 
+− 
+− 
+ 
+ 
+ 
+ 
+the alignment is produced using a series of successive estimations. 
+To process a corpus with a larger quantity of sentences, the process 
+takes several hours. The alignment method’s outcomes establish 
+a connection between the target and source words. 
+• Reordering It is the process of restructuring the word order of 
+one natural language sentence to make it more similar to the word 
+order of another natural language sentence. It is a critical task 
+in transcription for languages with different syntactic structures. 
+The Moses system learns different reordering possibilities for each 
+phrase during the training process. Instead of default reordering, 
+the model uses the distance reordering model (Kumawat et al 
+(2014)). 
+– Distance-Based Reordering: The reordering of the tar- get 
+output phrases is represented by the relative distortion 
+probability distribution re (St, Et 1). Here, St refers to the 
+starting position of the source phrase that is interpreted into 
+the t   1 th target phrase.   The reordering distance (St - Et 
+1) is calculated as follows: When taking source words out of 
+sequence, the reordering distance is the number of words ig- 
+nored (either forward or backward). If two phrases are trans- 
+lated in sequence, then t = Et 1 +1; that is, the first word of 
+the phrase immediately follows the last word of the pre- 
+vious phrase. A reordering cost of re(0) is used in this case. 
+The distance-based model assigns a linear cost to reordering 
+distance, implying that the movement of phrases over long 
+distances is more expensive. 
+6. Fine tuning: It is the process of determining the best configuration file 
+settings for a translation model when it is used for a specific pur- pose. 
+It uses a translation model to translate all 15 ILs source language phrases 
+in the tuning set. Then, it compares the model’s output to a set of 
+reference (human translations) and adjusts the settings to improve 
+translation quality. This procedure is repeated several times. The 
+tuning process repeats the steps with each iteration until the transla- 
+tion quality is optimized. The model is fine-tuned on the preprocessed 
+Flores-200 dev dataset. 
+7. Translation: The final model is used to translate the preprocessed 
+
+18 
+ 
+ 
+ 
+ 
+ 
+Flores-200 devtest dataset from the source to the target language. 
+8. Postprocessing and Detokenization: Redundant punctuation marks 
+(quotation marks, apostrophes, and commas) are removed, and the 
+translation file is detokenized using the Moses detokenizer. 
+9. Evaluation: The evaluation metrics use for our experiments are ME- 
+TEOR (Banerjee et al (2005)), RIBES (Wołk et al (2016)), and BLEU 
+(Papineni et al (2002)). 
+ 
+4 Essential metrics for MT translation evalu- 
+ation 
+The most crucial phase of any MT system is MT evaluation. Both automatic 
+and manual methods can be applied to analyze MT tasks. The effective- 
+ness of a system’s output can be evaluated either directly through human 
+assessments, or indirectly using reading cases, other downstream activities, 
+and even through estimating the amount of effort necessary to rectify the 
+output. A better outcome is obtained through manual evaluation, which 
+includes task-based evaluations, fluency and adequacy scores, human vot- ing 
+for translations task, post-editing measures, etc. However, the major 
+challenges of manual evaluation are time-intensiveness, absence of repeata- 
+bility and high cost. In order to evaluate the effectiveness of MT output, 
+different automated approaches are there such as Metric for Evaluation of 
+Translation with Explicit Ordering (METEOR), Bilingual Evaluation Un- 
+derstudy(BLEU), Levenshtein, Rank-based Intuitive Bilingual Evaluation 
+Score(RIBES), Word Error Rate (WER) and NIST exist. Several intuitive 
+advantages exist for automated metrics that can give points for synonyms or 
+paraphrases. A few of the evaluation metrics which are used in our work are 
+discussed below 
+1. Bilingual Evaluation Understudy (BLEU): The most widely used 
+method for evaluating machine translation (MT) is known as BLEU. 
+This method, first introduced in 2002 (Papineni et al (2002)) exam- ines 
+one or more reference translations to the hypothetical translation. When 
+the hypothetical translation matches numerous strings with the 
+reference translation, the MT evaluation gives it a higher score. The 
+BLEU system assigns a translation a score from 0 to 1. However, it is 
+ 
+
+19 
+ 
+1 
+ 
+ 
+ 
+usually represented as a percentage value. The nearer the translation is 
+to 1, the more it corresponds to the reference translation. This match- 
+ing of translation is conducted word-by-word in the same word order 
+in both datasets. SacreBLEU is used to calculate the BLEU scores of 
+baseline models. 
+2. Rank-based Intuitive Bilingual Evaluation Score (RIBES): It 
+is calculated by incorporating a rank correlation coefficient before uni- 
+gram matches, eliminating the necessity for higher-order n-gram matches. 
+This metric is concerned with word order. To compare SMT and ref- 
+erence translations, it employs Kendall’s tau coefficient (τ ) based on 
+word order to indicate rank differences (Wołk et al (2016)). To assure 
+positive values, the coefficient is normalized as shown below: 
+ 
+ 
+Normalized Kendall’s τ (NKT) = τ + 1 
+2 
+(5) 
+This coefficient can be paired with unigram-precision p1 and Brevity 
+Penalty BP and changed to prevent overestimation of the correlation 
+between only relevant words in SMT and reference translations. 
+ 
+ 
+RIBES = NKT.(pα).(BPβ ) 
+(6) 
+Here, α and β are parameters between 0 and 1. 
+
+20 
+ 
+ 
+ 
+ 
+ 
+3. Metric for Evaluation of Translation with Explicit Ordering 
+(METEOR): Meteor scores a translation depending on explicit word- 
+by-word similarities between both the translation and a provided ref- 
+erence translation (Banerjee et al (2005)). It is specifically created to 
+generate sentence-level scores that are highly correlated with human 
+evaluations of translation quality. Meteor utilizes and highlights recall 
+in combination with precision, a feature that numerous measures have 
+verified as crucial for a strong correlation with human judgments. It also 
+intends to address the problem of imprecise reference translations by 
+utilizing adaptable word matching in consideration with synonyms and 
+morphological variances. To achieve a score of 1, the words of the 
+machine-generated output should be present in the reference and each 
+of the words of the reference is in the machine-generated output. 
+ 
+5 Results and Discussion 
+In this work, the evaluation metrics used are METEOR (Banerjee et al 
+(2005)), RIBES (Wołk et al (2016)), and BLEU (Papineni et al (2002)). All 
+the evaluation metrics used in our work are prominent metrics for deter- 
+mining the quality of the machine-translated text. 
+Table 3 displays the translation of all the 15 ILs to English and vice versa 
+using SMT without fine-tuning. Evaluation metrics of SMT with finetuning 
+using the Flores-200 dev dataset are shown in Table 4. RIBES and METEOR 
+range is 0-1. For EN-IL and IL-EN language using SMT, the BLEU score 
+lies between 0.46 to 13.09 and 0.49 to 15.41 respectively. The RIBES score for 
+EN-IL and IL-EN is between 0.04 to 0.63 and 0.14 to 0.61 respectively. 
+METEOR scores lie between 0.01 to 0.28 for EN-IL and 0.02 to 0.28 for 
+IL-EN. SMT models using distance reordering techniques are giving better 
+BLEU Scores for languages BN, PA, UR, HI, and GU than the rest. With- 
+out fine-tuning, SI performs the worst in terms of all three metrics of all 
+languages in both directions, whereas with fine-tuning EN-SI and TA-EN 
+perform worse than all other EN-IL and IL-EN models respectively with all 
+
+21 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Table 3: Evaluation Metrics Result of SMT without Finetuning 
+ 
+Languages 
+Pairs 
+BLEU 
+RIBES 
+METEOR 
+AS 
+EN-AS 
+1.90 
+0.50 
+0.09 
+AS-EN 
+3.21 
+0.46 
+0.11 
+ML 
+EN-ML 
+3.79 
+0.27 
+0.08 
+ML-EN 
+4.59 
+0.43 
+0.12 
+BN 
+EN-BN 
+6.41 
+0.62 
+0.17 
+BN-EN 
+3.06 
+0.45 
+012 
+MR 
+EN-MR 
+3.17 
+0.43 
+0.09 
+MR-EN 
+3.62 
+0.43 
+0.09 
+GU 
+EN-GU 
+7.62 
+0.56 
+0.16 
+GU-EN 
+10.14 
+0.59 
+0.21 
+KN 
+EN-KN 
+5.06 
+0.40 
+0.11 
+KN-EN 
+7.17 
+0.51 
+0.16 
+HI 
+EN-HI 
+13.09 
+0.63 
+0.28 
+HI-EN 
+15.41 
+0.64 
+0.28 
+OR 
+EN-OR 
+3.92 
+0.59 
+0.14 
+OR-EN 
+6.41 
+0.52 
+0.17 
+PA 
+EN-PA 
+7.22 
+0.63 
+0.18 
+PA-EN 
+11.7 
+0.61 
+0.24 
+TE 
+EN-TE 
+8.16 
+0.42 
+0.12 
+TE-EN 
+5.77 
+0.52 
+0.18 
+SD 
+EN-SD 
+1.29 
+0.39 
+0.08 
+SD-EN 
+2.48 
+0.35 
+0.09 
+SI 
+EN-SI 
+0.93 
+0.05 
+0.02 
+SI-EN 
+0.49 
+0.14 
+0.05 
+NE 
+EN-NE 
+6.00 
+0.58 
+0.16 
+NE-EN 
+8.29 
+0.53 
+0.19 
+TA 
+EN-TA 
+2.78 
+0.16 
+0.05 
+TA-EN 
+2.64 
+0.31 
+0.07 
+UR 
+EN-UR 
+9.43 
+0.62 
+0.24 
+UR-EN 
+11.35 
+0.61 
+0.23 
+ 
+
+22 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Table 4: Evaluation Metrics Result of SMT with Finetuning 
+ 
+Languages 
+Pairs 
+BLEU 
+RIBES 
+METEOR 
+AS 
+EN-AS 
+2.17 
+0.50 
+0.08 
+AS-EN 
+3.21 
+0.42 
+0.10 
+ML 
+EN-ML 
+2.05 
+0.23 
+0.06 
+ML-EN 
+1.84 
+0.27 
+0.06 
+BN 
+EN-BN 
+8.26 
+0.63 
+0.19 
+BN-EN 
+12.16 
+0.60 
+0.23 
+MR 
+EN-MR 
+2.43 
+0.39 
+0.08 
+MR-EN 
+2.49 
+0.36 
+0.07 
+GU 
+EN-GU 
+5.82 
+0.52 
+0.14 
+GU-EN 
+3.56 
+0.45 
+0.01 
+KN 
+EN-KN 
+3.35 
+0.09 
+0.14 
+KN-EN 
+3.67 
+0.41 
+0.10 
+HI 
+EN-HI 
+8.64 
+0.57 
+0.22 
+HI-EN 
+5.38 
+0.49 
+0.14 
+OR 
+EN-OR 
+5.25 
+0.58 
+0.15 
+OR-EN 
+2.22 
+0.39 
+0.11 
+PA 
+EN-PA 
+5.71 
+0.60 
+0.15 
+PA-EN 
+7.75 
+0.55 
+0.19 
+TE 
+EN-TE 
+4.4 
+0.38 
+0.10 
+TE-EN 
+3.34 
+0.44 
+0.12 
+SD 
+EN-SD 
+1.59 
+0.41 
+0.09 
+SD-EN 
+2.53 
+0.38 
+0.09 
+SI 
+EN-SI 
+0.46 
+0.04 
+0.01 
+SI-EN 
+3.11 
+0.37 
+0.11 
+NE 
+EN-NE 
+4.00 
+0.55 
+0.14 
+NE-EN 
+5.25 
+0.49 
+0.13 
+TA 
+EN-TA 
+1.86 
+0.16 
+0.05 
+TA-EN 
+1.03 
+0.08 
+0.02 
+UR 
+EN-UR 
+6.34 
+0.56 
+0.19 
+UR-EN 
+7.07 
+0.54 
+0.18 
+
+23 
+ 
+ 
+ 
+three metrics. HI and BN languages have qualitative, large, and less noisy 
+datasets compared to other languages. Hence, HI performs the best among all 
+languages without fine-tuning in all three metrics in both directions, and BN 
+performs the best among all languages with fine-tuning in both direc- tions 
+with respect to BLEU and RIBES. In addition, UR and PA also produce good 
+RIBES metrics than other languages. RIBES score for PA is 0.63(for EN-
+PA) and 0.61(PA-EN), and for UR, RIBES score is 0.62(EN-UR) and 
+0.61(UR-EN). 
+Even though SI has a good amount of corpus, the corpus does not have reli- 
+able translations compared to other languages. For example, the sentence in 
+English “Heb. 11:32-34; Judg. 16:18-21, 28-30 Jehovah’s spirit operated on 
+Samson in a unique way because of unusual circumstances” has been trans- 
+lated to Sinhala in the corpus as “11:32-34; Gනි.”, which only translates 
+“Heb. 11:32-34;”. Hence, SI does not perform well compared to other lan- 
+guages. Similarly, in the EN-TA corpus, the sentence “He’s my boss” has 
+been translated to “அவர் எனது ே மலாளர் மட்டும்தான் .” which ac- 
+tually means “He is only my manager”. From the example, it is clear that 
+EN-TA corpus also has ambiguity. Additionally, even though the ILs-English 
+and English-ILs systems are trained using the same corpus, a significant dis- 
+crepancy in the BLEU scores is observed. This is due to the significant 
+morphological diversity of ILs and the relative difficulty of translating from 
+English to ILs. It has been observed that SI has a high number of lines (8.68 
+M) but performs poorly as compared to languages like PA (2.42 M) and GU 
+(3.05 M). It is also observed that languages with very steep slopes tend to 
+have low scores. For example, EN-TA and EN-ML have 60% sentences with 
+less than 4 tokens, and they have not-so-good scores as shown in Figure 1. 
+In contrast, languages with good scores, like HI and BN have more gentle 
+slopes. So, length of sentences is a contributing factor. EN-SD is an ex- 
+ception which has a gentle slope but does not give good scores, because the 
+corpus does not have good translation quality. Therefore, the quality of the 
+corpus matters more than the size of the dataset. 
+ 
+6 Conclusion and Future Work 
+This paper has presented the MT work for 15 ILs to English and vice versa 
+using SMT. It also describes the linguistic features of all 15 ILs. A tailor- 
+made preprocessing approach has been incorporated into this work. The 
+
+24 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 1: Less than ogive for number of tokens in a sentence for all fifteen 
+language corpora 
+
+LessthanOgiveforNumberoflokens
+1.2
+given
+numberoftokenslessthan
+1
+0.8
+0.6
+ofsentences
+0.4
+Ratio
+0.2
+0
+0
+1
+2
+m
+4
+5
+6
+7
+8
+9
+Numberoftokens
+EN-AS(AS)
+EN-SI(SI)
+EN-ML(ML)
+EN-MR(MR)
+EN-OR(OR)
+EN-SD(SD)
+EN-UR(UR)
+—EN-BN(BN)
+EN-GU(GU)
+EN-HI(HI)
+EN-KN(KN)
+EN-NE(NE)
+EN-PA(PA)
+EN-TA(TA)
+EN-TE(TE)25 
+ 
+ 
+ 
+ 
+ 
+model has utilized the grow-diag-final-and alignment model and distance re- 
+ordering model. For checking the quality of translation, different evaluation 
+Metrics such as BLEU, RIBES, and METEOR are utilized in this work. 
+From the result, it is observed that the proposed SMT model is quite satis- 
+factory for some of the ILs. However, the level of performance is not at par 
+with the rest of the ILs and there lies the need of improvement to be made. 
+Due to the scarcity and quality of parallel corpus, the metrics obtained are 
+quite low. 
+It has been observed that the translations of some of the languages are not 
+sufficiently accurate. Measures of validating corpus quality shall be explored 
+in order to observe the corpus quality and remove inaccurate lines. Dravid- ian 
+languages are in general agglutinative languages (words are made up of 
+morphemes, with each morpheme contributing to the meaning of the word). 
+In future, means to infer translations from the breakdown of words in these 
+languages shall also be explored. 
+Interestingly, in some of the ILs, finetuning schemes are hampering the qual- 
+ity. The causes of this phenomenon shall be analyzed and mitigated via 
+techniques such as noise reduction, corpus cleaning, and finetuning schemes 
+for those languages to ensure better quality. In addition, more language 
+pairs and corpora can be analyzed and evaluated using various other meth- 
+ods. Other techniques, such as hybridized SMT-NMT systems and the usage 
+of other alignment and reordering models will be studied for further course 
+of research. 
+ 
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+arXiv:2301.01023v1  [physics.comp-ph]  3 Jan 2023
+Performance investigation of supercapacitors
+with PEO-based gel polymer & ionic liquid
+electrolytes: Molecular Dynamics Simulation
+Nasrin Eyvazi,† Davood Abbaszadeh,† Morad Biagooi,‡ and SeyedEhsan Nedaaee
+Oskoee∗,†,¶
+†Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan
+45137-66731, Iran
+‡Intelligent Data Aim Ltd (IDA Ltd), Science and Technology Park of Institute for
+Advanced studies in Basic Sciences, Zanjan 45137-65697, Iran
+¶Research Center for Basic Sciences & Modern Technologies (RBST), Institute for
+Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran
+E-mail: [email protected]
+Phone: (+98) 241-415-2217. Fax: (+98) 241-415-2104
+Abstract
+Due to the importance of using supercapacitors in electronic storage devices, im-
+proving their efficiency is one of the topics that has attracted the attention of many
+researchers. Choosing the proper electrolyte for supercapacitors is one of the most sig-
+nificant factors affecting the performance of supercapacitors. In this paper, two classes
+of electrolytes, i.e.
+liquid electrolyte (ionic liquid electrolyte) and solid electrolyte
+(polymer electrolyte) are compared by molecular dynamics simulation. We consider
+the polymer electrolyte in linear and network configurations. The results show that
+1
+
+although ionic liquid-based supercapacitors have a larger differential capacitance, since
+they have a smaller operation voltage, the amount of energy stored is less than poly-
+mer electrolyte-based supercapacitors. Also, our investigations indicate that polymer
+electrolyte-based supercapacitors have more mechanical stability. Therefore, they can
+be considered a very suitable alternative to liquid electrolyte-based supercapacitors that
+do not have known liquid electrolyte problems and display better performance.
+Introduction
+Supercapacitors (SCs), also known as Electric Double Layer Capacitors (EDLCs), have re-
+cently attracted much attention in the field of electrical energy storage. The SCs fill the gap
+between batteries and conventional capacitors in terms of energy and power density. They
+consist of two porous electrodes immersed in an electrolyte. Due to the potential differ-
+ence between the electrodes, the charged electrodes repel the co-ions in the electrolyte while
+attracting their counter-ions, resulting in charge separation and charge storage.
+The SCs have higher energy density in comparison with conventional capacitors due
+to their porous electrodes with large surface areas and small charge separation distances.1
+Also, compared to batteries, SCs have the advantages of higher power density induced by
+a fast charging/discharging rate (in seconds), a long cycle life (4,100,000 cycles), and high
+power density.1 Despite their higher power density, they cannot store the same amount of
+energy as batteries.1 Extensive efforts and research have been devoted to increasing the
+energy density of SCs to 20-30 Wh/L to solve the problems and satisfy the performance
+demands.2–4 According to relation E = 1
+2CV 2, the energy density (E) of SCs is proportional
+to the capacitance (C) and the square of the voltage (V ). Therefore, increasing either the
+capacitance or the voltage of a cell can be an effective way to achieve high energy density.2
+The efficiency of SCs depends mainly on both electrolyte and electrode structure. The
+pore size and surface area of electrodes, ionic conductivity, and electrolyte operating voltage
+window play a significant role in developing high-performance and flexible SCs.5 Especially
+2
+
+in the case of liquid electrolytes, they have some disadvantages for use in flexible SCs like
+being toxic and corrosive,2 requiring high-cost packaging to fabricate, and are associated
+with leakage problems.2,6 In general, the critical features of an ideal electrolyte include: (1)
+a wide voltage and temperature window; (2) a high ionic conductivity; (3) a high chemical
+and mechanical stability; (4) well-matched with the electrolyte materials; (5) low volatility
+and flammability; (6) safety; and (7) simple processing with low cost.6–8
+To overcome the limitations of liquid electrolytes, polymer electrolytes (PEs) were in-
+troduced. In 1970, Armand first used PE in Lithium Ion Batteries (LIBs) and proposed
+LIBs with improved efficiency and energy density.9 PEs consist of a macromolecule matrix
+dissolved in a low viscosity and high dielectric constant organic solvent.9 PEs have many
+advantages such as avoiding liquid leakage and corrosion problems, good ionic conductivity,
+high chemical and mechanical stability, high energy density, safety, solvent-free condition,
+being light in weight, low cost, and simple manufacturing process.6–9
+Due to the advantages of PEs, they are ideal candidates for use in SCs as electrolytes.
+PEs for SCs can be classified into three categories: (1) solid polymer electrolytes (SPEs),
+(2) gel polymer electrolytes (GPEs), and (3) polyelectrolytes. The SPE is composed of a
+polymer (e.g., PEO) and a salt (e.g., LiCl), without any solvents. The ions in the SPE are
+transported through the polymer2,5 and the polymer works as a host matrix for ion move-
+ment.2,10 In contrast, the GPE consists of a polymer host (e.g., PVA) and a liquid electrolyte
+or a conducting salt dissolved in a solvent.2,11 The polymer in GPE is swollen by the solvent
+and acts as a dynamic moving matrix. The conductivity of ions occurs through the solvent
+instead of the polymer phase.2,11 In GPEs, the liquid electrolyte generally provides free ions
+that participate in conductivity enhancement and also acts as a conductive medium. In
+addition, the polymer provides perfect mechanical stability by increasing the viscosity of
+the electrolyte.5 Recently, researchers have shown that using Ionic Liquids (ILs) can im-
+prove ionic conductivity and cell voltage, resulting to improvement in the electrochemical
+performance of GPE.5 By its softening effect on the polymer chains, IL can increase the
+3
+
+electrolyte’s ionic conductivity and facilitate ion transfer.5 GPE based on ILs and linear
+polymers usually exhibit poor mechanical properties, including both strength and flexibil-
+ity, because of their few polymer chain entanglements separated by small molecules.12 To
+generate polymer networks, cross-linking strategies have been proposed in recent years. The
+behaviors and performance of polymer gels are largely determined by the structure of the
+polymer network that makes up the gel. This is due to the interaction between the net-
+work and the solvent.5 Gels generally have high mobility because the polymer networks are
+dissolved by a large amount of entrapped solvent.5
+In the polyelectrolyte, ionic conductivity is created by charged polymer chains.12 As
+it turns out, each type of these solid-state electrolytes has its advantages and disadvan-
+tages. Typically, GPEs have the highest ionic conductivity among the three types of solid-
+state electrolytes.5,12 Due to the liquid phase in the GPE, its ionic conductivity is signif-
+icantly higher than dry SPE. Therefore, GPE-based SCs currently dominate the products
+of solid electrolyte-based SCs.
+Several polymer matrices have been explored for prepar-
+ing GPEs in the role of host polymer including: poly (vinyl alcohol) (PVA), poly (acrylic
+acid) (PAA), potassium polyacrylate (PAAK), poly (ethyl oxide) (PEO), poly (methyl-
+methacrylate) (PMMA), poly (ether ether ketone) (PEEK), and poly (vinylidene fluoride-
+co-hexafluoro-propylene) (PVDF-HFP).2
+Beyond all of the experimental achievements,6,13–16 modeling the EDLCs under differ-
+ent physical conditions would provide a lot of insight into the system’s physics. Modeling
+the microstructure will reveal how the related dynamics for charge carriers happen. So far,
+numerous types of research have been done on liquid electrolyte-based SCs to explore the
+performance of different kinds of liquid electrolyte-based SCs and the effect of electrode struc-
+ture and its pore size.17–20 Our previous work,21 models the third classification of polymer
+electrolytes (polyelectrolytes). In this work, considering the advantages of GPEs compared
+to other polymer electrolytes and their useful applications, we investigated the behavior of
+GPE-based SCs using the molecular dynamics simulation method. Here, we compared IL-
+4
+
+based and GPE-based SCs in order to find out the effects of adding host polymers to the
+IL electrolyte. In the first section, we describe the method of simulation under different
+conditions. Based on the results we get from our model, we discuss the pros and cons of
+using GPEs and IL electrolytes in SCs.
+Method
+Systems’ model and force field parameters
+Molecular Dynamics (MD) simulations have been widely used to describe the behavior of SCs.
+Using MD simulations, we compared liquid electrolyte-based SCs with PE-based SCs under
+various conditions. The simulations were carried out by the CAVIAR software package.22
+We have investigated the performance of three different systems with similar electrodes but
+different electrolytes. The electrolytes were confined between two electrodes with single slit-
+pore geometry, placed at a distance of 15 nm from each other. This model called a slit-pore
+model was introduced by Breitsprecher et al18,23 for simulating porous media.
+The slit-pore length was set to 15 nm, in order to be compared with the bulk region and
+the width of the pore is 2.5 nm (larger than the size of two ionic particles) so that at least
+two ionic particles can pass through the pore.24
+To compare the electrolyte role we have simulated the above mentioned pore structure
+with different electrolyte systems; liquid electrolyte (system A), linear polymer electrolyte
+(system B), and polymer network electrolyte (system C). By comparing the results, we can
+discuss the effects and the function of different electrolytes.
+System A: Liquid electrolyte-based supercapacitors
+Our first model contains IL electrolytes confined between those described electrodes. We
+used Coarse-Grained (CG) models of ILs, where ions are soft spheres with diameters dcation
+= danion = 1 nm and valency q = ± 1, in units of the elementary charge. The mass of
+5
+
+cations and anions are mcation = 117.17 g
+mol and manion = 86.81 g
+mol corresponding to EMIM+
+and BF4-.
+All the particles interact with the non-bonding Lennard-Jones (LJ) potential:
+VLJ(rij) =
+
+
+
+
+
+4εij[( σij
+rij )12 − ( σij
+rij )6]
+;
+r ≤ rcut
+0
+;
+r ≥ rcut
+(1)
+Where rij is the relative distance between each pair of particles, ε and σ are the length
+and energy parameters. In calculating the LJ potential, rcut =
+6√
+2 σ is the cutoff length
+to ensure short-range repulsive force at any distance. In addition, charged particles also
+interact through the electrostatic non-bonding potential:
+VC(rij) =
+�
+i
+�
+i<j
+zizje2
+4πǫ|ri − rj|
+(2)
+The ǫ and e are the electric permittivity and the electrical charge. The z = ±1 is the valency
+of ions. For the electrostatic interactions we used the electric permittivity of ǫ = 10, a typical
+value for ILs at 300 ◦K.17
+System B: Linear polymer electrolyte-based supercapacitors
+In system B, we added polymer hosts to system A to simulate GPE-based SCs. By com-
+paring these two systems’ behavior, we can find out the advantages of one over another.
+Poly(ethylene oxide) (PEO) with the formula H3C-O-(CH2-CH2-O)n-CH3,25 the most com-
+mon polymer with a broad range of applications in polymer chemistry,26 biotechnology,27
+and medical science,9,28 is used as a host polymer in our simulations. We used MARTINI-like
+CG models for PEO simulation.25,28 Fig.1 represents the CG model for PEO2.
+Generally, the MARTINI potential consists of bond, angle, LJ, electrostatic, and torsional
+terms. In our system, two consecutive beads in each chain are connected by a harmonic
+6
+
+Figure 1: PEO2 and its CG model.
+stretching spring whose potential is taken to be:
+Vbond(r) = 1
+2Kb(r − r0)2
+(3)
+Where Kb is the bond force constant, r is the instantaneous bond length, and r0 is the
+equilibrium length of the bond. Bond angle is the angle formed between three atoms across
+at least two bonds which can be described as:
+Vangle(θ) = 1
+2Kθ(cosθ − cosθ0)
+(4)
+and the torsion angle (also called dihedral angle) is defined by 3 consecutive bonds involving
+4 atoms:
+Vdihedral(φ) =
+4
+�
+n=1
+Kφ,n(1 + cos(nφ − φn))
+(5)
+Where n and φ are the multiplicities and offsets of the n individual dihedral terms. The
+force constants for these bonded interactions are listed in Table.1.25
+Table 1: Parameters for bonded interactions for modeling CG PEO.
+Bond
+Angle
+Dihedral
+r0(Å)
+Kb(
+kJ
+mol nm2)
+θ0 (deg)
+Kθ ( kJ
+mol)
+φn (deg)
+Kφ ( kJ
+mol)
+n
+3.30
+17000
+130
+85
+180
+1.96
+1
+0
+0.18
+2
+0
+0.33
+3
+0
+0.12
+4
+7
+
+PEO2
+PEO2 (CG)The LJ parameters in Eq.1 for CG PEO beads were set to σij = 5 Å and εij = 3.375
+kJ
+mol.
+Each chain in CG PEO modeling has 25 spherical beads connected with the mass mPEO =
+60.05376
+g
+mol.
+System C: Network polymer electrolyte-based supercapacitors
+Nowadays, polymer networks have received much attention due to their characteristics and
+have been used in various applications.29 In network polymer structures, all polymer chains
+are directly or indirectly linked to each other.
+The cross-linking of polymer chains into
+complex networks is a promising strategy to improve the mechanical strength of a GPE and
+provide dimensional stability at high temperatures30 which has been widely investigated.29
+As a model for the polymer network, we consider a cross-link of linear chains as follows.
+First, we highlighted some monomers as cross-links. As the cross-links are closer than the
+distance r0 = 3.30 Å, a new permanent bond is formed between the cross-linkers, acting
+like cross-linked monomers.29 Once equilibrium is reached, the polymer network can be used
+instead of linear polymers in our simulations. Fig. 2a demonstrates the linear polymers we
+consider in our simulation as system B and Fig. 2b displays cross-linked polymers in system
+C.
+Simulation details and electrodes model:
+The simulations were performed using the Langevin thermostat to keep the temperature
+constant.
+Thus, the equation of motion of the system was calculated by the Langevin
+relation:
+m¨ri = −∇iU({ rj(t)}) − γm˙ri + Fi.
+(6)
+The first term describes the deterministic forces between particles (the force acting on atom
+i due to the interaction potentials), and the last two terms implicitly consider the effect of
+the solvent by coupling the system to a Langevin thermostat which maintains a constant
+8
+
+(a)
+(b)
+Figure 2: (a) Linear polymers distribution in system B and (b) Cross-linked polymers in
+system C.
+average temperature of the system. The parameter γ is the friction coefficient and Fi is a
+Gaussian distributed random force with31
+⟨Fi(t)⟩ = 0
+⟨Fi(t)Fj(t′)⟩ = 6kBTmγδijδ(t − t′).
+(7)
+Our simulation box included 630 charged particles, where half of which are cations and
+the remains are anions, and 1575 monomers. The density of the bulk region was set to
+1.07
+1
+nm3 and the pore size in three defined systems was equal. To simplify the units, the
+reduced LJ unit was used. The length was scaled with ion size ˜l = σ = 5 Å, and the mass
+unit was ˜m = 144
+g
+mol. The energy unit and the charge unit were ˜ε = 1
+kJ
+mol and ˜q = e.
+By applying ˜t =
+�
+˜mσ2
+˜ε , the time unit was obtained ˜t = 5 ps. In addition, temperature
+9
+
+and voltage were scaled as ˜T =
+˜ε
+kB
+= 120.267 ◦K and ˜V = ˜q
+˜ε = 0.01036 V . Temperature
+was kBT = 2.5 ε which corresponded to 300 ◦K in real unit. The friction coefficient in the
+Langevin equation was set to 1
+γ = 1
+˜t and the time step was ∆t = 0.0005 ˜t equal to 6 fs.
+The electrodes were built from carbon atoms with the following parameters: σC = 3.37
+Å and εC = 1
+kJ
+mol.
+Here, using the Poisson to Laplace Transformation (PLT) method,
+which is recently developed in the CAVIAR package,22 the electrodes are surfaces with a
+constant potential. Periodic boundary condition applied in the XY plane and the long-range
+coulombic interaction performed using a 1D Ewald algorithm32 with RC = 15 σ as the cutoff
+distance for electrostatic interaction. The systems were simulated at 17 various potential
+differences between electrodes: 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25,
+3.5, 3.75, 4 V. Calculations were done after the system reached equilibrium.
+Results and Discussion
+We begin our discussion with a snapshot of three defined systems. Fig.3 illustrates a snap-
+shot from cross-section of the systems configuration and ions separation in the equilibrium
+condition for ∆V = 2 V . In order to find out the charge separation process and the behavior
+of these three systems, it is necessary to conduct numerous calculations and measurements.
+We will discuss this below.
+Charging and Differential Capacitance
+Initially, we discuss the results of the charging process. Fig.4 demonstrates average charge
+density accumulation on the surface of the electrodes for the three systems as a function of the
+applied potential difference between the electrodes. According to the figure, increasing the
+potential difference results in a high concentration of counterions on the electrodes’ surface.
+Therefore average induced charge grows by increasing the applied potential until reaching
+saturation. For system A, the induced charge is greater than in the two other systems. Based
+10
+
+15 nm
+4.825 nm
+4.825 nm
+2.5 nm
+8.1 nm
+(a)
+10 nm
+10 nm
+15 nm
+15 nm
+2.5 nm
+(b)
+2.5 nm
+15 nm
+15 nm
+10 nm
+10 nm
+(c)
+Figure 3: The systems configuration at ∆V = 2V after equilibration for (a) System A (b)
+System B and (c) System C.
+on the figure, the saturation voltages for system A is ∆V = 2 V , and for systems B and C is
+∆V = 3.5 V . Since IL-based SCs are saturated at a lower voltage than polymer-based SCs,
+their operating voltage window is also smaller.
+For systems B and C, the induced charges on the electrodes for voltages ∆V < 1.25 V ,
+are almost the same. In voltages, ∆V > 1.25 V , induced charges in system C, are greater
+than system B until both systems reach saturation in ∆V = 3.5 V .
+On the other hand, charge storage in SCs is described by the critical quantities of the
+Differential Capacitance, Cd(V ), and the energy density. The Cd(V ) is the derivative of
+the electrodes surface charge with respect to the applied potential difference between the
+electrodes:33
+Cd(V ) = dq
+dV .
+(8)
+According to the Eq.8, the Cd(V ) is the derivative of the curves in Fig.4. Fig.5, which is
+obtained by the numeric derivation of smooth curves of q(V ), shows the Cd(V ) plots for
+three systems.
+Based on Kornyshev theory for most ILs, Cd(V ) displays a bell-shape, with a maximum at
+11
+
+ 0
+ 5
+ 10
+ 15
+ 20
+ 25
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+Charge Density (µC/cm2)
+Voltage (V)
+System A
+System B
+System C
+Figure 4: Mean induced charge density on the electrodes surface. The points in the figure
+display the simulation data and the solid lines in the plots represent the smoothed data.
+the Potential of Zero Charge (PZC) or a camel-shape with two peaks. In both cases, Cd(V )
+decreases at large potential difference. The reason is, the ions are only allowed to pack up to a
+given maximum density in the double layer. Therefore, by increasing the potential difference,
+the ions concentration near the electrodes’ surface reaches their maximum value and the
+effective diffuse layer thickness actually grows larger, leading to a decrease in Cd(V ).34,34
+Fig.5 is a plot of the Cd(V ) for three systems, which display camel-shaped figures. There
+is limited change in the Cd(V ) curve for system B, indicating the surface charge density
+changes less rapidly in system B with increasing potential. In contrast, there is a peak at
+∆V = 1.5 V in the Cd(V ) plot of system C which points out that more counter-ions can be
+condensed on the electrodes’ surface of system C in comparison with system B. The Cd(V )
+peak for system A is the highest among systems B and C. The result of Korneyshev theory
+is observed in Fig.4 and Fig.5. In systems A, B and C, the saturation voltage is 2, 3 and 3.5
+12
+
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+Differential Capacitance (µF/cm2)
+Voltage (V)
+System A
+System B
+System C
+Figure 5: Differential Capacitance for three systems.
+V . Due to this, in the Cd(V ) plot, the DC drops at voltages above these values.
+As mentioned before, energy density is another useful parameter that shows the efficiency
+of SCs. SC with a higher energy density is more efficient in electrical devices. The stored
+energy density in SCs is obtained:35
+E(V ) =
+� V
+0
+V Cd(V ) dV
+(9)
+Based on the above equation, the plot of energy density for the three systems is calculated
+and plotted in Fig.6. According to this plot, for ∆V < 2 V the energy density of system
+A is greater than systems B and C. For ∆V > 2 V systems B and C have higher energy
+density which indicates that the GPEs-based SC has more efficiency than IL-based SCs in
+higher potential difference.
+13
+
+ 0
+ 5
+ 10
+ 15
+ 20
+ 25
+ 30
+ 35
+ 40
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+Energy Density (µW/cm2)
+Voltage (V)
+System A
+System B
+System C
+Figure 6: The stored energy density in three systems.
+Structural investigation: Ion density profile
+Further insight into the ion structure near the electrodes’ surface is gained in this section.
+Due to the constant density in all three systems, system A has a smaller bulk region between
+its electrodes than systems B and C. It is expected that the ion density profiles will oscillate
+near the charged surface in IL-based SCs,34 as shown in Fig.7a for ∆V = 2 V . Fig.7b and
+Fig.7c depict ion density profiles for systems B and C at ∆V = 2 V after reaching equilibrium.
+Similar to ILs, the ion density profile of these systems exhibits layers and oscillations near
+electrode surfaces. While the conductive electrodes repel co-ions and attract counter-ions,
+co-ions can still be observed inside the pores. The highest peak in Fig.7a can be seen at the
+entrance and end of the pores, which contributes to the high induced charge on the surface
+of the electrodes. Fig.7c, the ion density profile of system C, displays a higher peak at the
+entrance of pores in analogy with the profile of system B. This leads to more induced charge
+14
+
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+-15
+-10
+-5
+ 0
+ 5
+ 10
+ 15
+ 20
+2 V
+0 V
+Ion Density Profile (nm-3)
+X (nm)
+(a)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+-15 -10 -5
+ 0
+ 5
+ 10  15  20  25  30
+2 V
+0 V
+Ion Density Profile (nm-3)
+X (nm)
+(b)
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+-15 -10 -5
+ 0
+ 5
+ 10  15  20  25  30
+2 V
+0 V
+Ion Density Profile (nm-3)
+X (nm)
+(c)
+Figure 7: Ion density profile of three systems along x axis for (a) System A (b) System B
+and (c) System C. Green and red graphs display positive and negative ions.
+on the electrodes’ surface and eventually higher energy density.
+The linear polymers can be placed near the electrode surfaces as well as in bulk space in
+system B, in contrast to system C, which places the polymer network only in bulk space. As
+a result, the accessible space for ions near electrode surfaces is reduced. Accordingly, system
+B exhibits a lower ion density at the pore entrance than systems A and C.
+Dynamical investigation: Mean Squared Displacement and
+Diffusion coefficient
+Gels with cross-linked polymers demonstrated higher mobility.5 Since the mobility of ions
+is proportional to the diffusion coefficient (D) we need to calculate the diffusion coefficient.
+The ionic diffusion coefficient is derived from the Mean-Squared Displacement (MSD) curve
+using the 3D diffusion relation:
+MSD ≡< ∆r(t)2 >= 1
+N
+N
+�
+i=1
+< ri(t)2 − ri(0)2 >,
+(10)
+where N is the number of particles. Diffusion coefficient can be calculate using the following
+equation;
+D = lim
+t→∞
+< ∆r(t)2 >
+6t
+.
+(11)
+15
+
+According to the above equation, also known as Einstein’s relation,36 the self-diffusion of
+particles is calculated from the slope of the MSD curve over time.
+We calculated MSD for clusters of linear polymers in system B and cross-linked polymers
+in system C. According to Fig.8, MSD for linear polymers and cross-linked polymers display
+superdiffusive motion. On the other hand, by taking the slope of the MSD into account, the
+diffusion coefficient for cross-linked polymers is greater than linear polymers and therefore
+the mobility of cross-linked polymers is higher. This result can also be seen in Fig.9. In this
+figure, two snapshots of both polymers are shown for equal time intervals ∆t = 10 timestep.
+Fig.9a shows lower mobility (for linear polymers) than Fig.9b (for cross-linked polymers).
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+~ t1.91
+~ t1.99
+Mean Squared Displacement (× 106 nm2)
+Time (ns)
+Linear polymers in system B
+Network polymer in system C
+ 1×10-5
+ 1×10-4
+ 1×10-3
+ 1×10-2
+ 1×10-1
+ 1×100
+ 1×101
+ 1×10-1
+ 1×100
+ 1×101
+Figure 8: Mean Squared Displacement of polymers in systems B and C in absence of electric
+field. Inner is the same result but in logarithmic scale, indicating on power-law behavior of
+MSD.
+In addition, we obtained MSD for cations and anions for all systems. Due to volume
+effects and the high mobility of cross-linked polymers, these polymers provide a limited path
+for charged particles in system C in comparison with linear polymers in system B. Therefore
+16
+
+(a)
+(b)
+Figure 9: Two different snapshots of polymers for (a) linear polymers in system B and (b)
+cross-linked polymers in system C.
+as shown in Fig.10, the charged particles’ diffusion in system C is lower than in systems A
+and B. Since a lower diffusion coefficient is associated with a larger viscosity, the viscosity
+of system C is greater than systems A and B. Consequently, increasing the viscosity of the
+electrolyte improves its mechanical stability. Accordingly, system C has higher mechanical
+stability than systems A and B.
+Furthermore, it is important to note that cations and anions differ slightly in their results
+due to their mass differences. Compared to a cation, an anion has a lower weight, so its
+MSD plot shows a higher value.
+17
+
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+D ~ 0.21538 × 10-4 (m2/s)
+D ~ 0.1375 × 10-4 (m2/s)
+D ~ 0.07313 × 10-4 (m2/s)
+Mean Squared Displacement (× 105 nm2)
+Time (ns)
+anion in system A
+cation in system A
+anion in system B
+cation in system B
+anion in system C
+cation in system C
+ 1×10-4
+ 1×10-3
+ 1×10-2
+ 1×10-1
+ 1×100
+ 1×101
+ 1×10-1
+ 1×100
+ 1×101
+Figure 10: Mean Squared Displacement of ions in all three systems at ∆V = 2 V . Inner plot
+is the same as outer but in logarithmic scale.
+Conclusion
+In summary, simulations demonstrate a difference between the performance of two types
+of electrolytes, i.e. liquid and solid electrolytes in supercapacitors. Solid electrolytes are
+used as an alternative to reduce the problems associated with liquid electrolytes. Therefore,
+simulation and comparison of these two categories can give us a clear insight into improving
+the efficiency of supercapacitors.
+Liquid electrolyte-based supercapacitors have a smaller operating voltage window there-
+fore, the amount of energy stored is less. Polymer electrolytes can be considered as a cluster
+of linear polymers or as cross-linked polymers in which linear polymers are connected and
+form a network. In linear polymers, since polymers can be present near the walls of the elec-
+trodes as well as inside the pores, the available space for ions in the vicinity of the electrodes
+is reduced. In contrast, in a polymer network, the movement of polymers is collective, and
+18
+
+therefore polymers can only be present in bulk space. Therefore, the accessible space for
+ions near the surfaces of the two electrodes and inside the pores increases. So the amount
+of charge stored in the electrodes and the supercapacitor efficiency increases.
+19
+
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+24
+

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+ 
+ 
+  
+The Field Structure of Free Photons 
+ 
+Anthony Rizzi  
+Institute for Advanced Physics, [email protected] 
+ 
+Abstract: Using a quantum field theoretic description of the photon it is shown that, as 
+intuitively expected but not before theoretically proven, the vector potential of a photon 
+has a likely amplitude associated with a discrete frequency and therefore energy, and 
+momentum. In particular, by finding the wave-functional for the vector potential, it is 
+shown that the likely absolute amplitude spectrum has delta function at a given 
+frequency. This analysis is extended to n-photon systems. It shows that such systems 
+have a vector potential distribution whose most likely element has a strong sinusoidal 
+component which has an amplitude corresponding to n-fold more energy than a single 
+photon system. An analogous result for photons of different energy is also derived. 
+Through the use of Parseval’s theorem for stochastic systems, the calculations and 
+associated analyses introduces a simple tool for exploring the nature of QFT Schrödinger 
+wave-functional generally.  
+ 
+Introduction 
+ 
+Photons are discussed everywhere from middle school, even grade school, to 
+graduate textbooks and beyond and are a core part of modern physics, but proving and 
+show their field structure, the focus of this article, has not been a part of these 
+discussions. Rules of thumb and intuitions from ordinary quantum mechanics and 
+quantum optics abound. But, as we know, these do not substitute for rigorous analysis 
+from first principles from what we know theoretically. And, what we know theoretically 
+about photons is contained in quantum field theory. Ordinary quantum mechanics and 
+quantum optics studies can only handle photons to the extent information is imported 
+from QFT. This paper for the first time directly applies the relevant QFT to obtain the full 
+probabilistic structure of the photon’s vector potential (which is what specifies what it is) 
+towards proving its key property, its discrete energy nature. In the process, a new 
+technique for probing such structures is introduced. Though QFT is a mature part of 
+physics, no one has yet shown the nature of a photon using QFT in the primal way done 
+here. Seeing this structure and how it connects to our previous intuitions is an essential 
+insight into the physics of photons, which, in turn, are at the center of modern physics.  
+ 
+Much previous work has been done in trying to probe the nature of the photon,1 
+beginning with the early investigations trying to “split” the photon.2 Free photons are 
+often discussed as if they were localized particles, but they are not. Of course, photons by 
+definition cause localized detections, but, as this article, proves their most likely field 
+structure is, in the idealized limit, an infinite length sine wave of undetermined phase. 
+This agrees with the natural understanding of a photon as a packet of definite energy (and 
+thus definite frequency), but makes it clear that even though when it interacts, it interacts 
+                                                 
+1 E.G., see special issue on nature of photon: Optics and Photonics News (vol. 14, October 2003). 
+2 E. O. Lawrence, J. W. Beams, On the Nature of Light, PNAS 13 No. 4, 207-212 (1927), G. P. Thomson, 
+Test of a Theory of Radiation, Proceedings of the Royal Society of London. Series A, Containing Papers of 
+a Mathematical and Physical Character 104 No. 724, 115-120 (1923).  
+
+2 
+  
+locally (e.g., with an atom), when it is free, before it begins to interact (with an atom for 
+example), its wave structure is spread out, not a localized wave packet.3 Giving (via 
+proofs) the structure of the photon offers us insights into simplest QFT structure, the free 
+photon, and thus can be a first glimpse of a deeper view of the nature of QFT itself as 
+well as a formalisms that has seldom been used, the Schrödinger wavefunctional and a 
+statistical formalism that has not been yet applied in QFT. 
+ 
+Up to this point, photon structure has been, indeed, taken for granted as issues 
+related to a wide range of important problems were discussed. These extend from trying 
+to find analogs of ordinary of wavefunctions for the spatially limited photon as well as its 
+associated probability for locating a photon, separately developing probability 
+distributions for detecting a photon based on an electric field operator, finding energy 
+distribution functions of such spatially limited photons to studying correlation functions 
+and coherent states created as superposition of Fock states and photon interference 
+phenomena as well as light squeezing.4 These problems are addressed using various 
+approaches, but seldom, if ever, use the QFT formalism of the Schrödinger 
+wavefunctional. The most natural approach to the photon field structure is this 
+Schrödinger formalism in combination with the commonly used Fock formalism. Now, 
+each formalism facilitates a different insight into the physics of a given system. Feynman 
+once said that every theorist should know many ways to solve the same problem.5 The 
+Schrödinger wavefunctional formalism has been somewhat neglected in this regard. In 
+ordinary quantum mechanics, the Schrödinger equation is usually preferentially used in 
+both teaching and applications. In ordinary QM, the S-matrix type approach and other 
+approaches (e.g. Green functions and path integrals)  have their place and are used, but do 
+not eclipse the Schrödinger equation. By contrast in QFT teaching and applications, the 
+analogous Schrödinger wave functional equation is left out almost completely (along 
+with all the intuitive advantages it affords). Few resources use or even mention it.6,7 
+Because of this, we first review those little known results of the analysis for our case and 
+then move to understand what they imply. Deriving those results is helpful in clearly 
+                                                 
+3 This is sometimes alluded to but usually vaguely but, in the literature as far as I could find, always 
+without proof. 
+4 See for example R. Loudon, The Quantum Theory of Light Third Edition (Oxford University Press, New 
+York, 2000) and L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge, NY, 1995) 
+5 “Every theoretical physicist who is any good knows six or seven different theoretical representations for 
+exactly the same physics.  He knows that they are all equivalent, and that nobody is ever going to be able 
+to decide which one is right at that level, but he keeps them in his head, hoping that they will give him 
+different ideas for guessing.” Feynman, MIT Press Character of Physical Law (1985), 168. 
+6 Quantum field theory textbooks that I am aware of do not mention it. For example, the following first rate 
+texts do not discuss Schrödinger wavefuntionals: F. Mandl and G. Shaw, Quantum Field Theory (2nd ed, 
+Wiley, West Sussex 2010), P. Ramond, Field Theory: A Modern Primer (2nd ed, Westview Press, Boulder 
+2001), C. Itzykson and J. Zuber, Quantum Field Theory (McGraw-Hill, USA 1980), M. Peskin and D. 
+Schroeder, Quantum Field Theory (Perseus Books, MA 1995), M. Kaku, Quantum Field Theory: A Modern 
+Introduction  (Oxford University Press, NY 1993), A. Zee, Quantum Field Theory in a Nutshell (Princeton 
+Univ. Press, NJ  2003). W. Greiner and J. Reinhardt, Field Quantization (Springer-Verlag, Berlin, 1996). S. 
+Weinberg, The Quantum Theory of Fields, Volume I and II (Cambridge University Press, NY 1995, 1996). 
+7 K. Symanzik, Schrodinger Representation and Casimir Effect in Renormalizable Quantum Field Theory, 
+Nucl. Phys. B 190, 1-44 (1981) showed the Schrödinger representation was renormalizable. These paper 
+discuss it: B. Hatfield, Quantum Field Theory of Point Particles and Strings (Perseus, Cambrdige MA, 
+1992), R. Jackiw, Analysis on Infinite-Dimensional Manifolds Schrödinger Representation for Quantized 
+Fields in Field Theory and Particle Physics, O. Eboli, et al. eds. (World Scientific, Brazil, 1989) 78-143p.  
+
+3 
+  
+understanding the approach’s fundamental nature and hopefully in spawning its further 
+use. 
+ 
+In section 1, first the basic formalism is introduced in the process of setting up the 
+calculation for the field structure of the  single photon wave functional. Next, Parseval’s 
+theorem is introduced to analyze that structure statistically, as it must be since the 
+quantum mechanical vector potential is fundamentally stochastic. This statistical 
+formalism is commonly used, for example, in electrical engineering where one looks, 
+amidst noise, for the spectral content of a signal.8 This is fundamentally is needed here.  
+The calculation shows that the amplitude of A for the single photon of momentum p is as 
+intuitively expected sinusoidal of the frequency 
+/ h
+ν = p
+. 
+ 
+In section 2, the formalism is generalized to the case of n identical free photons; 
+the full explicit form of the wavefunctional is given up to 4 photons and then the core 
+form of the n-photon wavefunctional is given. In section 3, the multi-photon case with 
+different momenta is resolved by giving the explicit results for the two photon different-
+momenta case. In each of these last two sections, the results prove that 1-photon 
+sinusoidal structure is found, as intuitively expected, also in the general n-photon 
+systems. 
+ 
+We take metric, 
+μν
+η
+, to have signature (
+)
+, , ,
++ − − −  and components {
+}
+0,1,2,3  and 
+1
+=
+�
+, and, ultimately, 
+1
+c = . 
+Single Photon Wave-functional 
+ 
+Assuming one is operating in the Lorenz gauge, where the classical equations of 
+motion are separable, we get the equations for the vector potential components in the 
+presence of a current source: 
+(1) 
+ 
+ 
+ 
+2
+2
+2
+2
+1
+4
+A
+A
+J
+c
+t
+c
+μ
+μ
+μ
+π
+∂
+− ∇
+=
+∂
+        
+ 
+ 
+We are not here interested in the production of the fields, so we consider only the 
+source-less case, which can arise from considering only regions sufficiently far from the 
+sources or, for conceptual simplicity, we can simply posit a given field of radiation as an 
+initial condition. Furthermore, since our interest is in radiation, for which the scalar 
+potential is not needed, we simply ignore it.9 This leaves taking 
+1
+c =  and using the 
+Einstein summation convention: 
+(2) 
+ 
+ 
+( )
+( )
+2
+0,
+j
+i
+i
+A
+x
+A
+x
+− ∂
+=
+��
+ where x is the 4-vector coordinate. 
+ 
+ 
+(this EOM is associated with 
+2
+2
+1
+1
+2
+2
+i
+i
+i
+A
+A
+m A
+ν
+μ
+=
+∂
+∂
+−
+L
+, the spatial Proca  
+ 
+ 
+ Lagrangian, with 
+0
+m =
+) 
+                                                 
+8 R. M. Howard, Principles of Random Signal Analysis and Low Noise Design: The Power Spectral Density 
+and Its Applications (John Wiley & Sons, 2002).  
+9 In this way, we sidestep problems with quantizing the full special relativistic field ( Aμ ) in the Lorentz 
+gauge. A more formal way to ignore the scalar potential is assume an arbitrarily small mass (rather than 
+zero mass) in the Proca Lagrangian and then use the Lorenz gauge condition to eliminate the scalar 
+potential (since in the massive case there are three independent degrees of freedom, rather than the two of 
+the massless field). 
+
+4 
+  
+ 
+Moving to second quantization operators, recalling the classical canonical 
+momentum 
+( )
+( ) /
+i
+i
+x
+A x
+t
+π
+≡ ∂
+∂ , this equation implies the following Hamiltonian operator 
+in Heisenberg picture. 
+(3) 
+ 
+ 
+(
+)
+3
+2
+1
+ˆ
+ˆ
+ˆ
+ˆ
+( , )
+( , )·
+( , ) ,
+2
+i
+i
+i
+H
+t
+A
+t
+A
+t
+d
+π
+=
++ ∇
+∇
+∫
+x
+x
+x
+x
+ 
+ 
+where we impose (equal time) canonical quantization:9      
+ 
+ 
+ 
+ˆ
+ˆ
+[
+( , ),
+( , )]
+(
+)
+i
+j
+ij
+A
+t
+t
+i
+π
+δ δ
+=
+−
+′
+x
+x
+x
+x
+. 
+Thus, the Schrödinger equation for the wave functional Ψ is: 10 
+(4) 
+ 
+(
+)
+(
+)
+(
+)
+2
+2
+3
+2
+1
+2
+ˆ
+i
+i
+i
+iA
+A
+i
+d x
+A
+t
+A
+δ
+δ
+∂Ψ
+⎛
+⎞
+=
+−
++ ∇
+Ψ
+⎜
+⎟
+∂
+⎝
+⎠
+∫
+,   
+Here we used: 
+(5) 
+ 
+ 
+ˆ
+ˆ
+i
+i
+i
+i
+i
+i
+A
+A
+A
+δ
+δ
+π
+δ
+δ
+⎛
+⎞
+=
+→ −
+=
+⎜
+⎟
+⎝
+⎠
+�
+ 
+The free solution (in the Heisenberg picture) for the vector potential operator is, taking 
+motion along z-axis, and assuming only one polarization (thereby also limiting A 
+eigenstates, A , to one polarization at a time 11):12,13  
+(6) 
+( ) (
+)
+(
+)
+( ) ( )
+( ) ( )
+(
+)
+3
+†
+3/2
+1
+ˆ
+ˆ
+ˆ
+,
+2
+2
+i
+i
+i
+i
+d p
+t
+a
+t e
+a
+t
+A
+e
+E
+λ
+λ
+λ
+λ
+δ
+π
+⋅
+−
+⋅
+=
++
+∫
+p x
+p x
+p
+p
+p
+x
+ 
+ 
+ 
+ 
+ 
+ 
+where: 
+( ) ( )
+( )
+ˆ
+ˆ
+iEt
+a
+t
+a
+e
+λ
+λ
+−
+=
+p
+p
+, 
+p
+E = p ,
+(
+)
+0,0, p
+=
+p
+ 
+ 
+ 
+ 
+ 
+ 
+          
+{
+}
+1,2
+λ ∈
+, (two independent polarizations) 
+From here on, we will usually drop the formal t dependence of the creation and 
+annihilation operators, so that by also dropping explicit polarization notation (by taking 
+i
+λ → ), we can write simply ˆi
+p
+a .  So, 
+ 
+ 
+with 
+†
+†
+ˆ
+ˆ
+ˆ
+ˆ
+,
+,
+0
+a
+a
+a
+a
+′
+′
+⎡
+⎤
+⎡
+⎤ =
+=
+⎣
+⎦
+⎣
+⎦
+p
+p
+p
+p
+, 
+(
+)
+†
+ˆ
+ˆ
+,
+a
+a
+δ
+′
+′
+⎡
+⎤ =
+−
+⎣
+⎦
+p
+p
+p
+p
+,  
+ 
+ 
+ 
+ 
+ 
+†
+3
+1
+ˆ
+ˆ ˆ
+2
+H
+E
+a a
+d p
+⎛
+⎞
+=
++
+⎜
+⎟
+⎝
+⎠
+∫
+p
+p
+p
+ . 
+Adding (6) to its time derivative and taking the Fourier transform gives, dropping the 
+polarization index from A:  
+(7) 
+ 
+(
+)
+(
+)
+(
+)
+(
+)
+3
+3/2
+1
+1
+ˆ
+,
+ˆ
+ˆ
+,
+2
+2
+i
+i
+i
+i
+a
+E
+t
+A
+A
+i
+t
+e
+d x
+E
+π
+−
+⋅
+=
++
+∫
+�
+p x
+p
+p
+p
+x
+x
+ 
+ 
+                                                 
+10 The equation is the field theory analog, for a given spectral mode of A at each point in space, of the 
+ordinary quantum mechanical oscillator. 
+11 Note this still preserves the commutation relations (since it becomes analogical to the scalar field case). 
+12 The equation of motion for the vector potential operator can be shown, via the quantum generalization of 
+Hamilton’s equations, to be the analog of the classical vector potential field’s equation of motion. 
+13 Note the measure in the integrand is not manifestly relativistic invariant, the manifestly Lorentz 
+invariant:
+(
+)
+4
+2
+2
+0
+(
+)
+d
+m
+u p
+p
+p
+δ
+−
+reduces, after integrating over p0 and with appropriate normalization, to 
+the 3D measure given in the text body. Note that a factor of 1 /
+2E  has been absorbed into the definition 
+of the creation operators to avoid a factor of 2E in the creation operator commutation relations.  
+
+5 
+  
+ 
+Note: x is the spatial coordinate. 
+ 
+ 
+Now, noting that the QFT creation operators are analogous to the creation and 
+annihilation operators of the quantum mechanical harmonic oscillator (QMHO) 
+( ˆ
+ˆ
+ˆ
+~
+a
+x
+p
++
+), we can apply the well known formal method of calculating the QMHO 
+energy eigenfunctions to calculate the wave-functionals of the n-photon (free) A-field. 
+We start with the one photon case.  
+ 
+In particular, using (5) in (7), we define the relevant wave functional as the 
+probability amplitude for finding a field which is prepared in the state called a (ideal) 
+“single photon of momentum p” (and described by the ket  
+† 0
+a
+=
+p
+p
+) in the state 
+iA
+, which is the eigenstate of 
+( )
+ˆ iA
+x . We get, suppressing the explicit time dependence 
+(by taking, e.g., 
+0
+t =
+): 
+(8) 
+( )
+(
+)
+†
+1
+0
+i
+i
+i
+i
+A
+A
+A
+A a
+Ψ
+=
+=
+=
+p
+p
+p
+x
+p
+ 
+  
+ 
+       
+ 
+ 
+ 
+ 
+ 
+  
+ 
+(
+)
+( )
+( )
+(
+)
+( )
+( )
+(
+)
+3
+0
+0
+3/2
+1
+1
+2
+2
+i
+i
+i
+i
+i
+E A
+A
+A
+e
+d x
+A
+E
+δ
+δ
+π
+⋅
+⎛
+⎞
+=
+Ψ
+−
+Ψ
+⎜
+⎟
+⎝
+⎠
+∫
+p x
+p
+p
+x
+x
+x
+x
+ 
+ 
+                   
+where: 
+( )
+(
+)
+( )
+0
+0
+0
+i
+i
+A
+A
+Ψ ≡ Ψ
+⋅
+≡
+⋅
+is the vacuum field wave functional which is 
+complex valued and not a function of x , the argument of A, despite its 
+notational presence  above (to distinguish between position and momentum 
+field representations).  
+So: 
+(9) 
+( )
+(
+)
+(
+)
+( )
+(
+)
+( )
+(
+)
+( )
+0
+0
+1
+2
+i
+i
+i
+i
+i
+A
+A
+E A
+A
+A
+E
+δ
+δ
+⎛
+⎞
+Ψ
+⎜
+⎟
+Ψ
+=
+−
+Ψ
+−
+⎜
+⎟
+⎝
+⎠
+�
+�
+p
+p
+p
+x
+x
+p
+x
+p
+, 
+where: 
+( )
+(
+)
+( )
+3
+3/2
+1
+2
+i
+i
+i
+A
+d p A
+e
+π
+⋅
+≡
+∫
+�
+p x
+x
+p
+, which gives: 
+( )
+( )
+/
+i
+i
+i
+A
+A
+e
+δ
+δ
+⋅
+=
+�
+p x
+x
+p
+, and allows us to take 
+( )
+iA
+x  to be an implicit 
+functional of 
+( )
+iA�
+p ). Also: 
+( )
+( )
+( )
+( )
+(
+)
+( )
+0
+3
+0
+i
+i
+i
+i
+i
+A
+A
+d x
+A
+A
+A
+δ
+δ
+δ
+δ
+δ
+δ
+Ψ
+Ψ
+= ∫
+�
+�
+x
+x
+x
+p
+p
+. 
+We need the form of the vacuum wave functional,
+(
+)
+0
+iA
+Ψ
+, which arises from noting: 
+(10)a  
+ˆ 0
+0
+i
+p
+A a
+=
+ 
+And, using (5) and (7) in this equation, we get: 
+(10)b  
+( )
+(
+)
+( )
+( )
+( )
+(
+)
+0
+3
+0
+0
+i
+i
+i
+i
+i
+A
+d x e
+E A
+A
+A
+δ
+δ
+−
+⋅ ⎛
+⎞
+Ψ
+⎜
+⎟
++
+Ψ
+=
+⎜
+⎟
+⎝
+⎠
+∫
+p x
+p
+x
+x
+x
+x
+ 
+Using, analogous to above, 
+( )
+(
+)
+( )
+3
+3/2
+1
+2
+i
+i
+i
+A
+d x A
+e
+π
+−
+⋅
+≡
+∫
+�
+p x
+p
+x
+, we can write:  
+
+6 
+  
+(11) 
+ 
+( )
+(
+)
+0
+0
+0
+i
+i
+E A
+A
+δ
+δ
+Ψ
++
+−
+Ψ =
+�
+�
+p
+p
+p
+. 
+So, that finally, the vacuum wave functional: is seen to be: 
+(12) 
+(
+)
+( )
+3
+0
+1
+exp
+2
+i
+i
+N
+d p E A
+A
+⎛
+⎞
+Ψ =
+−
+⎜
+⎟
+⎝
+⎠
+∫
+�
+�
+p
+-p
+p
+ 
+where we drop its 
+( )
+iA
+x  dependence on Ψ0 because we are here 
+displaying its implicit dependence on 
+( )
+iA�
+p  and N is the 
+normalization of the wave functional.  
+ 
+ 
+Now, we can get the wave functional of the ideal single photon of momentum 
+p  by substituting (12) in (9), ignoring the normalization of the wave-functional:  
+(13) 
+ 
+ 
+( )
+(
+)
+(
+)
+( )
+3
+1
+2
+2
+i
+i
+d
+A
+p
+i
+e
+A
+A
+2
+−
+Ψ
+=
+∫
+−
+�
+p
+p
+p
+x
+p
+p
+, 
+ 
+           
+where we used the fact that ( )
+A x  is real implies
+( )
+(
+)
+*
+A
+A
+=
+−
+�
+�
+p
+p . 
+ 
+From this, we get the probability of finding the field in a state with a Fourier 
+transform in a small region of functions space 
+i
+DA�  near 
+( )
+iA�
+p is:                 
+(14) 
+( )
+(
+)
+(
+)
+( )
+( )
+3
+2
+2
+2
+i
+d
+i
+i
+i
+p
+A
+p
+P A
+A
+A
+e
+2
+−∫
+= Ψ
+=
+�
+�
+p
+p
+p
+p
+p
+. 
+ 
+We need to find the magnitude of 
+( )
+iA
+x , or equivalently the “energy” 
+,
+2
+A , that is the most likely.14 Parseval’s theorem tells us that the “energy” 
+spectral density of a spatial function 
+( )
+iA
+x  is 
+( )
+( )
+2
+i
+i
+D
+A
+≡ �
+p
+p
+, so in these 
+terms, (14) can be recast as: 
+(15) 
+ 
+ (
+)
+( )
+( )
+(
+)
+(
+)
+( ) 3
+2
+2
+2
+i
+i
+i
+D
+d p
+i
+D
+D
+P D
+e
+A
+−∫
+=
++
+= Ψ p
+p
+p
+p
+-p
+p
+ 
+where, for the multiplicative factor, we have replaced D by the 
+given sum to make explicit the reflective symmetry of D in p ,15,16 
+which we need to see the consequences of. (Note: such a 
+replacement does not change anything in the integral in the 
+exponent). 
+ 
+We need to extremize this probability distribution, so we set the first variation to 
+zero giving:  
+                                                 
+14 Note we are here interested in A, not E or B. We want to know the most likely vector potential structure. 
+15 D(p) must be even in p for the autocorrelation function to be positive, as it must be. 
+16 Note: even though, for simplicity, we have set p = (0,0,p), no generality was lost in so doing; so we may 
+at any moment let p point in any direction we like. 
+
+7 
+  
+(16) 
+ 
+( )
+(
+)
+( )
+( )
+(
+)
+(
+)
+3
+2
+Dd p
+P D
+D
+D
+e
+D
+D
+δ
+δ
+δ
+δ
+−
+⎛
+⎞
++
+∫
+∝
+⎜
+⎟
+⎜
+⎟
+⎝
+⎠
+p
+p
+p
+-p
+p
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(
+)
+(
+)
+(
+)
+( )
+3
+3
+0
+2
+p Dd p
+Dd p
+e
+D
+e
+δ
+δ
+−
+−
+−
++
++
+∫
+∫
+=
+−
+=
+p
+p
+p
+p
+p
+p
+p
+, 
+where we have dropped the component index, i, to simplify the notation. 
+Thus, it is seen that the most likely energy spectral density is:                                                         
+(17) 
+ 
+ 
+( )
+(
+)
+(
+)
+(
+)
+max
+2
+D
+δ
+δ
+−
++
++
+=
+p
+p
+p
+p
+p
+p
+. 
+This result agrees with the obvious fact that the maximum of 
+(
+)
+P D  occurs when the 
+integral in the exponent is minimal and the multiplicative factor term is maximal, for at 
+the
+= ±
+p
+p , the factor is infinite, while the exponent is finite.  
+ 
+Using (17) gives the autocorrelation function:17  
+(18) 
+ 
+ 
+3
+max
+cos
+( )
+[ ( ) (
+)]
+( )
+i
+AA
+R
+A
+A
+D
+e
+d p
+∞
+−∞
+⋅
+⋅
+=
++
+=
+=
+∫
+E
+p x
+p x
+x
+y
+y
+x
+p
+p
+. 
+    
+Hence, we see the strong sinusoidal dependence is present in the stochastic A-
+field that characterizes a single (ideal) photon.   
+N-Photon Wavefunctionals 
+ 
+Extending what we did in equation (8), and in analogy to the generation of the 
+energy eigenstate wave functions for  the harmonic oscillator in ordinary quantum 
+mechanics,18 we construct the wavefunctional corresponding to n photons: 19 
+(19) 
+ 
+( )
+(
+)
+(
+)
+†
+0
+n
+i
+i
+i
+i
+n
+A
+A n
+A n
+A
+a
+Ψ
+=
+=
+=
+p
+p
+p
+x
+p
+, 
+Substituting (7) and generalizing from (8): 
+(20)   
+( )
+(
+)
+(
+)
+(
+)
+( )
+( )
+3
+3 /2
+/2
+1
+1
+ˆ
+0
+2
+2
+n
+i
+i
+i
+i
+n
+n
+i
+A
+A
+e
+E A
+d x
+A
+E
+δ
+δ
+π
+⋅
+⎛
+⎞
+⎛
+⎞
+Ψ
+=
+−
+⎜
+⎟
+⎜
+⎟
+⎜
+⎟
+⎝
+⎠
+⎝
+⎠
+∫
+p x
+np
+p
+p
+x
+x
+x
+ 
+ 
+Note: we will need to use: 
+( )
+( )
+i
+A
+e
+A
+δ
+δ
+−
+⋅
+=
+�
+p x
+p
+x
+ 
+We have already done 
+0
+n =
+and 
+1
+n =  case; we now calculate the next three cases (see 
+Appendix A). As a result, we can write the wavefunctionals for 0,1,2, 3 and 4 photons 
+(leaving aside complicating multiplicative factors): 
+(21) 
+ 
+( )
+(
+)
+( )
+3
+1
+2
+0
+iA
+i
+d p
+e
+A
+2
+−
+Ψ
+∝
+∫ p
+p
+x
+ 
+                                                 
+17 In relation to conventional notation, we have: 
+3
+( )
+( )
+( )
+( )
+AA
+A
+i
+A
+A
+A
+D
+S
+R
+x e
+d x
+R
+−
+−∞
+⋅
+∞
+≡
+=
+=
+∫
+�
+p x
+p
+p
+p . 
+18 The analogy is fundamentally:
+( )
+( )
+,
+x
+A x
+p
+A p
+↔
+↔ �
+ 
+19 Note that the Fourier transform of the general energy eigenstate, is proportional to a state with the exact 
+same formal structure except with x replaced by p, thus in some extended sense it is also an eigenstate of 
+the Fourier operator.  
+
+8 
+  
+(22) 
+ 
+( )
+(
+)
+(
+)
+( )
+3
+1
+2
+2
+i
+i
+d
+A
+p
+i
+e
+A
+A
+2
+−
+Ψ
+∝
+∫
+−
+�
+p
+p
+p
+x
+p
+p
+ 
+(23)  
+ 
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+( )
+3
+1
+2
+2
+2
+2
+2
+iA
+d p
+i
+i
+A
+e
+A
+δ
+2
+−
+Ψ
+∝
+−
+−
+∫
+�
+p
+p
+2p
+x
+p
+p
+p
+p
+  
+(24) 
+ 
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+) (
+)
+(
+)
+( )
+3
+1
+3
+2
+2
+4
+2
+3
+2
+iA
+p
+i
+i
+i
+d
+e
+A
+A
+A
+δ
+2
+−
+Ψ
+∝
+−
+−
+−
+∫
+�
+�
+p
+p
+3p
+x
+p
+p
+p
+p
+p
+ 
+(25) 
+ 
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+( )
+3
+1
+4
+2
+2
+2
+2
+2
+4
+12
+2
+4
+3
+2
+i
+i
+i
+A
+p
+i
+d
+A
+e
+A
+A
+δ
+δ
+2
+−
+⎛
+⎞
+−
+−
+⎜
+⎟
+Ψ
+∝
+⎜
+⎟
++
+⎝
+⎠
+∫
+�
+�
+p
+p
+4p
+p
+p
+p
+-p
+p
+x
+p
+p
+ 
+Now, we can drop the ( )
+δ
+p terms, as we have obviously chosen 
+0
+≠
+p
+; so, for example, 
+for the 4-photon case we get:  
+(26) 
+ 
+( )
+(
+)
+(
+)
+(
+)
+( )
+3
+4
+4
+1
+2
+16
+iA
+p
+i
+i
+d
+e
+A
+A
+2
+−
+Ψ
+∝
+−
+∫
+�
+p
+p
+4p
+x
+p
+p
+  
+(
+0
+≠
+p
+) 
+ 
+We see the pattern. Using the formal analogy to the ordinary quantum oscillator,20 
+or induction, we can write the first try at the solution as, dropping the component 
+superscript, i, and the multiplicative factors: 
+(27) 
+1st try:  
+( )
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+3
+3
+3
+2
+2
+1
+2
+.
+1
+i
+n
+d
+d p
+A
+p
+A
+A
+n
+d p
+n
+n
+A
+e
+e
+e
+A
+δ
+δ
+2
+−
+−
+−
+−
+Ψ
+∝
+∫
+∫
+∫
+−
+−
+�
+�
+�
+p
+p
+p
+p
+p
+p
+p
+x
+p
+ 
+This first try, however, includes the terms that correspond to the delta functions which, as 
+just stated, we don’t need; so, we write: 
+(28)
+( )
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+( )
+3
+3
+1
+1
+2
+2
+.
+1
+2
+2
+i
+i
+d
+d
+A
+p
+A
+p
+n
+n
+n
+n
+n
+A
+A
+e
+e
+A
+2
+2
+−
+−
+∫
+Ψ
+∝
+−
+−
+−
+∫
+−
+=
+�
+�
+p
+p
+p
+p
+p
+x
+p
+p
+p
+p
+ 
+ 
+This means the corresponding probability distribution is, dropping constant 
+multiplicative factors: 
+(29) 
+ 
+(
+)
+( )
+( )
+( )
+( )
+3
+3
+2
+i
+i
+d
+n
+A
+p
+A
+p
+n
+d
+n
+A
+D
+P
+D
+e
+e
+2
+2
+−
+−
+∝
+=
+∫
+∫
+�
+p
+p
+p
+p
+p
+p
+p
+, 
+ 
+where we introduce 
+( )
+( )
+(
+)
+(
+) / 2
+D
+D
+D
+≡
++
+−
+p
+p
+p
+ to make D’s 
+symmetry in p  explicit for the minimization differentiation. 
+Minimizing gives: 
+(30) 
+ 
+( )
+(
+)
+( )
+( )
+(
+)
+( )
+3
+2
+d
+n
+D
+np
+p
+P
+D
+D
+D
+D
+D
+e
+δ
+δ
+δ
+δ
+−
+⎛
+⎞
++
+−
+⎛
+⎞
+⎜
+⎟
+∝
+⎜
+⎟
+⎜
+⎟
+⎝
+⎠
+⎝
+∫
+⎠
+p
+p
+p
+p
+p
+p
+ 
+ 
+( )
+(
+)
+(
+)
+( )
+( )
+( )
+3
+3
+1
+0
+2
+n
+D
+D
+n
+d p
+d
+n
+p
+e
+e
+nD
+D
+δ
+δ
+−
+−
+−
+−
++
++
+⎛
+⎞
+−
+∫
+∫
+=
+=
+⎜
+⎟
+⎝
+⎠
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+which implies, dropping the explicit separation of D in the first term: 
+                                                 
+20 The Fourier transform of the energy eigenstates (
+2
+exp(
+)
+/ 2)
+(
+n
+n
+x
+H
+x
+ψ
+=
+−
+) in ordinary quantum 
+mechanics, 
+that 
+we 
+use 
+here 
+by 
+analogy 
+is: 
+(
+)
+2
+2
+exp(
+/
+( )
+(
+x
+2
+/
+)
+e p(
+)
+)
+n
+n
+n
+n
+p
+p
+p
+i
+d
+dp
+ψ
+=
+−
+ 
+Note:
+2
+2
+exp
+( )
+(
+( 1)
+exp(
+/
+)
+)
+n
+n
+n
+n
+H
+x
+x
+x
+x
+d
+d
+= −
+−
+.  
+
+9 
+  
+(31) 
+ 
+( )
+(
+)
+(
+)
+( )
+1
+2
+n
+n
+nD
+D
+δ
+δ
+−
+−
++
++
+⎛
+⎞ =
+⎜
+⎟
+⎝
+⎠
+p
+p
+p
+p
+p
+p
+p
+ 
+So we have: 
+(32) 
+ 
+( )
+(
+)
+(
+)
+2
+D
+n δ
+δ
+⎛
+⎞
+−
++
++
+=
+⎜
+⎟
+⎝
+⎠
+p
+p
+p
+p
+p
+p
+ 
+ 
+So n photons have n times the energy spectral density as a single one, but the 
+same concentration of that energy into a sinusoidal form. This is a very nice result that 
+agrees with ones intuition; one can think of the field of a photon as mostly having sine-
+like behavior, but on top of a stochastic background. (Of course, more information can be 
+extracted from the wavefunctional, such as the moments of the distribution.) 
+ 
+Many-photon Wave-functionals with Different Momenta 
+ 
+The above formalism is generalizable to the photons of different momenta. The 
+basic principle can be illustrated by the two photon case, one with momentum 
+1p  the 
+other with momentum 
+2p . 
+ We can extend the general formalism to this case by applying the creation operator twice 
+(
+)
+(
+)
+†
+†
+1
+2
+2
+1
+,
+0
+p
+p
+a
+p
+a
+p
+=
+); we get, temporarily dropping the factor 
+( )
+3
+d p
+D
+e
+−∫ p
+p
+: 
+(33) 
+(
+) (
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+1
+2
+1
+2
+1
+2
+1
+2
+1
+2
+1
+1
+2
+*
+,
+,
+,
+*
+*
+1
+2
+1
+2
+1
+1
+2
+4
+2
+4
+2
+A
+A
+P
+A
+A
+δ
+ψ
+ψ
+δ
+⎛
+⎞
+−
++
+×
+⎜
+⎟
+=
+∝ ⎜
+⎟
+−
++
+⎝
+⎠
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+(
+)(
+)
+(
+)
+(
+)
+(
+)
+2
+2
+*
+*
+1
+2
+1
+2
+1
+2
+1
+2
+1
+2
+1
+2
+1
+1
+2
+8
+2
+4
+D D
+A A
+A A
+δ
+δ
+=
+−
++
++
++
++
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+ 
+ 
+First, we consider the most general and most interesting case (for these idealized 
+photons), which is also the simplest to calculate. 
+Case I:  
+1
+2
+≠ −
+p
+p .  
+This means the delta functions in (33) are not in play, so we have, in terms of the 
+previously defined D and reinserting the exponential term:  
+(34) 
+ 
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+( )
+2
+3
+1
+2
+2
+,
+1
+2
+1
+2
+1
+2
+2
+,
+16
+d
+D
+p
+p
+p
+P
+D p
+D p
+D
+D
+e
+−∫
+∝
+p
+p
+p
+p
+p
+p
+ 
+We want: 
+1
+2
+,
+1
+0
+p
+p
+P
+D
+δ
+δ
+=
+  and    
+1
+2
+,
+2
+0
+p
+p
+P
+D
+δ
+δ
+=
+. 
+Since this is formally same as calculation previously done, we get: 
+(
+)
+(
+)
+2
+i
+i
+i
+i
+D
+δ
+δ
+−
++
+−
+=
+p
+p
+p
+p
+p
+. So, as expected, we get one field structure with strong 
+sinusoidal structure of frequency associated with 
+1p and one associated with 
+2p . Note 
+equation (34) and this last result reduce to the same result for the case in which 
+1
+2
+≡
+=
+p
+p
+p ; to see the latter, compare equations (34) and (29).  This again verifies our 
+intuition of two photon systems.  
+ 
+We now move to the second, more limited case. 
+Case II: 
+1
+2
+= −
+p
+p  
+
+10 
+  
+ 
+Given our analysis in the previous sections, we would expect the two photons 
+might cancel each other out in our (idealized photon) case. To do the calculation, we take 
+the following definitions: 
+1
+2
+≡
+= −
+p
+p
+p , 
+(
+) (
+)
+( )
+2
+2
+i
+i
+i
+i
+D
+D
+A
+A
+A
+A
+≡
+≡
+=
+−
+=
+p
+p
+p
+and 
+recall 
+( )
+(
+)
+*
+A
+A
+=
+−
+p
+p . 
+Equation (33) becomes in the limit that 
+1
+2
+→ −
+p
+p : 
+(35)
+(
+)
+( )
+(
+) ( )
+( ) (
+)
+(
+)
+(
+)
+( )
+(
+)
+1
+2
+2
+2
+2
+2
+2
+,
+1
+2
+,
+8
+2
+0
+4
+0
+p
+p
+P
+D D
+D
+A
+A
+A
+A
+δ
+δ
+∝
+−
+−
+−
+−
++
+p
+p
+p
+p
+p
+p
+p
+ 
+When finding the maximum, it is the highest order delta functions that matter. Hence, 
+only the last term needs to be considered (the second term cannot contribute a second 
+order delta function because the piece of that term in parenthesis goes to zero) so we 
+have, recalling the exponential term: 
+ 
+(
+)
+(
+)
+(
+)
+( )
+3
+1
+2
+2
+2
+,
+1
+2
+1
+2
+,
+4
+D
+d p
+p
+p
+P
+D D
+e
+δ
+−∫
+∝
++
+p
+p
+p
+p
+p
+   (
+1
+2
+≡
+= −
+p
+p
+p ) 
+Finding the extremum, we get: 
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+( )
+( )
+3
+3
+1
+2
+2
+2
+,
+1
+2
+1
+2
+0
+D
+d p
+D
+d p
+p
+p
+i
+i
+P
+e
+D
+e
+D
+D
+δ
+δ
+δ
+δ
+δ
+δ
+−
+−
+∫
+∫
+∝
++
+= −
++
+=
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+So, 
+( )
+0
+D
+=
+p
+, which means that there is no average energy in this mode in the most 
+likely A amplitude distribution. 
+Conclusion 
+ 
+We have here given the structure of the photon vector potential distribution and 
+theoretically derived, for the first time, its key feature, thus enlightening long held, 
+though often confused, intuitions about photons. 
+ 
+Like all references to quantum mechanical states, the term photon refers to a 
+system statistically, not individually; quantum mechanics only predicts probable 
+outcomes, so needs many experiments on different systems (experiments on an 
+“ensemble” of systems) in the same state (or approximately the same state) to verify the 
+prediction. This paper proves that a photonic state is a state which has a stochastic vector 
+potential (A) filling space whose most likely amplitude (magnitude) structure (across the 
+ensemble of systems) has a sinewave component of frequency 
+/ h
+p
+ for each photon of 
+momentum p in the system. Getting this insight into the free field states, being 
+fundamental, can build intuition into other more complicated (e.g., interactional) QFT 
+problems. 
+ 
+Recognizing the stochastic character of the QFT states can lead one to introduce 
+Parseval’s theorem (and other statistical theorems) which along with the Schrödinger 
+formalism are intuitive tools and are required to get these results. The QFT analog of the 
+Schrödinger equation allows one to carry all the intuition one gained in ordinary quantum 
+mechanics about waves and particles over to QFT, including into Yang Mill’s theory.  It 
+has been used in attempts to use QFT in general relativity in a 3+1 space time setting, 
+which also points to its draw back that it is not explicitly relativistically invariant.21 It is 
+                                                 
+21 D.V. Long, G.M. Shore, The Schrödinger Wave Functional and Vacuum States in Curved Spacetime, 
+Nucl. Phys. B 530, 247-278 (1998) D.V. Long, G.M. Shore, The Schrodinger Wave Functional and 
+Vacuum States in Curved Spacetime II: Boundaries and Foliations, Nucl. Phys. B 530, 279-303 (1998). 
+P.R. Holland, The De Broglie-Bohm Theory of Motion and Quantum Field Theory, Phys. Rept. 224 No. 3, 
+
+11 
+  
+also required and used in interesting ways in De Broglie-Bohm (dBB) formalism of 
+QFT.22 Though this work fills a lacunae and is foundational (further demonstrating the 
+nature of the fundamental entity “photon”), it requires the relatively infrequently used 
+QFT Schrödinger formalism and this seems to be part of why it has been overlooked till 
+now. Hopefully, this filling of the lacunae will help spur the use of the Schrödinger 
+formalism and thereby make available the attending insights offered by its distinct 
+perspective on QFT.23,24  
+ 
+ 
+Appendix A 
+Calculation of the wave-functionals for 2, 3 and 4 photon systems. 
+(36) 
+ 
+( )
+(
+)
+(
+)
+†
+0
+n
+i
+i
+i
+i
+n
+A
+A n
+A n
+A
+a
+Ψ
+=
+=
+=
+p
+p
+p
+x
+p
+, 
+Substituting (7) and generalizing from (8): 
+(37)   
+( )
+(
+)
+(
+)
+(
+)
+( )
+( )
+3
+3 /2
+/2
+1
+1
+ˆ
+0
+2
+2
+n
+i
+i
+i
+i
+n
+n
+i
+A
+A
+e
+E A
+d x
+A
+E
+δ
+δ
+π
+⋅
+⎛
+⎞
+⎛
+⎞
+Ψ
+=
+−
+⎜
+⎟
+⎜
+⎟
+⎜
+⎟
+⎝
+⎠
+⎝
+⎠
+∫
+p x
+np
+p
+p
+x
+x
+x
+ 
+ 
+Note: we will need to use: 
+( )
+( )
+i
+A
+e
+A
+δ
+δ
+−
+⋅
+=
+�
+p x
+p
+x
+ (and perhaps
+( )
+( )
+i
+A
+e
+A
+δ
+δ
+⋅
+=
+�
+p x
+x
+p
+) 
+Dropping multiplicative factors to reduce space and focus on form of the equations, we 
+can write the generic equation that we will use below: 
+(38) 
+( )
+(
+)
+(
+)
+( )
+(
+)
+( )
+(
+)
+1
+i
+i
+i
+n
+p
+i
+A
+A
+A
+A
+δ
+δ
+−
+⎛
+⎞
+Ψ
+==
+−
+−
+Ψ
+⎜
+⎟
+⎝
+⎠
+�
+�
+np
+x
+p
+p
+x
+p
+ 
+Using this and recalling
+( )
+(
+)
+(
+)
+( )
+3
+1
+2
+2
+i
+i
+d
+A
+p
+i
+e
+A
+A
+2
+−
+Ψ
+=
+∫
+−
+�
+p
+p
+p
+x
+p
+p
+, we now calculate the 2, 
+3 and 4 photon wavefunctionals.  
+ 
+n=2 case : 
+                                                                                                                                                 
+95-150 (1993), N. Pinto-Neto, J. C. Fabris, Quantum Cosmology from the De Broglie-Bohm Perspective, 
+Class. Quantum Grav. 30 No. 143001, 57 (2013). 
+22 P.R. Holland, The De Broglie-Bohm Theory of Motion and Quantum Field Theory, Phys. Rept. 224 No. 
+3, 95-150 (1993), N. Pinto-Neto, J. C. Fabris, Quantum Cosmology from the De Broglie-Bohm 
+Perspective, Class. Quantum Grav. 30 No. 143001, 57 (2013).others P. Roser, A. Valentini, Classical and 
+Quantum Cosmology with York Time, Class. Quantum Grav. 31 No. 245001 (2014). See other papers by 
+P.R. Holland which use Schrödinger approach relative to the dBB interpretation of quantum mechanics, 
+and A. Valentini’s papers on dBB applied to quantum cosmology. 
+23 C. Kiefer, Functional Schrodinger Equation for Scalar QED, Phys. Rev. D 45 No. 6, 2044-2056, (1992) 
+24 Also, the use of functionals cross fertilizes with their use in the path integral formulation as well as, 
+through the functional Schrödinger equation, evokes the functional domain analog of the path from the 
+Schrödinger equation to path integrals. 
+
+12 
+  
+(39)
+( )
+(
+)
+(
+)
+( )
+(
+)
+( )
+( )
+(
+)
+0
+i
+i
+i
+i
+i
+i
+A
+A
+A
+A
+A
+A
+δ
+δ
+δ
+δ
+⎛
+⎞⎛
+⎞
+Ψ
+=
+−
+−
+−
+−
+Ψ
+⎜
+⎟⎜
+⎟
+⎝
+⎠⎝
+⎠
+�
+�
+�
+�
+2p
+x
+p
+p
+p
+p
+x
+p
+p
+(
+)
+( )
+( )
+(
+)
+i
+i
+i
+A
+A
+A
+δ
+δ
+⎛
+⎞
+=
+−
+−
+Ψ
+⎜
+⎟
+⎝
+⎠
+�
+�
+p
+p
+p
+x
+p
+(
+)
+( )
+(
+)
+( )
+(
+)
+( )
+i
+i
+i
+i
+A
+A
+A
+A
+δ
+δ
+Ψ
+=
+−
+Ψ
+−
+�
+�
+p
+p
+x
+p
+p
+x
+p
+ 
+ 
+(
+)
+(
+)
+( )
+(
+)
+( )
+( )
+3
+1
+3
+2
+1
+2
+2
+i
+i
+d
+A
+p
+d
+e
+A
+p
+i
+i
+i
+i
+e
+e
+A
+A
+A
+A
+δ
+δ
+− ∫
+2
+2
+−
+⎛
+⎞
+⎛
+⎞
+−
+⎜
+⎟
+⎜
+⎟
+⎝
+⎠
+⎜
+⎟
+=
+−
+−
+−
+⎜
+⎟
+⎜
+⎟
+⎜⎝
+∫
+⎟⎠
+�
+�
+�
+�
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+( )
+3
+1
+2
+2
+2
+2
+2
+iA
+d p
+i
+i
+A
+e
+A
+δ
+2
+−
+Ψ
+=
+−
+−
+∫
+�
+p
+p
+2p
+x
+p
+p
+p
+p
+ 
+ 
+ 
+n=3 case: 
+(40) 
+ 
+( )
+(
+)
+(
+)
+( )
+( )
+(
+)
+i
+i
+i
+i
+A
+A
+A
+A
+δ
+δ
+⎛
+⎞
+Ψ
+=
+−
+−
+Ψ
+⎜
+⎟
+⎜
+⎟
+⎝
+⎠
+�
+�
+3p
+2p
+x
+p
+p
+x
+p
+ 
+ 
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+( )
+3
+2
+1
+2
+2
+2
+2
+iA
+d p
+i
+i
+i
+A
+A
+e
+A
+δ
+δ
+δ
+2
+−
+⎛
+⎞
+=
+−
+−
+−
+−
+⎜
+⎟
+⎜
+⎟
+⎝
+∫
+⎠
+�
+�
+�
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+ 
+ 
+(
+)
+(
+)
+(
+) (
+)
+(
+)
+(
+) (
+)
+(
+)
+(
+)
+(
+) (
+)
+(
+)
+(
+)
+( )
+3
+2
+3
+1
+3
+2
+2
+2
+2
+4
+2
+2
+2
+i
+i
+i
+i
+d
+A
+p
+i
+i
+e
+A
+A
+A
+A
+A
+δ
+δ
+δ
+2
+−
+⎛
+⎞
+−
+−
+−
+⎜
+⎟
+=
+⎜
+⎟
+⎜
+⎟
+−
+−
+−
+−
+−
+−
+⎝
+⎠
+∫
+�
+�
+�
+�
+�
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+ 
+So, 
+ 
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+) (
+)
+(
+)
+( )
+3
+1
+3
+2
+2
+4
+2
+3
+2
+iA
+p
+i
+i
+i
+d
+e
+A
+A
+A
+δ
+2
+−
+Ψ
+=
+−
+−
+−
+∫
+�
+�
+p
+p
+3p
+x
+p
+p
+p
+p
+p
+ 
+ 
+n=4 case: 
+ 
+(41) 
+( )
+(
+)
+(
+)
+( )
+( )
+(
+)
+i
+i
+i
+i
+A
+A
+A
+A
+δ
+δ
+⎛
+⎞
+Ψ
+=
+−
+−
+Ψ
+⎜
+⎟
+⎝
+⎠
+�
+�
+4 p
+3 p
+x
+p
+p
+x
+p
+ 
+ 
+
+13 
+  
+ 
+       
+(
+)
+(
+)
+(
+)
+(
+)
+(
+) (
+)
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+(
+) (
+)
+(
+)
+( )
+( )
+3
+3
+3
+2
+3
+1
+2
+1
+2
+2
+3
+2
+4
+2
+3
+2
+i
+i
+d
+A
+p
+i
+i
+i
+A
+i
+i
+d p
+i
+e
+e
+A
+A
+A
+A
+A
+A
+δ
+δ
+δ
+δ
+2
+2
+−
+−
+⎛
+⎞
+−
+−
+−
+−
+⎜
+⎟
+⎜
+⎟
+⎜
+⎟
+⎛
+⎞
+=
+−
+−
+−
+⎜
+⎟
+⎜
+⎟
+⎝
+⎠
+⎜
+∫
+∫
+⎟
+−
+⎜
+⎟
+⎝
+⎠
+�
+�
+�
+�
+�
+�
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+ 
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+) (
+)
+(
+)
+(
+)
+( )
+3
+4
+2
+2
+2
+2
+1
+2
+2
+3
+2
+3
+2
+4
+6
+2
+3
+2
+2
+3
+2
+i
+i
+i
+A
+p
+i
+i
+i
+d
+i
+A
+A
+A
+A
+A
+A
+e
+δ
+δ
+δ
+δ
+2
+−
+⎛
+⎞
+−
+−
+⎜
+⎟
+=
+⎜
+⎟
+−
+−
+−
+−
+−
+−
+−
+−
+⎜
+⎟
+⎝
+⎠
+∫
+�
+�
+�
+�
+�
+�
+p
+p
+p
+p
+p
+-p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+p
+ 
+So, 
+( )
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+( )
+3
+4
+2
+2
+2
+2
+1
+2
+4
+4
+12
+2
+3
+2
+iA
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+ 
+

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@@ -0,0 +1,1110 @@
+Average Is Not Enough: Caveats of Multilingual Evaluation
+Matúš Pikuliak and Marián Šimko
+Kempelen Institute of Intelligent Technologies
+[email protected]
+Abstract
+This position paper discusses the problem of
+multilingual evaluation. Using simple statis-
+tics, such as average language performance,
+might inject linguistic biases in favor of domi-
+nant language families into evaluation method-
+ology. We argue that a qualitative analysis in-
+formed by comparative linguistics is needed
+for multilingual results to detect this kind of
+bias. We show in our case study that results in
+published works can indeed be linguistically
+biased and we demonstrate that visualization
+based on URIEL typological database can de-
+tect it.
+1
+Introduction
+The linguistic diversity of NLP research is grow-
+ing (Joshi et al., 2020; Pikuliak et al., 2021) thanks
+to improvements of various multilingual technolo-
+gies, such as machine translation (Arivazhagan
+et al., 2019), multilingual language models (Devlin
+et al., 2019; Conneau and Lample, 2019), cross-
+lingual transfer learning (Pikuliak et al., 2021) or
+language independent representations (Ruder et al.,
+2019). It is now possible to create well-performing
+multilingual methods for many tasks. When deal-
+ing with multilingual methods, we need to be able
+to evaluate how good they really are, i.e. how effec-
+tive they are on a wide variety of typologically di-
+verse languages. Consider the two methods shown
+in Figure 1 (a). Without looking at the particular
+languages, Method A seems better. It has better re-
+sults for the majority of languages and its average
+performance is better as well. However, the trio
+of languages, where Method A is better, are in fact
+all very similar Iberian languages, while the fourth
+language is Indo-Iranian. Is the Method A actually
+better, or is it better only for Iberian? Simple av-
+erage is often used in practice without considering
+the linguistic diversity of the underlying selection
+of languages, despite the fact that many corpora
+and datasets are biased in favor of historically dom-
+inant languages and language families.
+Additionally, as the number of languages in-
+creases, it is harder and harder to notice phenomena
+such as this. Consider the comparison of two sets
+of results in Table 1. With 41 languages it is cog-
+nitively hard to discover various relations between
+the languages and their results, even if one has the
+necessary linguistic knowledge.
+In this position paper, we argue that it is not
+the best practice to compare multilingual methods
+only with simple statistics, such as average. Com-
+monly used simple evaluation protocols might bias
+research in favor of dominant languages and in turn
+hurt historically marginalized languages. Instead,
+we propose to consider using qualitative results
+analysis that takes linguistic typology (Ponti et al.,
+2019) and comparative linguistics into account as
+an additional sanity check. We believe that this
+is an often overlooked tool in our research toolkit
+that should be used more to ensure that we are
+able to properly interpret results from multilingual
+evaluation and detect various linguistic biases and
+problems. In addition to this discussion, which
+we consider a contribution in itself, we also pro-
+pose a visualization based on URIEL typological
+database (Littell et al., 2017) as an example of such
+qualitative analysis, and we show that it is able to
+discover linguistic biases in published results.
+2
+Related Work
+Linguistic biases in NLP.
+Bender (2009) postu-
+lated that research driven mainly by evaluation in
+English will become biased in favor of this lan-
+guage and it might not be particularly language
+independent. Even in recent years, popular tech-
+niques such as word2vec or Byte Pair Encoding
+were shown to have worse performance on morpho-
+logically rich languages (Bojanowski et al., 2017;
+Park et al., 2020). Similarly, cross-lingual word
+embeddings are usually constructed with English
+arXiv:2301.01269v1  [cs.CL]  3 Jan 2023
+
+Spanish
+Catalan
+Portugese
+Persian
+Average
+Performance
+76.0
+79.0
+74.0
+52.0
+70.25
+63.0
+62.0
+59.0
+74.0
+64.5
+(a)
+Method A
+Method B
+Atlantic-Congo
+Afro-Asiatic
+Semitic
+Pama-Nyungan
+Otomanguean
+Italic
+Slavic
+Germanic
+Uralic
+Indo-Iranian
+Dravidian
+Turkic
+Sino-Tibetan
+Austroasiatic
+Austronesian
+(b)
+(c)
+Figure 1: (a) Comparison of two methods on unbalanced set of languages. (b) Visualization of URIEL languages
+with certain language families color-coded. (c) Comparison of two methods from Rahimi et al. This uses the same
+map of languages as b, but the view is zoomed.
+Language
+afr
+arb
+bul
+ben
+bos
+cat
+ces
+dan
+deu
+ell
+eng
+spa
+est
+pes
+fin
+fra
+heb
+hin
+hrv
+hun
+ind
+Method A
+74
+54
+54
+60
+77
+79
+72
+79
+64
+34
+57
+76
+71
+52
+69
+73
+46
+58
+77
+69
+61
+Method B
+59
+64
+61
+70
+63
+62
+62
+62
+58
+61
+47
+63
+64
+74
+67
+57
+53
+68
+61
+59
+67
+Language
+ita
+lit
+lav
+mkd
+zlm
+nld
+nor
+pol
+por
+ron
+rus
+slk
+slv
+alb
+swe
+tam
+tgl
+tur
+ukr
+vie
+AVG
+Method A
+76
+75
+67
+48
+63
+78
+77
+77
+74
+74
+36
+76
+76
+76
+69
+25
+57
+67
+49
+48
+64.5
+Method B
+60
+62
+68
+67
+66
+59
+65
+61
+59
+66
+53
+62
+64
+69
+69
+54
+66
+61
+60
+55
+62.1
+Table 1: Comparison of two methods from Rahimi et al. (2019).
+as a default hub language, even though this might
+hurt many languages (Anastasopoulos and Neubig,
+2020). Perhaps if the practice of research was less
+Anglocentric, different methods and techniques
+would have become popular instead. Our work
+is deeply related to issues like these. We show that
+multilingual evaluation with an unbalanced selec-
+tion of languages might cause similar symptoms.
+Benchmarking.
+Using benchmarks is a practice
+that came under a lot of scrutiny in the NLP com-
+munity recently. Benchmark evaluation was said
+to encourage spurious data overfitting (Kavumba
+et al., 2019), encourage metric gaming (Thomas
+and Uminsky, 2020) or lead the research away from
+general human-like linguistic intelligence (Linzen,
+2020). Similarly, benchmarks are criticized for be-
+ing predominantly focused on performance, while
+neglecting several other important properties, e.g.
+prediction cost or model robustness (Ethayarajh
+and Jurafsky, 2020). Average in particular was
+shown to have several issues with robustness that
+can be addressed by using pair-wise instance evalu-
+ation (Peyrard et al., 2021). To address these issues,
+some benchmarks refuse to use aggregating scores
+and instead report multiple metrics at the same time
+leaving interpretation of the results to the reader.
+Gehrmann et al. (2021) is one such benchmark,
+which proposes to use visualizations to help the in-
+tepretation. In this work, we also use visualizations
+to similar effect.
+3
+Multilingual Evaluation Strategies
+When comparing multilingual methods with non-
+trivial number of languages, it is cognitively hard
+to keep track of various linguistic aspects, such
+as language families, writing systems, typologi-
+cal properties, etc. Researchers often use various
+simplifying strategies instead:
+Aggregating
+metrics.
+Aggregating
+metrics,
+such as average performance or a number of
+languages where a certain method achieves the
+best results provide some information, but as
+we illustrated in Figure 1 (a), they might not
+tell the whole story.
+By aggregating results
+we lose important information about individual
+languages and language families.
+Commonly
+used statistics usually do not take underlying
+linguistic diversity into account. This might lead
+to unwanted phenomena, such as bias in favor
+of dominant language families.
+The encoded
+values of the aggregating metrics might not align
+with the values we want to express. Average is
+an example of utilitarianist world view, while
+using minimal performance might be considered
+to be a prioritarianist approach (Choudhury and
+Deshpande, 2021). Even though analyzing the
+
+values encoded in metrics is a step towards a fairer
+evaluation, they still miss a more fine-grained
+details of the results.
+Aggregated metrics for different groups.
+An-
+other option is to calculate statistics for certain
+linguistic families or groups. These are steps in
+the right direction, as they provide a more fine-
+grained picture, but there are still issues left. It is
+not clear which families should be selected, e.g.
+should we average all Indo-European languages
+or should we average across subfamilies, such as
+Slavic or Germanic. This selection is ultimately
+opinionated and different selections might show us
+different views of the results. In addition, aggregat-
+ing across families might still hide variance within
+these families. Grouping languages by the size of
+available datasets (e.g. low resource vs. high re-
+source) shows us how the models deal with data
+scarcity, but the groups might still be linguistically
+unbalanced.
+Balanced language sampling.
+Another option
+is to construct a multilingual dataset so that it is
+linguistically balanced. This process is called lan-
+guage sampling (Rijkhoff et al., 1993; Miestamo
+et al., 2016). In practice, this means that a small
+number of representative languages is selected for
+each family. The problem with dominant fami-
+lies is solved because we control the number of
+languages per family. However, selecting which
+families should be represented and then selecting
+languages within these families is again an opin-
+ionated process. Different families and their sub-
+families might have different degrees of diversity.
+Different selections might favor different linguistic
+properties and results might vary between them. It
+is also not clear, how exhaustive given selection is,
+i.e. how much of the linguistic variety has been
+covered. Some of the existing works mention their
+selection criteria: Longpre et al. (2020) count how
+many speakers the selection covers, Clark et al.
+(2020) use a set of selected typological proper-
+ties, Ponti et al. (2020) use the so called variety
+language sampling. Publishing the criteria allows
+us to do a post-hoc analysis in the future to evaluate,
+how well did these criteria work.
+Qualitative analysis
+In this paper, we argue that
+qualitative analysis is an often overlooked, yet ir-
+replaceable evaluation technique. In the following
+section, we will present our case study of how to
+perform qualitative analysis.
+4
+Case Study: Qualitative Analysis
+through Visualization
+In this section we show how to perform a quali-
+tative analysis of multilingual results with a visu-
+alization technique based on URIEL typographic
+database.
+We show that using this we can (1)
+uncover linguistic biases in the results, and (2)
+make sense of results from non-trivial number of
+languages. As case study, we study results from
+Rahimi et al. (2019). Our goal is not to evaluate
+particular methods from this paper, but to demon-
+strate how linguistically-informed analysis might
+help researchers gain insights into their results. We
+analyze the results from this paper not because we
+want to criticize it, but because it is a well-written
+paper that actually attempts to do multilingual eval-
+uation for non-trivial number of languages with
+significantly different methods. The linguistic bi-
+ases we uncover are already partially discussed in
+the paper. Here, we only show how to effectively
+perform qualitative analysis and uncover these bi-
+ases with appropriate visualization. Appendix A
+shows similar analysis for another paper (Heinzer-
+ling and Strube, 2019) where linguistic biases are
+visible.
+We use URIEL, a typological language database
+that consists of 289 syntactic and phonological bi-
+nary features for 3718 languages. We use UMAP
+feature reduction algorithm (McInnes and Healy,
+2018) to create a 2D typological language space.
+This map is shown in Figure 1 (b). The map is inter-
+active and allows for dynamic filtering of languages
+and families, as well as inspection of individual
+languages and their properties.1 Each point is one
+language and selected language families are color-
+coded in the figure. Even though URIEL features
+used for dimensionality reduction do not contain in-
+formation about language families, genealogically
+close languages naturally form clusters in our vi-
+sualization. Certain geographical relations are cap-
+tured as well, e.g. Sudanic and Chadic languages
+are neighboring clusters, despite being from differ-
+ent language families. This evokes the linguistic
+tradition of grouping languages according to the
+regions and macroregions. This shows that our vi-
+sualization is able to capture both intrafamiliar and
+interfamiliar similarities of languages and is thus
+appropriate for our use-case.
+We visualize results from Rahimi et al. (2019)
+on this linguistic map. Rahimi et al. use Wikipedia-
+1Code available at GitHub
+
+based corpus for NER, and they compare various
+cross-lingual transfer learning algorithms for 41
+languages. They use an unbalanced set of lan-
+guages, where the three most dominant language
+families – Germanic, Italic and Slavic – make up
+55% of all languages. See Appendix A for more
+details about the paper. We use our URIEL map to
+visualize a comparison between a pair of methods
+on all 41 languages from Table 1. In Figure 1 (c) we
+compare two methods – Method A – cross-lingual
+transfer learning methods using multiple source lan-
+guages (average performance 64.5), and seemingly
+worse Method B – a low-resource training with-
+out any form of cross-lingual supervision (average
+performance 62.1). We use the same URIEL map,
+but we superimpose the relative performance of the
+two methods as colored columns. Orange columns
+on this map show languages where Method A per-
+forms better, while blue columns show the same for
+Method B. Height of each column shows how big
+the relative difference in performance is between
+the two methods. I.e. taller orange columns mean
+dominant A, taller blue columns mean dominant B.
+We can now clearly see that there is a pattern in
+the location of the colored columns. Using aver-
+age as evaluation measure, Method A seems better
+overall. Here we can see that it is only better in
+one particular cluster of languages – the cluster of
+orange columns. All these are related European
+languages. Most of them are Germanic, Italic or
+Slavic, with some exceptions being languages that
+are not Indo-European, but are nevertheless geo-
+graphical neighbors, such as Hungarian. On the
+other hand, all the non-European languages actu-
+ally prefer Method B. These are the blue columns
+scattered in the rest of the space that consists of
+languages such as Arabic (Semitic), Chinese (Sino-
+Tibetan) or Tamil (Dravidian).
+This shows important fact about the two methods
+that was hidden by using average. Cross-lingual su-
+pervision seemed to have better performance, but it
+has better performance only in the dominant cluster
+of similar languages where the cross-lingual super-
+vision is more viable. Other languages, would actu-
+ally prefer using monolingual low-resource learn-
+ing, as they are not able to learn from other lan-
+guages that easily. In this case, average is overesti-
+mating the value of cross-lingual learning for non-
+European languages. This overestimation might
+cause harm to these languages.
+We can also see that there are some exceptions –
+the blue columns in the orange cluster. These ex-
+ceptions are Greek, Russian, Macedonian, Bulgar-
+ian and Ukrainian – all Indo-European languages
+that use non-Latin scripts. In this case, different
+writing systems are probably cause of additional
+linguistic bias. It might be hard to notice this pat-
+tern by simply looking at the table of results, but
+here we can quickly identify the languages as out-
+liers and then it is easy to realize what they have in
+common.
+Note that we do not expect to see this level of
+linguistic bias in most papers and we have cherry-
+picked this particular methods from this particu-
+lar paper because they demonstrate the case when
+the linguistic bias in the results is the most obvi-
+ous. This is caused mainly by unbalanced selection
+of languages on Wikipedia and in a sense unfair
+comparison of cross-lingual supervision with low
+resource learning.
+5
+Conclusions
+Multilinguality in NLP is becoming more common
+and methodological practice is sometimes lagging
+behind (Artetxe et al., 2020; Keung et al., 2020;
+Bender, 2011). Making progress will be inherently
+hard without proper evaluation methodology. In
+this work, we argue for necessity for qualitative
+results analysis and we showed how to use such
+analysis to improve the evaluation with interactive
+visualizations. In our case study, we were able to
+uncover linguistic biases in published results.
+Considering the practice in machine learning and
+NLP, it might be tempting to reduce a multilingual
+method performance to a single number. However,
+we believe that intricacies of multilingual evalua-
+tion can not be reduced so easily. There are too
+many different dimensions that need to be taken
+into consideration and NLP researchers should un-
+derstand these dimensions. We believe that appro-
+priate level of training in various linguistic fields,
+such as typology or comparative linguistics, is nec-
+essary for proper understanding of multilingual
+results and for proper qualitative analysis. We ar-
+gue that qualitative analysis is an oft overlooked
+approach to results analysis that should be utilized
+more to prevent various distortions in how we un-
+derstand linguistic implications of our results.
+6
+Ethical Considerations
+Much of current NLP research is focused on only a
+small handful of languages. Communities of some
+
+language users are left behind, as a result of data
+scarcity. We believe that our paper might have
+positive societal impact. It focuses on the issues
+of these marginalized languages and communities.
+Following our recommendations might lead to a
+more diverse and fair multilingual evaluation both
+in research and in industry. This might in turn led to
+better models, applications and ultimately quality
+of life changes for some.
+Acknowledgments
+This research was partially supported by DisAi,
+a project funded by Horizon Europe under GA No.
+101079164.
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+ter Kahrel. 1993.
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+2019. A survey of cross-lingual word embedding
+models. Journal of Artificial Intelligence Research,
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+lem with metrics is a fundamental problem for AI.
+CoRR, abs/2002.08512.
+
+A
+Details of Analysed Papers
+In this appendix, we provide additional information
+about papers we analysed.
+A.1
+Rahimi et al.
+This is the paper we used for demonstration in the
+main paper in Section 4. We use results reported in
+Table 4 in their paper. The languages they use are
+listed here in Table 2. We can see the apparent dom-
+inance of Indo-European languages. There are 14
+different methods listed in their paper. We compare
+the results for these methods in Figure 2. There we
+can see how the average results for individual meth-
+ods compare with the average results for non-GIS
+(Germanic-Italic-Slavic) languages. The numbers
+correspond to the order of methods listed in the
+original paper. The two methods compared in Fig-
+ure 1 (c) are shown as blue and orange, respectively.
+The orange Method A is BEAtok in the original pa-
+per. The blue Method B is called LSup. We can see
+the linguistic bias with this simplistic view as well.
+All the cross-lingual learning based methods have
+worse non-GIS results than methods that do not use
+cross-lingual learning (methods 1 and 2). However,
+this analysis can not replace the visualization we
+propose in Section 4. It provides a GIS-centered
+view, but it can not capture other sources of bias.
+For example, it does not show various outliers that
+were seen in the visualization, such as Uralic lan-
+guages that behave similarly to GIS languages, or
+Slavic languages with Cyrilic alphabet that behave
+differently than other Slavic languages.
+A.2
+Heinzerling and Strube
+Similar linguistic biases can be seen in Heinzer-
+ling and Strube as well. They evaluate various
+representations performance on POS tagging and
+NER. In Figure 3 we compare POS accuracy of a
+multilingual model with a shared embedding vocab-
+ulary (average performance 96.6, MultiBPEmb
++char +finetune in the original paper) and a
+simple BiLSTM baseline with no transfer super-
+vision (average performance 96.4, BiLSTM in the
+original paper). Orange columns are for languages
+that prefer the multilingual model, blue columns
+prefer the baseline. In this case, almost all orange
+columns are in fact GIS languages. Other lan-
+guages are having significantly worse results with
+this method and most of them actually prefer the
+simple baseline with no cross-lingual supervision.
+This shows the limitations of proposed multilingual
+ISO
+Language
+Subfamily
+Family
+bul
+Bulgarian
+Slavic
+Indo-European
+bos
+Bosnian
+ces
+Czech
+hrv
+Croatian
+mkd
+Macedonian
+pol
+Polish
+rus
+Russian
+slk
+Slovak
+slv
+Slovenian
+ukr
+Ukrainian
+afr
+Afrikkans
+Germanic
+dan
+Danish
+deu
+German
+nld
+Dutch
+nor
+Norwegian
+swe
+Swedish
+cat
+Catalan
+Italic
+fra
+French
+ita
+Italian
+por
+Portugese
+rom
+Romanina
+spa
+Spanish
+ben
+Bengali
+Indo-Iranian
+hin
+Hindi
+pes
+Iranian Persian
+lit
+Lithuanian
+Baltic
+lav
+Latvian
+ell
+Greek
+alb
+Albanian
+est
+Estonian
+Uralic
+fin
+Finnish
+hun
+Hungarian
+ind
+Indonesian
+Austronesian
+tgl
+Tagalog
+zlm
+Malay
+arb
+Standard Arabic
+Afro-Asiatic
+heb
+Hebrew
+vie
+Vietnamese
+Austroasiatic
+tam
+Tamil
+Davidian
+tur
+Turkish
+Turkic
+Table 2: Languages used in Rahimi et al..
+ISO
+Language
+Subfamily
+Family
+dan
+Danish
+Germanic
+Indo-European
+deu
+German
+eng
+English
+nld
+Dutch
+nor
+Norwegian
+swe
+Swedish
+bul
+Bulgarian
+Slavic
+ces
+Czech
+hrv
+Croatian
+pol
+Polish
+slv
+Slovenian
+fra
+Frech
+Italic
+ita
+Italian
+por
+Portugese
+spa
+Spanish
+hin
+Hindi
+Indo-Iranian
+pes
+Iranian Persian
+eus
+Basque
+Isolate
+fin
+Finnish
+Uralic
+heb
+Hebrew
+Afro-Asiatic
+ind
+Indonesian
+Austronesian
+Table 3: Languages used in Heinzerling and Strube.
+
+50
+60
+70
+80
+90
+Total average
+40
+50
+60
+70
+80
+90
+Non-GIS average
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+Rahimi et al.
+95.25 95.50 95.75 96.00 96.25 96.50 96.75
+Total average
+94.5
+95.0
+95.5
+96.0
+Non-GIS average
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Heinzerling and Strube
+Figure 2: Comparison of method performance. The re-
+lation between global average and average on non-GIS
+languages is shown. Each point represents one method
+from the papers.
+supervision for outlier languages.
+We use results reported in Table 5 in their paper.
+The languages they use are listed here in Table 3.
+Again, we can see an apparent dominance of GIS
+languages. There are 11 different methods listed in
+their paper. We omitted results for additional 6 low
+resource languages reported in Table 7, because
+only 4 out of 11 methods were used there. We
+compare the results for these methods in Figure 2,
+similarly as in the previous paper. The orange point
+is the multilingual model, the blue point is the base-
+line. Now we can see that the BiLSTM baseline is
+actually the best performing method for non-GIS
+languages.
+B
+Hyperparameters
+We use UMAP python library2 with the following
+hyperparameters:
+2umap-learn.readthedocs.io
+Figure 3: Comparison of two methods from Heinzer-
+ling and Strube.
+• Number of neighbours (n_neighbors): 15
+• Distance metric (metric): cosine
+• Minimal distance (min_dist): 0.5
+• Random see (random_state): 1
+C
+Additional Visualizations
+In this Section we show several additional possibil-
+ities of using URIEL map of languages to visualize
+results from multilingual evaluation. Our goal here
+is to propose additional techniques that can be used
+for qualitative analysis apart from the comparison
+of two methods used in Figure 1 in the main body
+of this paper. This is not an exhaustive list of vi-
+sualizations. We believe that many other types of
+visualization can be done using this type of qualita-
+tive analysis, based on the needs and requirements
+of the user.
+In Figure 4 we show how to compare more than
+two methods by visualizing the performance for
+each method separately. We have created a sepa-
+rate plot for three methods and we can compare
+their performance visually. We can see that HSup
+method has overall stable high performance. LSup
+has worse performance, but its still quite balanced.
+Finally, BWET has similar performance as LSup,
+but we can see that there are regions where it fails,
+e.g. the languages in the rightmost part of the figure
+have visibly worse performance.
+In Figure 5 we show yet another type of visu-
+alization. In this case, we simply visualize what
+method is the best performing for each language.
+
+HSup
+20
+40
+60
+80
+LSup
+BWET
+Figure 4: Comparison of multiple methods using size to mark method performance for individual languages. HSup,
+LSup and BWET are methods reported in (Rahimi et al., 2019).
+LSup
+RaReuns
+BEAent
+unsx2
+BEAent
+uns
+Figure 5: The best performing methods for various lan-
+guages.
+We compare methods using crosslingual super-
+vision and low-resource training (LSup). From
+seven methods, only four achieved the best per-
+formance for at least one language and those are
+shown in the Figure. Again, we can see similar
+picture as before. One method (BEAent
+uns×2) is
+the best performing method taking average into
+account. However, in this visualization we can
+see that it is actually the best performing method
+only in the dominant cluster of European languages.
+Elsewhere, other methods perform better.
+

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+Particle-Number Threshold for Non-Abelian Geometric Phases
+Julien Pinske, Vincent Burgtorf, and Stefan Scheel∗
+Institut für Physik, Universität Rostock, Albert-Einstein–Straße 23-24, D-18059 Rostock, Germany
+(Dated: January 31, 2023)
+When a quantum state traverses a path, while being under the influence of a gauge potential, it
+acquires a geometric phase that is often more than just a scalar quantity. The variety of unitary
+transformations that can be realised by this form of parallel transport depends crucially on the
+number of particles involved in the evolution. Here, we introduce a particle-number threshold (PNT)
+that assesses a system’s capabilities to perform purely geometric manipulations of quantum states.
+This threshold gives the minimal number of particles necessary to fully exploit a system’s potential
+to generate non-Abelian geometric phases.
+Therefore, the PNT might be useful for evaluating
+the resource demands of a holonomic quantum computer. We benchmark our findings on bosonic
+systems relevant to linear and nonlinear quantum optics.
+I.
+INTRODUCTION
+The evolution of a quantum state,
+in the pres-
+ence of some potential, is completely determined by
+Schrödinger’s equation which incorporates aspects such
+as the system’s spectrum, or the overall evolution time. If
+the system undergoes slow (adiabatic) changes, the evolv-
+ing state remains unaffected by these dynamical contribu-
+tions. Instead, its wave function acquires a phase factor
+that only depends on the geometry of the path the quan-
+tum state has traversed. This was first noticed by Berry
+[1] who pointed out that, unlike dynamical phases, a ge-
+ometric phase cannot be removed by a rescaling of the
+energy (gauge transformation).
+A famous example for
+this is the Aharonov-Bohm effect [2], in which the wave
+function of an electron traveling around a solenoidal mag-
+netic field picks up a phase proportional to the magnetic
+flux through the surface enclosed by the trajectory of the
+electron. Pancharatnam studied the phenomenon in the
+context of classical optics [3], where it manifests itself in
+states of polarisation. It was pointed out by Simon [4]
+that this purely geometric signature of a quantum evo-
+lution has to be attributed to parallel transport of the
+state vector along a path in a (projective) Hilbert space.
+If a quantum system supports a d-fold degenerate sub-
+space H0 with eigenstates |ψa⟩ (a = 1, . . . , d), an initially
+prepared wave packet generically evolves into a superpo-
+sition of the |ψa⟩ when undergoing adiabatic changes,
+that is without population transfer to states of different
+energy [5]. Wilzeck and Zee [6] associated such degen-
+eracy of the spectrum with the possibility of emerging
+non-Abelian (i.e., noncommuting) gauge potentials. In
+this case, the state after a time period T does not only
+acquire a (scalar) geometric phase but differs from the
+initial one by a unitary d × d matrix.
+If the Hamiltonian of the system is expressed through a
+set of physically accessible parameters {κµ}µ that change
+cyclically, i.e., κµ(0) = κµ(T), the time evolution is asso-
+ciated with a closed path γ in the parameter space M .
+∗ [email protected]
+The time evolution then takes the form of a quantum
+holonomy (non-Abelian geometric phase) [6]
+UA(γ) = ˆPexp
+� ˛
+γ
+A
+�
+,
+(1)
+where A = �
+µ Aµdκµ is the adiabatic connection (non-
+Abelian gauge potential).
+Depending on the physical
+platform, the {κµ}µ might include external driving fields,
+subsystem couplings, or hopping probabilities between
+different states. Due to the generally noncommuting na-
+ture of the connection, i.e., [Aµ, Aν] ̸= 0, the integration
+in Eq. (1) has to be performed with respect to the path
+ordering ˆP. The matrix-valued components of A can be
+directly calculated from the eigenstates of the system,
+i.e.,
+(Aµ)ab = ⟨ψb| ∂µ |ψa⟩ ,
+∂µ = ∂/∂κµ.
+(2)
+By traversing different loops in M one can potentially ac-
+cess a variety of different unitaries UA(γ). The set of all
+such transformations spans the holonomy group Hol(A).
+It is a subset of the unitary group U(d). In addition to
+their frequent occurrence in lattice-gauge theory [7] and
+loop-quantum gravity [8], holonomy groups turn out to
+be a crucial ingredient for geometric [9, 10] and topologi-
+cal quantum computation [11], where they constitute the
+fundamental gate set from which quantum algorithms are
+to be implemented.
+The question of how many different unitaries can be
+harnessed by driving loops through M is therefore closely
+related to computational universality [12], which holds
+if Hol(A) = U(d). Not only does this require a d-fold
+degenerate subspace, but a large parameter space as well
+[9]. More recently, it was observed that the number of
+particles prepared in the subspace H0 might drastically
+alter the form of the holonomy UA(γ) [13–15]. This is
+because the corresponding eigenstates |ψa⟩ can differ in
+their particle number. In this work, quantum holonomies
+are studied in relation to the number of particles involved
+in the evolution. In the following, this issue is motivated
+through an illustrative example.
+arXiv:2301.11999v1  [quant-ph]  27 Jan 2023
+
+2
+A.
+Λ-scheme of bosonic modes
+Consider a chain of three bosonic modes [Fig. 1 (a)].
+The outer modes ˆa± experience complex next-neighbour
+couplings κ± to the central mode ˆac. The Hamiltonian
+of the system reads
+ˆH = κ+ˆa+ˆa†
+c + κ−ˆacˆa†
+− + H.c..
+(3)
+Here, ˆa†
+k and ˆak denote the bosonic creation and anni-
+hilation operators, respectively, and H.c. stands for the
+Hermitian conjugate. The Hamiltonian (3) is the bosonic
+counterpart of an atomic three-level system in Λ configu-
+ration [16]. Such systems are of practical interest as they
+describe linear-optical multiport systems [17] and can be
+designed, for instance, in terms of integrated photonic
+waveguides [18, 19].
+Suppose a single photon is injected into one of the
+outer modes of the optical setup, with couplings κ±(t)
+varying slowly compared to the minimal energy gap
+�
+|κ+|2 + |κ−|2 > 0 (level crossing neglected).
+In the
+adiabatic limit, the photon remains in the zero-eigenvalue
+eigenstate (aka dark state)
+|D⟩ = sin θ |1+⟩ − cos θeiϕ |1−⟩ ,
+where tan θ = |κ−|/|κ+|, ϕ = arg(κ+), and |1±⟩ = ˆa†
+± |0⟩
+with |0⟩ denoting the three-mode vacuum. Here, the con-
+nection Aϕ = i cos2 θ is Abelian (while Aθ = 0). After
+traversing a closed path γ in the (θ, ϕ) plane, the output
+state |Ψ(T)⟩ = eiφ(γ) |Ψ(0)⟩ picks up a geometric phase
+φ(γ) =
+¨
+D
+sin(2θ)dϕdθ,
+(4)
+which depends on the area D enclosed by the loop γ.
+Interestingly, injecting a second (indistinguishable)
+photon into the setup, leads to the two dark states
+|D1⟩ = sin2 θ |2+⟩ −
+√
+2 sin θ cos θeiϕ |1+1−⟩
++ cos2 θe2iϕ |2−⟩ ,
+|D2⟩ =
+1
+√
+2
+�
+sin2 θ |2−⟩ + cos2 θe−2iϕ |2+⟩ − |2c⟩
+�
++ 2 sin θ cos θe−iϕ |1+1−⟩ .
+Consequently,
+Aϕ is now a matrix-valued quantity.
+Naively, one might expect that this enables the gener-
+ation of non-Abelian holonomies. However, a direct eval-
+uation of the Eq. (1) leads to
+UA(γ) =
+�
+e2iφ(γ)
+0
+0
+e−2iφ(γ)
+�
+.
+(5)
+It is immediately clear from Eq. (5) that the transforma-
+tions UA(γ) and UA(γ′), induced by two arbitrary loops
+γ and γ′ in M , always commute. Hence, even though de-
+generacy of the system would allow for the generation of
+non-Abelian transformations, the actual holonomy group
+is still Abelian. This phenomenon remains present when
+subjecting even more photons to the system [20], i.e.,
+while degeneracy scales up, the resulting holonomies are
+always commuting.
+The phenomenon that a system’s degeneracy increases
+under the exposure to multiple photons is by no means
+a property unique to the Hamiltonian (3). Adding an
+additional mode to the Λ-scheme leads to a tripod struc-
+ture [Fig. 1 (b)] that allows for any U(2) transforma-
+tion between its single-photon dark states [21–23]. Con-
+sidering two photons, the dark subspace becomes four-
+dimensional. However, as it was noticed in Refs. [14, 24]
+not all elements of the group U(4) can be designed in that
+way (one of the eigenstates decouples). Only, recently
+were these two-particle dynamics verified experimentally
+[15].
+FIG. 1.
+Graph representation of (bilinear) Hamiltonians, in
+which particle number exchange between the modes (vertices)
+is resembled by a connecting edge. (a) Schematic represen-
+tation of three planarly arranged bosonic modes experiencing
+complex next-neighbour coupling κ±. (b) Term scheme of the
+bosonic tripod structure in which the mode c exclusively cou-
+ples to the outer modes µ = ±, 0 via κµ. (c) A four-mode fully
+connected graph, where each side can experience a different
+coupling κµ. (d) Triangular graph of modes with coupling κµ,
+µ = ±, 0.
+B.
+Aim of the article
+This simple introductory example hints at a more gen-
+eral question. What is the number of particles N injected
+into a given setup in order to generate the most versatile
+set of quantum holonomies? After reviewing properties
+of the holonomy group in Sec. II, we address this issue
+by introducing the particle-number threshold (PNT) in
+Sec. III. The PNT of a quantum system gives the min-
+imal number of particles necessary to fully exploit the
+system’s potential for designing non-Abelian holonomies.
+We discuss the basic properties of PNTs and present a
+number of different examples relevant to linear and non-
+
+3
+linear quantum optics. Finally, Sec. IV is reserved for
+a summary of the article as well as some concluding re-
+marks.
+II.
+CURVATURE AND UNIVERSALITY
+If the composition of loops in M allows for the genera-
+tion of any unitary on the lth eigenspace Hl of a Hamil-
+tonian ˆH, the connection Al is said to be irreducible, and
+the holonomy group
+Hol(Al) =
+�
+UAl(γ) | γ(0) = γ(T)
+�
+coincides with U(dl). If the eigenspace additionally pos-
+sesses a multi-partite structure (dl = 2k), then Hl may
+be viewed as a k-qubit quantum code [25, 26] on which
+universal manipulation of quantum information is possi-
+ble in terms of holonomic gates UAl(γ) only.
+A convenient measure of how close the group Hol(Al)
+comes to span the entire unitary group, is given in terms
+of the local curvature Fl (the non-Abelian field-strength
+tensor).
+It describes changes of the eigenstates in Hl
+under variation of the parameters κµ. Its antisymmetric
+components (Fl,µν = −Fl,νµ) are calculated from [27]
+Fl,µν = ∂µAl,ν − ∂νAl,µ + [Al,µ, Al,ν].
+(6)
+According to a statement from differential geometry, the
+number of (linear-independent) components {Fl,µν}µν
+gives a lower bound to the dimension of Hol(Al). Here,
+dimension refers to the degrees of freedom that com-
+pletely specify an element in a matrix group. For exam-
+ple, a unitary in U(dl) is completely determined by spec-
+ifying d2
+l real numbers. Hence, we write dim U(dl) = d2
+l .
+This implies, if there are d2
+l linear-independent matrices
+Fl,µν, it is possible to realise any element of the unitary
+group in terms of Eq. (1) [28, 29], i.e., Hol(Al) = U(dl).
+A more accurate bound on the dimension of Hol(Al)
+can be obtained by including higher-order covariant
+derivatives
+∇l,σFl,µν, ∇l,δ∇l,σFl,µν, ∇l,ϵ∇l,δ∇l,σFl,µν, . . . .
+(7)
+Here, the covariant derivative operator
+∇l,σ = ∂σ + [Al,σ, · ]
+generally is different for each eigenspace, thus depending
+on the index l. The number of linearly independent ma-
+trices in Eqs. (6) and (7) equals the dimension of Hol(Al)
+[30, 31].
+Clearly,
+if
+the
+components
+Al,µ
+are
+Abelian,
+then ∇l,σ
+=
+∂σ,
+and the span of the matrices
+{Fl,µν, ∂σFl,µν, . . . }µνσ... is one-dimensional. In follows
+that Hol(Al) is an Abelian subgroup of U(dl).
+Note
+that, even though the above statements do not provide
+an explicit recipe for designing specific transformations,
+there existential nature makes them suitable for estimat-
+ing the general potency of a quantum system to generate
+holonomies. The dimension of the holonomy group acts
+as a natural measure of this potency.
+A.
+Four-mode fully-connected graph
+In order to illustrate the, rather abstract techniques in-
+troduced in the previous section, we give an example of
+a four-mode fully-connected graph, shown in Fig. 1 (c).
+Fully-connected graphs constitute the most general type
+of graphs. Hence, it is not expected that their Hamilto-
+nians possess degenerate eigenvalues when arbitrary con-
+figurations κ = (κµ)µ are considered. Nevertheless, one
+can always construct specific configurations that lead to
+degenerate subspaces. This is done as follows.
+Let ˆH0 be a time-independent Hamiltonian with some
+fixed degeneracy structure.
+Consider the isospectral
+Hamiltonian
+ˆH(κ) = ˆV(κ) ˆH0 ˆV†(κ),
+(8)
+parameterised over points κ = (θ, ϕ) in M . For the four-
+mode system, let ˆH0 = ˆn1 + ˆn2 − ˆn4 (with ˆnk = ˆa†
+kˆak)
+and
+ˆV(θ, ϕ) = ˆV12(θ1, ϕ1) ˆV23(θ2, ϕ2) ˆV34(θ3, ϕ3)
+(9)
+is our unitary of choice. Here, ˆVkk+1(θk, ϕk) creates a
+mixing between the modes k and k+1. More specifically,
+we define
+ˆVkk+1ˆa†
+k ˆV †
+kk+1 = cos θkeiϕkˆa†
+k + sin θkˆa†
+k+1,
+ˆVkk+1ˆa†
+k+1 ˆV †
+kk+1 = cos θke−iϕkˆa†
+k+1 − sin θkˆa†
+k,
+(10)
+which describes a general SU(2) transformation.
+The
+transformation (9) is not the most general unitary, but
+is chosen such that the Hamiltonian (8) is still bilinear
+in the creation and annihilation operators. Thus, it can
+be represented by the graph in Fig. 1 (c).
+If a single particle is subjected to the system, the
+Hamiltonian has a 4 × 4 matrix representation ˆH|F1.
+Here, F1 denotes the first Fock layer, which contains
+the single-particle states |1k⟩ = ˆa†
+k |0⟩. In this Fock layer
+the system has only a single dark state
+|D⟩ = eiϕ3 cos θ3(sin θ1 sin θ2 |11⟩ + e−iϕ1 cos θ1 sin θ2 |12⟩
++ e−iϕ2 cos θ2 |13⟩) − sin θ3 |14⟩ .
+(11)
+A straight-forward calculation of the corresponding con-
+nection [cf. Eq. (2)] reveals (we omit the index l = 0 for
+notational ease)
+Aϕ1 = −i cos2 θ1 sin2 θ2 cos2 θ3,
+Aϕ2 = −i cos2 θ2 cos2 θ3,
+Aϕ3 = i cos2 θ3,
+and Aθ1 = Aθ2 = Aθ3 = 0.
+The curvature is readily
+calculated from to Eq. (6). Its nonvanishing components
+
+4
+are
+Fϕ1θ1 = −2i sin θ1 cos θ1 sin2 θ2 cos2 θ3,
+Fϕ1θ2 = 2i cos2 θ1 sin θ2 cos θ2 cos2 θ3,
+Fϕ1θ3 = −2i cos2 θ1 sin2 θ2 sin θ3 cos θ3,
+Fϕ2θ2 = −2i sin θ2 cos θ2 cos2 θ3,
+Fϕ2θ3 = −2i cos2 θ2 sin θ3 cos θ3,
+Fϕ3θ3 = 2i sin θ3 cos θ3.
+It follows that Abelian holonomies (i.e., Berry phases)
+can be designed by adiabatically traversing loops in M ,
+i.e., Hol(A) = U(1).
+Next, consider the second Fock layer F2 spanned by
+the two-particle states
+|21⟩ , |1112⟩ , |1113⟩ , |1114⟩ , |22⟩ ,
+|1213⟩ , |1214⟩ , |23⟩ , |1314⟩ , |24⟩ .
+The matrix ˆH|F2 supports a three-fold degenerate dark
+subspace with states |Dk⟩, for k = 1, 2, 3 (explicit form
+in Appendix A). The connection on this subspace is
+Aϕ1|θ2=0
+θ1=θ3= π
+4 = i
+2
+�
+�
+0
+0
+0
+0
+1
+ei(ϕ1−ϕ2)
+0 e−i(ϕ1−ϕ2)
+−1
+�
+� ,
+Aϕ2 = i cos2 θ2 cos2 θ3
+�
+�
+−2
+0
+0
+0
+−1 0
+0
+0
+1
+�
+� ,
+Aϕ3 = i cos2 θ3
+�
+�
+2
+0
+0
+0 −1 0
+0
+0
+1
+�
+� ,
+Aθ1 = cos θ2
+�
+�
+0
+0
+0
+0
+0
+−ei(ϕ2−ϕ1)
+0 ei(ϕ2−ϕ1)
+0
+�
+� ,
+and Aθ2 = Aθ3 = 0. Calculating the curvature (6) and its
+first order covariant derivative gives rise to (only linearly
+independent components are shown)
+Fϕ1θ1|κ0 =
+�
+�
+−i
+0
+0
+0
+i
+2
+−
+i
+2
+√
+2
+0
+−
+i
+2
+√
+2
+− i
+2
+�
+� ,
+Fϕ1θ2|κ0 =
+�
+�
+i
+0
+0
+0
+i
+2
+i
+√
+2
+0
+i
+√
+2 − i
+2
+�
+� ,
+Fϕ2θ1|κ0 =
+�
+�
+0 0 0
+0 0 i
+0 i 0
+�
+� ,
+∇ϕ1Fθ1θ2|κ0 =
+�
+�
+0
+0
+0
+0
+i
+−
+i
+2
+√
+2
+0 −
+i
+2
+√
+2
+−i
+�
+� ,
+∇θ1Fϕ2θ1|κ0 =
+�
+�
+0
+0
+0
+0 −i 0
+0
+0
+i
+�
+� ,
+(12)
+evaluated at the point κ0, with ϕk = 0 and θk = π/4.
+The matrices in Eq. (12) are the (infinitesimal) genera-
+tors [32] of a 5-dimensional Lie group. This constitutes
+a lower bound to the dimension of Hol(A). Nevertheless,
+the analysis illustrates that the two-particle case enables
+the generation of more intriguing holonomies than the
+single-particle case. More precisely, the two-particle dark
+states led to a non-Abelian holonomy group Hol(A), that
+is a proper subgroup of U(3).
+The key observation is that, increasing the number of
+particles significantly improved the computational capac-
+ity (from Abelian to non-Abelian holonomies) to gener-
+ate unitaries on the dark subspace. Intuitively, it is clear
+that, the dimension of Hol(A) cannot increase continually
+when the particle number becomes larger, as this would
+result in arbitrarily high computational power, while hav-
+ing only limited physical resources comprised in M . This
+leads us to an interesting question.
+How far can one increase the dimension of the holon-
+omy group by subjecting a larger number of particles to a
+system?
+This question will be addressed in the following section
+by means of a particle-number threshold, which consti-
+tutes a formal answer to the issue.
+III.
+PARTICLE-NUMBER THRESHOLD
+The previously presented benchmark system revealed
+a dependence of a system’s holonomy group on the par-
+ticle number N. Firstly, this is due to the fact that the
+spectral properties (in particular degeneracy) of a quan-
+tum system vary when the corresponding Hamiltonian ˆH
+is limited to act on different Fock layers
+FN =
+�
+|n1, n2 . . .⟩
+���
+�
+k nk = N
+�
+.
+Secondly, we noticed that even if the degeneracy in-
+creases, this does not necessarily mean that it is possible
+to generate a more useful (i.e., higher dimensional) sub-
+group of unitaries.
+Therefore, it is a natural question
+to ask, what is the particle number N at which one of
+the holonomy groups {Hol(Al)}l reaches its maximal di-
+mension and is therefore most suitable for designing a
+versatile set of unitaries. We refer to the number of par-
+ticles necessary for this endeavour as the particle-number
+threshold (PNT) Nt.
+Definition. Let ˆH be the Hamiltonian of a quantum sys-
+tem in second quantisation that evolves adiabatically in
+time. The particle-number threshold Nt denotes the min-
+imum number of particles necessary for one of the sys-
+tem’s holonomy groups to reach its maximum potential
+(maximal dimension) for the generation of holonomies.
+In the language of holonomic quantum computation
+[9, 10] the PNT of a quantum system ˆH gives the number
+of particles to be prepared, in order to come as close
+as possible to the desirable notion of universality.
+In
+
+5
+contrast to previous examples, where the focus was on the
+dark subspace, the above definition demands an analysis
+of the holonomy groups of each eigenspace Hl, in order
+to determine dim Hol(Al) for all l. Then, there exists an
+subspace with index l′ such that for all other l
+dim Hol(Al′) ≥ dim Hol(Al)
+holds.
+In other words, the PNT Nt gives the num-
+ber of particles necessary to populate any state in the
+eigenspace Hl′.
+A.
+Properties of PNTs
+The PNT Nt of a (bosonic) quantum system ˆH is, in
+general, hard to calculate, as it demands for a calculation
+of the connection Al for each eigenspace (there could be
+infinitely many). Nevertheless, some general remarks can
+still be made. Consider a quantum system that consists
+of a collection of noninteracting subsystems, i.e., ˆH =
+�
+a ˆHa. Suppose, the PNT N (a)
+t
+for each subsystem ˆHa
+is known and that, Hol
+�
+A(a)
+l′
+�
+denotes its holonomy group
+with maximal dimension. The composite system ˆH then
+has PNT Nt = �
+a N (a)
+t
+.
+This becomes evident when
+noting that the highest-dimensional holonomy group
+Hol(Al′) =
+�
+a
+Hol
+�
+A(a)
+l′
+�
+(13)
+is just the tensor product of the holonomy groups
+Hol
+�
+A(a)
+l′
+�
+of each individual subsystem. The holonomy
+group (13) of the composite system acts on the subspace
+with energy �
+a ε(a)
+l′ , where ε(a)
+l′
+denotes the eigenenergy
+of the subspace on which the group Hol
+�
+A(a)
+l′
+�
+acts.
+Next,
+consider
+a
+Hamiltonian
+with
+isospectral
+parametrisation, that is
+ˆH(κ) = ˆV(κ) ˆH0 ˆV†(κ),
+(14)
+with ˆH0 being a Hamiltonian having fixed degeneracy
+structure {dl}l and eigenstates {|ψl,a⟩}l,a. Suppose there
+is a sufficiently large parameter space M so that ˆV(κ) is
+the most general unitary operator. Adiabatic evolution
+in the lth eigenspace is then governed by the most general
+connection
+(Al,µ)ab = ⟨ψl,b| ˆV†∂µ ˆV |ψl,a⟩ .
+(15)
+In the above, we made use of the fact that ˆV(κ) |ψl,a⟩
+are the eigenstates of (14).
+By construction, one has
+Hol(Al) = U(dl). For such a general parametrisation, it
+is indeed the eigenspace with largest degeneracy dl′ ≥ dl,
+that is the one most desirable for the generation of non-
+Abelian holonomies. Hence, Nt is the number of particles
+necessary to populate any state in the most degenerate
+eigenspace Hl′.
+What happens when ˆV is not an arbitrary unitary,
+but is limited to some smaller set of physically accessi-
+ble operations? For concreteness, consider the two-mode
+Hamiltonian associated with a nonlinear Kerr medium
+ˆH0 = ˆn1(ˆn1 − ˆ1) + ˆn2(ˆn2 − ˆ1).
+Here, the unitary ˆV(α, β, ξ, ζ) is a product of single and
+two-mode displacement
+ˆDk(α) = exp
+�
+αˆa†
+k − α∗ˆak
+�
+,
+ˆK(β) = exp
+�
+βˆa†
+1ˆa2 − β∗ˆa1ˆa†
+2
+�
+,
+(16)
+as well as single and two-mode squeezing
+ˆSk(ξ) = exp
+�
+ξ(ˆa†
+k)2 − ξ∗ˆa2
+k
+�
+,
+ˆ
+M(ζ) = exp
+�
+ζˆa†
+1ˆa†
+2 − ζ∗ˆa1ˆa2
+�
+,
+(17)
+respectively [33].
+By driving coherent displacement
+(α, β) and squeezing parameters (ξ, ζ) through a closed
+loop in M = C4, holonomies on the eigenspaces of ˆH
+are obtained. In Ref. [34] it was shown that this enables
+arbitrary U(4) transformations over the zero-eigenvalue
+eigenspace H0. The subspace is spanned by the number
+states |0102⟩, |1102⟩, |0112⟩, and |1112⟩, i.e., two photons
+are necessary to fully occupy the subspace.
+An extended study (up to N = 50) of the curvature
+Fl shows that, even though further increasing the parti-
+cle number (N > 2) populates subspaces with increased
+degeneracy (up to dl = 10 for some eigenspaces), their
+holonomy groups do not offer a computational advantage.
+By that we mean
+dim Hol(Al) ≤ dim Hol(A0)
+verified for all eigenspaces Hl with index l ≤ 352 (cf.
+Tab. I). We did so by explicitly calculating the compo-
+nents Fl,µν of the curvature and their covariant deriva-
+tives up to order 3, (these are to large to be displayed
+here). The computed dimension of the groups {Hol(Al)}l
+did not increase further after the first order derivatives,
+thus giving us good confidence that the dimension was
+determined accurately.
+There is an intuitive explanation for the fact that
+eigenspaces involving higher particle numbers (N > 6)
+lead to less useful holonomy groups. The Gaussian op-
+erations (16) and (17) contribute to the evolution only
+via the connection Al.
+The derivative ∂µ in Eq.
+(15)
+that acts on the operators (16) and (17), leads to cre-
+ation and annihilation operators of (at most) quadratic
+order. Hence, Fock states with larger differences in their
+photon numbers cannot be transformed into each other
+by a quantum holonomy, even when they lie in the same
+subspace.
+In summary, the subspace H0 (containing at most two-
+particle states) should be preferred when the system is
+utilised in a holonomic quantum computation.
+There-
+fore, the PNT of the two-mode Kerr Hamiltonian is
+
+6
+Nt = 2.
+Moreover, it was shown that restricting the
+parametrisation of the Hamiltonian (14) to unitaries ˆV
+that can be implemented by Gaussian operations (16)
+and (17), led to most of the system’s eigenspaces hav-
+ing reducible connections Al, i.e., Hol(Al) ⊂ U(dl).
+Hence, degeneracy became a quantity of secondary in-
+terest. In Tab. I the spectral properties of the two-mode
+Kerr Hamiltonian ˆH are listed together with their ca-
+pacity to generate holonomies on the eigenspaces Hl (for
+l = 0, . . . , 352).
+Note that subspaces with degeneracy
+dl ≤ 4 are not listed in in Tab. I, as it is already clear
+that their holonomy groups cannot exceed the dimension
+of Hol(A0) = U(4).
+B.
+PNTs of coupled harmonic oscillators
+While the exact calculation of a PNT can be a daunting
+task, given a collection of coupled harmonic oscillators,
+certain specialisations arise that can simplify calculations
+drastically. In Fig. 1 such systems were represented as
+graphs. The calculation of PNTs for such systems would
+be relevant, for instance, to the geometric manipulation
+of multi-photon states in linear optics [14] as well as linear
+optical quantum computation by holonomic means [20].
+Population transfer between different Fock layers FN
+does not occur in these systems, as the total number of
+particles stays conserved throughout an evolution. From
+a mathematical viewpoint, this implies that the system’s
+Hamiltonian reveals a block-matrix structure, i.e.,
+ˆH =
+�
+N∈N
+ˆH|FN .
+In addition, there always exists a spectral decomposition
+ˆH = �
+l εl ˆΠl, with ˆΠl denoting the projector onto the
+eigenspace Hl. It follows that the eigenspaces themselves
+admit a similar decomposition, that is
+ˆΠl =
+�
+N(l)
+ˆΠl|FN(l),
+(18)
+where summation is carried out over those particle num-
+bers N(l) at which the corresponding energy εl occurs.
+As an example, the Hamiltonian (3) of the Λ-scheme
+[Fig. 1. (a)] does not possess single-particle eigenstates
+with energy 2
+�
+|κ+|2 + |κ−|2. In other words the eigen-
+value does not lie in the spectrum of ˆH|F1, but it is an
+eigenvalue of the matrix ˆH|FN for N ≥ 2. In this case,
+the sum in Eq.
+(18) corresponds to an infinite series
+starting with N(l) = 2, 3, . . . .
+If additionally the evolution is assumed to be adia-
+batic, population transfer occurs within each eigenspace
+separately. Hence, the decomposition (18) is inherited to
+the time-evolution operator (quantum holonomy)
+UAl(γ) =
+�
+N(l)
+UAl(γ)|FN(l).
+(19)
+l
+εl
+dl
+≤ N
+dim{Fl,µν}µν dim Hol(Al)
+0
+0
+4
+2
+14
+16
+1
+2
+4
+3
+14
+16
+5
+12
+5
+6
+9
+9
+16
+42
+6
+10
+9
+9
+26
+72
+6
+13
+12
+12
+37
+110
+6
+15
+9
+9
+45
+132
+6
+17
+9
+9
+54
+162
+6
+19
+6
+6
+60
+182
+6
+20
+9
+9
+70
+212
+6
+21
+3
+3
+78
+240
+6
+21
+9
+9
+87
+272
+6
+24
+9
+9
+99
+312
+5
+26
+3
+3
+108
+342
+6
+27
+9
+9
+113
+362
+6
+27
+3
+3
+130
+420
+5
+30
+9
+9
+131
+422
+6
+30
+3
+3
+141
+462
+8
+31
+9
+9
+157
+512
+6
+33
+6
+6
+168
+552
+10
+34
+9
+9
+199
+662
+6
+36
+3
+3
+208
+702
+6
+38
+9
+9
+215
+722
+6
+39
+6
+6
+222
+756
+6
+38
+9
+9
+225
+762
+6
+40
+3
+3
+238
+812
+10
+41
+9
+9
+266
+912
+6
+42
+3
+3
+274
+942
+8
+44
+3
+3
+285
+992
+6
+45
+9
+9
+306
+1062
+8
+47
+3
+3
+320
+1112
+6
+48
+3
+3
+323
+1122
+6
+48
+9
+9
+346
+1202
+8
+50
+3
+3
+349
+1212
+6
+49
+3
+3
+352
+1232
+8
+50
+3
+3
+TABLE I.
+Holonomy groups of the two-mode nonlinear
+Kerr medium parameterised by the Gaussian operations (16)
+and (17).
+The table contains the degeneracy dl of the lth
+eigenspace (with energy εl). N denotes the number of parti-
+cles necessary to fully occupy the corresponding eigenspace.
+The number of linear-independent curvature components
+Fl,µν as well as the dimension of the holonomy group Hol(Al).
+Covariant derivatives were calculated up the order of 3.
+Remarkably, the connection will always be reducible for
+such a system, because it is not possible to generate
+transformations between different Fock layers. The best
+one can hope for, is to find is a highly-degenerate N-
+particle block in the eigenspace Hl such that the holon-
+omy UAl(γ)|FN realises any unitary transformation on
+the subspace Hl|FN .
+This is nothing but a geometric
+
+7
+incarnation of the well-known fact that networks of cou-
+pled oscillators (by themselves) do not allow for univer-
+sal quantum computation [35], but must be supported by
+additional resources, such as measurement-induced non-
+linearities [36, 37].
+Note that, even though the quantum holonomy (19)
+can have an infinite-dimensional matrix representation,
+it might still be commuting, that is UAl(γ)UAl(γ′) =
+UAl(γ′)UAl(γ) for any two loops γ and γ′ in M .
+For
+the purpose of illustration, consider the Hamiltonian (3)
+of the Λ-scheme [cf. Fig. 1 (a)] which gives rise to an
+infinite-dimensional dark subspace. For a single photon,
+the matrix ˆH|F1 has only one dark state. Given two or
+three photons in the setup, ˆH|F2 and ˆH|F3 both have
+two dark states. Subjecting four photons to the system
+leads to a Hamiltonian matrix ˆH|F4 having three dark
+states.
+Even though degeneracy further increases, the
+quantum holonomy
+UA0 =
+�
+��
+UA0|F1
+UA0|F2
+...
+�
+��
+will remain Abelian, because the N-particle block
+UA0(γ)|FN = diag
+�
+eiNφ(γ), e−iNφ(γ), eiNφ(γ), . . .
+�
+,
+is itself a diagonal matrix [cf. Eq. (5) for N = 2]. Here,
+φ(γ) is the geometric phase defined in Eq. (4). The above
+analysis illustrates, that increasing the particle number in
+the photonic Λ-scheme does not increase the holonomy
+group’s dimension, i.e., it stays Abelian. Similar argu-
+ments hold for the other eigenspaces of the system, and
+thus a single photon is sufficient to generate any phase
+in U(1). Hence, the PNT is Nt = 1.
+1.
+Three-mode fully-connected graph
+Consider a setup containing three oscillator modes ˆa±
+and ˆa0. Coupling between the modes is described by the
+parameters κ± and κ0, respectively. The system corre-
+sponds to the three-mode fully-connected graph shown
+in Fig. 1 (d).
+For simplicity, its Hamiltonian is considered to be in
+the configuration
+ˆH(θ, ϕ) = ˆV(θ, ϕ) ˆH0 ˆV†(θ, ϕ),
+(20)
+with ˆH0 = ˆn+ − ˆn−. In the above,
+ˆV(θ, ϕ) = ˆV+0(θ+, ϕ+) ˆV0−(��−, ϕ−),
+with the operator ˆVkk+1 defined in Eq. (10). The 3 × 3
+matrix ˆH|F1 possesses single-particle eigenstates
+|B+⟩ = cos θ+eiϕ+ |1+⟩ − sin θ− |10⟩ ,
+|D⟩ = cos θ−eiϕ−�
+cos θ+e−iϕ+ |1+⟩ − sin θ+ |10⟩
+�
+− sin θ− |1−⟩ ,
+|B−⟩ = cos θ−e−iϕ− |1−⟩
+− sin θ−
+�
+cos θ+e−iϕ+ |10⟩ − sin θ+ |1+⟩
+�
+,
+with eigenenergies ε± = ±1 and ε0 = 0. The connec-
+tion for each eigenvalue is readily calculated via Eq. (2),
+leading to the nonvanishing components
+A+,ϕ+ = i cos2 θ+,
+A0,ϕ± = i cos2 θ±,
+A−,ϕ+ = −i sin2 θ+ cos2 θ+,
+A−,ϕ− = −i cos2 θ−.
+When given more than just a single particle, the Hamil-
+tonian (20) gives rise to degenerate subspaces, e.g., con-
+sidering two-particles in the system, the 6 × 6 matrix
+ˆH|F2 possesses two dark states. However, in the follow-
+ing it will be shown that the resulting holonomies are
+still Abelian for arbitrary particle numbers N. It is a
+well known fact for coupled-mode systems, that knowing
+the single-particle evolution is equivalent to knowing the
+evolution of the modes ˆak(t) in the Heisenberg picture
+[33], viz.
+ˆa†
+±(T) = e∓iT e
+¸
+A±ˆa†
+±(0),
+ˆa†
+0(T) = e
+¸
+A0ˆa†
+0(0).
+Subsequently, the evolution of any N-particle state can
+be given explicitly. It follows the state can only attain a
+Berry phase as well. The initial N-particle state
+|Ψ(0)⟩ =
+1
+�
+n+!n0!n−!
+�
+ˆa†
++
+�n+�
+ˆa†
+0
+�n0�
+ˆa†
+−
+�n− |0⟩
+(N = n+ + n0 + n−) adiabatically evolves into
+|Ψ(T)⟩ = en+
+¸
+A+en0
+¸
+A0en−
+¸
+A− |n+⟩ ⊗ |n0⟩ ⊗ |n−⟩ ,
+accumulating a (scalar) geometric phase.
+We this conclude that, independent of the provided
+particle number N, the holonomy group can only be
+Abelian.
+Hence, the PNT of the system is Nt = 1,
+as moving beyond the single-particle case did not lead
+to more versatile groups of holonomies but just higher-
+dimensional representations of the group U(1).
+The
+above argument is the special case of a more general
+bosonic-operator framework, which we devised in Ref.
+[20]. This formalism enables a photon-number indepen-
+dent description of holonomies, and thus might be useful
+for the calculation of PNTs in coupled-mode systems.
+C.
+PNTs of fermionic systems
+So far, all considered quantum systems were bosonic
+in nature. Nevertheless, the definition of a PNT is appli-
+cable to any quantum system given in second quantisa-
+tion (cf. Sec. III). Fermionic modes are associated with
+
+8
+creation and annihilation operators satisfying canonical
+anticommutation relations.
+Because of this, the most
+prominent difference to the bosonic setups studied previ-
+ously, is that fermions have to obey the Pauli principle,
+i.e., two fermions cannot occupy the same mode simulta-
+neously. This drastically reduces the number of possible
+states in a system and in particular, the corresponding
+Hilbert space (Fock space) is finite dimensional. Hence,
+the calculation of the PNT of a fermionic system be-
+comes much more manageable in comparison to bosonic
+systems.
+PNTs can also be calculated for systems comprising
+both bosonic and fermionic modes. As an elementary ex-
+ample, consider the Jaynes-Cummings Hamiltonian de-
+scribing the interaction between an incident light field
+and a single atomic energy level at resonance. Within
+the rotating wave approximation, the Hamiltonian reads
+[33]
+ˆHJC = ωAˆσ+ˆσ− + ωcˆn + κ
+�
+ˆa†ˆσ− + ˆaˆσ+�
+,
+with resonance frequency ωA of the atom, ωc being the
+frequency of the incident light field, and κ describing the
+strength of the light-matter interaction. The atomic lad-
+der operators ˆσ− and ˆσ+ = (ˆσ−)† shift an electron from
+the ground to the excited state, and vice versa. The sys-
+tem possesses a nondegenerate spectrum {εn±}n∈N with
+corresponding eigenstates
+|n+⟩ = sin θ |g, n + 1⟩ + cos θ |e, n⟩ ,
+|n−⟩ = cos θ |g, n + 1⟩ − sin θ |e, n⟩ ,
+where tan(2θ) = 2κ√n + 1/(ωc − ωA) and n being the
+photon number. This form of the eigenstates highlights
+that the underlying parameter space does not possess any
+curvature, i.e., Fn±,θθ = 0 for all photon numbers n ∈ N.
+Hence, the system is not suitable for the generation of
+quantum holonomies, and this is reflected in the PNT,
+i.e., Nt = 0.
+IV.
+DISCUSSION
+In this article we studied quantum holonomies in rela-
+tion to the particle number in a system. It was shown
+that, increasing the number of particles can lead to a
+higher-dimensional holonomy group, thus improving the
+capabilities of the system to generate useful unitaries.
+We introduced the PNT of a quantum system, which de-
+notes the minimal number of particles necessary to fully
+exploit the systems capacity for generating a versatile set
+of quantum holonomies. Besides some general statements
+that could be made about PNTs, we illustrated the the-
+ory in terms of benchmark examples relevant to linear
+and nonlinear quantum optics. We saw that for systems
+of coupled oscillators only the Nt-particle block of an
+eigenspace contributes to its holonomy group relevantly,
+because the particle number Nt subjected to the system
+does not change throughout the propagation. This result
+appears to be relevant to linear optical quantum compu-
+tation by adiabatic means. We argued that the results
+presented are applicable to both bosonic and fermionic
+systems of interest. Our general investigation hints at the
+utility of the concept in assessing the capabilities of dif-
+ferent quantum systems to perform holonomic quantum
+computations in terms of holonomies. PNTs might also
+be relevant to the simulation of gauge groups in terms of
+adiabatic parameter variations.
+Currently, there is a lack of analytical tools to compute
+PNTs. While this is a straight forward task for fermionic
+systems, it becomes a challenging issue for bosonic sys-
+tems, where there is no bound on the particle number.
+Estimations of the PNT up to some finite particle num-
+ber, might be sufficient for most practical purposes, but
+general strategies for calculating PNTs can be relevant
+for a deeper understanding of many-particle physics in
+an adiabatic setting.
+ACKNOWLEDGMENTS
+Financial support by the Deutsche Forschungsgemein-
+schaft (DFG SCHE 612/6-1) is gratefully acknowledged.
+Appendix A: Two-particle dark states of the
+four-mode fully-connected graph
+In this appendix, we give the two-particle dark states
+of the Hamiltonian matrix ˆH|F2 of the four-mode fully-
+connected graph shown in Fig. 1 (c). The isospectral
+Hamiltonian of the system is given in Eq. (8). The three
+dark states of the system read
+|D1⟩ =
+1
+√
+2
+�
+eiϕ3 sin θ1 sin θ2 cos θ3ˆa†
+1 + ei(ϕ3−ϕ1) cos θ1 sin θ2 cos θ3ˆa†
+2 + ei(ϕ3−ϕ2) cos θ2 cos θ3ˆa†
+3 − sin θ3ˆa†
+4
+�2 |0⟩ ,
+|D2⟩ =
+�
+sin θ1 sin θ2 sin θ3ˆa†
+1 + e−iϕ1 cos θ1 sin θ2 sin θ3ˆa†
+2 + e−iϕ2 cos θ2 sin θ3ˆa†
+3 + e−iϕ3 cos θ3ˆa†
+4
+�
+×
+�
+eiϕ1 cos θ1ˆa†
+1 − sin θ1ˆa†
+2
+�
+|0⟩ ,
+|D3⟩ =
+�
+sin θ1 sin θ2 sin θ3ˆa†
+1 + e−iϕ1 cos θ1 sin θ2 sin θ3ˆa†
+2 + e−iϕ2 cos θ2 sin θ3ˆa†
+3 + e−iϕ3 cos θ3ˆa†
+4
+�
+×
+�
+eiϕ1 sin θ1 cos θ2ˆa†
+1 + ei(ϕ2−ϕ1) cos θ1 cos θ2ˆa†
+2
+�
+|0⟩ ,
+
+9
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