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+{"text":"\\section{Special values of the Lommel functions $s_{\\protect\\mu ,\\protect\\nu\n}(z)$ and $S_{\\protect\\mu ,\\protect\\nu }(z)$}\n\n\\ \\ \\ \\ The Lommel functions $s_{\\mu ,\\nu }(z)$ and $S_{\\mu ,\\nu }(z)$ with\nunrestricted $\\mu ,\\nu $ satisfy the relations \\cite{Magnus1}\n\n\\begin{quote}\n\\begin{equation}\n\\frac{2\\nu }{z}S_{\\mu ,\\nu }(z)=(\\mu +\\nu -1)S_{\\mu -1,\\nu -1}(z)-(\\mu -\\nu\n-1)S_{\\mu -1,\\nu +1}(z),  \\label{eq1}\n\\end{equation}\n\n\\begin{equation}\n\\lbrack (\\mu +1)^{2}-\\nu ^{2}]S_{\\mu ,\\nu }(z)+S_{\\mu +2,\\nu }(z)=z^{\\mu +1},\n\\label{eq2}\n\\end{equation\n\\begin{equation}\n\\frac{dS_{\\mu ,\\nu }(z)}{dz}+\\frac{\\nu }{z}S_{\\mu ,\\nu }(z)=(\\mu +\\nu\n-1)S_{\\mu -1,\\nu -1}(z),  \\label{eq3}\n\\end{equation\nand the symmetry propert\n\\begin{equation*}\nS_{\\mu ,-\\nu }(z)=S_{\\mu ,\\nu }(z).\n\\end{equation*}\n\nIn the case where $\\mu $ is an integer $k$ and $\\nu =1\/2$, the recurrence\nrelation (2) viewed as a second-order difference equation is \n\\begin{equation}\n(2k+1)(2k+3)S_{k,1\/2}(z)+4S_{k+2,1\/2}(z)=4z^{\\,k+1},  \\label{eq4}\n\\end{equation\nand can be reduced to first-order equations in the cases of even or odd\nvalues of $k.$ \\ \n\n\\bigskip\n\\end{quote}\n\n\\subsection{Case $k=2m$}\n\n\\begin{quote}\nWith $k=2m$ equation (4) can be written \n\\begin{equation}\n(4m+1)(4m+3)\\,f_{m}(z)+4\\,f_{m+1}(z)=4z^{\\,2m+1},  \\label{eq5}\n\\end{equation}\n\nwhere $f_{m}(z)=S_{2m,1\/2}(z)$ or $s_{2m,1\/2}(z).$ It has the solutio\n\\begin{equation*}\nf_{m}(z)=(-1)^{m}{\\Gamma (2}m+1\/2{)}\\left[ \\frac{f_{0}(z)}{\\sqrt{\\pi }\n-z\\sum_{j=0}^{m-1}\\frac{(-z^{2})^{\\,j}}{\\Gamma (2j+5\/2)}\\right] ,\n\\end{equation*\n\\ the initial values of $\\ f_{0}(z)$ being either $S_{0,1\/2}(z)$ or \ns_{0,1\/2}(z).$ \\ Using those relations the solutions to (5) become\n\n\\begin{subequations}\n\\label{0}\n\\begin{align}\nS_{2m,1\/2}(z)& =(-1)^{m}\\Gamma (2m+1\/2)[\\tfrac{S_{0,1\/2}(z)}{\\sqrt{\\pi }\n-z\\sum_{j=0}^{m-1}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+5\/2)}],  \\label{eq6a} \\\\\ns_{2m,1\/2}(z)& =(-1)^{m}\\Gamma (2m+1\/2)[\\tfrac{s_{0,1\/2}(z)}{\\sqrt{\\pi }\n-z\\sum_{j=0}^{m-1}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+5\/2)}],  \\label{eq6b}\n\\end{align\nwhere $S_{0,1\/2}(z),$ and $s_{0,1\/2}(z)$ have been given by Magnus, \\textit\net al} \\cite{Magnus2} as \n\\end{subequations}\n\\begin{eqnarray*}\n\\frac{1}{\\sqrt{\\pi }}S_{0,1\/2}(z) &=&\\sqrt{\\frac{2}{z}}\\left\\{ \\cos (z)\n\\frac{1}{2}-S(\\chi )]-\\sin (z)[\\frac{1}{2}-C(\\chi )]\\right\\} , \\\\\n\\frac{1}{\\sqrt{\\pi }}s_{0,1\/2}(z) &=&\\sqrt{\\frac{2}{z}}\\left\\{ \\sin\n(z)C(\\chi )-\\cos (z)S(\\chi )\\right\\} ,\n\\end{eqnarray*\nwit\n\\begin{equation*}\n\\chi =\\sqrt{\\frac{2z}{\\pi }},\n\\end{equation*}\n\nand where $S$ and $C$ are the Fresnel sine and cosine integrals \\cite{NBS\n\\begin{eqnarray*}\nS(z) &=&\\int_{0}^{z}\\sin (\\frac{1}{2}\\pi t^{2})\\,dt, \\\\\nC(z) &=&\\int_{0}^{z}\\cos (\\frac{1}{2}\\pi t^{2})\\,dt.\n\\end{eqnarray*}\n\\end{quote}\n\n\\subsection{Case $k=2m+1$}\n\n\\begin{quote}\n\\ \\ \\ \\ In the case where $k=2m+1,$ the difference equation (2) become\n\\begin{equation}\n(4m+3)(4m+5)\\,f_{m}(z)+4\\,f_{m+1}(z)=4z^{\\,2m+2},  \\label{eq7}\n\\end{equation}\n\nwhere $f_{m}(z)=S_{2m+1,1\/2}(z)$ or $s_{2m+1,1\/2}(z).$ Here the solution is \n\\begin{equation*}\nf_{m}(z)=(-1)^{m}\\Gamma (2m+3\/2)\\left[ \\frac{2\\,f_{0}(z)}{\\sqrt{\\pi }\n-z^{2}\\sum_{j=0}^{m-1}\\frac{(-z^{2})^{\\,j}}{\\Gamma (2j+7\/2)}\\right] .\n\\end{equation*\nIn the special case $\\mu =-1$ and $\\nu =1\/2$ in (2) the functions\\thinspace\\ \n$f_{0}(z)\n\\begin{equation*}\nf_{0}(z)=S_{1,1\/2}(z)=1+\\frac{1}{4}S_{-1,1\/2}(z),\n\\end{equation*\no\n\\begin{equation*}\nf_{0}(z)=s_{1,1\/2}(z)=1+\\frac{1}{4}s_{-1,1\/2}(z).\n\\end{equation*\nWe hav\n\\begin{eqnarray*}\nS_{1,1\/2}(z) &=&1+\\sqrt{\\frac{\\pi }{2z}}\\{\\cos (z)[\\frac{1}{2}-C(\\chi\n)]+\\sin (z)[\\frac{1}{2}-S(\\chi )]\\}, \\\\\ns_{1,1\/2}(z) &=&1-\\sqrt{\\frac{\\pi }{2z}}\\{\\sin (z)S(\\chi )+\\cos (z)C(\\chi\n)\\},\n\\end{eqnarray*\nwhere values of $S_{-1,1\/2}(z),$ $s_{-1,1\/2}(z)$ have been given by Magnus, \n\\textit{et al }as\n\n\\begin{eqnarray*}\n\\frac{\\,1}{2\\sqrt{\\pi }}S_{-1,1\/2}(z) &=&\\sqrt{\\frac{2}{z}}\\left\\{ \\cos (z)\n\\frac{1}{2}-C(\\chi )]+\\sin (z)[\\frac{1}{2}-S(\\chi )]\\right\\} , \\\\\n\\frac{\\,1}{2\\sqrt{\\pi }}s_{-1,1\/2}(z) &=&-\\sqrt{\\frac{2}{z}}\\left\\{ \\sin\n(z)S(\\chi )+\\cos (z)C(\\chi )\\right\\} .\n\\end{eqnarray*}\n\\end{quote}\n\nUsing those relations, the solutions to (7) become after resumming \n\\begin{subequations}\n\\begin{eqnarray}\nS_{2m+1,1\/2}(z) &=&(-1)^{m}\\Gamma (2m+3\/2)[\\frac{S_{-1,1\/2}(z)}{2\\sqrt{\\pi }\n+\\sum_{j=0}^{m}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+3\/2)}],  \\label{eq8a} \\\\\ns_{2m+1,1\/2}(z) &=&(-1)^{m}\\Gamma (2m+3\/2)[\\frac{s_{-1,1\/2}(z)}{2\\sqrt{\\pi }\n+\\sum_{j=0}^{m}\\tfrac{(-z^{2})^{\\,j}}{\\Gamma (2j+3\/2)}].  \\label{eq8b}\n\\end{eqnarray}\n\n\\subsection{Integrals with values containing the Fresnel functions}\n\nThe integrals \n\\end{subequations}\n\\begin{eqnarray*}\n&&\\int_{0}^{1}z^{2k}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{2k}\\sin (\\lambda z^{2})dz,\n\\end{eqnarray*\nwhich contain even powers of the variable, can be expressed in terms of the\nLommel functions. \\ \n\nIn the first instance, Maple gives \n\\begin{eqnarray}\n\\int_{0}^{1}z^{2k}\\cos (\\lambda z^{2})dz &=&[1-\\frac{s_{k+1,1\/2(\\lambda )}}\n\\lambda ^{k}}]\\frac{\\cos (\\lambda )}{(2k+1)}  \\label{eq9} \\\\\n&&+[(2k-1)s_{k,3\/2}+(2\/\\lambda )s_{k+1,1\/2}]\\frac{\\sin (\\lambda )}{2\\lambda\n^{k}(2k+1)}.  \\notag\n\\end{eqnarray\nUsing (1) and (2) we get the simplified for\n\\begin{equation}\n\\int_{0}^{1}z^{2k}\\cos (\\lambda z^{2})dz=\\frac{1}{4\\lambda ^{k}}[(2k-1)\\cos\n(\\lambda )\\,s_{k-1,1\/2}(\\lambda )+2\\sin (\\lambda )\\,s_{k,1\/2}(\\lambda )].\n\\label{eq10}\n\\end{equation\nThe values of the integral in the case where $k=2m$ can be obtained from the\nLommel expressions above in (6b) and (8b) with $m$ replaced with $m-1$. \\ We\nge\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m}\\Gamma (2m+1\/2)}\n2\\lambda ^{2m}}\\{\\sqrt{\\frac{2}{\\lambda }}C(\\chi ) \\\\\n&&-\\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-\\lambda ^{2})^{\\,j}}{\\Gamma\n(2j+3\/2)} \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-\\lambda ^{2})^{\\,j}}\n\\Gamma (2j+5\/2)}\\}.\n\\end{eqnarray*\nIn the case $k=2m+1$ we hav\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m+1}\\Gamma (2m+3\/2\n}{2\\lambda ^{2m+1}}\\{\\sqrt{\\frac{2}{\\lambda }}S(\\chi ) \\\\\n&&+\\lambda \\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-\\lambda ^{2})^{\\,j}}\n\\Gamma (2j+5\/2)} \\\\\n&&-\\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-\\lambda ^{2})^{\\,j}}{\\Gamma (2j+3\/2)\n\\}.\n\\end{eqnarray*\nWith these results we see that all cases of cosine integrals with even\npowers of the variable have been obtained in terms containing the Fresnel $S$\nfunction. \\ \n\nFor the corresponding sine integrals, integration by parts give\n\\begin{equation*}\n\\int_{0}^{1}z^{2k}\\sin (\\lambda z^{2})dz=\\frac{\\sin (\\lambda )}{2k+1}-\\frac\n2\\lambda }{2k+1}\\int_{0}^{1}z^{2(k+1)}\\cos (\\lambda z^{2})dz.\n\\end{equation*\nUsing (10) we hav\n\\begin{equation*}\n\\int_{0}^{1}z^{2k}\\sin (\\lambda z^{2})dz=\\frac{1}{4\\lambda ^{k}}[(2k-1)\\sin\n(\\lambda )s_{k-1,1\/2}(\\lambda )-2\\cos (\\lambda )s_{k,1\/2}(\\lambda )].\n\\end{equation*\nWith $k=2m$ we ge\n\\begin{equation*}\n\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m}}[(4m-1)\\sin\n(\\lambda )s_{2m-1,1\/2}(\\lambda )-2\\cos (\\lambda )s_{2m,1\/2}(\\lambda )],\n\\end{equation*\nwhich then become\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}\\Gamma (2m+1\/2)}\n2\\lambda ^{2m}}\\{\\sqrt{\\frac{2}{\\lambda }}S(\\chi ) \\\\\n&&-\\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{\\Gamma\n(2j+3\/2)} \\\\\n&&+\\lambda \\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n\\Gamma (2j+5\/2)}\\}.\n\\end{eqnarray*\nFor $k=2m+1$ the sine integrals are given by Maple a\n\\begin{equation*}\n\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz=\\frac{1}{4\\lambda ^{2m+1}}[\\sin\n(\\lambda )(4m+1)s_{2m,1\/2}(\\lambda )-2\\cos (\\lambda )s_{2m+1,1\/2}(\\lambda )].\n\\end{equation*\nUsing the values of Lommel functions given above we hav\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}\\Gamma (2m+3\/2)}\n2\\lambda ^{2m+1}}\\{\\sqrt{\\frac{2}{\\lambda }}C(\\chi ) \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n\\Gamma (2j+5\/2)} \\\\\n&&-\\cos (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{\\Gamma\n(2j+3\/2)}\\}.\n\\end{eqnarray*\nWith these results we see that all values of the sine integrals which\ncontain \\textit{even} powers of $z$ requires the presence of the Fresnel\nintegrals $C(\\chi )$.\n\n\\section{Integrals with values containing elementary functions}\n\nWe next consider integrals which contain \\textit{odd} powers of $z$ i.e\n\\begin{eqnarray*}\n&&\\int_{0}^{1}z^{2k+1}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{2k+1}\\sin (\\lambda z^{2})dz.\n\\end{eqnarray*\nIn the first instance Maple give\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{2k+1}\\cos (\\lambda z^{2})dz &=&\\frac{1}{2(k+1)}\\{(1-\\frac\ns_{k+3\/2,1\/2}(\\lambda )}{\\lambda ^{k+1\/2}})\\cos (\\lambda ) \\\\\n&&+(\\,k\\,s_{k+1\/2,3\/2}(\\lambda ))+\\frac{s_{k+3\/2,1\/2}(\\lambda )}{\\lambda }\n\\frac{\\sin (\\lambda )}{\\lambda ^{k+1\/2}}\\},\n\\end{eqnarray*\nwhich reduces t\n\\begin{equation}\n\\int_{0}^{1}z^{2k+1}\\cos (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{k+1\/2}}[k\\cos\n(\\lambda )s_{k-1\/2,1\/2}(\\lambda )+\\sin (\\lambda )s_{k+1\/2,1\/2}(\\lambda )]\n\\label{eq11}\n\\end{equation\nusing (1) and (2). \\ Where $k=2m$ we hav\n\\begin{equation}\n\\int_{0}^{1}z^{4m+1}\\cos (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m+1\/2}\n[2m\\cos (\\lambda )s_{2m-1\/2,1\/2}(\\lambda )+\\sin (\\lambda\n)s_{2m+1\/2,1\/2}(\\lambda )].  \\label{eq12}\n\\end{equation\nWe see that this expression requires the Lommel functions \ns_{2m-1\/2,1\/2}(\\lambda )$ and $s_{2m+1\/2,1\/2}(\\lambda )$. \\ In the first\ncase we have from (2) and the initial condition $s_{3\/2,1\/2}(\\lambda )=\\sqrt\n\\lambda }\\left[ 1-\\sin (\\lambda )\/\\lambda \\right] $, the Lommel function \ns_{2m-1\/2,1\/2}(\\lambda )$ as \n\\begin{equation*}\ns_{2m-1\/2,1\/2}(\\lambda )=\\frac{(-1)^{m}(2m-1)!}{\\sqrt{\\lambda }}\\left[ \\sin\n(\\lambda )-\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j+1}}{(2j+1)!}\\right] .\n\\end{equation*\n\\ The function $s_{2m+1,1\/2}(\\lambda )$ then can be obtained from the\ndifferential-difference equation in (3). \\ That is to sa\n\\begin{equation}\n\\frac{d\\,s_{2m+1\/2,1\/2}(\\lambda )}{d\\lambda }+\\frac{1}{2\\lambda \ns_{2m+1\/2,1\/2}(\\lambda )=2m\\,s_{2m-1\/2,1\/2}(\\lambda ).  \\label{eq13}\n\\end{equation\nWe get with the initial condition $s_{2m+1\/2,1\/2}(0)=0,$ the solution to\n(13) a\n\\begin{equation}\ns_{2m+1\/2,1\/2}(\\lambda )=\\frac{2m}{\\sqrt{\\lambda }}\\int_{0}^{\\lambda }\\sqrt{\n}s_{2m-1\/2,1\/2}(z)\\,dz,  \\label{eq14}\n\\end{equation\nwhich immediately gives \n\\begin{equation*}\ns_{2m+1\/2,1\/2}(\\lambda )=\\frac{(-1)^{m+1}(2m)!}{\\sqrt{\\lambda }}\\left[ \\cos\n(\\lambda )-\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)!}\\right] .\n\\end{equation*\nWe have obtained in these cases closed forms for the Lommel functions which\ncontain only elementary functions. As a result, we ge\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+1}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m}(2m)!}{2\\lambda\n^{2m+1}}\\{\\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)!}\n\\\\\n&&-\\lambda \\cos (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n(2j+1)!}\\}.\n\\end{eqnarray*\nThe cosine integral containing powers $4m+3$ is given b\n\\begin{equation*}\n\\int_{0}^{1}z^{4m+3}\\cos (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m+3\/2}\n[(2m+1)\\cos (\\lambda )\\,s_{2m+1\/2,1\/2}(\\lambda )+\\sin (\\lambda\n)s_{2m+3\/2,1\/2}(\\lambda )].\n\\end{equation*\nIn the latter expression the quantity $s_{2m+3\/2,1\/2}(\\lambda )$ can be\nobtained from (3) and we have \n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+3}\\cos (\\lambda z^{2})dz &=&\\frac{(-1)^{m+1}(2m+1)!}\n2\\lambda ^{2m+2}}\\{1-\\cos (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda\n^{2j}}{(2j)!} \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n(2j+1)!}\\}.\n\\end{eqnarray*\nAn expression for the sine integrals with odd powers i.e\n\\begin{equation*}\n\\int_{0}^{1}z^{2k+1}\\sin (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{k+1\/2}}[k\\sin\n(\\lambda )\\,\\,s_{k-1\/2,1\/2}(\\lambda )-\\cos (\\lambda )s_{k+1\/2,1\/2}(\\lambda\n)],\n\\end{equation*\nhas been obtained using integration by parts of the corresponding $\\cos\n(\\lambda z^{2})$ integral together with the latter's integrated form. \\ Then\nin the case where $\\ k=2m$ we have \n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+1}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}(2m)!}{2\\lambda\n^{2m+1}}\\{1-\\cos (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)\n} \\\\\n&&-\\lambda \\sin (\\lambda )\\sum_{j=0}^{m-1}\\frac{(-1)^{\\,j}\\lambda ^{2j}}\n(2j+1)!}\\}.\n\\end{eqnarray*\nWhere $k=2m+1$ we hav\n\\begin{equation*}\n\\int_{0}^{1}z^{4m+3}\\sin (\\lambda z^{2})dz=\\frac{1}{2\\lambda ^{2m+3\/2}\n[(2m+1)\\sin (\\lambda )\\,\\,s_{2m+1\/2,1\/2}(\\lambda )-\\cos (\\lambda\n)s_{2m+3\/2,1\/2}(\\lambda ).\n\\end{equation*\nIn the latter expression the quantity $s_{2m+3\/2,1\/2}(\\lambda )$ can also be\nobtained from (3) and the integral being sought has the value\n\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+3}\\sin (\\lambda z^{2})dz &=&\\frac{(-1)^{m}(2m+1)!}\n2\\lambda ^{2m+2}}\\{-\\lambda \\cos (\\lambda )\\sum_{j=0}^{m}\\frac\n(-1)^{\\,j}\\lambda ^{2j}}{(2j+1)!} \\\\\n&&+\\sin (\\lambda )\\sum_{j=0}^{m}\\frac{(-1)^{\\,j}\\lambda ^{2j}}{(2j)!}\\}.\n\\end{eqnarray*\n\\ \n\n\\section{Asymptotic forms for the integrals containing $S(\\protect\\chi )$\nand $C(\\protect\\chi )$}\n\nIt is useful to provide values of the integrals evaluated above in section 2\nwhere the parameter $\\lambda $ is large. \\ Initially we consider the case of\nthe integrals with even powers of the integration variable $z$ i.e\n\\begin{eqnarray*}\n&&\\int_{0}^{1}z^{4m}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{4m+2}\\cos (\\lambda z^{2})dz, \\\\\n&&\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz,\n\\end{eqnarray*\nwhich we have seen contain the Fresnel integrals. The asymptotic forms for\nthe functions $S\\left( \\chi \\right) $ and $C(\\chi )$ can be obtained from\nthe expressions \\cite{NIST} \n\\begin{eqnarray*}\nS\\left( \\sqrt{\\frac{2\\lambda }{\\pi }}\\right) &=&\\frac{1}{2}-f\\,(\\sqrt{\\frac\n2\\lambda }{\\pi }})\\cos (\\lambda )-g(\\sqrt{\\frac{2\\lambda }{\\pi }})\\sin\n(\\lambda ), \\\\\nC\\left( \\sqrt{\\frac{2\\lambda }{\\pi }}\\right) &=&\\frac{1}{2}+f\\,(\\sqrt{\\frac\n2\\lambda }{\\pi }})\\sin (\\lambda )-g(\\sqrt{\\frac{2\\lambda }{\\pi }})\\cos\n(\\lambda ),\n\\end{eqnarray*\nwhere $f(z)$ and $g(z)$ are the Fresnel auxiliary functions. \\ Their\nasymptotic forms with (cut of\\thinspace f $N\\geq 1$) are given b\n\\begin{eqnarray*}\nf\\,(\\sqrt{\\frac{2\\lambda }{\\pi }}) &\\thicksim &\\frac{1}{\\sqrt{2}\\,\\pi\n\\lambda ^{1\/2}}\\sum_{j=0}^{N-1}\\frac{(-1)^{j}}{\\lambda ^{2j}}\\Gamma (2j+1\/2),\n\\\\\ng\\,(\\sqrt{\\frac{2\\lambda }{\\pi }}) &\\thicksim &\\frac{1}{\\sqrt{2}\\,\\pi\n\\lambda ^{3\/2}}\\sum_{j=0}^{N-1}\\frac{(-1)^{j}}{\\lambda ^{2j}}\\Gamma (2j+3\/2).\n\\end{eqnarray*\nWe get for the integrals in question, having used the relatio\n\\begin{equation*}\n\\frac{1}{\\Gamma (-z)}=-\\frac{\\Gamma (z+1)}{\\pi }\\sin (\\pi z),\n\\end{equation*\nin the sine and cosine sums, the expression\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\cos (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+1\/2)}{\n}\\{\\frac{(-1)^{m}}{\\sqrt{2}\\lambda ^{2m+1\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j}}\\Gamma (2j-2m-1\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j-1}}\\Gamma (2j-2m-3\/2)\\},\n\\end{eqnarray*\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m}\\sin (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+1\/2)}{\n}\\{\\frac{(-1)^{m}}{\\sqrt{2}\\lambda ^{2m+1\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j-1}}\\Gamma (2j-2m-3\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j}}\\Gamma (2j-2m-1\/2)\\},\n\\end{eqnarray*\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\cos (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+3\/2\n}{2}\\{\\frac{(-1)^{m+1}}{\\sqrt{2}\\lambda ^{2m+3\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j}}\\Gamma (2j-2m-3\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N+1}\\frac{(-1)^{\\,j}}{\\lambda\n^{2j-1}}\\Gamma (2j-2m-5\/2)\\},\n\\end{eqnarray*\n\\begin{eqnarray*}\n\\int_{0}^{1}z^{4m+2}\\sin (\\lambda z^{2})dz &\\thicksim &\\frac{\\Gamma (2m+3\/2\n}{2}\\{\\frac{(-1)^{m}}{\\sqrt{2}\\lambda ^{2m+3\/2}} \\\\\n&&+\\frac{\\cos (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j+1}}\\Gamma (2j-2m-1\/2) \\\\\n&&+\\frac{\\sin (\\lambda )}{\\pi }\\sum_{j=1}^{m+N}\\frac{(-1)^{\\,j+1}}{\\lambda\n^{2j}}\\Gamma (2j-2m-3\/2)\\}.\n\\end{eqnarray*\nWe see that the terms in these asymptotic expressions contain the sine and\ncosine sums along with the additional and significant contributions from the\nFresnel integrals.\n\n\\subsection{Asymptotic values for Fresnel related integrals}\n\n\\bigskip\n\nIn extensions of the Schwinger-Englert semi-classical theory \\cite{englert}\\\nof atomic structure, asymptotic forms for the integral\n\\begin{eqnarray*}\n&&\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz, \\\\\n&&\\int_{0}^{1}\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz,\n\\end{eqnarray*\narise with $\\lambda \\rightarrow \\infty $ and $0<a<1.$ \\ Here we give\nestimates for those forms. \\ Expanding the denominators of the integrals\nabove we have \n\\begin{equation*}\n\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz=\\sum_{k=0}^{\\infty\n}a^{2k}\\int_{0}^{1}z^{4k}\\sin (\\lambda z^{2})-a\\sum_{k=0}^{\\infty\n}a^{2k}\\int_{0}^{1}z^{4k+2}\\sin (\\lambda z^{2}),\n\\end{equation*\nusing the results obtained above we get (where $z!!$ is the double factorial\nfunction and $N$ is the order of truncation\n\\begin{eqnarray*}\n\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz &\\thicksim &\\frac{1}\n2}\\sqrt{\\frac{\\pi }{2\\lambda }}\\sum_{k=0}^{\\infty }(-1)^{k}\\left( \\frac{a}\n2\\lambda }\\right) ^{2k}[(4k-1)!!-\\frac{a}{2\\lambda }(4k+1)!!] \\\\\n&&+\\frac{\\cos (\\lambda )}{2}\\{\\sum_{k=0}^{\\infty\n}a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac{(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda\n\\right) ^{2j-1}} \\\\\n&&+\\frac{a}{\\lambda }\\sum_{k=0}^{\\infty }a^{2k}(4k+1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-3)!!}{\\left( 2\\lambda \\right) ^{2j}}\\} \\\\\n&&+\\frac{\\sin (\\lambda )}{2}\\{\\sum_{k=0}^{\\infty\n}a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac{(-1)^{j}(4j-4k-3)!!}{\\left( 2\\lambda\n\\right) ^{2j}} \\\\\n&&+\\frac{a}{\\lambda }\\sum_{k=0}^{\\infty }a^{2k}(4k+1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda \\right) ^{2j-1}}\\}.\n\\end{eqnarray*\nIn lowest order in $a$ and $1\/\\lambda $ we hav\n\\begin{equation}\n\\int_{0}^{1}\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}dz\\thicksim \\frac{1}{4\n\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1-\\frac{a}{2\\lambda }\\right) -\\frac{\\cos\n(\\lambda )}{2\\lambda }(1-a)-\\frac{\\sin (\\lambda )}{\\left( 2\\lambda \\right)\n^{2}}(1+a)+\\cdots ,  \\label{eq15}\n\\end{equation\nand in the case of the cosine integral we hav\n\\begin{eqnarray*}\n\\int_{0}^{1}\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz &\\thicksim &\\frac{1}\n2}\\sqrt{\\frac{\\pi }{2\\,\\lambda }}\\sum_{k=0}^{\\infty }(-1)^{k}\\left( \\frac{a}\n2\\lambda }\\right) ^{2k}[(4k-1)!!+\\frac{a}{2\\lambda }(4k+1)!!] \\\\\n&&+\\cos (\\lambda )\\{\\sum_{k=0}^{\\infty }a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-3)!!}{\\left( 2\\lambda \\right) ^{2j}} \\\\\n&&+\\sum_{k=0}^{\\infty }a^{2k+1}(4k+1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda \\right) ^{2j}}\\} \\\\\n&&-\\sin (\\lambda )\\{\\sum_{k=0}^{\\infty }a^{2k}(4k-1)!!\\sum_{j=1}^{k+N}\\frac\n(-1)^{j}(4j-4k-5)!!}{\\left( 2\\lambda \\right) ^{2j-1}} \\\\\n&&+\\sum_{k=0}^{\\infty }a^{2k+1}(4k+1)!!\\sum_{j=1}^{k+N+1}\\frac\n(-1)^{j}(4j-4k-7)!!}{\\left( 2\\lambda \\right) ^{2j-1}}\\}.\n\\end{eqnarray*\nIn lowest order in $a$ and $1\/\\lambda $ we hav\n\\begin{equation}\n\\int_{0}^{1}\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz\\thicksim \\frac{1}{4\n\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1+\\frac{a}{2\\lambda }\\right) -\\frac{\\cos\n(\\lambda )}{\\left( 2\\lambda \\right) ^{2}}(1+a)+\\frac{\\sin (\\lambda )}\n2\\lambda }(1-a)+\\cdots  \\label{eq16}\n\\end{equation\nIt is interesting to note that the integrals with infinite range i.e\n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}=-\\frac{\\pi }{\n\\sqrt{a}}\\sin (\\lambda \/a)+\\frac{\\pi }{2\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\nC(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\sin (\\frac{\\lambda }{a})+\\cos \n\\frac{\\lambda }{a})\\right] \\\\ \n+S(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\sin (\\frac{\\lambda }{a})-\\cos \n\\frac{\\lambda }{a})\\right\n\\end{array\n\\right\\} ,  \\label{eq17}\n\\end{equation\nan\n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}=\\frac{\\pi }{\n\\sqrt{a}}\\cos (\\lambda \/a)-\\frac{\\pi }{2\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\nC(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\cos (\\frac{\\lambda }{a})-\\sin \n\\frac{\\lambda }{a})\\right] \\\\ \n+S(\\sqrt{\\frac{2\\lambda }{\\pi a}})\\left[ \\cos (\\frac{\\lambda }{a}))+\\sin \n\\frac{\\lambda }{a})\\right\n\\end{array\n\\right\\} ,  \\label{eq18}\n\\end{equation\nwhich have the exact values shown above, posses\\ the same forms for large \n\\lambda $ as the integrals (albeit without oscillations) with finite range.\n\\ That is to say \n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\sin (\\lambda z^{2})}{(1+a\\,z^{2})}\\thicksim \\frac{\n}{4}\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1-\\frac{a}{2\\lambda }\\right) +\\cdots\n,  \\label{eq19}\n\\end{equation\n\\begin{equation}\n\\int_{0}^{\\infty }\\frac{\\cos (\\lambda z^{2})}{(1+a\\,z^{2})}dz\\thicksim \\frac\n1}{4}\\sqrt{\\frac{2\\pi }{\\lambda }}\\left( 1+\\frac{a}{2\\lambda }\\right) +\\cdots\n\\label{eq20}\n\\end{equation\n\\ \\ \\ \n\nIt is also worth noting that asymptotic values of the integrals which\ncontain higher powers of the trigonometric function described above and with\nhigher powers of the denominators i.e. \n\\begin{equation*}\n\\int_{0}^{1}\\frac{\\sin ^{n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }},\\hspace\n0.25in}\\int_{0}^{1}\\frac{\\cos ^{n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }},\n\\end{equation*\nan\n\\begin{equation*}\n\\int_{0}^{\\infty }\\frac{\\sin ^{n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }}\n\\hspace{0.25in}\\int_{0}^{\\infty }\\frac{\\cos ^{n}(\\lambda z^{2})}\n(1+a\\,z^{2})^{\\,\\nu }},\n\\end{equation*\ncan be expressed in terms of the forms given in (15,16) and (19,20). \\ By\nway of example, writing \n\\begin{equation*}\nI_{\\nu }^{\\,(\\eta )}(a,\\lambda )=\\int_{0}^{\\infty }\\frac{\\cos ^{\\,\\eta\n}(\\lambda z^{2})}{(1+a\\,z^{2})^{\\,\\nu }},\n\\end{equation*\n(noting the scaling property\n\\begin{equation*}\nI_{\\nu }^{\\,(\\eta )}(a,\\lambda )=\\frac{1}{\\sqrt{a}}I_{\\nu }^{\\,(\\eta\n)}(1,\\lambda \/a),\n\\end{equation*\nwe get the differential-difference equation\n\n\\begin{equation}\nI_{\\,\\nu +1\\,}^{\\,(\\eta )}(a,\\lambda )=I_{\\nu }^{\\,(\\eta )}(a,\\lambda\n)+\\left( \\frac{a}{\\nu }\\right) \\frac{d\\,I_{\\nu }^{\\,(\\eta )}(a,\\lambda )}{da\n.  \\label{eq21}\n\\end{equation\nIn the case where $\\eta =1$ and $\\nu =1\/2$ we hav\n\\begin{eqnarray*}\nI_{1\/2}^{\\,(1)}(a,\\lambda ) &=&\\int_{0}^{\\infty }\\frac{\\cos (\\lambda z^{2})}\n(1+a\\,z^{2})^{\\,1\/2}} \\\\\n&=&\\frac{\\pi }{4\\sqrt{a}}\\left\\{ \\sin (\\frac{\\lambda }{2a})\\,J_{0}(\\frac\n\\lambda }{2a})-\\cos (\\frac{\\lambda }{2a})Y_{0}(\\frac{\\lambda }{2a})\\right\\} ,\n\\end{eqnarray*\na closed form expression where $J_{0}(z)$ and $Y_{0}(z)$ are Bessel\nfunctions of the first and second kind. \\ The value for this integral for\nlarge $\\frac{\\lambda }{2a}$ using Hankel's \\cite{Hankel}\\ \\ asymptotic\nexpressions for the Bessel functions i\n\\begin{equation*}\nI_{1\/2}^{\\,(1)}(a,\\lambda )\\thicksim \\frac{1}{4}\\sqrt{\\frac{2\\pi }{\\lambda }\n\\left\\{ 1+\\frac{1}{4}\\left( \\frac{a}{\\lambda }\\right) -\\frac{9}{32}\\left( \n\\frac{a}{\\lambda }\\right) ^{2}-\\frac{75}{128}\\left( \\frac{a}{\\lambda \n\\right) ^{3}+\\cdots \\right\\} .\n\\end{equation*\nIn the case of the integral $I_{1}^{\\,(2)}(a,\\lambda )$, its exact value is \n\\begin{eqnarray*}\nI_{1}^{\\,(2)}(a,\\lambda ) &=&\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2}\n}{(1+a\\,z^{2})} \\\\\n&=&\\frac{\\pi }{2\\sqrt{a}}\\cos ^{2}(\\lambda \/a)-\\frac{\\pi }{4\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\nC(\\sqrt{\\frac{4\\lambda }{\\pi a}})\\left[ \\cos (\\frac{2\\lambda }{a})-\\sin \n\\frac{2\\lambda }{a})\\right] \\\\ \n+S(\\sqrt{\\frac{4\\lambda }{\\pi a}})\\left[ \\cos (\\frac{2\\lambda }{a}))+\\sin \n\\frac{2\\lambda }{a})\\right\n\\end{array\n\\right\\} ,\n\\end{eqnarray*\nor rewriting it in terms of the Fresnel auxiliary functions \\cite{Aux}$\\\nf\\,(\\chi ^{\\prime })$ and $g(\\chi ^{\\prime })$ referred to above (here with \n\\chi ^{\\prime }=2\\sqrt{\\lambda \/\\pi a}$ ) we get \n\\begin{equation*}\n\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2})}{(1+a\\,z^{2})}=\\frac{\\pi }{\n\\sqrt{a}}\\left\\{ 1+f\\,(\\chi ^{\\prime })+g(\\chi ^{\\prime })\\right\\} .\n\\end{equation*\nThe asymptotic values of $\\ f\\,(\\chi ^{\\prime })$ and $\\ g(\\chi ^{\\prime })$\nfor large $\\lambda $ \\cite{asymptotic} give the resul\n\\begin{eqnarray*}\n\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2})}{(1+a\\,z^{2})} &\\thicksim \n\\frac{\\pi }{4\\sqrt{a}}+\\frac{1}{8}\\sqrt{\\frac{\\pi }{\\lambda }}\\left\\{ 1\n\\frac{3}{16}\\left( \\frac{a}{\\lambda }\\right) ^{2}+\\cdots \\right\\} \\\\\n&&+\\frac{1}{32}\\sqrt{\\frac{\\pi }{\\lambda }}\\left\\{ \\frac{a}{\\lambda }-\\frac\n15}{16}\\left( \\frac{a}{\\lambda }\\right) ^{3}+\\cdots \\right\\} .\n\\end{eqnarray*\nThe first of the higher order integrals i.e. $\\ I_{2}^{\\,(2)}$ has the\nclosed form expression \n\\begin{eqnarray*}\nI_{2}^{\\,(2)} &=&\\int_{0}^{\\infty }\\frac{\\cos ^{2}(\\lambda z^{2})}\n(1+a\\,z^{2})^{2}} \\\\\n&=&\\frac{\\pi }{8\\sqrt{a}}\\left\\{ \n\\begin{array}{c}\n2\\cos ^{2}(\\frac{\\lambda }{a})+4(\\frac{\\lambda }{a})\\sin (\\frac{2\\lambda }{a\n)+\\sqrt{\\frac{4\\lambda }{\\pi a}} \\\\ \n+C(\\sqrt{\\frac{4\\lambda }{\\pi a}})[\\sin (\\frac{2\\lambda }{a})(1-4\\frac\n\\lambda }{a})-\\cos (\\frac{2\\lambda }{a})(1+4\\frac{\\lambda }{a})] \\\\ \n-S(\\sqrt{\\frac{4\\lambda }{\\pi a}})[\\sin (\\frac{2\\lambda }{a})(1+4\\frac\n\\lambda }{a})+\\cos (\\frac{\\lambda }{a})(1-4\\frac{2\\lambda }{a}\n\\end{array\n\\right\\} .\n\\end{eqnarray*}\n\nThe higher order integrals i.e. those containing powers of the cosine\nfunction i.e. $\\cos ^{\\,\\eta }(\\lambda x^{2})$ are expressible in terms of\nthe integrals in (18). \\ Using the trigonometric relations \n\\begin{equation*}\n\\cos ^{\\eta }(\\lambda x^{2})=\\frac{1}{2^{\\eta }}\\sum_{j=0}^{\\eta }\\binom\n^{\\eta }}{j}\\,\\cos ([\\eta -2j]\\lambda x^{2}),\n\\end{equation*\nwe have \n\\begin{eqnarray*}\nI_{\\nu }^{\\,(2\\eta )}(a,\\lambda ) &=&\\sqrt{\\frac{\\pi }{a}}\\frac{\\Gamma\n(2\\eta -1\/2)}{2^{\\,2\\eta }\\,\\eta !\\,(\\eta -1)!}+\\frac{1}{2^{2\\eta -1}\n\\sum_{j=1}^{\\eta }\\binom{2\\,\\eta }{\\eta -j}\\,I_{\\nu }^{\\,(1)}(a,2j\\,\\lambda\n), \\\\\nI_{\\nu }^{\\,(2\\eta +1)}(a,\\lambda ) &=&\\frac{1}{2^{2\\eta }}\\sum_{j=0}^{\\eta \n\\binom{2\\eta +1}{\\eta -j}\\,I_{\\nu }^{\\,(1)}(a,(2j+1)\\,\\lambda ).\n\\end{eqnarray*\nWith results from Maple, it is possible using an incomplete form of\ninduction to give general expressions for the integrals $I_{\\nu\n}^{\\,(1)}(a,\\lambda )$ with even and odd values of $\\nu .$ The integrals are\nfound to contain the Anger functions \\cite{Anger} $\\mathbf{J}_{1\/2}(z)$ and \n\\mathbf{J}_{3\/2}(z)$ i.e\n\\begin{equation*}\nI_{2n}^{\\,(1)}(a,\\lambda )=\\frac{1}{\\sqrt{a}\\,2^{\\,\\epsilon (n)+n}}\\left\\{ \n\\begin{array}{c}\n2\\sqrt{\\frac{2\\pi a}{\\lambda }}\\sum_{k=0}^{2n-1}a_{k,2n}\\left( \\frac{\\lambda \n}{a}\\right) ^{k} \\\\ \n+2\\pi \\cos (\\lambda \/a)\\sum_{k=0}^{n-1}c_{k,2n}\\left( \\frac{\\lambda }{a\n\\right) ^{2k} \\\\ \n+4\\pi \\sin (\\lambda \/a)\\sum_{k=0}^{n-1}d_{k,2n}\\left( \\frac{\\lambda }{a\n\\right) ^{2k+1} \\\\ \n-\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{1\/2}(\\lambda\n\/a)\\sum_{k=0}^{n}e_{k,2n}\\left( \\frac{\\lambda }{a}\\right) ^{2k} \\\\ \n+\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{3\/2}(\\lambda\n\/a)\\sum_{k=0}^{n-1}f_{k,2n}\\left( \\frac{\\lambda }{a}\\right) ^{2k+1}\n\\end{array\n\\right\\}\n\\end{equation*\nwhere the Greubel eta function $\\epsilon (n)$ is defined here as \n\\begin{equation*}\n\\epsilon (n)=\\lfloor \\sqrt{2}n\\rfloor +\\lfloor \\sqrt{3\/2}n\\rfloor \\,\\ ,\n\\end{equation*\nand \\ $\\lfloor z\\rfloor $ is the floor function. \\ The sequence of integers\nproduced by the eta function has been studied by G. C. Greubel and others\n\\cite{Greubel}. The expression for $I_{2n+1}^{\\,(1)}(a,\\lambda )$ i\n\\begin{equation*}\nI_{2n+1}^{\\,(1)}(a,\\lambda )=\\frac{1}{\\sqrt{a}\\,2^{\\,\\epsilon (n+1)+n}\n\\left\\{ \n\\begin{array}{c}\n2\\sqrt{\\frac{2\\pi a}{\\lambda }}\\sum_{k=0}^{2n-1}a_{k,2n+1}\\left( \\frac\n\\lambda }{a}\\right) ^{k} \\\\ \n+2\\pi \\cos (\\lambda \/a)\\sum_{k=0}^{n}c_{k,2n+1}\\left( \\frac{\\lambda }{a\n\\right) ^{2k} \\\\ \n+2\\pi \\sin (\\lambda \/a)\\sum_{k=0}^{n-1}d_{k,2n+1}\\left( \\frac{\\lambda }{a\n\\right) ^{2k+1} \\\\ \n-\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{1\/2}(\\lambda\n\/a)\\sum_{k=0}^{n}e_{k,2n+1}\\left( \\frac{\\lambda }{a}\\right) ^{2k} \\\\ \n+\\pi \\sqrt{\\frac{2\\pi a}{\\lambda }}\\mathbf{J}_{3\/2}(\\lambda\n\/a)\\sum_{k=0}^{n}f_{k,2n+1}\\left( \\frac{\\lambda }{a}\\right) ^{2k+1\n\\end{array\n\\right\\} .\n\\end{equation*\nAlternatively, in keeping with results given above the Anger functions can\nbe written in terms of the Fresnel functions appearing in $I_{1}^{\\,(2)},$\nand $\\ I_{2}^{\\,(2)}$, using the relation\n\\begin{eqnarray*}\n\\mathbf{J}_{1\/2}(z) &=&\\sqrt{\\frac{2}{\\pi z}}[C(\\sqrt{\\frac{2z}{\\pi }\n)\\{\\sin (z)+\\cos (z)\\}+S(\\sqrt{\\frac{2z}{\\pi }})\\{\\sin (z)-\\cos (z)\\}], \\\\\n\\mathbf{J}_{3\/2}(z) &=&-\\frac{2}{\\pi z}+\\sqrt{\\frac{2}{\\pi z}}C(\\sqrt{\\frac\n2z}{\\pi }})\\{\\sin (z)\\,(1+1\/z)-\\cos (z)\\,(1-1\/z)\\} \\\\\n&&-\\sqrt{\\frac{2}{\\pi z}}S(\\sqrt{\\frac{2z}{\\pi }})\\{\\sin (z)\\,(1-1\/z)+\\cos\n(z)\\,(1+1\/z)\\}.\n\\end{eqnarray*}\n\nThe coefficients $a_{k,\\nu },c_{k,\\nu },d_{k,\\nu },e_{k,\\nu },f_{k,\\nu }$\noccurring in the $I_{\\nu }^{\\,(1)}$ integrals are interrelated by (21) from\nwhich we obtain the connection formula\n\\begin{eqnarray*}\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}a_{2n,2n+1} &=&-e_{n,2n},\\hspace{0.25in\n(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}a_{2k,2n+1}+(4k-4n)\\,a_{2k,2n}\n&=&-e_{k,2n},\\hspace{0.25in}(k<n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}a_{2k+1,2n+1}+(4k+2-4n)\\,a_{2k+1,2n}\n&=&f_{k,2n},\\hspace{0.25in}(k<n), \\\\\n&& \\\\\n\\frac{n}{2^{\\Delta \\,\\epsilon (n)}}c_{n,2n+1} &=&-d_{n-1,2n},\\hspace{0.25in\n(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}c_{k,2n+1}+(4k+1-4n)\\,c_{k,2n}\n&=&-4\\,d_{k-1,2n},\\hspace{0.25in}(k<n), \\\\\n&& \\\\\n2n\\,d_{k,2n+1}+(4k+3-4n)\\,d_{k,2n} &=&c_{k,2n,}\\hspace{0.25in}(k\\leq n-1), \\\\\n&& \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}e_{0,2n+1} &=&(4n-1)e_{0,2n},\\hspace\n0.25in}(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}e_{k,2n+1}+(4k+1-4n)\\,e_{k,2n}\n&=&2f_{k-1,2n},\\hspace{0.25in}(k>0), \\\\\n&& \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}f_{n,2n+1} &=&-2e_{n,2n},\\hspace{0.25in\n(k=n), \\\\\n\\frac{4n}{2^{\\Delta \\,\\epsilon (n)}}f_{k,2n+1}+(4k-1-4n)\\,f_{k,2n}\n&=&-2e_{k,2n},\\hspace{0.25in}(k<n),\n\\end{eqnarray*\nwith $\\Delta \\epsilon (n)=\\epsilon (n+1)-\\epsilon (n).$ \\ These relations do\nnot appear to be useful except as an internal check on the values of the\ncoefficients.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIndividual neutral atoms trapped in optical tweezers\\,\\cite{Saffman10,Saffman2016,barredo_atom-by-atom_2016,endres_atom-by-atom_2016,barredo_synthetic_2018,Browaeys20,Henriet2020quantum,Morgado20,Beterov20,Wu21} have emerged as a powerful platform for quantum information processing. In those systems, one uses the strong interaction between atoms excited to a \\emph{Rydberg state} to generate entanglement. This interaction can be exploited for various tasks, including the creation of fast and robust quantum gates\\,\\cite{Jaksch00,Isenhower10,levine_high-fidelity_2018,Levine19} for digital quantum information processing, the generation of tunable spin models for the study of many-body quantum physics\\,\\cite{labuhn2016tunable,Bernien17,Lienhard18,leseleuc_observation_2019,omran_generation_2019}, or the deterministic creation of large highly entangled GHZ states with up to 20 atoms\\,\\cite{omran_generation_2019}.\n\n\nQuantum computing and quantum simulation schemes for neutral-atom Quantum Processing Units (QPUs) are extremely flexible, in part due to the fact that, unlike for most other platforms, the QPU (and therefore its qubit connectivity) is not hard-wired during the manufacturing process. In fact, in neutral-atom optical traps, the connectivity of the QPU can be reprogrammed at \\textit{every single run}. This opportunity opens up new avenues for exploration on these devices but also requires providing the user with a lower-level description of the physical setup.\n\nWe here introduce \\emph{Pulser}~\\cite{Pulser}, an open-source software library for designing pulse sequences for neutral-atom QPUs. The main goal of this library is to serve as an interface between experienced users and neutral-atom quantum hardware. Using Pulser, users can control all the relevant physical parameters of a pulse-level quantum program. This program can subsequently be sent to QPUs via external tools and executed on the hardware after a device-specific compilation step. Providing users with a pulse-level access to QPUs will allow them to explore advanced pulse-shaping techniques facilitating the design of new quantum gate protocols, the implementation of optimal control and error mitigation techniques, or the exploration of mixed digital-analog algorithms\\,\\cite{Parra-rodriguez_20}. Exposing the low-level details of the hardware will also enable algorithm developers to design software procedures while keeping the particularities of the hardware in mind. Besides providing an interface to a QPU, Pulser includes a built-in emulator that faithfully reproduces the hardware behavior. This is intended to make the design of experiments easier for both the theoretician and the experimentalist.\n\n\nPulser is an open-source Python library licensed under the Apache License 2.0 and intended for the community of quantum scientists and engineers working on neutral-atom devices. The development and use of open-source software has enabled the flexible and efficient simulation of quantum systems \\cite{Johansson_2012,Johansson_2013}, the connection of quantum processors to the cloud \\cite{Zeng_2017_Nature, Karalekas_2020} and their use to launch jobs running quantum hardware. These tasks appear as quantum circuits expressed in intermediate representation, or in the more recent framework of hybrid classical-quantum algorithms, which require a fast and tailored interoperability between quantum and classical computing units with classical optimization routines \\cite{McClean_2016_NJP,Bharti_2021,Endo_2021}. More recently, specific tools and frameworks have also put more focus on the pulse-level description of quantum controls \\cite{Alexander_2020_QST,Ball_2020,Wittler_2021, QUA-libs, li2021qutip-qip}, and their optimization to mitigate the impact of noise on the coherence of the intended protocols \\cite{Goerz_2019_SciPost} in the so-called era of noisy-intermediate scale quantum computing (NISQ) \\cite{Preskill_2018_Quantum}. However, no existing open-source tool has yet specialized its features to the physics of neutral-atom quantum processors, which, due to the peculiarities of their physical properties, provide largely untapped opportunities both for quantum simulation and quantum computing tasks.\n\nIn this paper, we start by briefly reviewing in Section \\ref{section:description} the main physical ingredients that determine the dynamics of Rydberg atom arrays. We then present in Section \\ref{section:sequences} a guided walk-through for designing pulse sequences with Pulser, followed by examples on how Pulser can be leveraged for some notable use-cases in Section \\ref{section:usecases}. Finally, we conclude and comment on the future directions and extensions for this tool.\n\n\\section{Description of the physical system}\n\\label{section:description}\nIn neutral-atom devices, atoms are trapped by arrays of optical tweezers\\,\\cite{Nogrette14} in arbitrary, customizable patterns. In Pulser, we call this ensemble of atoms and their specific positions the \\texttt{Register}. For each atom in a \\texttt{Register}, the quantum information is encoded in specific electronic energy levels, which are reached through optical addressing techniques. In Pulser, the components responsible for driving these transitions are called \\texttt{Channel}s and they do so through the emission of \\texttt{Pulse}s.\n\n\n\\subsection{Driving two-level transitions}\n\nWe define a pulse as the modulation of a channel's output amplitude, detuning, and phase over a finite duration $\\tau$. For a channel targeting the transition between energy levels $a$ and $b$, with resonance frequency $\\omega_{ab} = |E_{a}-E_{b}|\/\\hbar$, the output amplitude determines the \\textit{Rabi frequency}\\footnote{The Rabi frequency determines the amplitude of a signal (in frequency units), not the frequency itself.} $\\Omega(t)$, and the detuning $\\delta(t)$ is defined relatively to $\\omega_{ab}$ and the frequency of the channel's output signal $\\omega(t)$, as $\\delta(t) = \\omega(t) - \\omega_{ab}$. Additionally, the phase $\\varphi$ of a pulse can be set to an arbitrary, constant value.\n\nA pulse-driven transition between two energy levels can be mapped to a spin-$1\/2$ system through the drive Hamiltonian:\n\\begin{equation}\\label{drive_ham}\nH^D(t)= \\frac{\\hbar}{2} \\mathbf{\\Omega}(t)\\cdot\\bm{\\sigma},\n\\end{equation}\nwhere $\\bm{\\sigma} = (\\sigma^x,\\sigma^y,\\sigma^z)^T$ is the Pauli vector and $\\bm{\\Omega}(t) =\n(\\Omega(t) \\cos(\\varphi),-\\Omega(t) \\sin(\\varphi),-\\delta(t))^T$ the rotation vector. Since there can be multiple transitions being addressed overall, we explicitly define the Pauli matrices for a transition between states $\\ket{a}$ and $\\ket{b}$ (where $E_b > E_a$) to be:\n\\begin{align}\n    \\sigma^x &= \\op{a}{b} + \\op{b}{a}\\\\\n    \\sigma^y &= i\\op{a}{b} - i\\op{b}{a} \\\\\n    \\sigma^z & = \\op{b}{b} - \\op{a}{a}\n\\end{align}\nIn the Bloch sphere representation, for each instant $t$, this Hamiltonian describes a rotation around the axis $\\mathbf{\\Omega}$ with angular velocity $\\Omega_{eff} = |\\bm{\\Omega}| = \\sqrt{\\Omega^2 + \\delta^2}$, as illustrated in Figure \\ref{fig:bloch_rotation}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.9\\columnwidth, trim=0 0.5cm 0 0.8cm, clip]{figures\/bloch_rotation.png}\n\\caption{Representation of the drive Hamiltonian's dynamics as a rotation in the Bloch sphere.}\n\\label{fig:bloch_rotation}\n\\end{figure}\n\n\n\n\\subsection{Rydberg states}\n\\label{sec:rydberg}\n\nIn neutral-atom devices, atoms are driven to Rydberg states as a way to make them interact over large distances. Depending on the electronic levels that are involved in the process, the atoms experience different types of interactions, translating into different Hamiltonians~\\cite{Browaeys20}.\\\\\n\nIn the so-called ``Ising'' configuration, which is obtained when the spin states are one of the ground states $\\ket{g}$ and a Rydberg state $\\ket{r}$~\\cite{schauss2015crystallization,labuhn2016tunable,Bernien17,leseleuc2018accurate}  (the \\texttt{ground-rydberg} basis in Pulser, see Fig. \\ref{Ising_config}.), the interaction adds a term to the drive Hamiltonian describing the transition between those states: \n\n\\begin{equation}\n\\mathcal H^{gr}(t) = \\sum_i \\left(H^D_i(t) + \\sum_{j<i}\\frac{C_6}{(R_{ij})^6}\\hat n_i \\hat n_j \\right),\n\\label{eq:ising_ham}\n\\end{equation}\n\n\\noindent where $\\hat n_i$ denotes the projector $\\ketbra{r}{r}_i$ on the $i$-th atom, $R_{ij}$ is the distance between atoms $i$ and $j$, and $C_6$ is a constant depending on the specific Rydberg level $\\ket{r}$. \n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.4]{figures\/two-level.png}\n\\caption{The excitation between a ground and Rydberg state is taken as two-level transition with  Rabi frequency $\\Omega$ and detuning $\\delta$. In Pulser, it is called the \\texttt{ground-rydberg} basis and is addressed by \\texttt{Rydberg} channels.}\n\\label{Ising_config}\n\\end{figure}\n\nThe effect of this interaction is expressed in the so-called \\textit{Rydberg blockade}, illustrated in Figure \\ref{blockade}. When driving two atoms from $\\ket{g}$ to $\\ket{r}$ with a resonant pulse ($\\delta=0$) of Rabi frequency $\\Omega$, the interaction prevents the simultaneous excitation of the two atoms in the state $ \\ket{r}$ if $\\hbar\\Omega \\ll C_6\/R^6$. Instead, the system evolves between the $\\ket{gg}$ and the entangled state $\\ket{\\psi_+}=(\\ket{gr}+\\ket{rg})\/\\sqrt{2}$\\footnote{Note that the state $\\ket{\\psi_-}=(\\ket{gr}-\\ket{rg})\/\\sqrt{2}$ is not coupled to $\\ket{gg}$ by the laser}, with an effective Rabi frequency of $\\sqrt{2}\\Omega$. Similarly, an atom's excitation from $\\ket{g}$ to $\\ket{r}$ is suppressed if there is another atom in the Rydberg state at a distance $R \\ll R_b$, where \n\\begin{equation}\\label{eq:blockade_radius}\n    R_b=\\left(\\frac{C_6}{\\hbar\\Omega}\\right)^{1\/6}\n\\end{equation}\nis the \\textit{Rydberg blockade radius}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/rydberg_computing_blockade.pdf}\n\\caption{Two nearby atoms interact via a van der Waals potential which shifts the energy level for the doubly excited state $|rr\\rangle$. If the laser excites simultaneously both atoms, the lower-energy entangled state $(|gr\\rangle + |rg\\rangle)\/\\sqrt{2}$ is produced instead.}\n\\label{blockade}\n\\end{figure}\n\n\n\n\nBy acting on all atoms with the same global pulse, addressing the $\\ket{g} \\leftrightarrow \\ket{r}$ transition with $\\varphi=0$, one obtains the Hamiltonian:\n\\begin{equation}\n    \\mathcal H(t) = \\sum_i \\left(\\frac{\\hbar\\Omega(t)}{2} \\sigma_i^x \n    -\\hbar \\delta(t) \\hat n_i +\\sum_{j<i} \\frac{C_6}{(R_{ij})^6}\\hat n_i \\hat n_j \\right)\n\\label{eq:global_ising}\n\\end{equation}\nSince $\\hat n_i=(1+\\sigma_i^z)\/2$, Eq. (\\ref{eq:global_ising}) represents a quantum Ising Hamiltonian with transverse field $\\Omega(t)$ and Ising couplings $J_{ij} \\propto 1\/R_{ij}^6$.\\\\\n\nOther configurations exist, but the corresponding channels are currently not implemented in Pulser. In the so-called ``XY'' configuration, which is obtained when the spin states $\\ket{\\downarrow}$ and $\\ket{\\uparrow}$ are two Rydberg states that are dipole-coupled, such as $\\ket{nS}$ and $\\ket{nP}$~\\cite{barredo2015coherent,orioli2018relaxation,leseleuc_observation_2019,Scholl2021}, the dipolar interaction generates an exchange-like term of the form\n\\begin{equation}\n\\mathcal{H}_{int}=  2\\sum_{i \\neq j}\\frac{C_3}{R_{ij}^3} \\left(\\sigma_i^x \\sigma_j^x + \\sigma_i^y \\sigma_j^y \\right),\n\\label{eq:XY_hamiltonian}\n\\end{equation}\nwhere $C_3$ is a coefficient which depends on the selected Rydberg states. In this configuration, the driving is induced by a microwave field instead of an optical laser field.\\\\\n\nThe continuous manipulation of a system's Hamiltonian is referred to as the \\textit{analog approach} to quantum computing. The very high level of control in this analog mode has several applications, the most direct one being the quantum simulation of interacting many-body systems. This includes observing the out-of-equilibrium dynamics of the system after a quench or a drive\\,\\cite{Bernien17,Bluvstein21}, or determining the ground-state of a specific Hamiltonian through adiabatic variation of its parameters\\,\\cite{Lienhard18,scholl2020programmable,ebadi2020quantum}. Additionally, one can use the blockade effect to reproduce a connectivity structure among the atoms, which can be exploited to solve combinatorial problems that can be mapped onto the Ising Hamiltonian\\,\\cite{pichler2018quantum,henriet2020robustness, dalyac2020qualifying}. Those applications will be studied in more detail in Section \\ref{section:usecases}.\\\\\n\n\n\n\\subsection{Digital addressing and quantum gates}\n\nIn opposition to the analog approach stands the \\textit{digital approach}, in which a system's state evolves through a series of discrete manipulations of its qubits' states, known as quantum gates. This is the underlying approach in quantum circuits and can be replicated on neutral-atom devices at the pulse-level. To achieve this, the qubit states are encoded in two hyperfine ground states of the system, named \\textit{ground}, $\\ket{g}\\equiv\\ket{0}$, and \\textit{hyperfine}, $\\ket{h}\\equiv\\ket{1}$. In Pulser, these states form the \\texttt{digital} basis, which is addressed by \\texttt{Raman} channels.\n\nFor these energy levels, the Rydberg blockade effect is not present, so the resulting dynamics from a pulse addressing the $\\ket{h} \\leftrightarrow \\ket{g}$ transition of an atom will be dictated by the driving Hamiltonian of eq. (\\ref{drive_ham}). For a pulse of duration $\\tau$, the time evolution of this system is dictated by the operator\n\\begin{equation}\nU(\\bm \\Omega, \\tau) = T \\exp \\left[ -  \\frac{i}{2} \\int_0^\\tau \\bm \\Omega(t) \\cdot \\bm \\sigma \\mathrm dt\\right],\n\\end{equation}\nwhere $T$ denotes the time-ordering operator. The unitary $U$ describes a rotation around the time-dependent axis $\\bm\\Omega(t)$. For a resonant pulse (i.e. one with $\\delta = 0$) of phase $\\varphi$, we get a rotation angle of\n\\begin{equation}\\label{rot_angle}\n    \\theta = \\int_0^\\tau \\Omega(t) \\mathrm dt\n\\end{equation}\naround the fixed axis \n\\begin{equation}\\label{rot_axis}\n   \\bm e(\\varphi) = (\\cos\\varphi, -\\sin\\varphi, 0),\n\\end{equation}\nsituated on the equator of the Bloch sphere. The corresponding unitary operator is\n\\begin{equation}\n\\begin{split}\\label{res_rotation}\n    R_{\\bm e(\\varphi)}(\\theta) &= e^{-i\\frac{\\theta}{2}(\\cos(\\varphi) \\sigma_x - \\sin(\\varphi) \\sigma_y)}\\\\\n    &=  e^{i\\frac{\\varphi}{2}\\sigma_z}e^{-i\\frac{\\theta}{2}\\sigma_x}e^{-i\\frac{\\varphi}{2}\\sigma_z}\\\\\n    &= R_{ z}(-\\varphi)R_{ x}(\\theta)R_{ z}(\\varphi)\n\\end{split}\n\\end{equation}\nwhich, as its decomposition shows, can be thought of as a rotation around the Bloch sphere's $x$-axis, conjugated by $z$-rotations (i.e. phase gates). By following this gate with another $z$-rotation (which can be achieved virtually through a shift in the phase reference frame \\cite{IBM_VZ-gates}), we can then construct any arbitrary single-qubit gate,\n\\begin{equation}\n\\begin{split}\n    U(\\gamma, \\theta, \\varphi) &= R_z(\\gamma + \\varphi) R_{\\bm e(\\varphi)}(\\theta) \\\\\n    &= R_z(\\gamma)R_x(\\theta)R_z(\\varphi),\n\\end{split}\n\\end{equation}\nwhich relies solely on resonant pulses and phase reference frame changes. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/CZ_full}\n    \\caption{CZ implementation protocol, where three consecutive pulses address the $\\ket{g} \\leftrightarrow \\ket{r}$ transition. Full (dashed) red lines illustrate if a given pulse is effective (not effective).}\n    \\label{fig:cz_protocol}\n\\end{figure}\n\nWith the ability to perform arbitrary single-qubit gates, all that is missing in order to have a universal quantum gate set is an appropriate two-qubit gate. In neutral-atom devices, the native gate is the controlled-Z (CZ); its implementation leverages the Rydberg blockade effect to flip the phase of a two-qubit subsystem, conditionally on its initial state. To achieve this, the atoms, depending on their initial state, are sequentially brought to an auxiliary Rydberg state and back by a sequence of three consecutive laser pulses. The effect of this pulse sequence on the four computational states $\\ket{00}$, $\\ket{01}$, $\\ket{10}$ and $\\ket{11}$ is shown in Figure \\ref{fig:cz_protocol}. The first $\\pi$-pulse attempts to bring the \\textit{control} atom into the Rydberg state, which will condition whether the \\textit{target} will be able to go to $\\ket{r}$ and back with the second $2\\pi$-pulse, due to the Rydberg blockade. In particular, the $\\ket{00}$ state does not experience two consecutive phase flips as it would if the atoms did not interact (see top right panel of Fig. \\ref{fig:cz_protocol}, where the Rydberg energy level of the target atom experiences a shift in energy $\\Delta E$ due to the control atom being in $\\ket{r}$). Finally, the third pulse brings the \\textit{control} back to its original state, in case it was excited. In the end, all states but $\\ket{11}$ experience a phase flip (see bottom left panel of Fig. \\ref{fig:cz_protocol}), and the process corresponds to an application of a CZ within a global phase of $\\pi$ (see table \\ref{tab:cz}).\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{c||c | c |c | c}\n       \n     \n        Initial state & $\\ket{00}$ & $\\ket{01}$ & $\\ket{10}$ & $\\ket{11}$ \\\\\n        \\hline\n        Final state &  $-\\ket{00}$ & $-\\ket{01}$ & $-\\ket{10}$ & $\\ket{11}$ \\\\\n    \\end{tabular}\n    \\caption{Effect of the CZ pulse sequence in the computational basis.}\n    \\label{tab:cz}\n\\end{table}\n\n\nAltogether, the space in which the atom arrays can evolve is here composed of three states: two non-interacting hyperfine states, $\\{ |g\\rangle , |h\\rangle \\}$ and an auxiliary Rydberg state $|r\\rangle $ which is only populated in the transient dynamics of the system during the application of multi-qubit gates. The Rydberg state thus plays both the role of an excited state in the \\texttt{ground-rydberg} basis in the analog approach, and of an auxiliary state for realizing multi-qubit gates in the \\texttt{digital} basis.\n\nSince the most general sequence, with channels addressing both transitions, involves three energy levels, the size of the Hilbert space is $3^N$, where $N$ is the number of atoms in the array. Typically, a personal computer can efficiently simulate the evolution of systems of size below $N \\sim 20$.\n\n\\subsection{Measurement}\n\nIn neutral-atom devices, the measurement process encompasses all the atoms in the register and necessarily terminates the computation. The process is based on the fluorescence imaging of the atom array, where ``bright'' and ``dark'' sites are linked to the measured states\\,\\cite{Fuhrmanek11}. Due to there being multiple addressable transitions, the final state of each atom can be in more than two levels and, therefore, the measurement's outcome will be basis-dependent. In Pulser, the convention is to associate the state that is exclusive to a given basis to $1$ and all other outcomes to $0$, as summarized in table \\ref{tab:measures}.\n\n\\begin{table}[h]\n    \\centering\n    \\begin{tabular}{c||c | c}\n        \\cline{2-3}\n     \n        &  \\lstinline[] $ground-rydberg$ & \\lstinline[] $digital$\\\\ \\hline\n        $\\ket{r}$ & 1 & 0 \\\\\n        $\\ket{g}$ & 0 & 0 \\\\\n        $\\ket{h}$ & 0 &  1 \\\\\n    \\end{tabular}\n    \\caption{Corresponding measurement outcome of each state, depending on the measurement basis.}\n    \\label{tab:measures}\n\\end{table}\n\n\n\n\\section{Designing sequences with Pulser}\n\\label{section:sequences}\n\nPulser is written in the Python language to facilitate adoption by a widespread audience. It is also Open Source Software and can be accessed and developed by any interested contributor. To install the latest stable version of the Pulser package, type into a terminal (where Python version 3.7 or greater is installed):\n\\begin{lstlisting}[numbers=none]\npip install pulser\n\\end{lstlisting}\n\n\n\\subsection{Code Architecture}\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/Pulser_Architecture_v2.png}\n\\caption{Relationship between the main Pulser classes. The central object is the \\texttt{Sequence}, which is linked to a \\texttt{Device}. The \\texttt{Device} holds the available \\texttt{Channels} --- which are selected and declared in the \\texttt{Sequence} --- and information of the hardware constraints. These constraints are enforced upon the \\texttt{Register}, where the neutral-atom array is defined, and upon the \\texttt{Pulses}. Each \\texttt{Pulse}, defined by its amplitude and detuning \\texttt{Waveforms} and a fixed phase, populates the declared channels alongside other commands like \\texttt{target} --- which points local addressing channels to specific qubits --- and \\texttt{delay} --- which idles the channel. The resulting \\texttt{Sequence} can then be sent for execution on the neutral-atom QPU or emulated through Pulser's \\texttt{Simulation} class.}\n\\label{fig:architecture}\n\\end{figure*}\n\nThe main components of Pulser are:\n\n\\begin{itemize}\n    \\item[(\\emph{i})] The \\texttt{Device}, which consists of a series of specifications that characterize the hardware, including the chosen Rydberg level, the ranges of the amplitudes and frequencies of the lasers, the minimal and maximal distance between the atoms, and the different channels that can be declared (see (\\emph{iii}) below).\n    \\item[(\\emph{ii})] The \\texttt{Register} stores the information about the coordinates of the atoms and their respective ID's, which serve to identify them when targeting specific operations.\n    \\item[(\\emph{iii})] The \\texttt{Channel}s represent the action of the lasers and are organized by addressing (local or global) and the type of transition (Rydberg or Raman).\n    \\item[(\\emph{iv})] \\texttt{Waveform}s are the basic building blocks of a pulse. They can have custom or predetermined shapes, like a ramp waveform or a Blackman waveform, all of them indicating the specific profile of the waveform and its duration.\n    \\item[(\\emph{v})] \\texttt{Pulse}s consist of waveforms for the amplitude and the detuning. They can be further shifted by a phase. Once a \\texttt{Pulse} is constructed it has to be added to the sequence indicating which atom(s) are targeted and what channel will implement it.\n    \\item[(\\emph{vi})] The \\texttt{Sequence} contains the schedule of the pulses in each channel. It is also linked with a \\texttt{Register} and the \\texttt{Device} in which it is to be executed. This is the information that can be sent to a real neutral-atom QPU or simulated on a classical computer.\n    \\item[(\\emph{vii})] \\texttt{Simulation} is included to emulate results for the application of sequences on small instances. In Pulser, we have made use of the QuTiP libraries\\,\\cite{Johansson_2012,Johansson_2013} for the simulation of quantum systems. Each simulation run returns a specialized object that holds the results and features methods for post-processing them.\n\\end{itemize}\n\n\\subsection{Pulse Sequence Creation: A guided walkthrough}\n\nWe will show, step-by-step, how a basic pulse sequence is created with Pulser by going through the paradigmatic protocol for creating the Bell state $\\ket{\\Phi^+} = \\frac{1}{\\sqrt{2}}(\\ket{00} + \\ket{11})$. \n\nStarting from the archetypal circuit in Figure \\ref{fig:bell_circuit}(a), we need to adapt it for execution on neutral-atom devices. The major and necessary change is the decomposition of the CNOT gate, which is not native in neutral-atom devices, to an equivalent construction based on the CZ gate. Additionally, we replace the Hadamard with an equivalent gate (when starting from $\\ket{0}$) that can be executed with a resonant pulse, which reduces the number of free parameters in the pulse, making it less error-prone. These changes result in the equivalent circuit of Figure \\ref{fig:bell_circuit}(b).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=1.\\columnwidth]{figures\/bell_eq.pdf}\n\\caption{ (a) Typical quantum circuit for generating the Bell state $\\ket{\\Phi^+}.$ (b) Equivalent circuit to that of Figure \\ref{fig:bell_circuit}(a), adapted for execution on a neutral-atom device. }\n\\label{fig:bell_circuit}\n\\end{figure}\n\nFrom this quantum circuit, we can create the corresponding pulse sequence in Pulser. For this effect, the following modules will be needed:\n\n \\begin{lstlisting}[firstnumber=1]\n import numpy as np\n import pulser\n from pulser import Pulse, Sequence, Register\n \\end{lstlisting}\n \n\n\\subsubsection{Creating the Register}\n\nThe \\texttt{Register} defines the positions of the atoms and their respective names. There are multiple ways of defining a \\texttt{Register}, the most customizable one being to create a dictionary that associates a name (the key) to a coordinate in micrometers (the value). In this case, we only need two qubits, so we will create a \\texttt{Register} with two atoms, 4 $\\mu$m apart, which we will name \\lstinline{'c'} and \\lstinline{'t'}.\n\n\\begin{lstlisting}[firstnumber=4]\n# Coordinates in micrometers\nqubits = {'c': (-2, 0), 't': (2, 0)}\nreg = Register(qubits)\n\\end{lstlisting}\n\nNote that, although we can arbitrarily specify the position of each qubit, the \\texttt{Device} with which we will associate it will impose some restrictions, limiting in particular the minimal distance between any two qubits and the maximal distance of each qubit from the center of the array.\n   \n\n\n\n\n\n\\subsubsection{Initializing the Sequence}\n\nTo create a \\texttt{Sequence}, one has to provide it with the \\texttt{Register} instance and the \\texttt{Device} in which the sequence will be executed. Each \\texttt{Device} object is associated with a physical device and holds its particular constraints. When linked to a \\texttt{Sequence}, it dictates whether the chosen register is valid and, as the sequence is created, validates all instructions.\nWe import the device (in this case, \\texttt{Chadoq2}, which is Pasqal's research and development prototype) from \\texttt{pulser.devices} and initialize our sequence with the freshly created register:\n\n\\begin{lstlisting}[firstnumber=7]\nfrom pulser.devices import Chadoq2\nseq = Sequence(reg, Chadoq2)\n\\end{lstlisting}\n\n\\subsubsection{Channel declaration}\n\nEach device has a set of available channels, which can be consulted through \\texttt{seq.available\\_channels}. Since we need local addressing on both the \\texttt{ground-rydberg} basis (for the CZ gate) and the \\texttt{digital} basis (for the single-qubit gates), we will declare the \\lstinline{'rydberg_local'} and \\lstinline{'raman_local'} channels, which we will name \\lstinline{'rydberg'} and \\lstinline{'digital'}, respectively.\n\n\\begin{lstlisting}[firstnumber=9]\nseq.declare_channel('digital', 'raman_local')\nseq.declare_channel('rydberg', 'rydberg_local',\n    initial_target='c')\n\\end{lstlisting}\n\nOn real devices, each channel can only be declared once and will no longer appear in the list of available channels after declaration, appearing instead in \\texttt{seq.declared\\_channels}.\n\n\\subsubsection{Making the pulses}\nThe different channels will be populated by \\textit{pulses}, which are specific objects that need to be created. A pulse is made up of an amplitude (i.e. Rabi frequency) waveform, a detuning waveform and a fixed phase, although special convenience methods exist when one or both waveforms are constant. In our example, all pulses will be resonant, so we will use \\texttt{Pulse.ConstantDetuning()} to conveniently set $\\delta=0$ without defining its waveform. \n\nFor the amplitude waveform, eq. (\\ref{rot_angle}) tells us that what defines the angle of rotation is simply the integral of the waveform. This ideal scenario does not distinguish between a rectangular waveform or something more complex, as long as its integral is the same. In practice, a properly-shaped waveform can mitigate unwanted modifications to the modulated signal, stemming from spectral leakage or noise sensitivity. Here, we will use the \\textit{Blackman} waveform, which is a tapering function designed to have close to minimum spectral leakage.\n\nStarting with the single-qubit gates, all of those involved have $\\theta=\\pi\/2$, so we declare a Blackman waveform with an area of $\\pi\/2$ and duration of $200$ ns, making up a relatively short pulse without exceeding the amplitude range allowed by the channel where it will be executed. \n\n\\begin{lstlisting}[firstnumber=12]\nfrom pulser.waveforms import BlackmanWaveform\nhalf_pi_wf = BlackmanWaveform(200, np.pi\/2)\n\\end{lstlisting}\n\nThe rotation axis is defined from eq. (\\ref{rot_axis}), from which we gather that a rotation around the $y$-axis corresponds to a phase $\\varphi = -\\pi\/2$. Inversely, since $R_y^\\dag(\\pi\/2) = R_{-y}(\\pi\/2)$, we will have $\\varphi=\\pi\/2$ defining a rotation around $-y$. The corresponding pulse declarations are:\n\n\n\\begin{lstlisting}[firstnumber=14]\nry_pulse = Pulse.ConstantDetuning(\n    amplitude=half_pi_wf,\n    detuning=0,\n    phase=-np.pi\/2,\n)\nry_dag_pulse = Pulse.ConstantDetuning(\n    amplitude=half_pi_wf,\n    detuning=0,\n    phase=np.pi\/2,\n)\n\\end{lstlisting}\n\nMoving on to the CZ gate, which follows the protocol of Figure \\ref{fig:cz_protocol}, we will need a $\\pi$-pulse and a $2\\pi$-pulse, which we can shape with a Blackman waveform as well. However, because we are now dealing with Rydberg interactions, we have to make sure that the Rydberg blockade radius resulting from the chosen waveform is well above the spacing between the atoms. This concern does not extend to all the pulses assigned to the \\texttt{Rydberg} channel, because a Rydberg blockade is only meaningful if multiple atoms are being excited to the Rydberg state simultaneously or if an atom is being excited after its neighbors. Thus, in this case, only the $2\\pi$-pulse will determine a relevant Rydberg blockade radius, which means the $\\pi$-pulses surrounding it can be kept short:\n\n\\begin{lstlisting}[firstnumber=24]\npi_wf = BlackmanWaveform(200, np.pi)\npi_pulse = Pulse.ConstantDetuning(pi_wf, 0, 0)\n\\end{lstlisting}\n\nTo figure out what is the Rabi frequency upper bound that corresponds to a certain Rydberg blockade radius, as dictated by eq. (\\ref{eq:blockade_radius}), we use the \\texttt{rabi\\_from\\_blockade()} method of our chosen device. We set a conservative Rydberg blockade radius of $8 \\mu$m, double the distance between the atoms. The corresponding maximum Rabi frequency value is then\n\n\\begin{lstlisting}[firstnumber=26]\nmax_val = Chadoq2.rabi_from_blockade(8)\n# Result: max_val = 19.10672378540039 rad\/us\n\\end{lstlisting}\nConversely, it is also possible to figure out what the Rydberg blockade radius is for a given Rabi frequency value through the \\texttt{rydberg\\_blockade\\_radius()} method.\n\nWith an upper bound on the amplitude, we can create the appropriate Blackman waveform and corresponding pulse using the following formulation (which leverages the fact that there is a simple relationship between the duration, area and maximum value of a Blackman waveform):\n\\begin{lstlisting}[firstnumber=28]\ntwo_pi_wf = BlackmanWaveform.from_max_val(\n    max_val=max_val,\n    area=2*np.pi,\n)\ntwo_pi_pulse = Pulse.ConstantDetuning(\n    amplitude=two_pi_wf,\n    detuning=0,\n    phase=0,\n)\n\\end{lstlisting}\n\nAll the pulse objects necessary to program the circuit of Figure \\ref{fig:bell_circuit}(b) have been created. As a remark, note that every \\texttt{Pulse} and \\texttt{Waveform} can be plotted by calling its \\texttt{draw()} method, which can be useful to visualize the shape of the waveforms, as well as inspect the range of values they span. As an example, Figure \\ref{fig:two_pi_draw} displays the plot of \\lstinline{two_pi_pulse}, showing both the amplitude and detuning waveforms in the same picture.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/two_pi_draw}\n    \\caption{The outcome of calling \\lstinline{two_pi_pulse.draw()}. Both the amplitude and detuning waveforms are displayed.}\n    \\label{fig:two_pi_draw}\n\\end{figure}\n\n\n\\subsubsection{Composing the Sequence}\n\nAs previously mentioned, the workings of neutral-atom devices ultimately influence the sequences designed in Pulser. In many aspects --- like the way single-qubit gates are realized as pulses --- neutral-atom devices are very similar to other popular implementations. However, when compared to superconducting qubit architectures for instance, there are some marked differences, the most striking being in the connection between channels and qubits. \n\nIn superconducting qubit devices, each qubit has multiple dedicated channels, each serving a specific function, like that of manipulating or measuring the state of the qubit. This means that if one is to apply a pulse on a given qubit, that pulse will be allocated to the channel exclusively serving that qubit. \n\nOn the other hand, a channel in a neutral-atom QPU can serve all the qubits in the register, be it simultaneously or sequentially. In practice, this means that a channel needs to know on which qubit (or qubits) to apply the pulses to, and while there are some channels --- specifically, those of the \\texttt{Global} type --- whose targets are fixed to be the entire register, the channels of type \\texttt{Local} have the ability to change their target throughout the sequence. When the state of a single-qubit is changed, a \\texttt{Local} channel targets it before applying the pulse, after which it can move on to target a different qubit onto which the next pulse will be applied. In this way, a \\texttt{Local} channel can individually manipulate the state of all the qubits in a register, though it will do it sequentially rather than in parallel.\n\nThus, every channel needs to start with a target. For \\texttt{Global} channels, all the atoms are the target by definition, but for \\texttt{Local} channels, their target has to be explicitly defined. This initial target can be set at channel declaration (see how \\lstinline{'rydberg'} was set to target the \\lstinline{'c'} atom), or it can be done by using the standard \\lstinline{target} method of \\lstinline{seq} --- the \\texttt{Sequence} instance previously created.\n\n\\begin{lstlisting}[firstnumber=37]\nseq.target('c', 'digital')\n\\end{lstlisting}\n\nNow both channels have an initial target, so we can compose the sequence by allocating the \\texttt{Pulse}s to the declared \\texttt{Channel}s.\n\nWe start with the single-qubit gates, $R_y(\\pi\/2)$ and $R_y^\\dag(\\pi\/2)$. Recall that these transitions are addressed by the \\texttt{Raman}-type channel, which we called \\lstinline{'digital'}. Thus, the first two gates of circuit \\ref{fig:bell_circuit} are added to the same channel as follows:\n\n\\begin{lstlisting}[firstnumber=38]\nseq.add(ry_pulse, 'digital')\nseq.target('t', 'digital')\nseq.add(ry_dag_pulse, 'digital')\n\\end{lstlisting}\nNotice how the target is changed to \\lstinline{'t'} in between additions, so that the $R_y^\\dag(\\pi\/2)$ gate acts on the correct qubit. \n\nNext, we implement the CZ gate by replicating the protocol of Figure \\ref{fig:cz_protocol}. This will involve the other declared channel, so we must clarify how pulses in different channels are aligned in time. By default, when adding a pulse to a channel (which is targeting one or more qubits), it will be added as soon as all targeted qubits are free. This means that, if existing pulses on other channels are acting on one of the targeted qubits, the channel will be idle until they are done. On the other hand, if there is no overlap between the different channels' targets, the pulses will be played in parallel; this is called the \\lstinline{'min-delay'} protocol. The other addition protocols are the \\lstinline{'wait-for-all'}, in which the pulse will only play after all previously added pulses are finished, and the \\lstinline{'no-delay'}, where a delay is never added. \n\nThere is an additional command, called \\texttt{align()}, which introduces delays that align a subset of channels with the one that finished the latest, such that the next pulse added to any of them will start right after the latest channel has finished. When all the declared channels are aligned, the outcome is identical to that of using the \\lstinline{'wait-for-all'} protocol; otherwise, when only some of the channels are aligned, it corresponds to a ``wait-for-some'' protocol.\n\nAs illustrated in Fig.\\,\\ref{fig:cz_protocol}, the CZ implementation starts with a $\\pi$-pulse on channel \\lstinline{'rydberg'}, which is targeted to the \\lstinline{'c'} qubit. The channel starts empty, but the pulse will not be added at $t=0$ by default because it would overlap with the first pulse of channel \\lstinline{'digital'}, which is also targeting \\lstinline{'c'}. Instead, it would wait for this first pulse to end and then start playing in parallel with the second pulse in the \\lstinline{'digital'} channel. Although this would be a valid implementation, we will instead use the \\lstinline{align} command to make the pulse start only after the \\lstinline{'digital'} channel is done with both pulses (the \\lstinline{'wait-for-all'} would work too because we only declared two channels, but this way can be generalized to the case were other channels are declared). This will avoid dead-times in the CZ implementation, thus keeping the atoms in the Rydberg state no longer than necessary (which is desirable, since the atoms are more sensitive to noise when in the Rydberg state). \n\nThus, the CZ is implemented through\n\\begin{lstlisting}[firstnumber=41]\nseq.align('digital', 'rydberg')\nseq.add(pi_pulse, 'rydberg')\nseq.target('t', 'rydberg')\nseq.add(two_pi_pulse, 'rydberg')\nseq.target('c', 'rydberg')\nseq.add(pi_pulse, 'rydberg')\n\\end{lstlisting}\n\nFinally, there is still a single-qubit gate on the \\lstinline{'t'} qubit that must be applied after the CZ is done (recall that channel \\lstinline{'digital'} was left targeting \\lstinline{'t'}),\n\n\\begin{lstlisting}[firstnumber=47]\nseq.align('digital', 'rydberg')\nseq.add(ry_pulse, 'digital')\n\\end{lstlisting}\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/bell_seq.pdf}\n\\caption{The final Bell state creation sequence, displayed by calling \\lstinline{seq.draw()}. The current target of each channel is displayed in the orange boxes and is part of the original output, but the gates corresponding to each part of the circuit were annotated on top of it. The blue boxes on the right specify the basis in which the system is measured at the end of the sequence.}\n\\label{fig:cz_seq}\n\\end{figure*}\n\n\\subsubsection{Measurement}\n\nAt last, the measurement signals the end of a sequence, so after it no more changes are possible. We can measure a sequence by calling:\n\\begin{lstlisting}[firstnumber=49]\nseq.measure(basis='digital')\n\\end{lstlisting}\nWhen measuring, one has to select the desired measurement basis\\footnote{Here, we choose the \\lstinline{'digital'} basis, not to be confused with the channel we named \\lstinline{'digital'} for addressing this transition.}. The available options depend on the device and can be consulted by calling the chosen device's \\texttt{supported\\_bases} attribute.\n\n\\subsubsection{Visualization}\n\nThroughout its creation process, the sequence can be visualized through the \\texttt{Sequence.draw()} method.\nAs an example, the final sequence is shown in Figure \\ref{fig:cz_seq}. Note that the phase of each pulse is not represented, only the amplitude and detuning modulations of each channel's output --- the latter being always zero in this example. To inspect the phase of each pulse, one can print out the sequence instead, through\n\\begin{lstlisting}[firstnumber=50]\nprint(seq)\n\\end{lstlisting}\nThis will list all the contents of each channel, alongside its start and finish times.\n\n\n\n\\subsubsection{Simulation}\nThe primary goal of Pulser is to serve as a front-end framework for designing pulse sequences for neutral-atom quantum processors. However, it is also possible to emulate the output of a sequence locally on a classical device through the \\texttt{Simulation} class. Pulser's emulator relies on the open-source package QuTiP. Incorporating an emulator in Pulser is valuable for several reasons. First, it enables users to carefully design their pulse sequences and understand their effects before sending the instructions to a real device. It also enables them to gain intuition with simple setups, and explore new possible applications on limited-size systems. Coming back to our example, we use the emulator on our sequence by calling:\n\\begin{lstlisting}[firstnumber=51]\nfrom pulser.simulation import Simulation\nsim = Simulation(seq)\nres = sim.run()\n\\end{lstlisting}\nEvery time the \\texttt{run()} method is called, a \\texttt{SimulationResults} object instance is returned, which holds the results of the simulation.\n\nThe simulation performed is a time evolution of the system at specific times, such that \\lstinline{res} holds the state of the system at each instant. All the states can be accessed through the \\texttt{states} attribute, the final entry being the state of the system at the end of the sequence (before measurement). \n\nIn this case, since the three energy levels of the two atoms were involved, the final state will be a nine-entry vector. However, the population of the Rydberg levels should be negligible, so it is possible to reduce the final state to the \\lstinline{'digital'} basis of interest. We do so by calling\n\\begin{lstlisting}[firstnumber=54]\nres.get_final_state(reduce_to_basis='digital')\n\\end{lstlisting}\nwhich returns a \\texttt{qutip.Qobj} instance holding the final state in the \\lstinline{'digital'} basis. For this example, the outcome is\n$$\n\\begin{pmatrix} \n    0.707+i0.006 \\\\\n    -3.88 \\times 10^{-05}+i0.006 \\\\\n    -2.99\\times 10^{-05}      \\\\\n     0.707       \n\\end{pmatrix},\n$$\nwhich is very close to what we expect for $\\ket{\\Phi^+}$. \n\nAnother way to probe the results is to emulate a real experiment, in which case the only information about the final state of the system is obtained in the measurement. For a given number \\texttt{N\\_samples} of repetitions of the experiment, we call\n\\begin{lstlisting}[firstnumber=55]\nres.sample_final_state(N_samples=1e4)\n\\end{lstlisting}\nDue to finite sampling errors, the outcome will vary slightly for each call. An example of an outcome is \\lstinline[] ${'00': 5005, '11': 4995}$.\n\n\\section{Pulser in action : exploration of usecases with the emulator}\n\\label{section:usecases}\n\nIn this section, we reproduce some results obtained experimentally using Pulser's built-in emulator. These examples, among several others, can be  accessed online at Pulser's documentation site\\footnote{https:\/\/pulser.readthedocs.io}, which includes the code for the optimization of parameters and the generation of the figures.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/afm_seq_ph_diag.pdf}\n\\caption{(a) Schematic phase diagram of a two-dimensional quantum Ising system, with respect to the ratio of applied Rabi frequency $\\hbar \\Omega\/U$ and detuning $\\hbar \\delta\/U$. The blue region indicates the antiferromagnetic (AFM) phase, and outside of this region a paramagnetic (PM) phase is found. (b) The pulse sequence for the adiabatic preparation of the antiferromagnetic state in the Ising model, displayed by calling \\lstinline{seq.draw()}.}\n\\label{AFM:phase_diagram_and_seq}\n\\end{figure*}\n\n\n\\subsection{Preparing an antiferromagnetic state and observing the produced correlations}\n\\label{adiabatic_preparation}\n\n\nFirst we illustrate how to build a sequence for adiabatically preparing an antiferromagnetic state from the Ising-like model of eq.\\eqref{eq:ising_ham}, as well as studying its correlations. These types of states have been studied in neutral-atom setups up to hundreds of qubits, showing remarkable precision and allowing measurements that are beyond the best available classical simulation algorithms \\cite{Bernien17,Lienhard18, scholl2020programmable, ebadi2020quantum}.\n\nThe atoms will be placed in a square array of interatomic distance $R$. We will shine a laser globally with a pulse sequence representing a path in the phase diagram of the quantum Ising model, that starts in the paramagnetic phase and ends in the antiferromagnetic dome, as shown in Fig. \\ref{AFM:phase_diagram_and_seq}(a). We shall evaluate the quality of the resulting state by calculating the spin-spin correlation function, defined as:\n\\begin{equation}\\label{eq:corr_fct}\ng^{(2)}(k,l) = \\frac{1}{N_{k,l}}\\sum_{\\{i,j\\}} \\Big(\\langle \\hat n_i \\hat n_j \\rangle - \\langle \\hat n_i \\rangle \\langle \\hat n_j \\rangle \\Big).\n\\end{equation}\nThe function $g^{(2)}$ measures the correlation between local operators $\\hat n_i, \\hat n_j$ at different positions, averaged over all pairs of atoms $i,j$ separated by a vector $(kR,lR)$ ($N_{k,l}$ is the total number of such pairs). The pulses will be constructed by indicating the values for the maximum Rabi frequency, and the initial and final detuning (as reported in \\cite{lienhard2018observing}). \n\\begin{lstlisting}[firstnumber=1]\n# Load basic packages and classes\nimport numpy as np\nimport pulser\nfrom pulser import Register, Pulse, Sequence\nfrom pulser.devices import Chadoq2\nfrom pulser.waveforms import RampWaveform\nfrom pulser.simulation import Simulation\n# Define setup parameters (in rad\/us and ns)\nOmega_max = 2.3 * 2*np.pi\nU = Omega_max \/ 2.3\ndelta_0 = -6 * U\ndelta_f = 2 * U\nt_rise = 250\nt_fall = 500\nt_sweep = (delta_f-delta_0)\/(2*np.pi*10)*1000\n\\end{lstlisting}\nSince we want to use the blockade effect for nearest-neighbors only, we tune $\\Omega_\\text{max}$ to be of the same order as the van der Waals interaction magnitude for nearby atoms, $\\Omega_{\\text{max}} \\sim U$. This will fix an interatomic distance $R_{\\text{interatomic}}$, which we use to define our register:\n\n\\begin{lstlisting}[firstnumber=16]\nR_interatomic = Chadoq2.rydberg_blockade_radius(U)\nreg = Register.square(3, R_interatomic, prefix='q')\n\\end{lstlisting}\n\n\n\\subsubsection{Creating the sequence}\n\nWe compose our pulse using the \\texttt{Pulse} and \\texttt{Sequence} classes:\n\n\\begin{lstlisting}[firstnumber=18]\n# Define each pulse\nrise = Pulse.ConstantDetuning(\n        RampWaveform(t_rise, 0., Omega_max),\n        delta_0,\n        0.)\nsweep = Pulse.ConstantAmplitude(\n        Omega_max,\n        RampWaveform(t_sweep, delta_0, delta_f),\n        0.)\nfall = Pulse.ConstantDetuning(\n        RampWaveform(t_fall, Omega_max, 0.),\n        delta_f,\n        0.)\n# Add to sequence\nseq = Sequence(reg, Chadoq2)\nseq.declare_channel('ising', 'rydberg_global')\nseq.add(rise, 'ising')\nseq.add(sweep, 'ising')\nseq.add(fall, 'ising')\n\\end{lstlisting}\nThe sequence is readily plotted with the \\texttt{draw()} method, as illustrated in Figure \\ref{AFM:phase_diagram_and_seq}(b).\n\n\n\\subsubsection{Simulation: Spin-Spin Correlation Function}\n\nWe run a simulation of the sequence with the \\texttt{run()} method of the \\texttt{Simulation} class. For simple pulse sequences like the one we are using here, we can use a reduced sampling rate for faster emulation. A progress estimate can also be included if desired, with the parameter \\lstinline{progress_bar=True}. Once the final state is obtained, we may sample from it using the \\texttt{sample\\_final\\_state()} method. Since there was no measurement basis (\\lstinline{'ground-rydberg'} or \\lstinline{'digital'}) specified during the composition of the sequence, we include it as an argument with \\texttt{meas\\_basis}:\n\n\\begin{lstlisting}[language=python, firstnumber=37]\nsimul = Simulation(seq, sampling_rate=0.02)\nresults = simul.run(progress_bar=True)\ncount = results.sample_final_state(\n                   meas_basis='ground-rydberg')\nmost_freq = {k:v for k,v in count.items() if v>10}\n\\end{lstlisting}\n\nFrom the output of the simulation, we can plot the histogram of the sampled results, shown in Figure \\ref{histogram_results}. Notice the peak corresponding to the antiferromagnetic state.\\\\\n\n\\begin{figure}[h]\\label{fig:AFM_hist}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/AFM_hist}\n\\caption{Most frequent results from the sampling of the final state.}\n\\label{histogram_results}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/figure_AFM_pulser_paper_2.pdf}\n\\caption{(a) Simulation results for the $g^{(2)}$ correlation function, eq.\\eqref{eq:corr_fct}. (b) Sweep results for the $S_{\\text{N\u00e9el}}$ score function, eq.\\eqref{eq:neel}.}\n\\label{fig:AFM}\n\\end{figure}\n\nFrom \\lstinline{results}, one can access the state of the system throughout the simulation (as a list of \\texttt{qutip.Qobj}s) by calling \\lstinline{results.states}. This can be useful for post-processing the simulation results and computing observables. Alternatively, one can also call \\lstinline{results.expect()}  to get the exact expectation value of a list of observables. For example, \\lstinline{results.expect( [qutip.basis(9, 0).proj()])} will return the probability of measuring the all-excited state $|r\\rangle^{\\otimes 9}$ for every state in \\lstinline{results}.\n\nWe show in Figure \\ref{fig:AFM}(a) the output of the simulation for the correlation function $g^{(2)}$, eq. \\eqref{eq:corr_fct}. Note that the correlation function is expected to decay exponentially (modulo finite-size effects), which is best observed at larger system sizes~\\cite{scholl2020programmable}.\n\nChanging the endpoint $\\delta_f$ of the pulse's path results in different qualities of the final state. In order to evaluate how well it represents an antiferromagnetic state, we will explore the $\\Omega = 0$ line on the phase diagram, Fig. \\ref{AFM:phase_diagram_and_seq}(a). Since the value of correlation function $g^{(2)}(k,l)$ (eq. \\eqref{eq:corr_fct}) can be positive or negative, we compensate with alternating signs to construct a single score for the state \\cite{lienhard2018observing}:\n\\begin{equation}\nS_{\\text{N\u00e9el}} = \\sum_{(k,l)\\neq (0,0)} (-1)^{|k| + |l|} g^{(2)}(k,l),\n\\label{eq:neel}\n\\end{equation}\nwhich should be largest when the state is closest to an antiferromagnetic product state. By sweeping over different values of detuning $\\delta_f$, we examine the region $ 0 < \\hbar \\delta_f\/U < 4$. In a few seconds, Pulser gives the results shown in Figure \\ref{fig:AFM} (b), showing where the antiferromagnetic phase is more strongly present.\n\n\n\\subsection{Variational algorithms with Pulser: solving a graph problem with neutral atoms}\n\n\nThe implementation of variational algorithms, such as the Quantum Approximation Optimization Algorithm\\,\\cite{Farhi14} (QAOA), is facilitated by the use of a \\textit{parametrized sequence} in Pulser. We illustrate this aspect here, by solving the \\emph{Maximum Independent Set} (MIS) problem on \\emph{Unit Disk} graphs, which can be naturally studied on neutral-atom devices \\cite{pichler2018quantum, henriet2020robustness, dalyac2020qualifying}. \n\n\n\\subsubsection{From a graph to an atomic register} \n\nAn \\emph{independent set} of a graph is a subset of vertices where any two elements of this subset are not connected by an edge. The MIS corresponds to the largest of such subsets. The so-called \\emph{Unit Disk} (UD) version of this problem corresponds to the instances where the graph under consideration lives in 2D and displays an edge between two nodes if they are within a unit length of each other.\\\\\n\nInterestingly, ensembles of interacting Rydberg atoms in 2D can be naturally represented by Unit Disk graph structures: the atoms are the nodes of the graphs and an edge in the graph corresponds to two atoms being within the blockade radius of each other. In this example, we will take $\\Omega$ fixed to a frequency of 1 $\\text{rad}\/\\mu$s, hence the blockade radius can be obtained by using \\lstinline{Chadoq2.rydberg_blockade_radius(1.)}. The following code block instantiates the atomic register, and displays the graph induced by their interactions with the \\lstinline{draw_graph=True} and \\lstinline{draw_half_radius=True} arguments passed to the \\lstinline{reg.draw()} method, as illustrated in Figure \\ref{fig:MIS_reg}.\n\n\n\\begin{lstlisting}[firstnumber=1]\n# Load basic packages and classes\nimport numpy as np\nimport pulser \nfrom pulser import Register, Pulse, Sequence\nfrom pulser.devices import Chadoq2\nfrom pulser.simulation import Simulation\n# Set Register\npos = np.array([[0., 0.], [-4, -7], [4,-7], [8,6], [-8,6]])\nreg = Register.from_coordinates(pos)\nRb = Chadoq2.rydberg_blockade_radius(1.)\n\nreg.draw(\n    blockade_radius=Rb,\n    draw_graph=True,\n    draw_half_radius=True\n    )\n\\end{lstlisting}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.6]{figures\/MIS_reg}\n\\caption{The register for the MIS graph problem. The shaded circles show the blockade half-radius, such that the intersecting regions indicate existing links.}\n\\label{fig:MIS_reg}\n\\end{figure}\n\nThe graph $G$ has two maximal independent sets: $(1,3,4)$ and $(2,3,4)$, respectively \\texttt{01011} and \\texttt{00111} in binary. One could try to prepare those states using an adiabatic approach such as the one illustrated in section \\ref{adiabatic_preparation}. Another approach is to use QAOA, as we illustrate below.\n\n\\subsubsection{Building the parametrized sequence}\n\nQAOA exploits both a quantum and a classical processor that exchange information within a feedback loop to solve an optimization problem. The role of the neutral-atom processor is to prepare and measure an $N$-atom parametrized wavefunction. The outcome of the measurement is then used by the classical processor in a standard classical optimization procedure, that updates the parameters for the next iteration.\\\\\n\nMore specifically, the preparation of the parametrized wavefunction is achieved through the successive application of two non-commutative Hamiltonians, with all atoms initially starting in the ground state $|00\\dots0\\rangle$ of the \\lstinline{ground-rydberg} basis. The first one is realized by taking $\\Omega = 1$ $\\text{rad}\/\\mu s$, and $\\delta = 0 $ $\\text{rad}\/\\mu s$ in Eq.(\\ref{eq:global_ising}). The second Hamiltonian is realized with $\\Omega = \\delta = 1$ $\\text{rad}\/\\mu s$. These two Hamiltonians are applied successively, for durations $t_j$ and $s_j$, respectively. The dimension of $\\bm t$ and $\\bm s$, i.e. the number of layers applied to the system, is referred to as the depth of the algorithm. The outcome of the measurement is then used as the objective function in a standard classical optimization procedure, that updates the parameters for the next iteration.\\\\\n\nWhen implementing a variational procedure, multiple sequences are created that differ only by the durations $t_j$ and $s_j$ of the pulses. This is conveniently handled in Pulser by turning a regular sequence into a \\textit{parametrized} sequence. A \\texttt{Sequence} is said to be parametrized once a \\textit{variable} is declared and used in a sequence-building call, from which point the sequence building process can continue as usual but the sequence is no longer built on the fly. Instead, all calls to a parametrized sequence are stored as a \\textit{blueprint} for generating a new \\texttt{Sequence}, which may depend on the value of the declared variables. Consequently, it is not possible to progressively monitor the creation process of a sequence that is parametrized (e.g. by drawing the state of a sequence as new pulses are added). The building of the sequence itself only happens when \\texttt{Sequence.build()} is called, at which point specific values for all the declared variables have to be specified as arguments.\\\\\n\nThe \\texttt{Variable} objects are obtained by calling \\texttt{Sequence.declare\\_variable()} and can be used throughout the \\texttt{Sequence} creation process (i) as parameters when creating new \\texttt{Waveform}s and \\texttt{Pulses}, or (ii) as arguments for standard sequence creation methods like \\texttt{add}, \\texttt{target}, \\texttt{align} and \\texttt{delay}. Moreover, they support basic arithmetic operations and, when they are of \\texttt{size}$>1$, iteration and item accessing. \\\\\n\nThe notion of parametrized sequence can become very handy with real world constraints such as those we find in cloud-based platforms. First, the bandwidth allocation of a program, variational or not, is greatly reduced thanks to this factorization. A user only needs to send the parametrized sequence that describes his program once, and then each new iteration only requires the associated set of parameters. Thanks to the high flexibility of Pulser, users can have a very fine-grained control over the waveforms that define their pulses. This results in needing larger objects to define these pulses, which makes such a factorization even more useful.\n\nIn addition, one area of improvement for cold atom platforms is the large calibration times for atom register configuration changes. This means that running randomly independent sequences on the same QPU can be very inefficient. The sequences of a given user's program on the other hand share some common parameters, including the register configuration. On the QPU side, we can treat the parametrized sequences as the basic block of the QPU scheduling to greatly reduce the latencies due to the calibration processes, make sure the different user programs are run efficiently and that all the sequences in a given program are run on the same QPU.\\\\\n\nThe example below shows how \\texttt{Sequence} can be used to create a parametrized QAOA sequence with two layers, with the variable duration of each layer stored in the \\lstinline{t_list} and \\lstinline{s_list} arrays. Notice that no value is given to the contents of \\lstinline{t_list} and \\lstinline{s_list}.\n \n\n\\begin{lstlisting}[firstnumber=17]\n# Parametrized sequence\nseq = Sequence(reg, Chadoq2)\nseq.declare_channel('ch0', 'rydberg_global')\n\nt_list = seq.declare_variable('t_list', size=2)\ns_list = seq.declare_variable('s_list', size=2)\n\nfor t, s in zip(t_list, s_list): \n    pulse_1 = Pulse.ConstantPulse(1000*t, 1., 0., 0) \n    pulse_2 = Pulse.ConstantPulse(1000*s, 1., 1., 0)\n    \n    seq.add(pulse_1, 'ch0')\n    seq.add(pulse_2, 'ch0')\n\\end{lstlisting}\n\nThe sequence \\lstinline{seq} above will only be built once the user provides specific values for \\lstinline{t_list} and \\lstinline{s_list} while calling the \\texttt{Sequence.build()} method. It thus enables to build a variety of sequences that share the same structure:\n\n\\begin{lstlisting}[numbers=none]\n# Build sequences with specific values\nmy_seq_1 = seq.build(t_list=[2,4], s_list=[3,6])\nmy_seq_2 = seq.build(t_list=[1,3], s_list=[2,5])\n\\end{lstlisting}\n\n\\subsubsection{Classical optimization and QAOA results}\n\nThe parametrized sequence above can then be used in conjunction with a  classical optimizer in order to determine a set of good values for \\lstinline{t_list} and \\lstinline{s_list}. When running the full QAOA procedure in closed loop, the optimizer is responsible for iteratively selecting the next set of parameters to be tested. The parametrized sequence can then be updated in Pulser and the new program sent externally to the quantum hardware. \n\nThe procedure can also be emulated locally: to try out the algorithm, we applied the non-gradient method Nelder-Mead for a few dozen function evaluations, initializing the parameters in a convenient point (the implementation can be found at Pulser's online documentation). \nThis is already sufficient in order to find some acceptable parameters $\\bm t$ and $\\bm s$. The performance of QAOA can be tested by sampling from the final state $|\\psi(t_f)\\rangle$ which returns both MISs of the graph with high probability. We show in Figure \\ref{fig:MIS_optim} the histogram of the recorded bitstrings.  \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/MIS_optim}\n\\caption{Histogram of most frequent bitstrings after the optimization loop. The MIS graphs (in red) are the ones that are observed most frequently.}\n\\label{fig:MIS_optim}\n\\end{figure}\n\n\n\\section{Conclusion}\n\nWith recent advances in the field, neutral-atom processors now provide unique opportunities for the exploration of uncharted territories in many-body quantum physics with hundreds of interacting quantum units\\,\\cite{ebadi2020quantum,scholl2020programmable,Bluvstein21}. By using distinct electronic transitions for encoding the quantum information, one can implement various Hamiltonians depending on the task at hand. Beyond analog computing, recent developments in qubit addressing have opened the door to high-fidelity quantum gates\\,\\cite{levine_high-fidelity_2018,Levine19}. In addition to the flexibility in terms of information encoding and processing, the geometry of the quantum register itself is also highly programmable\\,\\cite{barredo_synthetic_2018,Schymik20}. This holds great promises for the implementation of graph-related algorithms\\,\\cite{Pichler18,zhou20,henriet2020robustness,dalyac2020qualifying,Henry21}, the exploration of lattice models in condensed-matter physics\\,\\cite{ebadi2020quantum,scholl2020programmable,leseleuc_observation_2019}, or the reduction of the gate-count for quantum circuits\\,\\cite{Henriet2020quantum}. \n\nIn this paper, we introduced Pulser, an open-source Python library allowing users to pilot such powerful devices at the pulse level. This tool will enable the exploration of new avenues in physics and quantum information science requiring an advanced control of the system. After their creation in Pulser, pulse sequences can be translated and read by an arbitrary waveform generator for implementation on a quantum processor. The architecture of the software makes it possible to use as an interface for any neutral-atom QPU as the specifics of the back-end can be encapsulated by creating new \\texttt{Device} objects. Pulser currently enables the control of laser pulses during the quantum information processing phase, but it could be extended in the future to include other controls either in the processing phase, during the assembly of the register or the measurement phase. Being a low-level tool, Pulser could be interfaced with higher-level application-oriented libraries and frameworks for digital quantum information processing, such as Cirq\\,\\cite{cirq_developers_2021_4586899}, Qiskit\\,\\cite{Qiskit}, Atos myQLM, Pennylane\\,\\cite{Pennylane} or tket\\,\\cite{tket}, which abstract away many hardware features. \n\nThe primary purpose of Pulser's built-in emulator is to reproduce faithfully what the output of a real experiment would look like, in order to facilitate the design of hardware runs. In that spirit, some developments are currently underway for the inclusion of realistic error models\\,\\cite{sylvain18}. Some work could also be done to extend the simulations capabilities in terms of the number of qubits that can be efficiently simulated, including the development of tensor-network based methods and GPU acceleration. \n\n\\section*{Acknowledgments}\nWe thank Lucas Beguin, Julien Br\u00e9mont, Antoine Browaeys, Mauro d'Arcangelo, Eric Giguere, Christophe Jurczak, Thierry Lahaye, Boxi Li, Georges-Olivier Reymond, Pascal Scholl, Adrien Signoles, and William J. Zeng for useful discussions.\n\n\n\n\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe condition (on the autocovariances $\\gamma_k$ of a stationary time series)\n\\begin{equation}\n\\lim_{k \\rightarrow \\infty} |\\gamma_k| \\; \\ln k = 0\n\\label{eq:berman}\n\\end{equation}\nwas introduced by \\citet[Theorem 3.1, page 510]{ber}.  It appears to\nhave been adopted as a fundamental sufficient condition in proving\nresults about extreme value distributions for correlated data.  It is\ncited for instance in \\citet[2.5.1, p.~444]{lead1}, \\citet[5.1,\np.~248]{lin}, \\citet[4.1.1, p.~80]{lead2}, \\citet[Theorem 3.8.2,\np.~169; see also p.~198]{gal}, and in \\citet[Theorem 4.4.8,\np.~217]{emb}, where it is described as being ``very weak''.  It\nappears to be effectively the weakest condition that one can assume\nand still obtain positive results in this context.\n\nIn \\citet*{cha} the authors found it necessary to assume, in addition\nto the Berman condition, another condition\n\\begin{equation}\n\\sum_{k=1}^{\\infty} \\frac{|\\gamma_k|}{k^{\\epsilon}} < \\infty\n\\mbox{~~for some~~} \\epsilon < 1.\n\\label{eq:sum.cond}\n\\end{equation}\nThis is given as condition (7) on page 598 of \\citeauthor{cha}.  In that\npaper the authors find it expedient to deal with the residuals from\nfitting an ARMA model to the time series $X_t$ under consideration.\nThey are thereby concerned with the innovation terms of such a model.\nSuppose that $X_t$ and $a_t$ are related via an ARMA model in which\nthe $a_t$ play the role of the innovations.  \\citeauthor{cha} remark\nthat if the time series $X_t$ is a fractionally integrated ARMA\n(``FARIMA'') time series (whence it satisfies the two conditions\nof interest (\\ref{eq:berman}) and (\\ref{eq:sum.cond})) then the\ninnovations $a_t$ also form a FARIMA series provided that the\nmodel is invertible.  Hence the $a_t$ satisfy the two conditions\nof interest as well.\n\n\\citeauthor{cha} assert that more is true:  if $X_t$ is \\emph{any}\nstationary time series satisfying (\\ref{eq:berman}) and\n(\\ref{eq:sum.cond}) and if $X_t$ and a series of innovations $a_t$\nare related by an invertible ARMA model, then the $a_t$ will also\nsatisfy these conditions.  In this note we present the proof of\nthat claim.\n\nWe now remark that interest is focussed on the $a_t$ and these\nquantities are thought of as being the output of a filter, with\nthe $X_t$ being the input.  However the phrasing of the claim,\nwith the $a_t$ being the innovations of an ARMA model, makes it\nappear as if the $a_t$ are the \\emph{input} to a filter, which is\nrather confusing.  The required condition of invertibility\nof the ARMA model is also somewhat disconcerting.  Finally,\nit turns out that a slightly stronger claim may be established.\nWe therefore re-phrase the assertion to be proven, in a stronger\nand less confusing form, and state the original claim as a corollary\nof the re-phrased assertion.\n\n\\section{The Main Result}\n\\label{sec:result}\n\nWe state the result to be proven as follows:\\\\\n\n\\noindent\n{\\bf Theorem:} Suppose that $X_t$ is a stationary time series with\nautocovariances $\\gamma_k$ satisfying conditions (\\ref{eq:berman})\nand (\\ref{eq:sum.cond}) and that the series $Y_t$ is the output of\na linear filter with input $X_t$ given as follows:\n\\[\nY_t = \\sum_{n=0}^{\\infty} \\psi_n X_{t-n}\n\\]\nSuppose that the $\\psi_n$ are summable (whence the $Y_t$ form a\nstationary time series).  Furthermore suppose that the $\\psi_n$\nsatisfy the condition\n\\begin{equation}\n|\\gamma^W_k| \\leq C r^{|k|} \\mbox{~~for all~~} k\n\\label{eq:bound}\n\\end{equation}\nfor some constants $C$ and $r$, $0 < r < 1$, where\n\\[\n\\gamma^W_k = \\sum_{n = -\\infty}^{\\infty} \\psi_n \\psi_{n+k}\n\\]\nand where we set $\\psi_n = 0$ for $n < 0$ (to simplify the notation).\nThen the autocovariances $\\gamma^Y_k$ of the series $Y_t$ satisfy\n(\\ref{eq:berman}) and (\\ref{eq:sum.cond}) as well.\\\\\n\n\\noindent {\\bf Proof:}\\\\\n\\begin{quote}\n\nWe remark that the $\\gamma^W_k$ are in fact the autocovariances of\na time series $W_t$ defined by\n\\[\nW_t = \\sum_{n=0}^{\\infty} \\psi_n b_{t-n}\n\\]\nwhere $b_t$ is white noise with variance 1.\n\nObserve that\n\\begin{eqnarray*}\n\\gamma^Y_k & = & \\sum_{n=-\\infty}^{\\infty} \\sum_{m=-\\infty}^{\\infty}\n                 \\psi_n \\psi_m \\gamma_{m-n+k} \\\\\n           & = & \\sum_{h=-\\infty}^{\\infty} \\sum_{n=-\\infty}^{\\infty}\n                 \\psi_n \\psi_{n+h} \\gamma_{k+h} \\\\\n           & = & \\sum_{h=-\\infty}^{\\infty} \\gamma^W_h \\gamma_{k+h}\n\\end{eqnarray*}\n\nTo show that the $\\gamma^Y_k$ satisfy condition\n(\\ref{eq:berman}) we write\n\\begin{eqnarray*}\n\\gamma^Y_k\n & = & \\sum_{h=-\\infty}^{-k-1} \\gamma^W_h \\gamma_{k+h} +\n                 \\sum_{h=-k}^{-1} \\gamma^W_h \\gamma_{k+h} +\n                 \\sum_{h=0}^\\infty \\gamma^W_h \\gamma_{k+h} \\\\\n & = & \\sum_{j=1}^{\\infty} \\gamma^W_{k+j} \\gamma_j +\n       \\sum_{j=0}^{k-1} \\gamma^W_{k-j} \\gamma_j +\n       \\sum_{j=0}^{\\infty} \\gamma^W_j \\gamma_{k+j} \\\\\n & = & \\xi_1(k) + \\xi_2(k) + \\xi_3(k) \\mbox{~~(say).}\n\\end{eqnarray*}\n\nTo deal with $\\xi_1(k)$ we observe that\n\\[\n|\\xi_1(k)| \\ln k\n\\leq \\sum_{j=1}^{\\infty} |\\gamma^W_{k+j}| |\\gamma_j| \\; \\ln k\n\\leq C \\; \\gamma_0 \\; r^k \\ln k \\; \\frac{r}{1-r}\n\\]\nusing (\\ref{eq:bound}).  This quantity $\\rightarrow 0$\nas $k \\rightarrow \\infty$ since $r < 1$.\n\nSimilarly\n\\[\n|\\xi_3(k)| \\ln k \\leq \n\\sum_{j=0}^{\\infty} |\\gamma^W_j| |\\gamma_{k+j}| \\ln (k+j) \\; \\leq\nC \\sum_{j=0}^{\\infty} r^j |\\gamma_{k+j}| \\ln (k+j)\n\\]\nTake $\\delta > 0$; for sufficiently large $k$, $|\\gamma_{k+j}| \\ln (k+j)\n\\leq \\delta$ for all $j \\geq 0$.  Hence\n\\[\n|\\xi_3(k)| \\ln k \\leq \\frac{\\delta \\times C}{1-r}\n\\]\nfor sufficiently large $k$, and since $\\delta$ is arbitrary,\n$ |\\xi_3(k)| \\ln k \\rightarrow 0 $ as $k \\rightarrow \\infty$.\n\nTo deal with the middle term $\\xi_2(k)$ we note that\n\\begin{eqnarray*}\n|\\xi_2(k)| & \\leq & C \\sum_{j=0}^{k-1} |\\gamma_j| r^{k-j} \\\\\n & = & C \\left [ \\sum_{j=0}^{[k\/2]} |\\gamma_j| r^{k-j}\n               + \\sum_{j=[k\/2]+1}^{k-1} |\\gamma_j| r^{k-j}\n         \\right ] \\\\\n & \\leq & C \\sum_{j=0}^{[k\/2]} \\gamma_0 r^{k-j}\n               + C \\sum_{j=[k\/2]+1}^{k-1} |\\gamma_j| r^{k-j} \\\\\n & \\leq & C \\gamma_0 \\frac{r^{k\/2}}{1-r}\n               + C \\gamma_{j^*(k)} \\frac{r}{1-r}\n\\end{eqnarray*}\nwhere $j^*(k) = \\mbox{argmax} \\{|\\gamma_j| : [k\/2]+1 \\leq j \\leq k-1 \\}$.\n\nHence\n\\begin{eqnarray*}\n\\ln k \\times |\\xi_2(k)| & \\leq & C \\left [ \\gamma_0 \\ln k \\frac{r^{k\/2}}{1-r}\n                            + (\\ln k\/2 + \\ln 2) \\left ( \\gamma_{j^*(k)}\n                               \\frac{r}{1-r} \\right ) \\right ] \\\\\n & \\leq & C \\left [ \\gamma_0 \\ln k \\frac{r^{k\/2}}{1-r}\n                            + (\\ln j^*(k) + \\ln 2) \\left ( \\gamma_{j^*(k)}\n                               \\frac{r}{1-r} \\right ) \\right ] \\\\\n\\end{eqnarray*}\nwhich $\\rightarrow 0$ as $k \\rightarrow \\infty$.\n\nWe have thus established that the autocovariances $\\gamma^Y_k$\nsatisfy (\\ref{eq:berman}).  We now proceed to show that condition\n(\\ref{eq:sum.cond}) is satisfied:\n\n\\begin{eqnarray*}\n\\sum_{k=1}^{\\infty} \\frac{|\\gamma^Y_k|}{k^{\\epsilon}}\n & = & \\sum_{k=1}^{\\infty} \\left | \\sum_{h=-\\infty}^{\\infty} \\gamma^W_h\n       \\frac{\\gamma_{k+h}}{k^{\\epsilon}} \\right |\n       \\leq \\sum_{h=-\\infty}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} \\\\\n & = & \\sum_{h=-\\infty}^{-1} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} +\n       \\sum_{h=0}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} \\\\\n & = & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} +\n       \\sum_{h=0}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{k^{\\epsilon}} \\\\\n & \\leq & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} +\n       \\sum_{h=0}^{\\infty} |\\gamma^W_h| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k+h}|}{(k+h)^{\\epsilon}}\n       \\left (\\frac{k+h}{k}\\right)^{\\epsilon} \\\\\n & \\leq & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} +\n       \\zeta \\sum_{h=0}^{\\infty} |\\gamma^W_h| (1+h)\n       \\mbox{, where~~} \\zeta = \\sum_{k=1}^{\\infty}\n       \\frac{|\\gamma_{k}|}{k^{\\epsilon}}\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\phantom{\\sum_{k=1}^{\\infty} \\frac{|\\gamma^Y_k|}{k^{\\epsilon}}}\n & = & \\sum_{j=1}^{\\infty} |\\gamma^W_j| \\left [ \\sum_{k=1}^{j}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} + \\sum_{k=j+1}^{\\infty}\n       \\frac{|\\gamma_{k-j}|}{k^{\\epsilon}} \\right ] +\n       \\zeta \\sum_{h=0}^{\\infty} |\\gamma^W_h| (1+h) \\\\\n & \\leq & \\sum_{j=1}^{\\infty} C r^j \\left [ \\gamma_0 \\sum_{k=1}^{j}\n       \\frac{1}{k^{\\epsilon}} +\n       \\sum_{\\ell=1}^{\\infty} \\frac{|\\gamma_{\\ell}|}{\\ell^{\\epsilon}}\n       \\left ( \\frac{\\ell}{\\ell+j} \\right)^\\epsilon\n       \\right ] + \\zeta \\sum_{h=0}^{\\infty} C r^h \\; (1+h) \\\\\n & \\leq & C \\sum_{j=1}^{\\infty} r^j \\left [ j \\gamma_0 +\n       \\sum_{\\ell=1}^{\\infty} \\frac{|\\gamma_{\\ell}|}{\\ell^{\\epsilon}}\n       \\right ] + \\zeta C \\sum_{h=0}^{\\infty} r^h \\;(1+h) \\\\\n & \\leq & C \\sum_{j=1}^{\\infty} r^j [ j \\gamma_0 + \\zeta] +\n            \\zeta C \\sum_{h=0}^{\\infty} r^h \\;(1+h) \\\\\n & = & C \\left [ \\gamma_0 \\sum_{j=1}^{\\infty} jr^j +\n         \\zeta \\sum_{j=1}^{\\infty}  r^j + \\zeta \\sum_{j=0}^{\\infty} r^j\n         + \\zeta \\sum_{j=1}^{\\infty} j r^j \\right ] \\\\\n & = & C \\left [ (\\gamma_0 + \\zeta) \\sum_{j=1}^{\\infty} j r^j +\n       \\zeta \\; \\frac{1+r}{1-r} \\right ] < \\infty\n\\end{eqnarray*}\nsince $\\sum_{j=1}^{\\infty} jr^j$ converges.  (The radius of convergence\nof this power series is 1, and by assumption $0 < r < 1$.)\n\\hfill \\rule{2mm}{3.5mm}\\\\\n\\end{quote}\n\n\\section{The Original Claim}\n\\label{sec:orig.claim}\n\nWe state the original claim as:\\\\\n\n\\noindent\n{\\bf Corollary:}  Suppose that $X_t$ satisfies conditions\n(\\ref{eq:berman}) and (\\ref{eq:sum.cond}) and that $X_t$ and $a_t$\nare related by the invertible ARMA model\n\\[\n\\phi(B)X_t = \\theta(B) a_t\n\\]\nwhere $\\phi(z)$ and $\\theta(z)$ are polynomials and $B$ is the\n``backshift'' operator.  Then the $a_t$ satisfy conditions\n(\\ref{eq:berman}) and (\\ref{eq:sum.cond}) as well.\\\\\n\n\\noindent {\\bf Proof:}\\\\\n\\begin{quote}\nThe invertibility of the model tells us that $a_t$ can be\nexpressed as\n\\[\na_t = \\sum_{n=0}^{\\infty} \\psi_n X_{t-n}\n\\]\nwhere\n\\[\n\\frac{\\phi(z)}{\\theta(z)} = \\psi(x) = \\sum_{n=0}^{\\infty} \\psi_n z^n\n\\]\nwith the coefficients $\\psi_n$ being summable.  (It is more usual in\nsuch a context to denote the coefficients of the series expansion\nas ``$\\pi_n$'' rather than ``$\\psi_n$'', but to make clear the\nrelationship to the main result we eschew the $\\pi_n$ notation.)\n\nNow if we set\n\\[\nW_t = \\sum_{n=0}^{\\infty} \\psi_n b_{t-n}\n\\]\nwhere $b_t$ is white noise (with variance 1) then basic results\nabout ARMA time series (see e.g. \\citet[Chapter 3, problem 3.11]{bro}) tell\nus that the autocovariances $\\gamma^W_k$ of $W_t$ satisfy\n(\\ref{eq:bound}).  Hence the claim follows by the theorem proven\nin section~\\ref{sec:result}.\\\\\n\\mbox{ } \\hfill \\rule{2mm}{3.5mm}\\\\\n\\end{quote}\n\n\\section*{Acknowledgments}\nThe authors express their warm thanks to an anonymous referee whose\ncomments substantially improved the exposition in this paper.  The\nfirst author's research is supported by a grant from the Natural\nSciences and Engineering Research Council of Canada.\n\n\\bibliographystyle{melvin}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main goals of the LEP experiments is to search for new\nparticles predicted by theories beyond the Standard\nModel. In this letter we report on searches for \nunstable  charginos and neutralinos.\nThese particles are predicted by supersymmetric theories (SUSY)\n\\cite{susy}. In SUSY theories with minimal particle\ncontent (MSSM) \\cite{mssm}, in addition to the ordinary particles, there is\na supersymmetric spectrum of particles with spins which differ by one\nhalf with respect to their Standard Model partners.\n\nCharginos (\\charginos{\\pm}), the supersymmetric partners of \\Wpm\\, \nand \\Hpm, are pair produced via $s$-channel \n$\\gamma\/\\mathrm{Z}$ exchange. The production cross section can be\nreduced by an order of magnitude when the $t$-channel scalar neutrino\n(\\susy{\\nu}) exchange is important.\nNeutralinos, the supersymmetric partners of Z, \\gam, and neutral\nHiggs bosons, are pair produced  \n$\\ee \\rightarrow \\neutralino{i}\\neutralino{j}~ (i,j=1, \\ldots ,4$;\nordered by their masses) via $s$-channel $\\mathrm{Z}$ exchange and their\n production cross section can be enhanced by $t$-channel exchange of a \nscalar electron (\\selectron{\\pm}). \n\nShort-lived supersymmetric particles are expected\nin R-parity conserving SUSY models. \nThe R-parity is a quantum number which \ndistinguishes ordinary particles from supersymmetric particles.\nIf R-parity is conserved, supersymmetric particles are \npair-produced and the lightest supersymmetric particle, \nthe lightest neutralino, \\neutralino{1}, is stable. The neutralino is weakly-interacting \nand escapes detection.\nIn this letter we assume R-parity conservation, which implies \nthat the decay chain of supersymmetric particles always contain,\nbesides standard particles, two invisible neutralinos\ncausing the missing energy signature.\n\nWhen the masses of the scalar leptons and the charged Higgs bosons (\\Hpm) are very large,\nthe \\chargino{\\pm} decays via \\ensuremath{\\mathrm{W^\\ast}}:  \n$\\chargino{\\pm} \\rightarrow \\neutralino{1}\\ensuremath{\\mathrm{W^\\ast}}\n                \\rightarrow \\neutralino{1}\\, f\\bar{f}^\\prime$.\nIf the \\slepton{\\pm} and \\susy{\\nu}\\, masses are comparable to \n$M_{\\W}$ the chargino also \ndecays via virtual scalar lepton or scalar neutrino \nand the leptonic branching fraction is enhanced.\nFinally for \\slepton{\\pm} and \\susy{\\nu}\\, lighter than the chargino, \nthe decay modes \n$\\chargino{\\pm} \\rightarrow \\slepton{\\pm} \\nu$ or\n$\\chargino{\\pm} \\rightarrow \\susy{\\nu} \\ell^\\pm$ become dominant. \nWhen the masses of the neutral SUSY Higgs bosons (\\ho, \\Ao) and of\nthe scalar leptons are very large, the heavier neutralinos \n($\\neutralino{j},\\ j \\ge 2$) decay via \\ensuremath{\\mathrm{Z^\\ast}}:\n$\\neutralino{j} \\rightarrow \\neutralino{k}\\ensuremath{\\mathrm{Z^\\ast}}\n                \\rightarrow \\neutralino{k}\\, f\\bar{f}$  with $k < j$.\nFor a chargino lighter than neutralinos, the latter decay via\n$\\ensuremath{\\mathrm{W^\\ast}}$ such as $\\neutralino{j} \\rightarrow \\chargino{\\pm} f\\bar{f}^\\prime$.\nIf the scalar lepton masses are comparable to the Z mass,\nthe neutralino decays also via a virtual scalar lepton, enhancing the leptonic \nbranching fraction.\nFinally, for \\susy{\\nu}\\, and \\slepton{\\pm} lighter than neutralinos the\ntwo-body decays \n$\\neutralino{j} \\rightarrow \\slepton{\\pm} \\ell^\\mp$ or\n$\\neutralino{j} \\rightarrow \\susy{\\nu} \\nu$ (j $\\ge 2$)\nbecome dominant.    \nThe radiative decays $\\neutralino{j} \\rightarrow \\neutralino{k} \\gamma $ \nare also possible via higher-order diagrams.\n\nPrevious results on chargino and neutralino searches have been \nreported by L3~\\cite{reflep1.5,susy_96,susy_97} \nand other LEP experiments~\\cite{charlep98}.\nIn this letter, new limits are presented on chargino \nand neutralino production cross sections.\nThese experimental results are interpreted in \nthe framework of the constrained MSSM. \nWithin these models lower limits on the masses of \nsupersymmetric particles are derived.\nFor these limits present experimental results are combined \nwith those obtained previously by L3 \nat the Z peak \\cite{oldsusyl3} and at energies up to\n$\\sqrt{s} = 183 \\gev{}$ \\cite{reflep1.5,susy_96,susy_97,nota99slep}.\n\n\\section{Data Sample and Simulation} \\label{dtsmcs}\n\nWe present the analysis of data collected by\nthe L3 detector \\cite{l3-detector} in 1998, \ncorresponding to an integrated luminosity of 176.3 pb$^{-1}$ at an\naverage centre-of-mass energy of 188.6 \\gev{}, denoted hereafter as \n$\\sqrt{s} = 189 \\gev{}$.\n\nStandard Model reactions are simulated with the following \nMonte Carlo generators:\n{\\tt PYTHIA}~\\cite{PYTHIA} for \n  $\\ee \\rightarrow \\mathrm{q\\bar{q}}$,\n  $\\ee \\rightarrow \\mathrm{Z}\\,\\ee$ and \n  $\\ee \\rightarrow \\gamma\\!\/\\mathrm{Z}\\,\\gamma\\!\/\\mathrm{Z} $;\n{\\tt EXCALIBUR}~\\cite{EXCALIBUR} for\n   $\\ee \\rightarrow \\mathrm{W^\\pm\\, e^\\mp \\nu}$;\n{\\tt KORALZ}~\\cite{KORALZ} for\n   $\\ee \\rightarrow \\mu^+\\mu^-$ and\n   $\\ee \\rightarrow \\tau^+\\tau^-$;\n{\\tt BHWIDE}~\\cite{BHWIDE} for \n   $\\ee \\rightarrow \\ee$;\n{\\tt KORALW}~\\cite{KORALW} for\n   $\\ee \\rightarrow \\mathrm{W^+ W^-}$;\ntwo-photon interaction processes have been simulated using \n{\\tt DIAG36}~\\cite{DIAG} ($\\ee \\rightarrow \\ee \\ell^+\\ell^-$) and\n{\\tt PHOJET}~\\cite{PHOJET} ($\\ee \\rightarrow \\ee\\, \\mathrm{hadrons}$), \nrequiring at least 3 \\gev{} for the invariant mass of the two-photon system.\nThe number of simulated events for each background process is \nequivalent to more than 100 times the statistics of the collected \ndata sample except for two-photon interactions for which it is more \nthan two times the data statistics.\n\nSignal events are generated with the Monte Carlo program \n{\\tt SUSYGEN}~\\cite{susygen2.2}, for masses of SUSY particles \n($M_{\\rm SUSY}$) ranging from 45 \\gev{} up to the kinematic limit and for\n$\\DELTAM$ values \n($\\ensuremath{\\Delta M} = M_{\\rm SUSY} - M_{\\neutralino{1}}$) between 3 \\gev{} and\n$M_{\\rm SUSY}-1 \\GeV{}$. \nThe explicit two-body decay branching ratios for charginos \n$\\chargino{\\pm} \\rightarrow \\susy{\\nu} \\ell^\\pm, \\slepton{\\pm} \\nu $ \nor  $\\neutralino{2,3,4} \\rightarrow \\susy{\\nu} \\nu , \\slepton{} \\ell$\nhave been estimated with {\\tt SUSYGEN}.\n\nThe detector response is simulated using the {\\tt GEANT} \npackage~\\cite{geant}. It takes into account effects of energy loss,\nmultiple scattering and showering in the detector materials and\nin the beam pipe. Hadronic interactions are simulated with the\n{\\tt GHEISHA} program~\\cite{gheisha}. Time dependent inefficiencies\nof the different subdetectors are also taken into account\nin the simulation procedure.\n\n\n\\section{Analysis Procedure}\n\n\\subsection{Signal topologies and optimisation procedure}\n\\label{sec:optimization}\n\n\nBesides the main characteristic of missing transverse momentum,\nsupersymmetric particle signals can be further specified according\nto the number of leptons or the multiplicity of hadronic jets in the\nfinal state. As mentioned in the introduction, chargino pair\nproduction gives final states similar to \\W\\W\\, production.\nFor neutralinos, we distinguish two classes of detectable processes: \n$\\ee \\rightarrow \\neutralino{1}\\neutralino{2}$ and \n$\\ee \\rightarrow \\neutralino{2}\\neutralino{2}$.\nFor these signals, final states are given by the Z branching ratios.\nBoth for charginos and neutralinos, the event energy\nis directly related to \\ensuremath{\\Delta M}\\, ($\\ensuremath{\\Delta M} = M_{\\rm SUSY} - M_{\\neutralino{1}}$).\n\nWe devise five types of selection criteria oriented to all decays \nof charginos, as follows:\nat least two acoplanar leptons (e,$\\mu$);  hadrons and at least one isolated lepton;\n at least two acoplanar taus;  hadrons and at least one isolated tau;\n purely hadronic final states with high multiplicity.\n$\\neutralino{2}\\neutralino{2}$ production gives rise to final states very similar\nto those of chargino pair production, even if with very different \nbranching ratios. Hence, chargino selections based on these five topologies are also effective\nto select $\\neutralino{2}\\neutralino{2}$ events.\n\nThe two-acoplanar-jets final state on the other hand deserves a\ndedicated selection since it accounts for 70\\% of the decays in\n$\\neutralino{1}\\neutralino{2}$ events, and 28\\% in\n$\\neutralino{2}\\neutralino{2}$ events.\n\nThe signal topologies and the associated background sources depend strongly \non \\ensuremath{\\Delta M}. Therefore all five selections are optimised separately for four\ndifferent \\ensuremath{\\Delta M}\\, ranges:  \nthe very low \\ensuremath{\\Delta M}\\, range at $3-5 \\gev{}$, \nthe low \\ensuremath{\\Delta M}\\, range at $10-30 \\gev{}$, \nthe medium \\ensuremath{\\Delta M}\\, range at $40-70 \\gev{}$ and the high range\n\\ensuremath{\\Delta M}\\, at $80-94 \\gev{}$.\nIn the low and very low \\ensuremath{\\Delta M}\\, ranges, the expected topologies for the signal are \ncharacterised by a low multiplicity and a low visible energy, and\n the background is dominated by two-photon interactions.\nFor medium and high \\ensuremath{\\Delta M}\\, ranges, the signal signatures are very similar to those of\nW-pair production; in particular for $\\ensuremath{\\Delta M} > 80 \\gev{}$ on-shell Ws are produced.\n\n\nThe cut values of each selection are {\\it a priori} optimised using\nMonte Carlo signal and background events. \nThe optimisation procedure varies all cuts simultaneously to maximise the\nsignal efficiency and the background rejection.\nIn fact, the average limit ($\\kappa^{-1}$) is minimised for an infinite  \nnumber of tries, \nassuming only background contributions. This is expressed mathematically\nby the following formula:\n\\begin{equation}  \n\\kappa=\\epsilon\/ \\Sigma_{n=0}^{\\infty} k(b)_{n} P(b,n)\n\\end{equation}\nwhere $k(b)_{n}$ is the 95\\% confidence level Bayesian upper limit, \n$P(b,n)$ is the Poisson distribution \nfor $n$ events with an expected background of $b$ events, and \n$\\epsilon$ is the signal efficiency.\n\n\n\n\\subsection{Event selection}\n\nLepton and photon identification, and isolation criteria in hadronic events\nare unchanged compared to our previous analysis \\cite{susy_96}.\nThe Durham algorithm~\\cite{durham} is used for the clustering of\nhadronic jets.\n\nEvents are first selected by requiring at least 3 \\gev{} of visible energy\nand 3 \\gev{} of transverse momentum. Beam-gas events are rejected by\nrequiring the visible energy in a cone of $30^\\circ$ \naround the beam\npipe ($E_{30^0}$) to be less than 90\\% of the total and the missing momentum\nvector to be at least $10^\\circ$ away from the beam pipe.\nTagged two-photon interactions are rejected by requiring the sum of\nthe energies measured in the lead-scintillator\nring calorimeter and in the luminosity monitors~\\cite{l3-detector}\nto be less than 10 \\GeV{}.\nThese two detectors cover the polar angle range \n$1.5^\\circ < \\theta <9^\\circ$ on both sides of the interaction \npoint.\n\n\n\\subsubsection{Leptonic final states} \\label{sec:selection_lep}\n\nFor the pure leptonic final states, dedicated selections have been\noptimised for the charginos, where the two leptons \nmay have a different flavour. Those selections are very  \nsimilar to the scalar lepton selections which are described in \nReference \\cite{slep99ref}. At the end, a combination of all\nthe leptonic selections, providing the optimal sensitivity, is done for\nthe chargino and the neutralino leptonic decays.\n\n\\subsubsection{Lepton plus hadrons final states} \\label{sec:selection_lephad}\n\nWe select events with at least one isolated electron, muon or tau for which \nthe energy, not associated to the lepton, in a cone of $30^\\circ$ \nhalf-opening angle around its direction is less than 2 \\GeV{}. \nThe following quantities are defined:\nthe energy depositions\n($E^{\\perp}_{25}$ and $E_{25}$)\nwithin $\\pm25^\\circ$ around the missing \nenergy direction in the R--$\\phi$ plane \n or in space, respectively. \nWe apply cuts on the number of tracks in the hadronic system\n($N_{tk} - N_{lep}$) and the number of calorimetric clusters ($N_{cl}$).\nFurthermore, cuts are applied on the missing energy direction isolation\n($\\theta_{miss}$ and $E^{\\perp}_{25}$),\nthe total transverse momentum ($p_{\\perp}$),\nthe energy of the isolated lepton ($E_{lep}$), \nthe recoil mass ($M_{rec}$),\nas well as on the acoplanarity angle between\nthe jet and the lepton. A cut is applied on the visible energy ($E_{vis}$)\nand $E_{TTJL}$\nwhich is defined as the absolute\nvalue of the projection of the total momentum of the jet and the \nlepton onto the\ndirection perpendicular to the lepton-jet thrust computed in the R-$\\phi$ \nplane. \nA cut on the invariant mass of the hadronic system ($M_{had}$) \nremoves most of the WW background.\n\nThe cut values at $\\sqrt{s} = 189 \\gev{}$, are shown in Table~\\ref{tab2} for the \ndifferent \\ensuremath{\\Delta M}\\ ranges.\n\n\n\\subsubsection{Purely hadronic final states} \\label{sec:selection_hadrons}\n\nThe list of cuts used at $\\sqrt{s} = 189 \\gev{}$ is reported in Table~\\ref{tab3} for the different \n\\ensuremath{\\Delta M}\\ ranges. \nAgain, we apply cuts on $N_{cl}$, $N_{tk}$, $p_{\\perp}$, $E_{vis}$,\nacollinearity and acoplanarity as well as on the missing energy \npolar angle ($\\theta_{miss}$) and isolation ($E^{\\perp}_{25}$, $E_{25}$).\nThe absolute value of the total momentum of the event \nalong the beam line normalised to the visible energy ($p_\\parallel$),\nthe recoil mass ($M_{rec}$) and the visible mass ($M_{vis}$) are also used\nin the selections.\n\nIn all the selections, but the very low \\ensuremath{\\Delta M},\na cut on the width of the two jets \nis applied. We define $y_{\\perp}$ as the ratio between the scalar sum of the\nparticle momenta transverse to the jet direction and the jet energy.  \nWe require $y_{\\perp}$ to be large in order to select four-jet-like events.\nIn the low \\ensuremath{\\Delta M}\\ range a cut on the ratio $E_{TTJ}$\/$p_\\perp$ \nis applied. \n$E_{TTJ}$ is equivalent to $E_{TTJL}$ using the momenta and the directions of\nthe two jets.    \n\n\n\\section{Results}\n\nThe results at $\\sqrt{s} = 189 \\gev{}$, for \nthe eighteen chargino selections and the four neutralino\nselections are shown in Table \\ref{tab8}. The results for the very low and low \\ensuremath{\\Delta M}\\, selections are shown together.\nA good agreement between the expected background from Standard Model\nprocesses and the selected data is observed.\n\nThe eighteen chargino selections\nfind 147 candidates in the data when expecting 148 events\nfrom the Standard Model processes. \nIn the low and\nvery low \\ensuremath{\\Delta M}\\ regions\n72 events are selected, 11 events in the medium \\ensuremath{\\Delta M}\\, region and\n67 events in the high \\ensuremath{\\Delta M}\\, region. \nIn the four neutralino selections\n50 candidates are found \nwhereas 48.1 events are expected\nfrom the Standard Model processes, \nmost of those events are selected by the \nlow $\\ensuremath{\\Delta M}$ selections.\n\nEach selection is parametrised as a function of a single parameter, $\\xi$,\nin the following manner: \ngiven a lower edge, $X_{loose}^i$, and an upper \nedge, $X_{tight}^i$,\nfor the cut on the variable $i$, the parameter $\\xi$ \nis equal to 0 when this cut is at the lower edge\n(many background events satisfy the selection)\nand 100 when it is at the upper edge\n(no or few background events pass the selection). All cuts  \n($i = 1, . . ., N$) are related to the parameter $\\xi$ as follows:\n$$X_{cut}^i=\nX_{loose}^i +\n(X_{tight}^i- X_{loose}^i)\n\\times \\frac{\\xi}{100}.\n$$\nThe parameter $\\xi$ is scanned around the optimal value ($\\xi$=50)\nto check the agreement between data and Monte Carlo at different background \nrejection stages.\nAs illustrated in Figure~\\ref{fig:xi_chaneu} \nfor the lepton and hadrons final state in chargino decays and \nthe pure hadronic final state in the neutralino decays, the data and\nMonte Carlo simulations are \nin good agreement for all the \\ensuremath{\\Delta M}\\ selections.\nThe vertical arrows show the $\\xi$ value corresponding to\nthe optimised cuts.\n\nFor intermediate \\ensuremath{\\Delta M}\\, values different from those chosen for\noptimisation we choose the combination of selections \nproviding the highest sensitivity \\cite{susy_96}.\nIn this combination procedure, we take into account the overlap\namong the selections within the \ndata and Monte Carlo samples.\n\n\nTypical selection efficiencies, as well as the number of background events\nexpected for a chargino mass of 94 \\gev{} for the purely leptonic final state\n(LL) or\nfor the $\\ensuremath{\\mathrm{W^\\ast}}\\neutralino{1}$ decay mode, are displayed in\nTable \\ref{tab4}. In \nthe latter case, a maximum efficiency of 47\\% is reached for a background\ncontamination of 7.5 events for $\\ensuremath{\\Delta M} = 30 \\gev{}$.\nIn the low \\ensuremath{\\Delta M}\\ region the efficiency decreases due to\nthe large contamination of two-photon interactions and due to the\nlower trigger acceptance.\nFor large \\ensuremath{\\Delta M}\\ it decreases because of the WW background.\n\nThe selection efficiencies, as well as the number of background events\nexpected for a sum of neutralino masses \n$M_{\\neutralino{1}} + M_{\\neutralino{2}} = 188 \\gev{}$ \nfor the pure leptonic decays and for the $\\ensuremath{\\mathrm{Z^\\ast}}\\neutralino{1}$ \ndecay mode are displayed in Table~\\ref{tab5b}.\nCompared to the chargino selection, the efficiencies are\nlower due to the invisible decays of the \\ensuremath{\\mathrm{Z^\\ast}}.\n\nSystematic errors on the signal efficiencies are evaluated as in\nReference \\citen{reflep1.5}, and they are typically 5\\% relative, \ndominated by Monte Carlo statistics. These errors\nare taken into account following the procedure explained \nin Reference \\citen{cal_limit}.\n\n\n\\section{Model independent upper limits on production cross sections}\nNo excess of events is observed\nand we set upper limits on the \nchargino and neutralino production cross sections \nin the framework of the MSSM.\nExclusion limits at 95\\% C.L. are derived taking into account background\ncontributions.\n\nTo derive the upper limits on production cross sections \nand for interpretations in the MSSM we combine the \n$\\sqrt{s} = 189 \\gev{}$ data sample\nwith those collected by L3 at lower centre-of-mass energies \n\\cite{reflep1.5,susy_96,susy_97}.\n\nThe contours of upper limits on the production cross sections for the process\n$\\ee \\rightarrow \\chargino{\\pm}\\chargino{\\mp}$ are shown in \nFigure~\\ref{fig:xsection_chargino} assuming \n$\\chargino{\\pm} \\rightarrow \\ensuremath{\\mathrm{W^\\ast}}\\neutralino{1}$ for the chargino decay \nwith standard W branching fractions, \nand for purely leptonic W decays.\nIn most of the kinematically accessible region, cross sections larger than \n0.2~pb are excluded for both scenarios.\n\nSimilarly, cross section limits for associated neutralino production \n$\\ee \\rightarrow \\neutralino{1}\\neutralino{2}$ are derived as shown in\nFigure~\\ref{fig:xsection_neutralino}\nassuming  $\\neutralino{2} \\rightarrow \\ensuremath{\\mathrm{Z^\\ast}}\\neutralino{1}$, \nwith standard Z branching fractions \nand for purely leptonic Z decays.\nIn most of the kinematically accessible region, cross sections larger than \n0.3~pb are excluded for both scenarios.\n\n\n\n\\section{Interpretation in the MSSM}\n\nIn the MSSM, with Grand Unification assumptions~\\cite{MSSM_GUT},\nthe masses and couplings of the SUSY\nparticles as well as their production cross sections,\nare entirely described~\\cite{mssm} once five parameters are fixed:\n$\\tan\\beta$, the ratio of the vacuum expectation values of the two Higgs \ndoublets, $M \\equiv M_2$, the gaugino mass parameter,\n$\\mu$, the higgsino mixing parameter,\n$m_0$, the common mass for scalar fermions at the GUT scale, and $A$, \nthe trilinear coupling in the Higgs sector.\nThe following MSSM parameter space is investigated: \n$$\n   \\begin{array}{rclcrcl}\n         0.7 \\leq& \\tan\\beta & \\leq 60 ,     &&\n0 \\leq &M_2 &\\leq 2000 \\gev{} ,\\\\\n -2000 \\gev{} \\leq& \\mu       & \\leq 2000 \\gev{} ,&& \n0 \\leq& m_0         &\\leq  500 \\gev{} .\n  \\end{array}\n$$\n\nTo derive the absolute limits on the masses \nof the lightest neutralino and of\nthe lightest chargino,\na scan in the MSSM parameter space is performed \nin steps of $0.2 \\gev{}$ for $M_2$,\n$1.0 \\gev{}$ for $\\mu$ and $0.5 \\gev{}$ for $m_0$. \n\nMass eigenstates of scalar quarks and leptons are in general a \nmixture of the weak eigenstates $\\susy{f}_R$ and $\\susy{f}_L$.\nThe mixing between these two states\nis proportional to the mass of the partner fermion.\nHence the mixing can be sizable only for particles of the\nthird generation. The mixing is governed by the parameters $A$, \n$\\mu$ and $\\tan\\beta$. Besides $\\mu$ and $\\tan\\beta$, also\na scan on $A$ is performed to check the validity of the following results.\n\nAll the limits on the cross sections previously shown, combined with\nthe results obtained at lower centre-of-mass energies and with the\nresults of scalar lepton searches obtained at $\\sqrt{s} = 189 \\gev{}$ \n\\cite{slep99ref}, can be \ntranslated into exclusion regions in the MSSM parameter space.\nTo derive limits in the MSSM, \nwe optimise the global selection for any different\npoint in the parameter space. This is obtained, choosing every time the\ncombination of selections providing the highest sensitivity, given \nthe production cross sections and the decay branching fractions which are \ncalculated with the generator {\\tt SUSYGEN}. When the mixing in the\nscalar tau sector is considered, masses and decay branching fractions\nare calculated with the generator {\\tt ISAJET} \\cite{isajet}.\n\n\n\\subsection{Limits on chargino and neutralino masses} \n\nIn the MSSM, while the cross sections and decay branching fractions\nof the\ncharginos and neutralinos depend on the masses of the scalar leptons, \ntheir masses depend only on \n$M_2$, $\\mu$ and $\\tan\\beta$. \nThe exclusions in the\nhigh $m_0$ range are derived from chargino and\nneutralino searches, while for low $m_0$ the searches for scalar\nleptons \\cite{slep99ref}, and for photons and\nmissing energy final states \\cite{papho97}, also contribute.\nWe also take into account all chargino and neutralino cascade decays:\n\\begin{itemize}\n\\item\n $\\chargino{\\pm} \\rightarrow \\neutralino{2}\\, \\ensuremath{\\mathrm{W^\\ast}}$: \n we observe a slight decrease of the efficiency relative to\n $\\chargino{\\pm} \\rightarrow \\neutralino{1}\\, \\ensuremath{\\mathrm{W^\\ast}}$\n depending on the masses of \\neutralino{2}, \\chargino{\\pm} and \n \\neutralino{1}.\n The lowest efficiency is then used for cascade decays.\n\\item\n $\\neutralino{3,4} \\rightarrow \\neutralino{2}\\, \\ensuremath{\\mathrm{Z^\\ast}}$:\n the efficiency is found to be larger than the efficiency obtained for the\n $\\neutralino{3,4} \\rightarrow \\neutralino{1}\\, \\ensuremath{\\mathrm{Z^\\ast}}$ channel, \n especially in the high \\ensuremath{\\Delta M}\\ region. \n The efficiencies obtained in the latter\n channel are used. \n\\item \n\n$\\neutralino{3,4} \\rightarrow \\susy{\\nu} \\nu$: when the $ \\susy{\\nu}$\nbecomes detectable through its cascade decays into $\\neutralino{2}$ or \n  $\\chargino{\\pm}$. This is especially relevant in the mixed region\n($\\mu\\sim -M_2$) for\nthe low $\\tan\\beta$ values.\n\\end{itemize}\n\n\nDepending on the neutralino-chargino field content, \none distinguishes the following \ncases for the determination of lower limits on the neutralino and chargino \nmasses:\n\\begin{itemize}\n\\item   \n Higgsino-like  \\neutralino{2} and \\chargino{\\pm}  ($M_2 \\gg |\\mu|$):\n in this case, the production cross sections do not depend on the scalar lepton\n masses, \\ensuremath{\\Delta M}\\ is low and decreases with increasing $M_2$.\nConsequently, the limits on the masses of the next-to-lightest neutralino\n and the lightest chargino decrease with $M_2$ as depicted in \n Figure~\\ref{fig:mass_m2}.\n For $\\tan\\beta = \\sqrt{2}$ and $M_2$ less than $500 \\gev{}$, \n$M_{\\neutralino{2}} \\leq 101 \\GeV{}$ and\n $M_{\\chargino{\\pm}} \\leq 93 \\GeV{}$\n are excluded.\n\\item\n Gaugino-like $\\chargino{\\pm}$  ($|\\mu| \\gg M_2$):\n the chargino cross section depends strongly on the scalar neutrino mass. \n For $50 \\gev{} \\leq M_{\\susy{\\nu}} \\leq 80 \\gev{}$ \nthe cross section is reduced by one order of magnitude compared to what is\n expected for $M_{\\susy{\\nu}} \\geq 500 \\gev{}$.\n  When the two body decay \n $\\chargino{\\pm} \\rightarrow \\ell^\\pm \\, \\susy{\\nu}$ \n is dominant, the relevant \\ensuremath{\\Delta M}\\ becomes \n $\\ensuremath{\\Delta M} = M_{\\chargino{\\pm}} - M_{\\susy{\\nu}}$.\nIf the $\\susy{\\nu}$ is mass degenerate with the \\chargino{\\pm} the \nacceptance is substantially reduced. However, when this occurs scalar \nleptons are light and the experimental sensitivity is recovered with these\nchannels.\n\n\\end{itemize}\n\nThe mass limit of the lightest\nchargino is shown in Figure \\ref{fig:mass_cha} as a function of \n$\\tan\\beta$ for all the different chargino field contents. \nAt large $\\tan\\beta$ values,\nthe lower mass limit of the lightest chargino is obtained\nwhen the lightest chargino and the $\\susy{\\nu}$ are mass degenerate\n(gaugino region). At \nlow $\\tan\\beta$ values the lower mass limit on the lightest \nchargino is obtained when the lightest chargino and \nthe $\\neutralino{1}$ (LSP) are mass\ndegenerate (higgsino region). Finally, for $M_2<2$ \\TeV, \nfor $\\tan\\beta \\leq 60$ and for any $m_0$ values, \nthe lower mass limit of the lightest chargino is:\n$$   M_{\\chargino{\\pm}} \\geq 67.7 \\gev{} .$$\n\nThe scalar tau can be much lighter than\nthe scalar electron and muon. This mass splitting \noccurs in particular for large $\\tan\\beta$ and $A$ values.\nWhen this happens, chargino and next-to-lightest neutralino decays \nare affected. Therefore, detection efficiencies are estimated for chargino and\nnext-to-lightest neutralino decays with 100\\% branching ratio\ninto $\\stau{}_{1}\\nu$ and $\\stau{\\pm}_{1}\\tau^{\\mp}$, respectively.\nIn particular, when the \n$\\stau{}_{1}$ and the LSP are mass degenerate the efficiencies \ndecrease substantially.\nHowever, the experimental sensitivity can be partially recovered\ntaking into account also the process\n$\\epem\\ra \\susy{\\nu}\\susy{\\nu}$, where \nthe $\\susy{\\nu}$ is visible through its cascade decays.\nIn particular, the limit on the chargino mass \nholds for any value of the mixing if $\\tan\\beta < 20$.\nFor higher $\\tan\\beta$ values this limit can be decreased\nat most by 10 \\gev. \n\nIndirect limits on the mass of the lightest neutralino\nare also derived as a function of $m_0$ and as a function of tan$\\beta$.\nIn the\nlow  $m_0$ region ($\\leq 65 \\gev{}$) the mass limit on the LSP\ncomes mainly from the scalar lepton searches.\nFor large $m_0$ values ($\\geq 200 \\gev{}$), only  \nthe chargino and neutralino searches contribute. At \nlow $\\tan\\beta$ the processes\n$\\epem\\ \\rightarrow \\neutralino{2} \\neutralino{3,4}$\ncontribute significantly and they are taken into account.\nThe lower mass limit is found at $\\tan\\beta = 1$, $\\mu = -70 \\gev{}$\nand $m_0 = 500 \\gev{}$, as shown in Figure~\\ref{fig:neutralino_mass_a}.\nFor these values of the parameters, \nthe chargino mass is at the kinematic limit and \nthe mass difference between the chargino \nand the LSP is maximal.\n\nFor intermediate $m_0$ values ($65 \\gev{} \\leq m_0 \\leq 95 \\gev{}$)\nthe production cross section for charginos is minimal\nand the $\\susy{\\nu}$ is light enough to allow\nthe following decay modes:\n$\\neutralino{2,3,4} \\rightarrow \\susy{\\nu}\\nu$ and\n$\\chargino{\\pm} \\rightarrow \\susy{\\nu} \\ell^{\\pm}$. \nThis is the region where the exclusion is due to the interplay of many\ndifferent searches. The limit on the lightest neutralino as a function\nof $m_0$, and for two extreme values of $\\tan\\beta$, \nis shown in Figure~\\ref{fig:neutralino_mass_b}. \nFor low $\\tan\\beta$ values\n($\\le \\sqrt{2}$), the minimum is found for\n$\\mu \\sim -70 \\gev{}$ and large $m_0$ values. \nBetter limits are obtained  \nfor intermediate $m_0$ values, where \nthe neutralino production cross sections are large and the \ntwo body decays of the $\\neutralino{3,4}$ into $\\susy{\\nu} \\nu $   \nare visible through the cascade decays of the\n$\\susy{\\nu}$. \nFor larger $\\tan\\beta$ values, the minimum is found in the gaugino \nregion ($-2000 \\gev{}  < \\mu < -200 \\gev{}$) and for \n$70 \\gev{} \\leq m_0 \\leq 80 \\gev{}$. In this region of the parameter space,   \nthe $\\susy{\\nu}$ and the chargino are \nmass degenerate, the heavier neutralinos  \ndecay invisibly and the experimental sensitivity \nis entirely due to the scalar lepton searches.\n\nFinally in Figure~\\ref{fig:neutralino_any_a}, the mass limit on the\nlightest neutralino as a function of $\\tan\\beta$ for any $m_0$ value\nis shown. For $\\tan\\beta \\geq 0.7$, the \nlower mass limit of the lightest neutralino is \n$$\n   M_{\\neutralino{1}} \\geq 32.5 \\gev{} .\n$$\n\nThe mass limit on the\nlightest neutralino is very little affected\nby the mixing in the scalar tau sector.\nThe limit holds for any value of the mixing if $\\tan\\beta < 20$\nand it can be reduced at most by 1.5 \\gev{} for higher $\\tan\\beta$ values.\nNevertheless, the absolute mass limit for the lightest neutralino\ndoes not change since the lowest value is still found at $\\tan\\beta =1$.\n\n\nThe mass limit on the lightest neutralino is also translated in\nan absolute limit on $M_2$. This is shown as a function \nof $\\tan\\beta$ for any $m_0$ and $\\mu$ as depicted \nin Figure ~\\ref{fig:neutralino_any_b}. \nValues of $M_2$ lower than $54.8 \\gev{}$ are now excluded at 95$\\%$ C.L.\n\n\n\\section*{Acknowledgments}\n\nWe  express our gratitude to the CERN accelerator divisions for the\nexcellent performance of the LEP machine. We also acknowledge\nand appreciate the effort of the engineers, technicians and support staff \nwho have participated in the construction and maintenance of this experiment.\n\n\\newpage\n\n\n\\bibliographystyle{l3stylem}\n\\begin{mcbibliography}{10}\n\n\\bibitem{susy}\nY.A. Golfand and E.P. Likhtman, \\JETP {\\bf 13} (1971) 323; \\\\ D.V. Volkhov and\n  V.P. Akulov, \\PL {\\bf B 46} (1973) 109; \\\\ J. Wess and B. Zumino, \\NP {\\bf B\n  70} (1974) 39;\\\\ P. Fayet and S. Ferrara, \\PRep {\\bf C 32} (1977) 249;\\\\ A.\n  Salam and J. Strathdee, \\FortP {\\bf 26} (1978) 57\\relax\n\\relax\n\\bibitem{mssm}\nH. P. Nilles, \\PRep {\\bf 110} (1984) 1;\\\\ H. E. Haber and G. L. Kane, \\PRep\n  {\\bf 117} (1985) 75;\\\\ R. Barbieri, \\NCim {\\bf 11} No. 4 (1988) 1\\relax\n\\relax\n\\bibitem{reflep1.5}\nL3 Collab., M. Acciarri \\etal, \\PL {\\bf B 377} (1996) 289\\relax\n\\relax\n\\bibitem{susy_96}\nL3 Collab., M. Acciarri \\etal, Eur. Phys. Journal {\\bf C 4} (1998) 207\\relax\n\\relax\n\\bibitem{susy_97}\nL3 Collab., M. Acciarri \\etal, contributed paper n. 493 to {\\it ICHEP98},\n  Vancouver, July 1998\\relax\n\\relax\n\\bibitem{charlep98}\nALEPH Collab., R. Barate \\etal, CERN-EP-99-014 (1999);\\\\ DELPHI Collab., P.\n  Abreu \\etal, \\PL {\\bf B 446} (1999) 75;\\\\ OPAL Collab., G. Abbiendi \\etal,\n  Eur. Phys. Journal {\\bf C 8} (1999) 255\\relax\n\\relax\n\\bibitem{oldsusyl3}\nL3 Collab., O. Adriani \\etal, \\PRep {\\bf 236} (1993) 1; \\\\ L3 Collab., M.\n  Acciarri \\etal, \\PL {\\bf B 350} (1995) 109\\relax\n\\relax\n\\bibitem{nota99slep}\nL3 Collab., M. Acciarri \\etal, \\PL {\\bf B 456} (1999) 283\\relax\n\\relax\n\\bibitem{l3-detector}\nL3 Collab., B. Adeva \\etal, Nucl. Instr. and Meth. {\\bf A 289} (1990) 35; \\\\ M.\n  Chemarin \\etal, Nucl. Instr. and Meth. {\\bf A 349} (1994) 345; \\\\ M. Acciarri\n  \\etal, Nucl. Instr. and Meth. {\\bf A 351} (1994) 300; \\\\ G. Basti \\etal,\n  Nucl. Instr. and Meth. {\\bf A 374} (1996) 293; \\\\ I.C. Brock \\etal, Nucl.\n  Instr. and Meth. {\\bf A 381} (1996) 236; \\\\ A. Adam \\etal, Nucl. Instr. and\n  Meth. {\\bf A 383} (1996) 342\\relax\n\\relax\n\\bibitem{PYTHIA}\nT. Sj{\\\"o}strand, ``PYTHIA~5.7 and JETSET~7.4 Physics and Manual'', \\\\\n  CERN--TH\/7112\/93 (1993), revised August 1995;\\\\ T. Sj{\\\"o}strand, \\CPC {\\bf\n  82} (1994) 74\\relax\n\\relax\n\\bibitem{EXCALIBUR}\n{\\tt EXCALIBUR} version 1.11 is used.\\\\ F.A.~Berends, R.~Kleiss and R.~Pittau,\n  Nucl. Phys. {\\bf B 424} (1994) 308; Nucl. Phys. {\\bf B 426} (1994) 344; Nucl.\n  Phys. (Proc. Suppl.) {\\bf B 37} (1994) 163; Phys. Lett. {\\bf B 335} (1994)\n  490; Comp. Phys. Comm. {\\bf 83} (1994) 141\\relax\n\\relax\n\\bibitem{KORALZ}\n{\\tt KORALZ} version 4.02 is used.\\\\ S. Jadach, B.F.L. Ward and Z. W\\c{a}s,\n  \\CPC {\\bf 79} (1994) 503\\relax\n\\relax\n\\bibitem{BHWIDE}\n{\\tt BHWIDE} version 1.01 is used.\\\\ S. Jadach \\etal, \\PL {\\bf B 390} (1997)\n  298\\relax\n\\relax\n\\bibitem{KORALW}\n{\\tt KORALW} version 1.33 is used.\\\\ M.~Skrzypek \\etal, \\CPC {\\bf 94} (1996)\n  216;\\\\ M.~Skrzypek \\etal, \\PL {\\bf B 372} (1996) 289\\relax\n\\relax\n\\bibitem{DIAG}\nF.A.~Berends, P.H.~Daverfeldt and R. Kleiss,\n\\newblock  Nucl. Phys. {\\bf B 253}  (1985) 441\\relax\n\\relax\n\\bibitem{PHOJET}\n{\\tt PHOJET} version 1.10 is used. \\\\ R.~Engel, \\ZfP {\\bf C 66} (1995) 203; \\\\\n  R.~Engel and J.~Ranft, \\PR {\\bf{D 54}} (1996) 4244\\relax\n\\relax\n\\bibitem{susygen2.2}\n{\\tt SUSYGEN} version 2.2 is used.\\\\ S. Katsanevas and P. Morawitz, \\CPC {\\bf\n  112} (1998) 227\\relax\n\\relax\n\\bibitem{geant}\nThe L3 detector simulation is based on GEANT Version 3.15.\\\\ See R. Brun \\etal,\n  ``GEANT 3'', CERN DD\/EE\/84-1 (Revised), September 1987\\relax\n\\relax\n\\bibitem{gheisha}\nH. Fesefeldt, RWTH Aachen Preprint PITHA 85\/02 (1985)\\relax\n\\relax\n\\bibitem{durham}\nS. Catani \\etal, \\PL {\\bf B 269} (1991) 432;\\\\ S. Bethke \\etal, \\NP {\\bf B 370}\n  (1992) 310\\relax\n\\relax\n\\bibitem{slep99ref}\nL3 Collab., M. Acciarri \\etal, {\\it Search for Scalar leptons in $\\ee$\n  collisions at $\\sqrt{s}$=189 \\gev}, contributed paper n. 7-46 to {\\it\n  EPS-HEP99}, Tampere, July 1999, and also submitted to Phys. Lett\\relax\n\\relax\n\\bibitem{cal_limit}\nR.D. Cousins and V.L. Highland, \\NIM {\\bf A 320} (1992) 331\\relax\n\\relax\n\\bibitem{MSSM_GUT}\nSee for instance:\\\\ L. Ibanez, Phys. Lett. {\\bf B 118} (1982) 73;\\\\ R.\n  Barbieri, S. Farrara and C. Savoy, Phys. Lett. {\\bf B 119} (1982) 343\\relax\n\\relax\n\\bibitem{isajet}\n{\\tt ISAJET} version 7.44 is used. \\\\ H.~Baer \\etal, BNL-HET-98-39, (1998),\n  hep-ph\/9810440\\relax\n\\relax\n\\bibitem{papho97}\nL3 Collab., M. Acciarri \\etal, \\PL {\\bf B 444} (1998) 503\\relax\n\\relax\n\\end{mcbibliography}\n\n \n\\newpage\n\\typeout{   }     \n\\typeout{Using author list for paper 186 -?}\n\\typeout{$Modified: Fri Sep 10 08:43:14 1999 by clare $}\n\\typeout{!!!!  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\nD.Luckey\\rlap,\\tute{\\mit}\\\nL.Lugnier\\rlap,\\tute\\lyon\\ \nL.Luminari\\rlap,\\tute\\rome\\\nW.Lustermann\\rlap,\\tute\\eth\\\nW.G.Ma\\rlap,\\tute\\hefei\\ \nM.Maity\\rlap,\\tute\\tata\\\nL.Malgeri\\rlap,\\tute\\cern\\\nA.Malinin\\rlap,\\tute{\\moscow,\\cern}\\ \nC.Ma\\~na\\rlap,\\tute\\madrid\\\nD.Mangeol\\rlap,\\tute\\nymegen\\\nP.Marchesini\\rlap,\\tute\\eth\\ \nG.Marian\\rlap,\\tute\\debrecen\\ \nJ.P.Martin\\rlap,\\tute\\lyon\\ \nF.Marzano\\rlap,\\tute\\rome\\ \nG.G.G.Massaro\\rlap,\\tute\\nikhef\\ \nK.Mazumdar\\rlap,\\tute\\tata\\\nR.R.McNeil\\rlap,\\tute{\\lsu}\\ \nS.Mele\\rlap,\\tute\\cern\\\nL.Merola\\rlap,\\tute\\naples\\ \nM.Meschini\\rlap,\\tute\\florence\\ \nW.J.Metzger\\rlap,\\tute\\nymegen\\\nM.von~der~Mey\\rlap,\\tute\\aachen\\\nA.Mihul\\rlap,\\tute\\bucharest\\\nH.Milcent\\rlap,\\tute\\cern\\\nG.Mirabelli\\rlap,\\tute\\rome\\ \nJ.Mnich\\rlap,\\tute\\cern\\\nG.B.Mohanty\\rlap,\\tute\\tata\\ \nP.Molnar\\rlap,\\tute\\berlin\\\nB.Monteleoni\\rlap,\\tute{\\florence,\\dag}\\ \nT.Moulik\\rlap,\\tute\\tata\\\nG.S.Muanza\\rlap,\\tute\\lyon\\\nF.Muheim\\rlap,\\tute\\geneva\\\nA.J.M.Muijs\\rlap,\\tute\\nikhef\\\nM.Musy\\rlap,\\tute\\rome\\ \nM.Napolitano\\rlap,\\tute\\naples\\\nF.Nessi-Tedaldi\\rlap,\\tute\\eth\\\nH.Newman\\rlap,\\tute\\caltech\\ \nT.Niessen\\rlap,\\tute\\aachen\\\nA.Nisati\\rlap,\\tute\\rome\\\nH.Nowak\\rlap,\\tute\\zeuthen\\                    \nY.D.Oh\\rlap,\\tute\\korea\\\nG.Organtini\\rlap,\\tute\\rome\\\nR.Ostonen\\rlap,\\tute\\seft\\\nC.Palomares\\rlap,\\tute\\madrid\\\nD.Pandoulas\\rlap,\\tute\\aachen\\ \nS.Paoletti\\rlap,\\tute{\\rome,\\cern}\\\nP.Paolucci\\rlap,\\tute\\naples\\\nR.Paramatti\\rlap,\\tute\\rome\\ \nH.K.Park\\rlap,\\tute\\cmu\\\nI.H.Park\\rlap,\\tute\\korea\\\nG.Pascale\\rlap,\\tute\\rome\\\nG.Passaleva\\rlap,\\tute{\\cern}\\\nS.Patricelli\\rlap,\\tute\\naples\\ \nT.Paul\\rlap,\\tute\\ne\\\nM.Pauluzzi\\rlap,\\tute\\perugia\\\nC.Paus\\rlap,\\tute\\cern\\\nF.Pauss\\rlap,\\tute\\eth\\\nD.Peach\\rlap,\\tute\\cern\\\nM.Pedace\\rlap,\\tute\\rome\\\nS.Pensotti\\rlap,\\tute\\milan\\\nD.Perret-Gallix\\rlap,\\tute\\lapp\\ \nB.Petersen\\rlap,\\tute\\nymegen\\\nD.Piccolo\\rlap,\\tute\\naples\\ \nF.Pierella\\rlap,\\tute\\bologna\\ \nM.Pieri\\rlap,\\tute{\\florence}\\\nP.A.Pirou\\'e\\rlap,\\tute\\prince\\ \nE.Pistolesi\\rlap,\\tute\\milan\\\nV.Plyaskin\\rlap,\\tute\\moscow\\ \nM.Pohl\\rlap,\\tute\\eth\\ \nV.Pojidaev\\rlap,\\tute{\\moscow,\\florence}\\\nH.Postema\\rlap,\\tute\\mit\\\nJ.Pothier\\rlap,\\tute\\cern\\\nN.Produit\\rlap,\\tute\\geneva\\\nD.O.Prokofiev\\rlap,\\tute\\purdue\\ \nD.Prokofiev\\rlap,\\tute\\peters\\ \nJ.Quartieri\\rlap,\\tute\\salerno\\\nG.Rahal-Callot\\rlap,\\tute{\\eth,\\cern}\\\nM.A.Rahaman\\rlap,\\tute\\tata\\ \nP.Raics\\rlap,\\tute\\debrecen\\ \nN.Raja\\rlap,\\tute\\tata\\\nR.Ramelli\\rlap,\\tute\\eth\\ \nP.G.Rancoita\\rlap,\\tute\\milan\\\nG.Raven\\rlap,\\tute\\ucsd\\\nP.Razis\\rlap,\\tute\\cyprus\nD.Ren\\rlap,\\tute\\eth\\ \nM.Rescigno\\rlap,\\tute\\rome\\\nS.Reucroft\\rlap,\\tute\\ne\\\nT.van~Rhee\\rlap,\\tute\\utrecht\\\nS.Riemann\\rlap,\\tute\\zeuthen\\\nK.Riles\\rlap,\\tute\\mich\\\nA.Robohm\\rlap,\\tute\\eth\\\nJ.Rodin\\rlap,\\tute\\alabama\\\nB.P.Roe\\rlap,\\tute\\mich\\\nL.Romero\\rlap,\\tute\\madrid\\ \nA.Rosca\\rlap,\\tute\\berlin\\ \nS.Rosier-Lees\\rlap,\\tute\\lapp\\ \nJ.A.Rubio\\rlap,\\tute{\\cern}\\ \nD.Ruschmeier\\rlap,\\tute\\berlin\\\nH.Rykaczewski\\rlap,\\tute\\eth\\ \nS.Saremi\\rlap,\\tute\\lsu\\ \nS.Sarkar\\rlap,\\tute\\rome\\\nJ.Salicio\\rlap,\\tute{\\cern}\\ \nE.Sanchez\\rlap,\\tute\\cern\\\nM.P.Sanders\\rlap,\\tute\\nymegen\\\nM.E.Sarakinos\\rlap,\\tute\\seft\\\nC.Sch{\\\"a}fer\\rlap,\\tute\\aachen\\\nV.Schegelsky\\rlap,\\tute\\peters\\\nS.Schmidt-Kaerst\\rlap,\\tute\\aachen\\\nD.Schmitz\\rlap,\\tute\\aachen\\ \nH.Schopper\\rlap,\\tute\\hamburg\\\nD.J.Schotanus\\rlap,\\tute\\nymegen\\\nG.Schwering\\rlap,\\tute\\aachen\\ \nC.Sciacca\\rlap,\\tute\\naples\\\nD.Sciarrino\\rlap,\\tute\\geneva\\ \nA.Seganti\\rlap,\\tute\\bologna\\ \nL.Servoli\\rlap,\\tute\\perugia\\\nS.Shevchenko\\rlap,\\tute{\\caltech}\\\nN.Shivarov\\rlap,\\tute\\sofia\\\nV.Shoutko\\rlap,\\tute\\moscow\\ \nE.Shumilov\\rlap,\\tute\\moscow\\ \nA.Shvorob\\rlap,\\tute\\caltech\\\nT.Siedenburg\\rlap,\\tute\\aachen\\\nD.Son\\rlap,\\tute\\korea\\\nB.Smith\\rlap,\\tute\\cmu\\\nP.Spillantini\\rlap,\\tute\\florence\\ \nM.Steuer\\rlap,\\tute{\\mit}\\\nD.P.Stickland\\rlap,\\tute\\prince\\ \nA.Stone\\rlap,\\tute\\lsu\\ \nH.Stone\\rlap,\\tute{\\prince,\\dag}\\ \nB.Stoyanov\\rlap,\\tute\\sofia\\\nA.Straessner\\rlap,\\tute\\aachen\\\nK.Sudhakar\\rlap,\\tute{\\tata}\\\nG.Sultanov\\rlap,\\tute\\wl\\\nL.Z.Sun\\rlap,\\tute{\\hefei}\\\nH.Suter\\rlap,\\tute\\eth\\ \nJ.D.Swain\\rlap,\\tute\\wl\\\nZ.Szillasi\\rlap,\\tute{\\alabama,\\P}\\\nT.Sztaricskai\\rlap,\\tute{\\alabama,\\P}\\ \nX.W.Tang\\rlap,\\tute\\beijing\\\nL.Tauscher\\rlap,\\tute\\basel\\\nL.Taylor\\rlap,\\tute\\ne\\\nC.Timmermans\\rlap,\\tute\\nymegen\\\nSamuel~C.C.Ting\\rlap,\\tute\\mit\\ \nS.M.Ting\\rlap,\\tute\\mit\\ \nS.C.Tonwar\\rlap,\\tute\\tata\\ \nJ.T\\'oth\\rlap,\\tute{\\budapest}\\ \nC.Tully\\rlap,\\tute\\prince\\\nK.L.Tung\\rlap,\\tute\\beijing\nY.Uchida\\rlap,\\tute\\mit\\\nJ.Ulbricht\\rlap,\\tute\\eth\\ \nE.Valente\\rlap,\\tute\\rome\\ \nG.Vesztergombi\\rlap,\\tute\\budapest\\\nI.Vetlitsky\\rlap,\\tute\\moscow\\ \nD.Vicinanza\\rlap,\\tute\\salerno\\ \nG.Viertel\\rlap,\\tute\\eth\\ \nS.Villa\\rlap,\\tute\\ne\\\nM.Vivargent\\rlap,\\tute{\\lapp}\\ \nS.Vlachos\\rlap,\\tute\\basel\\\nI.Vodopianov\\rlap,\\tute\\peters\\ \nH.Vogel\\rlap,\\tute\\cmu\\\nH.Vogt\\rlap,\\tute\\zeuthen\\ \nI.Vorobiev\\rlap,\\tute{\\moscow}\\ \nA.A.Vorobyov\\rlap,\\tute\\peters\\ \nA.Vorvolakos\\rlap,\\tute\\cyprus\\\nM.Wadhwa\\rlap,\\tute\\basel\\\nW.Wallraff\\rlap,\\tute\\aachen\\ \nM.Wang\\rlap,\\tute\\mit\\\nX.L.Wang\\rlap,\\tute\\hefei\\ \nZ.M.Wang\\rlap,\\tute{\\hefei}\\\nA.Weber\\rlap,\\tute\\aachen\\\nM.Weber\\rlap,\\tute\\aachen\\\nP.Wienemann\\rlap,\\tute\\aachen\\\nH.Wilkens\\rlap,\\tute\\nymegen\\\nS.X.Wu\\rlap,\\tute\\mit\\\nS.Wynhoff\\rlap,\\tute\\aachen\\ \nL.Xia\\rlap,\\tute\\caltech\\ \nZ.Z.Xu\\rlap,\\tute\\hefei\\ \nB.Z.Yang\\rlap,\\tute\\hefei\\ \nC.G.Yang\\rlap,\\tute\\beijing\\ \nH.J.Yang\\rlap,\\tute\\beijing\\\nM.Yang\\rlap,\\tute\\beijing\\\nJ.B.Ye\\rlap,\\tute{\\hefei}\\\nS.C.Yeh\\rlap,\\tute\\tsinghua\\ \nAn.Zalite\\rlap,\\tute\\peters\\\nYu.Zalite\\rlap,\\tute\\peters\\\nZ.P.Zhang\\rlap,\\tute{\\hefei}\\ \nG.Y.Zhu\\rlap,\\tute\\beijing\\\nR.Y.Zhu\\rlap,\\tute\\caltech\\\nA.Zichichi\\rlap,\\tute{\\bologna,\\cern,\\wl}\\\nF.Ziegler\\rlap,\\tute\\zeuthen\\\nG.Zilizi\\rlap,\\tute{\\alabama,\\P}\\\nM.Z{\\\"o}ller\\rlap.\\tute\\aachen\n\\newpage\n\\begin{list}{A}{\\itemsep=0pt plus 0pt minus 0pt\\parsep=0pt plus 0pt minus 0pt\n                \\topsep=0pt plus 0pt minus 0pt}\n\\item[\\aachen]\n I. Physikalisches Institut, RWTH, D-52056 Aachen, FRG$^{\\S}$\\\\\n III. Physikalisches Institut, RWTH, D-52056 Aachen, FRG$^{\\S}$\n\\item[\\nikhef] National Institute for High Energy Physics, NIKHEF, \n     and University of Amsterdam, NL-1009 DB Amsterdam, The Netherlands\n\\item[\\mich] University of Michigan, Ann Arbor, MI 48109, USA\n\\item[\\lapp] Laboratoire d'Annecy-le-Vieux de Physique des Particules, \n     LAPP,IN2P3-CNRS, BP 110, F-74941 Annecy-le-Vieux CEDEX, France\n\\item[\\basel] Institute of Physics, University of Basel, CH-4056 Basel,\n     Switzerland\n\\item[\\lsu] Louisiana State University, Baton Rouge, LA 70803, USA\n\\item[\\beijing] Institute of High Energy Physics, IHEP, \n  100039 Beijing, China$^{\\triangle}$ \n\\item[\\berlin] Humboldt University, D-10099 Berlin, FRG$^{\\S}$\n\\item[\\bologna] University of Bologna and INFN-Sezione di Bologna, \n     I-40126 Bologna, Italy\n\\item[\\tata] Tata Institute of Fundamental Research, Bombay 400 005, India\n\\item[\\ne] Northeastern University, Boston, MA 02115, USA\n\\item[\\bucharest] Institute of Atomic Physics and University of Bucharest,\n     R-76900 Bucharest, Romania\n\\item[\\budapest] Central Research Institute for Physics of the \n     Hungarian Academy of Sciences, H-1525 Budapest 114, Hungary$^{\\ddag}$\n\\item[\\mit] Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n\\item[\\debrecen] Lajos Kossuth University-ATOMKI, H-4010 Debrecen, Hungary$^\\P$\n\\item[\\florence] INFN Sezione di Firenze and University of Florence, \n     I-50125 Florence, Italy\n\\item[\\cern] European Laboratory for Particle Physics, CERN, \n     CH-1211 Geneva 23, Switzerland\n\\item[\\wl] World Laboratory, FBLJA  Project, CH-1211 Geneva 23, Switzerland\n\\item[\\geneva] University of Geneva, CH-1211 Geneva 4, Switzerland\n\\item[\\hefei] Chinese University of Science and Technology, USTC,\n      Hefei, Anhui 230 029, China$^{\\triangle}$\n\\item[\\seft] SEFT, Research Institute for High Energy Physics, P.O. Box 9,\n      SF-00014 Helsinki, Finland\n\\item[\\lausanne] University of Lausanne, CH-1015 Lausanne, Switzerland\n\\item[\\lecce] INFN-Sezione di Lecce and Universit\\'a Degli Studi di Lecce,\n     I-73100 Lecce, Italy\n\\item[\\lyon] Institut de Physique Nucl\\'eaire de Lyon, \n     IN2P3-CNRS,Universit\\'e Claude Bernard, \n     F-69622 Villeurbanne, France\n\\item[\\madrid] Centro de Investigaciones Energ{\\'e}ticas, \n     Medioambientales y Tecnolog{\\'\\i}cas, CIEMAT, E-28040 Madrid,\n     Spain${\\flat}$ \n\\item[\\milan] INFN-Sezione di Milano, I-20133 Milan, Italy\n\\item[\\moscow] Institute of Theoretical and Experimental Physics, ITEP, \n     Moscow, Russia\n\\item[\\naples] INFN-Sezione di Napoli and University of Naples, \n     I-80125 Naples, Italy\n\\item[\\cyprus] Department of Natural Sciences, University of Cyprus,\n     Nicosia, Cyprus\n\\item[\\nymegen] University of Nijmegen and NIKHEF, \n     NL-6525 ED Nijmegen, The Netherlands\n\\item[\\caltech] California Institute of Technology, Pasadena, CA 91125, USA\n\\item[\\perugia] INFN-Sezione di Perugia and Universit\\'a Degli \n     Studi di Perugia, I-06100 Perugia, Italy   \n\\item[\\cmu] Carnegie Mellon University, Pittsburgh, PA 15213, USA\n\\item[\\prince] Princeton University, Princeton, NJ 08544, USA\n\\item[\\rome] INFN-Sezione di Roma and University of Rome, ``La Sapienza\",\n     I-00185 Rome, Italy\n\\item[\\peters] Nuclear Physics Institute, St. Petersburg, Russia\n\\item[\\salerno] University and INFN, Salerno, I-84100 Salerno, Italy\n\\item[\\ucsd] University of California, San Diego, CA 92093, USA\n\\item[\\santiago] Dept. de Fisica de Particulas Elementales, Univ. de Santiago,\n     E-15706 Santiago de Compostela, Spain\n\\item[\\sofia] Bulgarian Academy of Sciences, Central Lab.~of \n     Mechatronics and Instrumentation, BU-1113 Sofia, Bulgaria\n\\item[\\korea] Center for High Energy Physics, Adv.~Inst.~of Sciences\n     and Technology, 305-701 Taejon,~Republic~of~{Korea}\n\\item[\\alabama] University of Alabama, Tuscaloosa, AL 35486, USA\n\\item[\\utrecht] Utrecht University and NIKHEF, NL-3584 CB Utrecht, \n     The Netherlands\n\\item[\\purdue] Purdue University, West Lafayette, IN 47907, USA\n\\item[\\psinst] Paul Scherrer Institut, PSI, CH-5232 Villigen, Switzerland\n\\item[\\zeuthen] DESY, D-15738 Zeuthen, \n     FRG\n\\item[\\eth] Eidgen\\\"ossische Technische Hochschule, ETH Z\\\"urich,\n     CH-8093 Z\\\"urich, Switzerland\n\\item[\\hamburg] University of Hamburg, D-22761 Hamburg, FRG\n\\item[\\taiwan] National Central University, Chung-Li, Taiwan, China\n\\item[\\tsinghua] Department of Physics, National Tsing Hua University,\n      Taiwan, China\n\\item[\\S]  Supported by the German Bundesministerium \n        f\\\"ur Bildung, Wissenschaft, Forschung und Technologie\n\\item[\\ddag] Supported by the Hungarian OTKA fund under contract\nnumbers T019181, F023259 and T024011.\n\\item[\\P] Also supported by the Hungarian OTKA fund under contract\n  numbers T22238 and T026178.\n\\item[$\\flat$] Supported also by the Comisi\\'on Interministerial de Ciencia y \n        Tecnolog{\\'\\i}a.\n\\item[$\\sharp$] Also supported by CONICET and Universidad Nacional de La Plata,\n        CC 67, 1900 La Plata, Argentina.\n\\item[$\\diamondsuit$] Also supported by Panjab University, Chandigarh-160014, \n        India.\n\\item[$\\triangle$] Supported by the National Natural Science\n  Foundation of China.\n\\item[\\dag] Deceased.\n\\end{list}\n}\n\\vfill\n\n\n\n\n\n\n\n\\newpage\n\n\n\n\n\\begin{table} [htbp]\n\\begin{center} \n\\begin{tabular}{|c|c|c|c|c|} \\hline  \n\\multicolumn{5}{|c|}{Electron\/Muon plus hadrons selections  } \\\\  \\hline\n           & Very Low \\ensuremath{\\Delta M} & Low \\ensuremath{\\Delta M} & Medium \\ensuremath{\\Delta M} & Large \\ensuremath{\\Delta M} \\\\  \\hline\nNo. of isolated leptons   $\\ge$ &   1   &   1     & 1      &  1    \\\\  \\hline\n$N_{tk} - N_{lep}$       $\\ge$ &  2    &  4      & 5      &  4    \\\\  \\hline\n$N_{cl}$                 $\\ge$ &  6    &  10     & 10     &  10   \\\\  \\hline\n$\\sin(\\theta_{miss})$    $\\ge$ &  0.74 &  0.38   & 0.23   &  0.28 \\\\  \\hline\n $E^{\\perp}_{25}$ (\\gev{})  $\\le$ &  0.52 &  --     & --     & 11.6  \\\\  \\hline\n $p_{\\perp}$ (\\gev{})       $\\ge$ &  3.24 &  5.62   & 8.65   & 9.84  \\\\  \\hline\n $E_{lep} $  (\\gev{})       $\\ge$ &  1.51 & 2.59    & 6.17   & 25.9  \\\\  \\hline\n $E_{lep} $  (\\gev{})       $\\le$ &  9.12 &  27.5   & 31.2   & 43.8  \\\\  \\hline\n $E_{TTJL} $  (\\gev{})       $\\ge$ &  1.27 &  0.95  & 1.44   &  --   \\\\  \\hline\n $M_{had} $  (\\gev{})       $\\le$ &  5.0   &  28.2  & 39.1   & 89.0  \\\\  \\hline\n $M_{rec} $  (\\gev{})      $\\ge$ &  144   &  130    & 107    & 57.0  \\\\  \\hline\n $E_{vis} $  (\\gev{})      $\\ge$ &  4.02   & 8.90   & 31.5   & 65.3  \\\\  \\hline\n $E_{vis} $  (\\gev{})     $\\le$ &  11.0  &  59.1   & 93.6   & 118   \\\\  \\hline\n\\end{tabular} \n\\caption{Values of the cuts for the lepton plus hadrons selections; they\n         are determined with    the optimisation procedure described in \n         Section~\\protect\\ref{sec:optimization}.\\label{tab2}}\n\\end{center}\n\\end{table}\n\\begin{table}[!]\n\\vspace{-0.1 cm}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|} \\hline  \n\\multicolumn{5}{|c|}{Chargino Hadronic selections  } \\\\  \\hline\n                  & Very Low \\ensuremath{\\Delta M}    & Low \\ensuremath{\\Delta M} & Medium \\ensuremath{\\Delta M} & Large \\ensuremath{\\Delta M} \\\\  \\hline\n$N_{cl}$             $\\ge$ &  14    & 14      &  14    &  14      \\\\  \\hline\n$N_{tk}$             $\\ge$ &  5     & 5       & 5      &  5       \\\\  \\hline\n $p_{\\perp} $ (\\gev{})   $\\ge$ &  3.72  & 10.0    & 11.5   & 11.4     \\\\  \\hline\n $p_{\\perp}\/E_{vis}$  $\\ge$ &  --    & 0.20    & 0.15   & 0.10     \\\\  \\hline\n$E_{vis}$ (\\gev{})      $\\le$ & 12.0   &  68.0   & 76.0   & 149    \\\\  \\hline\n Acollinearity (rad)  $\\le$ & 2.00   & --      & --     &  3.02  \\\\  \\hline\n Acoplanarity (rad)   $\\le$ & 2.18   & 2.89    &  2.92  &  3.11 \\\\  \\hline\n sin($\\theta_{miss}$) $\\ge$ &  0.56  & 0.46    &  0.20  &  0.61  \\\\  \\hline\n$E^{\\perp}_{25} $ (\\gev{}) $\\le$ &  0.21& 5.80    &  5.80  & 3.25     \\\\  \\hline\n $E_{25}$ (\\gev{})        $\\le$ &  --  & --      &  --    & 2.53     \\\\  \\hline\n $p_{\\parallel}\/E_{vis}$  $\\le$ &  --  & 0.53    &  0.95  &  0.55  \\\\  \\hline\n $E_{lep}^{max}$ (\\gev{}) $\\le$ &  9.12 &  27.5   & 31.2   & 43.8  \\\\  \\hline\n $M_{vis}$ (\\gev{})       $\\ge$ &  2.85& 9.3    &  35.4  &  --      \\\\  \\hline\n $M_{rec}$ (\\gev{})       $\\ge$ &   -- & 124     &   67.2 & --     \\\\  \\hline\n $E_{vis}\/\\sqrt{s}$     $\\ge$ &   -- & --      &   --   & 0.60     \\\\  \\hline\n $E_{30^0}\/E_{vis}$     $\\le$ &  --  & 0.22    & 0.40   & 0.65      \\\\  \\hline\n $E_{TTJ}\/p_\\perp$      $\\ge$ &  0.24& --      &  0.24  & --       \\\\  \\hline\n $y_{\\perp}$            $\\ge$ &  --  & 0.28    &  0.28  &  0.40   \\\\  \\hline\n\\end{tabular} \n\\caption{Values of the cuts for the purely hadronic selections\n         which  are determined with the optimisation procedure\n         described in Section~\\protect\\ref{sec:optimization}.\\label{tab3}}\n\n\\end{center}\n\\end{table}\n\n\\begin{table} [htbp]\n\\begin{center} \n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|} \\hline  \n & \\multicolumn{2}{|c|}{Low \\ensuremath{\\Delta M}}& \\multicolumn{2}{|c|}{Medium \\ensuremath{\\Delta M}}& \n\\multicolumn{2}{|c|}{High \\ensuremath{\\Delta M}}& \\multicolumn{2}{|c|}{Combined} \\\\  \\hline\n  & $N_{data}$ & $N_{exp}$ & $N_{data}$ & $N_{exp}$ & $N_{data}$ & $N_{exp}$ \n& $N_{data}$ & $N_{exp}$ \\\\  \\hline\n\\chinop & 72 & 66.9 & 11 & 10.9 & 67 & 76.7 & 147 & 148. \\\\ \\hline\n\\chinonn & 43 & 39.3 & 6 & 7.78 & 3 & 2.45 & 50 & 48.1 \\\\ \\hline\n\\end{tabular} \n\\caption[cascade]{Results for charginos and neutralinos:\n           $N_{data}$ is the number of observed events and\n           $N_{exp}$ is the number of expected events from Standard \n           Model processes for the total integrated\n           luminosity collected at $\\sqrt{s} = 189 \\gev{}$.\n\\label{tab8}}\n\\end{center} \n\\end{table} \n\\begin{table} [htbp] \n\\begin{center} \n\\begin{tabular}{|c|r|r|r|r|} \n\\hline  \n{ }   &\n            \\multicolumn{2}{|c|}{~~LL~~} &\n            \\multicolumn{2}{|c|}{~~$\\neutralino{1}\\, \\ensuremath{\\mathrm{W^\\ast}}$~~} \\\\ \\hline\n{ \\ensuremath{\\Delta M}{} (\\gev{}) }  & $\\epsilon $ (\\%) & $~N_{exp}$ &\n                    $\\epsilon $ (\\%) & $N_{exp}$   \\\\\n\\hline \n3   &  1.6  & 20.3           &  1.9   & 39.4 \\\\ \n5   &  6.5   & 20.3          &  16.8  & 76.8  \\\\\n10  &  19.2   & 27.7            &  8.5   & 2.9  \\\\\n20  &  25.0 &   7.3           &  38.2  & 10.4  \\\\\n30  &  30.0   & 7.3            &  46.6  & 7.5  \\\\ \n40  & 28.9   & 7.3            &  44.2  & 4.9  \\\\\n50  & 26.9   & 7.3            &  25.3  & 4.9   \\\\ \n60  & 21.6   & 7.3            &  16.6  & 4.9  \\\\\n75  & 34.5   & 34.6            &  20.8  & 55.5  \\\\\n90  & 31.8   & 34.6            &  7.8   & 55.5  \\\\\n\\hline\n\\end{tabular} \n\\caption[cascade]{Optimised chargino efficiencies ($\\epsilon$)  \n        for the purely leptonic (LL) \n        and for the\n        $\\neutralino{1}\\,\\ensuremath{\\mathrm{W^\\ast}}$ decay mode.\n        $N_{exp}$ is the number of events expected from  \n        Standard Model processes.\n        Results are given as a function of \n        $\\ensuremath{\\Delta M}$\n        for $M_{\\chargino{\\pm}} = 94 \\gev{}$ at $\\sqrt{s} = 189 \\gev{}$.\n\\label{tab4}}\n\\end{center}\n\\end{table}\n\n\\begin{table} [htbp]\n\\begin{center}\n\\begin{tabular}{|c|r|r|r|r|}\n\\hline  \n{ }   &\n            \\multicolumn{2}{|c|}{~~LL~~} &\n            \\multicolumn{2}{|c|}{~~$\\neutralino{1}\\, \\ensuremath{\\mathrm{Z^\\ast}}$~~} \\\\ \\hline\n{ \\ensuremath{\\Delta M}{} (\\gev{}) }  & $\\epsilon $ (\\%)  &~$N_{exp}$ &\n                    $\\epsilon $ (\\%) & $N_{exp}$ \n                    \\\\\n\n\\hline\n6  & 9.1    & 9.5    & 3.5   & 35.9   \\\\\n10  & 10.9  & 10.2   & 10.9  & 35.9    \\\\\n20  & 27.3  & 4.2    & 9.2   & 3.4    \\\\\n40  & 30.8  & 2.8    & 25.9  & 3.4    \\\\\n60  & 34.1  & 19.2   & 35.2  & 7.7     \\\\\n80  & 35.5  & 19.2   & 35.2  & 7.7    \\\\\n100 & 29.3  & 21.4   & 20.7  & 7.7    \\\\\n140 & 17.7  & 25.5   & 9.4  & 2.4    \\\\\n180 & 10.5  & 25.5   & 8.7   & 2.4  \\\\\n\\hline \n\\end{tabular} \n\\caption[cascade]{Optimised neutralino efficiencies ($\\epsilon$)\n        for the purely leptonic (LL)\nfinal states and for the $\\neutralino{1}\\, \\ensuremath{\\mathrm{Z^\\ast}}$ decay mode. \n         Results are given as a function of         $\\ensuremath{\\Delta M}$\n        for $M_{\\neutralino{2}} + M_{\\neutralino{1}} = 188 \\gev{}$\nat $\\sqrt{s} = 189 \\gev{}$.\n \n\\label{tab5b}}\n\\end{center} \n\\end{table} \n\n\\pagebreak \n\n\\begin{figure}\n\\begin{tabular}{cc}\n\\psfig{file=xi_ljjl.eps,width=8.0cm} &\n\\psfig{file=xi_ljjm.eps,width=8.0cm} \\\\\n\\psfig{file=xi_nhhv.eps,width=8.0cm} &\n\\psfig{file=xi_nhhh.eps,width=8.0cm} \n\\end{tabular}\n \\caption{\nNumber of events selected in data (points),\n in Monte Carlo simulation of standard processes (solid line)\nand signal sensitivity (dashed line)\nas a function of selection cuts with increasing background rejection\npower. The vertical arrows show the $\\xi$ value corresponding to\nthe optimised cuts.\nThe distributions are shown for the \nchargino lepton-jets low \\ensuremath{\\Delta M}\\ a), the \nchargino lepton-jets medium \\ensuremath{\\Delta M}\\ b), the\nneutralino jet-jet very low \\ensuremath{\\Delta M}\\ c) and the \nneutralino jet-jet high  \\ensuremath{\\Delta M}\\ d)\nselections, respectively\n         }\n \\label{fig:xi_chaneu}\n\\end{figure}\n\n\\begin{figure}[hbtp]\n\\begin{center}\n  \\mbox{\\epsfxsize=10.0cm \\epsffile{upper_char_w_pap.eps}} \\\\ \n  \\mbox{\\epsfxsize=10.0cm \\epsffile{upper_char_ll_pap.eps}}\n  \\caption{Upper limits on the $\\ee \\rightarrow \\chargino{+}\\chargino{-}$\n production cross section\nup to $\\sqrt{s} = 189 \\gev{}$ in \nthe $M_{\\neutralino{1}} - M_{\\chargino{\\pm}}$ plane.\n           Exclusion limits are obtained assuming\n  standard W branching\nratios in the chargino decay a) or purely leptonic W \n           decays b), \n           $\\chargino{\\pm} \\rightarrow  \\neutralino{1} \\ell^\\pm \\nu $\n           ($\\ell=$e, $\\mu$, $\\tau$).\n          }\n  \\label{fig:xsection_chargino}\n \\end{center}\n\\end{figure} \n\\begin{figure}[hbtp]\n \\begin{center}\n  \\mbox{\\epsfxsize=10.0cm \\epsffile{pap_neu.eps}} \\\\\n  \\mbox{\\epsfxsize=10.0cm \\epsffile{pap_neu_lep.eps}}\n  \\caption{Upper limits on the $\\ee \\rightarrow \\neutralino{1}\\neutralino{2}$\n production cross section\nup to $\\sqrt{s} = 189 \\gev{}$ \n           in the $M_{\\neutralino{1}} - M_{\\neutralino{2}}$ plane.\n           Exclusion limits are obtained \nassuming standard Z branching\n ratios in the next-to-lightest neutralino decay \n$\\neutralino{2} \\rightarrow \\ensuremath{\\mathrm{Z^\\ast}} \\neutralino{1}$ a)\nor assuming purely leptonic Z \n           decays b),   \n            $ \\neutralino{2}\\rightarrow  \\neutralino{1} \\ell^+ \\ell^- $\n           ($\\ell=$e, $\\mu$, $\\tau$).\n          }\n  \\label{fig:xsection_neutralino}\n \\end{center}\n\\end{figure} \n\n\n\\begin{figure}[hbtp]\n \\begin{center} \\vspace*{-1.5cm}\n \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.50cm \\epsffile{higgsino_189_neu2.eps}} \\\\\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.50cm \\epsffile{higgsino_189_char.eps}}\n\\vspace{-.5cm}\n \\caption {Lower mass limits as a function of $M_2$ for\n           the next-to-lightest neutralino a)\nand the lightest chargino b).\n The limits are shown for $\\tan\\beta = \\sqrt{2}$ and for $\\mu>0$ and $\\mu<0$. \n          } \n  \\label{fig:mass_m2}\n \\end{center}\n\\end{figure} \n\n\n\n\n\\begin{figure}[hbtp]\n \\begin{center}\n\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=18cm \\epsffile{limchar_189_last.eps}}\n  \\caption{\nLower limit on $M_{\\chargino{\\pm}}$ as a function of\n           $\\tan\\beta$ and for any value of $m_0$.\nThe solid line (gaugino region) shows the lower limit obtained for\nlight scalar neutrinos (also small $M_2$ values), which corresponds to the \nabsolute lower limit for large $\\tan\\beta$ values. The \ndashed line (higgsino region) shows the lower limit obtained for very small $\\ensuremath{\\Delta M}$ values. This line corresponds to the absolute lower limit for \nsmall $\\tan\\beta$ values.\n  \\label{fig:mass_cha}}\n \\end{center}\n\\end{figure} \n\n\n\n\n\\begin{figure}\n \\begin{center}\n\\vspace{-1.5cm}\n  \\hspace*{-2.0cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{limneu_m0500_189_last.eps}}\n\\vspace{-0.5cm}\n \\caption{ Lower limit on the lightest neutralino mass, $M_{\\neutralino{1}}$,\n           as a function of $\\tan\\beta$ for  \n           $m_0=500 \\gev{}$, when combining all chargino and neutralino searches.}\n \\label{fig:neutralino_mass_a}\n\\vspace{-.5cm}\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{tan_beta_pap.eps}}\n\\vspace{0.0cm}\n  \\caption{Lower limit on the lightest neutralino mass, $M_{\\neutralino{1}}$,\n            as a function of\n            $m_0$ for two values of $\\tan\\beta$. Scalar lepton searches\ncontribute in the low $m_0$ region. Chargino searches contribute mainly\nin the high $m_0$ region. For the low $\\tan\\beta$ values, the neutralino\nsearches give additional contribution in the intermediate $m_0$ region.\n          }\n  \\label{fig:neutralino_mass_b}\n \\end{center}\n\\end{figure} \n\n\\begin{figure}\n \\begin{center}\n\\vspace{-1.cm}\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{limneu_anym0_189_last.eps}}\n\\vspace{-.5cm}\n  \\caption{Lower limit on $M_{\\neutralino{1}}$\n            as a function of\n  $\\tan\\beta$ and for any value of $m_0$,  when combining\nthe chargino, neutralino and scalar lepton searches.\n          }\n  \\label{fig:neutralino_any_a}\n\\vspace{-.5cm}\n  \\hspace*{-1.5cm}\\mbox{\\epsfxsize=11.5cm \\epsffile{limm2_anym0_189_last.eps}}\n  \\caption{Lower limit on $M_2$\n            as a function of\n           $\\tan\\beta$ and for any value of $m_0$,  when combining\nthe chargino neutralino and scalar lepton searches.          }\n  \\label{fig:neutralino_any_b}\n \\end{center}\n\\end{figure} \n\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nRobots that can operate in unconstrained environments and collaborate with humans must be capable of learning about new objects they may encounter. Pointing at an object with our hand is a natural way to communicate with the robot about a new object. In this paper, we consider the problem of teaching robots novel objects by pointing at the new object. Once we show the robot a new object, it can generate and store a feature vector corresponding to that object and then re-use it for one-shot localization of the object in new scenes (Fig.~\\ref{fig:hero}).\n\nNeural networks have been used in recent years to learn fully differentiable visuomotor policies that directly map pixels to actuator commands~\\cite{levine2016end,zhang2018deep,yu2018one,rahmatizadeh2018virtual,giusti2015machine}. The neural network architecture typically used for such policies can be decomposed into vision layers and control layers. The vision layers comprise of several convolutional and pooling layers followed by a spatial attention mechanism that attends to the objects of interest in the image. We propose modulating the spatial attention so as the network is able to attend to the object that the hand is pointing at (see Fig.~\\ref{fig:hero}) while ignoring the other distracting objects in the scene.\n\nIn this work, we assume that only the location of the object of interest is available for training and that the position and orientation of the pointing hand are unavailable. So, this is a weakly supervised learning problem where the neural network must figure out as part of the learning process that the pointing hand in the image is salient and then learn to attend to the object being pointed at. On the other hand, this assumption makes the process of data acquisition with a real robot easier by reducing the labeling effort\n\nUnlike other papers on object detection~\\cite{redmon2016you,ren2015faster}, we are primarily interested in teaching robots new objects. This means that we are interested in objects not seen by the neural network during training. We accomplish this using Siamese networks~\\cite{koch2015siamese,venkatesh2019one,bertinetto2016fully}, which are twin neural networks with shared weights. The idea is to use the neural network to obtain from the image a feature vector representing the object of interest rather than classifying the contents of the image as is usually done (Fig.~\\ref{fig:hero}). This vector can be subsequently used in new environments to find the novel object of interest.\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/hero.PNG}\n    \\caption{One-shot localization of novel object selected by the pointing hand. The feature vector of the object (blue bottle cap) that the hand is pointing at is extracted using the proposed attention modulation mechanism. This is then used to localize the object in new scenes. Note that the pointing finger is at considerable distance from the object of interest.}\n    \\label{fig:hero}\n\\end{figure}\n\nOur contributions are as follows:\n\\begin{itemize}\n    \\item We propose a spatial attention modulation mechanism that endows the neural network with the ability to selectively attend to the object that is being pointed at while ignoring other distracting objects in the scene.\n    \\item We show that the proposed method can be combined with Siamese networks to teach robots novel objects.\n\\end{itemize}\n\nThe proposed network architecture is trained on synthetic data constructed from a dataset of emojis. We demonstrate the proposed method on the Dobot Magician robot arm. We show that the robot learns new objects that we point at and can find them in new scenes.\n\nThe rest of this paper is organized as follows. In the next section, papers related to this work are discussed. In Section~3, the proposed model architecture is described. Section~4 details experimental results, and Section~5 concludes the paper.\n\n\n\\section{RELATED WORK}\n\\subsection{Hand Recognition}\nOne of the ways to design a system that can infer the object of interest in a similar scenario is to use a pipelined approach. For example, one could employ deep learning models trained to localize human hands in an image~\\cite{openpose, doosti2019hand}, extract the most relevant keypoints of the hand (say the joints of the index finger) and then fit a line that passes through these keypoints. An object recognition module could then be used to localize each object in the scene, project these points on the line and pick the object corresponding to the closest point to the the hand as the object of interest. However, training such a system requires a strong level of supervision such as the positions of all objects in the scene, and possibly even the keypoints of the hand if one would like to fine-tune the hand localization models. However, this approach will not be feasible in a weak supervision setting as outlined in this paper where only the location of the object of interest is given. Moreover, it has been shown across a wide range of problems that using an end-to-end approach leads to better performance as compared to using a pipelined approach for a given task~\\cite{gupta2016synthetic, end_to_end_speech, zhang2018deep}.\n\n\\subsection{Spatial Attention}\nThe architecture of typical end-to-end networks for visuomotor tasks can be broadly grouped into two sets of layers. The initial group of layers form the vision layers that help in localizing the relevant objects in the image. The remaining layers form what is known as the control layers which are responsible for coming up with the appropriate control actions required to perform the task at hand. A key component in such end-to-end networks is some form of a spatial attention mechanism that learns to attend to the relevant object of interest in the scene. The work presented in~\\cite{zhang2018deep} demonstrates the use of imitation learning for teaching a PR2 robot to perform simple tasks such as pick-and-place. The authors developed a virtual reality based system to teleoperate the robot and collect training data. The data was then used to train an end-to-end network that maps image pixels directly to robot joint velocities. The network consists of an initial set of convolution layers that generates a feature map. The feature map is passed through a spatial softargmax layer to output a feature vector. The resulting feature vector is then passed through a few fully connected layers to predict the joint velocities of the robot. The spatial softargmax layer serves as a simple spatial attention mechanism where the attention weight corresponding to each pixel of the feature map depends on the degree of activation.\n\n\\subsection{One Shot Learning}\nApart from inferring the object of interest in an image in the presence of other objects, another goal of this paper is to enable robots to recognize objects that they have never encountered before by training it on only a few examples involving the novel object. Broadly speaking, meta learning and Siamese networks are two approaches one can take to achieve this. We review both approaches below.\n\n\\subsubsection{Meta Learning}\nIn Meta learning, also known as learning to learn, a distribution of tasks are provided as training data. Typically only a few examples for each task are provided in the training data. The weights of the network after the training process completes serves as a good initialization for the network to learn to perform any new previously unseen task. Only a few training examples involving the novel task and a few gradient descent steps are required for the network to converge to an optimal set of weights~\\cite{maml}. Recently, meta learning and imitation learning has been combined to enable robots to perform novel tasks such as pick-and-place by training on just a single example \\cite{osiml, daml}.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{figures\/archext.PNG}\n    \\caption{The proposed neural network architecture for one-shot localization of the object selected by the pointing hand. The convolutional layers in the ``Conv\" block are Conv3x3(16)-ELU-Conv3x3(32)-ELU-Conv3x3(64)-ELU-MaxPool2x2-Conv3x3(64)-ELU-Conv3x3(128)-ELU-MaxPool2x2-Conv3x3(256)-ELU-Conv3x3(512)-ELU-Conv3x3(1024)-ELU. All convolutions are ``valid\" convolutions that do not use padding so that the feature vector for the object is the same regardless of whether it is near the edge of the image or at the center. The receptive field of the ``Conv\" block is 34~px.}\n    \\label{fig:arch}\n\\end{figure*}\n\n\\subsubsection{Siamese Networks}\nSiamese networks are used to address the similarity learning problem where it is desirable to infer if a pair of images (referred to exemplar and search images) are similar to each other or not. This is done by using twin convolution neural networks with shared weights that transform the images $x_1$ and $x_2$ into feature embeddings $\\phi(x_1)$ and $\\phi(x_2)$, respectively. The embedding pair is then combined using a transformation $g$ that can be used to make suitable predictions depending on the task at hand. For example, in the context of image classification~\\cite{koch2015siamese} the transformation $g$  is a distance metric that can be used to measure the similarity score between the object in the exemplar image and the search image. The system is trained on several examples of similar and dissimilar pairs of images. Once training is complete a database of images is built with one image corresponding to each object of interest. At test time the similarity of the search image is tested against each image in the database to determine the object class of the search image. Siamese networks have been used in face recognition systems as well~\\cite{deepface}. However, in both these papers the comparison is possible only if the exemplar and search images are of the same dimension. \n\nThe authors of~\\cite{bertinetto2016fully} used fully convolutional neural networks to enable comparison of images of dissimilar dimensions with Siamese networks. Their architecture was adapted successfully for object tracking in videos. Here the user provides the exemplar image by cropping out the object of interest from the first frame of the video which is then compared against each subsequent frame using the Siamese network. More recently, the authors in~\\cite{venkatesh2019one} combined fully convolutional Siamese networks with spatial attention to enable object localization for robot pick-and-place tasks. The paper explores specifying the object of interest by using visual cues instead of requiring the user to provide a cropped image of the object. Given a group of objects in a scene the user indicates to the robot the object of interest by shining a laser beam directly on it. Although the authors talk of localizing novel objects using a laser beam as a visual cue, the network designed by them should work for any other kind of visual cue (such as a stick or even a hand) so long as it is in very close proximity with the object of interest. However, a human merely has to point at an object from a distance to convey that it is of interest and an observer infers and localizes the object being referred to by looking in the direction of the pointing hand. We would like to design systems that communicate intent to robots much like how humans communicate with each other via visual cues or using natural language~\\cite{blocks, touchdown}. Learning to localize an object of interest from natural language instructions requires a different architecture design compared to the one presented in this paper. We will restrict our focus to learning to localize the object of interest by using a visual cue provided such as a hand pointing at the object from a distance.\n\n\n\n\n\\section{NETWORK ARCHITECTURE}\n\n\n\\subsection{Localizing the Object of Interest}\n\n\n\n\n\n\n\n\nThe proposed neural network for one-shot localization is shown in Fig.~\\ref{fig:arch}. Let the exemplar image (denoted as $x$) correspond to the image that contains the novel object that is being pointed at by the hand. Let the search image (denoted as $\\hat{x}$) be the image of the new scene in which the same object must be localized. The network outputs the locations of the object in the exemplar image and the search image which are denoted as $(p_x, p_y)$ and $(\\hat{p}_x, \\hat{p}_y)$, respectively. The mean squared error loss is used to train the network.\n\nThe localization of the object is performed in a similar fashion as described in \\cite{venkatesh2019one} except for the attention modulation block. The exemplar image is passed through the CNN to obtain a feature map $x_f$. This is then passed through a bottleneck convolutional layer (conv$1\\times1$) to obtain $x_o$. Let us ignore for now how the attention modulation map $x_m$ is generated. We will describe the generation of the attention map $x_m$ in Section~\\ref{sec:attn-mod}. A spatial attention map $x_{o^*}$ is generated  by adding the attention modulation map $x_m$ to $x_o$ and the resulting sum is passed through a spatial soft-argmax layer whose output is the predicted location of the object of interest $(p_x, p_y)$ in the exemplar image $x$ (see Eqns. \\eqref{eqn:x_ostart}, \\eqref{eqn:softargmax_x} and \\eqref{eqn:softargmax_y}). The spatial attention map $x_{o^*}$ is used to obtain the feature vector $f$ corresponding to the object of interest from the feature map $x_f$ (see Eqn. \\eqref{eqn:feat_vec}). Note that $\\odot$ in Fig.~\\ref{fig:arch} corresponds to the element-wise multiplication operation (with the appropriate broadcasting done to account for the different number of channels present in $x_f$ and $x_{o^*}$).\n\n\n\\begin{equation}\n    x_{o^*_{i, j}} = softmax_{i, j} \\big(x_{o_{i, j}} + x_{m_{i, j}}\\big)\n    \\label{eqn:x_ostart}\n\\end{equation}\n\n\\begin{equation}\n    p_x = \\sum_{i, j} x_{o^*_{i, j}} i\n    \\label{eqn:softargmax_x}\n\\end{equation}\n\\begin{equation}\n    p_y = \\sum_{i, j} x_{o^*_{i, j}} j\n    \\label{eqn:softargmax_y}\n\\end{equation}\n\n\\begin{equation}\n    f = \\sum_{i, j} x_{o^*_{i, j}} x_{f_{i, j}}\n    \\label{eqn:feat_vec}\n\\end{equation}\n\nThe localization of the object in the search image $\\hat{x}$ is done by first passing $\\hat{x}$ through the CNN to obtain $\\hat{x}_f$. Then the feature vector $f$ is used like a matched filter (or equivalently as a conv1$\\times$1 layer with $f$ as the weights) to generate $\\hat{x}_{o^*}$. The location of the object $(\\hat{p}_x,\\hat{p}_y)$ in the search image $\\hat{x}$ is then determined by passing $\\hat{x}_{o^*}$ through a spatial soft-argmax layer (similar to the operations in Eqns.~\\eqref{eqn:softargmax_x} and \\eqref{eqn:softargmax_y}).\n\n\n\n\\subsection{Generating the Attention Modulation Map}\n\\label{sec:attn-mod}\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=1.0\\linewidth]{figures\/attn_mod.PNG}\n    \\caption{Beam \/ Cone like attention modulation maps for different positions and orientations of the pointing hand. The dark regions correspond to a value of -2.0 which suppresses peaks corresponding to objects in $x_o$ (Fig.~\\ref{fig:arch}), whereas the bright regions correspond to value 0.0 which allows values in that area in $x_o$ to pass through unchanged. The beam width is \\ang{30}, and the step size is \\ang{15}.}\n    \\label{fig:attn_mod}\n\\end{figure}\n\nWhen there are multiple objects in the scene, we expect multiple bright spots each corresponding to an object in $x_o$ (Fig.~\\ref{fig:sample_non_siamese}). We would like to suppress the peaks in $x_o$ corresponding to objects that are not being pointed at. Had we known the location and orientation of the hand, we could have directly suppressed the irrelevant peaks. However, since we do not have labels corresponding to the pose of the pointing hand in the scene, the neural network must learn to attend to the hand and then use this to  suppress the irrelevant peaks in $x_o$. To enable this, we use a ``soft\" or differentiable way to compute the position and orientation of the hand which is then employed to suppress irrelevant objects in $x_o$.\n\nThe feature map $x_f$ is passed through two independent bottleneck layers to produce maps $x_{hp}$ and $x_{ho}$ corresponding to the position and orientation of the pointing hand respectively in $x$. Similar to Eq.~\\eqref{eqn:feat_vec}, spatial attention is used to attend to the pointing hand and to obtain the orientation of the hand $x_{ho^*}$. The final position and orientation of the hand ($x_h$) is used to ``soft select\" a pre-defined attention modulation map $x_m$ (see Fig.\\ref{fig:attn_mod}). The set of pre-defined attention modulation maps include beams from all possible locations and orientations of the pointing hand as shown in Fig.~\\ref{fig:attn_mod}. Each modulation map is constructed by drawing a beam emanating from the position of the hand and in the direction the hand is pointing at. We use an orientation step size of $\\ang{15}$ with a beam width of $\\ang{30}$. Thus, there are 30$\\times$30$\\times$24 $=$ 21600 such maps. Note that no explicit loss function is used to learn $x_{hp}$ and $x_{ho}$. Rather, network learns to predict appropriate values for $x_{hp}$ and $x_{ho}$ that result in the ``selection\" of an appropriate attention map, which is possible only by correctly recognizing the position and orientation of the hand. The modulation map thus obtained, $x_m$, is added to $x_o$ to highlight the object being pointed at while suppressing the irrelevant ones. Thus, the pixels in $x_o$ that lie inside the beam are passed as is whereas the pixels that lie outside the beam are suppressed. Note that the entire attention modulation scheme is differentiable and hence can be learned through back-propagation.\n\nThe proposed way of creating attention modulation maps is most suitable for top view images (Fig.~\\ref{fig:sample_non_siamese}). For perspective views where the depth of the object is more relevant, it may be necessary to generate maps in 3D by casting a cone of rays and using the perspective projection. We leave this for future work.\n\n\n\n\n\\section{EXPERIMENTAL RESULTS}\n\n\n\nTo evaluate the proposed neural network, we first train it on a synthetic dataset and compare it with alternative architectures. The trained network is deployed on a robot arm to demonstrate its real world performance.\n\n\\subsection{Localization Performance}\nA dataset of 5000 training images and 1000 test images is created by placing emojis (Fig.~\\ref{fig:emoji}) at non-overlapping positions against a backdrop as shown in Fig.~\\ref{fig:sample_non_siamese}. A hand emoji (Fig.~\\ref{fig:hand_emoji}) is placed at a random location pointing to an object. One or more distracting objects are placed at random locations not on the line segment between the pointing hand and the object. The label for each sample is the position of the object that the hand is pointing at.\n\nTo evaluate the proposed spatial attention modulation mechanism, we first consider only localization of the object in the image containing the pointing hand ($x$) while ignoring the other input ($\\hat{x}$) and the output of Siamese network $(\\hat{p}_x, \\hat{p}_y)$. Table~\\ref{table:compare} compares the proposed approach with two baselines. The FC layers baseline refers to using fully connected layers\\footnote{The fully connected layers used are FC1024-ELU-FC256-ELU-FC2.} to predict $(p_x, p_y)$ from $x_f$. The Conv layers baseline uses convolutional layers\\footnote{Conv3x3(2048)-ELU-MaxPool2x2-Conv3x3(2048)-ELU-Conv3x3(2048)-ELU-MaxPool2x2-Conv3x3(2048)-ELU-Conv3x3(2048)-ELU-FC2.} to predict the position of the object. The networks are trained with mean squared error loss with weight decay 1e-8 using the Adam optimizer\\cite{kingma2014adam} with learning rate 1e-4. The evaluation metric is accuracy where we consider the prediction to be accurate if there is sufficient overlap between the ground truth and predicted bounding box. Specifically, the IOU (intersection-over-union) between the ground truth bounding box and the predicted bounding box has to be at least 0.5. All the three networking achieve low training error (accuracy over 99\\%), but the test error varies, and the proposed approach generalizes the best. A sample output is shown in Fig.~\\ref{fig:sample_non_siamese} where we observe that the spatial attention modulation mechanism is working as one might expect.\n\nA second dataset containing images corresponding to a new environment ($\\hat{x}$) where the object highlighted by the pointing is present along with distracting objects is constructed as before (Fig.~\\ref{fig:sample_siamese_synth}). The proposed architecture in its entirety with the Siamese network to process $\\hat{x}$ and predict $(\\hat{p}_x, \\hat{p}_y)$ is trained on this dataset. The accuracy on this dataset drops only marginally to 95.31\\%. Table~\\ref{table:compare2} compares performance on this dataset and shows that attention modulation is essential to localize the desired object. The sample output in Fig.~\\ref{fig:sample_siamese_synth} shows that the desired object in $\\hat{x}$ is being attended to.\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/emoji.png}\n      \\caption{A few sample objects used for training and evaluation. The set of emojis is divided into 2075 for training and 703 for testing.}\n      \\label{fig:emoji}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.9\\linewidth]{figures\/hand_emoji.png}\n      \\caption{A few sample hand images used for training and evaluation. The set of hand emojis is divided into 47 for training and 8 for testing.}\n      \\label{fig:hand_emoji}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/sample_nonsiamese2.png}\n      \\caption{Sample prediction of the proposed architecture. The network has properly localized the pointing hand and chosen a suitable attention modulation map. The activation corresponding to the object that is not pointed at has been appropriately suppressed in $x_{o^*}$.}\n      \\label{fig:sample_non_siamese}\n\\end{figure}\n\n\\begin{table}[!t]\n\\caption{Comparison of the proposed approach with different baselines\n\\label{table:compare}\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline\nNeural Network Architecture & Accuracy\\\\\n\\hline\nFC layers & 11.72\\%\\\\\nConv layers & 41.41\\%\\\\\nProposed approach & 96.88\\%\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!t]\n\\caption{Comparison of localization performance of the Siamese network on novel objects with and without attention modulation\n\\label{table:compare2}\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline\nNeural Network Architecture & Accuracy\\\\\n\\hline\nWithout Attention Modulation\\cite{venkatesh2019one} & 12.5\\%\\\\\nProposed approach (with modulation) & 95.31\\%\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/sample_siamese_synth.png}\n      \\caption{A sample prediction from the proposed architecture on synthetic data.}\n      \\label{fig:sample_siamese_synth}\n\\end{figure}\n\n\\subsection{Evaluation on Robot Arm}\nWe demonstrate the proposed neural network using the Dobot Magician, a 3-DoF robot arm (Fig.~\\ref{fig:dobot}). The objects used for evaluation with the robot are shown in Fig.~\\ref{fig:real_objs}. To convert the localized object in pixel space to the robot co-ordinate space, a chessboard calibration pattern is used (Fig.~\\ref{fig:calib}), and OpenCV is used for calibration. Figure~\\ref{fig:sample_siamese} shows a sample predicted from the proposed neural network. We see that the pointing hand has been localized, and the network has learnt to predict an appropriate attention modulation map that selects the object being pointed at (blue bottle cap in Fig.~\\ref{fig:sample_siamese}). We also see that the activation corresponding to the distracting object in $x_{o^*}$ has been successfully suppressed. With the feature vector $f$  corresponding to the bottle cap extracted, the Siamese net successfully attends to the same bottle cap in a new scene ($\\hat{x}$). In this manner, 20 trials were performed. The proposed network localizes the desired object to within 1~cm in all the trials. A video of the robot in operation is available at \\url{https:\/\/youtu.be\/bJ5HKllhqLg}.\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.85\\linewidth]{figures\/dobot_1.png}\n      \\caption{A sample demonstration with the Dobot Magician robot arm.}\n      \\label{fig:dobot}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.85\\linewidth]{figures\/real_objs.jpg}\n      \\caption{Objects used for evaluating the proposed approach on the Dobot arm.}\n      \\label{fig:real_objs}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.85\\linewidth]{figures\/calib2.png}\n      \\caption{The chessboard calibration pattern used to convert pixels to robot co-ordinates.}\n      \\label{fig:calib}\n\\end{figure}\n\n\\begin{figure}[!t]\n      \\centering\n      \\includegraphics[width=0.99\\linewidth]{figures\/sample_siamese.png}\n      \\caption{A sample prediction from the proposed architecture.}\n      \\label{fig:sample_siamese}\n\\end{figure}\n\\section{CONCLUSIONS}\n\nWe have proposed a spatial attention modulation method that endows a neural network with the ability to attend to a hand pointing at an object in an image and to focus on the object that is being pointed at. The proposed approach generalizes significantly better compared to architectures that use only fully connected or convolutional layers for localization. Furthermore, this approach can be combined with a Siamese network to localize objects that were not present in the training dataset. This network architecture can be used in building robots that can interact naturally with humans and learn about new objects over time.\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\\section*{ACKNOWLEDGMENT}\n\nThis project was supported by the Robert Bosch Center for Cyber-Physical Systems.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}