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+{"text":"\\section{Introduction}\n\nScene understanding from sensory measurements is an essential task for intelligent agents, and has been widely investigated in recent years. The advances of deep learning brought scene understanding into a new era, especially for semantic image understanding tasks starting from object detection \\cite{girshick_rich_2014} to panoptic segmentation \\cite{kirillov_panoptic_2018}. It can be argued that the aforementioned tasks provide the scene representations majorly in the context of pixels, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot bounding boxes, per-pixel semantic labels, \\emph{etc}\\onedot} \\def\\vs{\\emph{vs}\\onedot, which are not the ideal data structures for high level reasoning and decision making by intelligent agents. An alternative option for representing geometric and semantic image content is a graph. Compared to pixel-based representations, a graph, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot OpenStreetMap \\cite{haklay_openstreetmap:_2008} or a scene graph as in \\cite{johnson_image_2015}, is a much better data structure to store scene information for high level reasoning and decision making and, at the same time, being an efficient representation in terms of memory and compute requirements. While the translation from image content (like bounding boxes \\emph{etc}\\onedot} \\def\\vs{\\emph{vs}\\onedot.) into a graph can be performed with hand-designed logic, such approaches are highly task-dependent and thus not scalable. Therefore, similar to our work, significant contemporary research is devoted into methods that can learn to translate image content in a (scene) graph.\n\nPractically all current state-of-the-art methods for this image-to-graph task, like \\cite{xu_scene_2017, herzig_mapping_2018, zellers_neural_2018, yang_graph_2018, li_factorizable_2018, qi_attentive_2019} and as discussed in Section \\ref{sect_related_work}, build on top of an object detection framework and rely on meticulously annotated datasets that contain the relationships between objects, as the ground truth for fully-supervised training. Although large scale datasets for this task are available \\cite{krishna_visual_2017}, the time required for the needed annotations is so significant that also these fully-supervised learning-based approaches do not scale well to task for which the required ground truth is not yet available. Therefore, we set the basic research goal to be able to learn the image-to-graph translation task in a self-supervised manner by extending the methodology of auto-encoding. While the current capability of our approach is limited to simple line drawings, we believe that it holds the key to developing image-to-graph methods that scale over many different scene understanding tasks and thus do not rely on hand-crafted logic or meticulously annotated datasets.\n\nThe details of our graph auto-encoding approach are provided in Section \\ref{sect_method}. At its core, it learns to translate image content into a graph that is represented by node and adjacency matrices. This learning is self-supervised using a fully-differential auto-encoder in which the decoder consists of several carefully designed neural networks and the decoder uses techniques from differentiable image drawing. The image-to-graph-to-image task is then learned in an auto-encoder fashion by minimizing the pixel loss between the input image and the image generated by the decoder. During inference, only the encoder is used, although the learning procedure can, in theory, also be used on-line to fine-tune the obtained graph.   \n\nTo demonstrate our approach, we use a synthetic dataset that contains several line drawings of simple shapes, each of which represents a graph with nodes and edges. The details of the dataset and its corresponding experimental results are presented in Section \\ref{sect_method} and Section \\ref{sect_exp}. Although the demonstrated experiments are still at an early stage and further work must be carried out to generalize to large scale real datasets like Visual Genome \\cite{krishna_visual_2017}, we believe our findings are worth to be shared with the community. Our contributions include:\n\\begin{itemize}\n\t\\item the, to the best of our knowledge, first neural network that can learn an image-to-graph translation task by only using self-supervision; \n\t\\item a comparison of our method to an fully-supervised equivalent baseline, which shows our approach exhibits comparable performance in terms of F1-score of triples matching;\n\t\\item several ablation studies that examine and reveal key properties of our method.\n\\end{itemize}\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{fig_overview}\n\t\\caption{The overview of the proposed approach. The auto-encoder is able to extract the canonical graph information from an input image at the bottleneck, with only the image-level self-reconstruction loss.}\n\\end{figure}\n\n\n\\section{Related work}\n\\label{sect_related_work}\n\n\\paragraph{Scene graph generation}\nThe task of scene graph generation, as pioneered by \\citet{johnson_image_2015}, is to estimate a graph structure representing the relevant information of the scene from sensory measurements, In the graph, the objects are nodes and their relationships are modeled as edges. Various approaches are proposed in recent years, enabled by the release of the Visual Genome dataset \\cite{krishna_visual_2017} with massive manual annotations. Most of the state-of-the-art approaches \\cite{xu_scene_2017, herzig_mapping_2018, zellers_neural_2018, yang_graph_2018, li_factorizable_2018, qi_attentive_2019} are formulated on top of bounding box-based detection frameworks, while using advances in graph convolutional networks (GCN) \\cite{yang_graph_2018} and graph attention mechanisms \\cite{qi_attentive_2019} to construct the scene graph. In \\citet{newell_pixels_2018} it is proposed to generate heat-maps for both objects (nodes) and relations (edges) simultaneously for further graph construction using associative embeddings. Recently, scene graph generation is also extended into 3D \\cite{armeni_3d_2019} and applied to input data other than images, such as point clouds. \\cite{wald_learning_2020}. \n\nIt is important to note that these mentioned state-of-the-art methods are, unlike our method, highly dependent on massive annotated ground truth for training. While annotating for object detection is rather straightforward, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, an object is present or not, annotating the relationships between objects is much more involved. It does not only depend on the image context but also on the task what relationships are relevant to be annotated and it is practically impossible to annotate all possible relationships between objects for all possible tasks. This is also why the main performance criteria for image-to-graph translation is currently based on recall but not on precision, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, methods typically estimate many redundant relationships that are not present in the ground truth annotations but that, in certain cases, can be considered as being correct. While the Visual Genome dataset \\cite{krishna_visual_2017} is extensive it surely does not cover the need for ground truth annotations for all possible tasks in computer vision that can benefit from obtaining graph representations from images. As such, to make available image-to-graph translation methods that can effortlessly scale to new tasks, we set the basic research goal to investigate and develop methods that require as little ground truth annotations as possible and preferably none.\n\n\\paragraph{Unsupervised scene representation learning}\nIn general, to reduce the dependency on expensive manual annotations, unsupervised approaches have been widely investigated. The learned representations include individual objects and their information, physical factors, \\emph{etc}\\onedot} \\def\\vs{\\emph{vs}\\onedot. \n\nIn \\citet{burgess_monet_2019} a variational auto-encoder is used and trained together with a recurrent attention network decomposing the objects in an image. This model is fully differentiable and can be trained with self-reconstruction loss in an end-to-end manner. More recently, the work in \\citet{yang_learning_2020} went one step further to decompose the image into objects and enable object manipulation without requiring object-level annotations. Also, in \\citet{wu_unsupervised_2020} the proposed encoder-decoder framework factors each input image into depth, albedo, viewpoint, and illumination, using only unsupervised reconstruction loss. This is achieved by interpreting the prior knowledge of symmetric structure in the design of the network's information flow.\n\nThe mentioned approaches achieve unsupervised learning of scene representation mainly by carefully designing the framework using human prior knowledge and training the models in an auto-encoding fashion with reconstruction supervision. Our approach has the same philosophy, but is more aggressive in the representation-shifting that is performed. In our approach, the self-learned representation (graph) is further away from the input representation (image) in terms of the format and the level of semantic information, than in other works where the self-learned representations are image-level objects.\n\n\\paragraph{Differentiable image synthesis}\nAn important part of our approach is the ability to decode a graph into an image. Conventional computer graphics (CG) methods generate realistic images by approximating the physical processes of light, which are typically hard-coded and non-differentiable. As such, they are not designed to be integrated with deep learning methods. To overcome this, differentiable renderers \\cite{loper_opendr_2014, kato_neural_2018} are proposed to mimic the classical CG process with differentiable operations, which enables various tasks of rendering in combination with using deep neural networks. These renderers are mainly conventional geometry-based approaches with novel differentiable extensions. Thus, they are not able to have the abstract semantic understanding of the scene.\n\nSignificant research is carried out on image synthesis using CNN-based generative networks, such as image-to-image translation \\cite{isola_image--image_2017, zhu_unpaired_2017}. CNN-based approaches are able to capture the complicated high level information of the scene to be generated, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot various background layouts, object classes and their attributions, \\emph{etc}\\onedot} \\def\\vs{\\emph{vs}\\onedot. However, these approaches are mostly data-driven and lack the explainability of the representations inside the CNNs. \n\nIn contrast, scene graphs are better able to provide high level explainable representations than conventional tensors or feature maps.\nMore close to our research, scene graphs are also used in the image synthesis tasks as input \\cite{johnson_image_2018}. Furthermore, image manipulation is also realized by interactively modifying the scene graph extracted from the input image in the bottleneck of an encoder-decoder network \\cite{dhamo_semantic_2020}. However, note that in \\cite{dhamo_semantic_2020} their explainable scene graph representation is first massively annotated by humans and then further learned by the model, unlike our approach that employs self-supervised learning and does not require any human labeling effort.\n\n\n\\section{Methodology}\n\\label{sect_method}\n\n\\subsection{Problem definition}\n\nWe start with the conventional problem definition of scene graph generation task and then discuss the similarities and differences in our setting. \n\nGiven an input image $I$, we assume that there exists a scene graph $G = (V, E)$ with $V$ being a set of nodes corresponding to localized object regions in image $I$, and $E$ being the edges representing the relationships between nodes $V$. Note that each element $v_i$ and $e_{ij}$ in $V$ and $E$ could have one of multiple semantic labels. Thus, the scene graph generation can be formulated as the mapping $f: I \\mapsto G$. Most approaches \\cite{xu_scene_2017, herzig_mapping_2018, zellers_neural_2018, yang_graph_2018, li_factorizable_2018, qi_attentive_2019} factorize it mainly into two sequential sub-mappings, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot $f = f_E \\cdot f_V$, where\n\\begin{equation}\n\\begin{array}{l}\nf_V: I \\mapsto V \\\\\nf_E: (I, V) \\mapsto E.\n\\end{array}\n\\end{equation}\n\n$f_V:I \\mapsto V$ is often accomplished by object detection frameworks, and $f_E: (I, V) \\mapsto E$ is the relationship classification. Both of these mapping processes are usually performed by learnable neural networks. In the conventional fully-supervised setting, for each training sample the ground truth nodes $\\bar{V}$ and edges $\\bar{E}$ are provided such that the neural networks performing the mapping $f: I \\mapsto G$ can be learned.\n\nIf the ground truth $\\bar{V}$ and $\\bar{E}$ are not available, the previous approaches cannot be applied, as they are discriminative models trained in a supervised manner. On the contrary, we opt to extend the task of scene graph generation to image-graph-image translation by using the concept of auto-encoding. The encoder $f: I \\mapsto G$ is expected to perform the same task as the scene graph generation \\cite{johnson_image_2015}, only in our case, the supervision is not provided. It is also composed of two sub-modules, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, $f_V: I \\mapsto V$ for node prediction and $f_E: (I, V) \\mapsto E$ for edge prediction. The decoder $g: G \\mapsto I$ takes the intermediate graph information as input, and re-generates the input image in a fully differentiable manner. By placing the encoder and decoder behind each other, techniques from auto-encoding can be used to learn the graph information.\n\nFormally, our goal is to have two mapping functions $f$ and $g$,\n\\begin{equation}\n\\begin{array}{l}\n\\tilde{G} = f(I) = f_E \\cdot f_V (I) \\\\\n\\tilde{I} = g(\\tilde{G}), \\\\\n\\end{array}\n\\end{equation}\nsuch that\n\\begin{equation}\n\\begin{array}{l}\n\\tilde{G} \\in \\mathcal{G}, \\\\\n\\tilde{I} \\in \\mathcal{I}, \\\\\nf,g = \\underset{f,g}{\\arg\\min}\\ \\mathcal{L}(I, \\tilde{I}), \\\\\n\\end{array}\n\\end{equation}\nwhere $\\mathcal{G}$ and $\\mathcal{I}$ are the collection of all possible graphs and images respectively, and $\\mathcal{L}(\\cdot, \\cdot)$ is the similarity measure of two images. \n\nIn the following sub-sections, we will detail the design and training protocol of the proposed approach. We take a simple toy task with a synthetic dataset as an example to illustrate the idea. The dataset contains a bunch of undirected graphs which are represented as images. The nodes in the graphs are the end-point of the visible edges and the edges are defined as the binary connectivity between two nodes. Note that ideally, the task and dataset can be extended with more complicated definitions, however as the first step, we opt to simplify the task and focus on the basic methodology. The details of the dataset will be discussed in Section~\\ref{sect_exp}.\n\n\\subsection{Network}\n\\label{sect_method_net}\n\n\\begin{figure}\n\t\\centering\t\\includegraphics[width=\\linewidth]{fig_network}\n\t\\caption{The overall framework of the proposed approach, with the upper part being the detailed pipeline and the lower part being the differentiable drawing module. The entire framework is trained end-to-end and using only self-reconstruction supervision. Please See Section \\ref{sect_method_net} for more details.}\n\t\\label{fig_network}\n\\end{figure}\n\nThe overall design of the auto-encoder for image-graph-image translation task is visualized in Figure~\\ref{fig_network}. It mainly consists of three differentiable modules, i.e. node prediction, edge classification and differentiable image synthesis. The three modules are sequentially connected and trained in an end-to-end manner using image self-reconstruction loss. \n\n\n\\subsubsection{Encoder: from image to graph}\n\nAs discussed previously, the encoder part of the network is to predict the graph information from the input image, which is formalized as the process $f = f_E \\cdot f_V :I \\mapsto G$. The encoder is composed of three sequential parts, namely node attention, coordinates transformation, and relation classification. The output of the encoder, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot the bottleneck of the auto-encoder, is the graph represented by two matrices, i.e. the node position matrix and the adjacency matrix.\n\n\\paragraph{Node attention} \n\nAt the early stage, the convolutional layers from ResNet-50 \\cite{he_deep_2016} are implemented as the feature extractor. Since our task is much simpler, we take features from the second residual block and reduce the output channels of two blocks to 128 and 64, respectively. To reduce the size of the extracted features, two max-pooling layers with kernel size and stride being 2 are also applied before each block. On top of the residual blocks we have two extra convolutional layers to predicts the node attention maps $M_{att}$, with the spacial size reduced by the factor of 4 due to the previous pooling operations, and with the number of channels being the pre-defined maximum number of nodes $N_{max}$. Each channel in node attention maps $M_{att}$ is passed through a 2-D softmax layer, such that the summation of all the elements strictly equals to 1, which is essential for the later computation of node coordinates. Note that here we expect the attention channels are able to provide channel-wise heat maps indicating the detected nodes. However, no supervision or extra information is supplied to the network, and the network is able to learn from the final reconstruction loss by gradient back-propagation. \n\n\n\n\\paragraph{Coordinates transformation} \n\nBeing different from other detection frameworks \\cite{girshick_rich_2014, girshick_fast_2015, ren_faster_2017}, in our setting the coordinates of nodes are not provided as the ground truth. Thus the differentiability is crucial at this stage, since without which the gradient back-propagation cannot be applied and the node attention module cannot learn anything. However, it is non-trivial to transform the positional information in the heat maps to numerical coordinates in a differentiable manner. Here we use differentiable spatial to numerical transform (DSNT) \\cite{nibali_numerical_2018} to perform the transformation. Due to the usage of the previously mentioned 2-D softmax layer, for each channel, the heat map can be seen as a 2-D probabilistic distribution of a certain node. Thus, one can create two fixed template maps containing the numerical coordinates value for each pixel, in horizontal and vertical directions, respectively. With element-wise production and summation, the numerical coordinates for each node can be computed. At this stage, there exist no trainable parameters, and the computation, which is detailed in \\cite{nibali_numerical_2018}, is fully deterministic and differentiable.\n\n\\paragraph{Edge prediction}\n\nOnce the coordinates of the nodes are computed, a set of regions of interest (ROIs) for edges can be constructed by combining arbitrary two node coordinates as a bounding box. Then ROI pooling \\cite{ren_faster_2017} is applied on the input image and creates a batch of local image patches of size $N_{max}^2 \\times 1 \\times 16 \\times 16$ for further edge prediction by a edge classifier. The classifier consists of two convolution layers (kernel size 3 and output channels 32 and 16, respectively) with Batch Normalization, ReLU activation and max-pooling. Then, the features are flattened and processed by two fully-connected layers with ReLU and Sigmoid activations, respectively.\n\nPlease note that in our toy task the edge represents the connectivity for the sake of simplicity, while it can be extended with semantic information with little extra effort. The predicted edge connectivities, together with the previously computed nodes, make up the output of the encoder, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot a graph with nodes position matrix and adjacency matrix.\n\n\\subsubsection{Decoder: from graph to image}\n\nThe task of the decoder $g: G \\mapsto I$ is to reconstruct the input image given the graph information provided from the encoder. The key is to ensure the reconstruction flow is fully differentiable such that the whole pipeline can be trained together end-to-end. Inspired by \\cite{johnson_image_2018}, we consider a template-based image synthesis approach, which is referred to as \\textit{differentiable drawing}. \n\nWe create a template set that contains all the pre-defined relationships that could exist in the dataset. For each forward-pass of the network, the differentiable drawing module walks through all the edges in the predicted adjacency matrix, and for the edges that need to be visualized in the image, spatial transformer network \\cite{jaderberg_spatial_2015} is implemented to copy and draw the corresponding edge template onto the canvas. The differentiable drawing module is able to create a coarse reconstructed image $\\tilde{I}_{\\text{coarses}}$ on a blank canvas. This process can be written as $g_{\\text{coarse}}: G \\mapsto \\tilde{I}_{\\text{coarse}}$. On top of the coarse image $\\tilde{I}_{\\text{coarses}}$, a refinement network $g_{\\text{refine}}: \\tilde{I}_{\\text{coarse}} \\mapsto \\tilde{I}_{\\text{refine}}$ is applied for further post-processing, which does not change the structural information but modifies and eliminates the textures and styles difference between the created coarse image and real input image. In our implementation, the refinement network contains three convolution layers with kernel size 3, intermediate channel size 16, and PReLU activations between every two layers.\n\n\n\\subsection{Training}\n\nThe entire framework is trained end-to-end without supervisions that require manual annotations. To achieve this goal, we apply two losses during training: the main image reconstruction loss for auto-encoding and the auxiliary node attention loss.\n\nAs the main image reconstruction loss, a structural similarity index measure (SSIM) \\cite{wang_image_2004} is used, which encourages the decoder to reconstruct the input image and thus encourages the encoder to understand the image in terms of graphs in an implicit manner. We apply SSIM losses with multi-scale setting on the refined images in the experiments, which is formulated as \n\\begin{equation}\n\t\\mathcal{L}_{\\text{main}} = \\text{MS-SSIM}(\\tilde{I}_{\\text{refine}}, I)\n\\end{equation} \nwhere $\\tilde{I}_{\\text{refine}}$ and $I$ are the reconstructed refined image and input image, respectively.\n\nThe node attention module is expected to predict the node positional attention maps from the input image solely relying on the gradient information from the reconstruction loss. Due to the lack of explicit supervision, the attention module occasionally tends to predict some nodes multiple times while ignoring other nodes. To overcome this behavior and stabilize the node attention training, we apply the \\textit{auxiliary loss} $\\mathcal{L}_{\\text{aux}}$. The idea is to penalize the overlay behavior between the predicted node attentions and encourage the attention module to discover as many different nodes as possible. On the other hand, since the pre-defined maximum number of nodes (attention channels) is fixed and is always larger than the real value, the overlay is expected to happen. We propose a penalty loss conditioned on the similarity measure for each sample, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, the penalty is applied to the samples only if the reconstruction is not good enough. The auxiliary loss can be written as \n\\begin{equation}\n\\mathcal{L}_{\\text{aux}} = \\left\\{\n\t\\begin{array}{ll}\n\t\\text{mean}([\\text{ReLU}(\\sum_{i}^{N_{\\text{max}}}M_{att}[i] - 1)]^2),& \\text{if MS\\_SSIM < mean MS\\_SSIM} \\\\ \n\t0, & \\text{if MS\\_SSIM > mean MS\\_SSIM}\n\t\\end{array}\\right.\n\\end{equation}\nwhere $M_{att}$ is the normalized predicted node attention maps with the maximum cell for each channel being 1. MS\\_SSIM and mean MS\\_SSIM are the similarity measures of each sample and the mean measure of the entire batch during training. All the operations are element-wise, and we skip the height and weight index for attention maps for the sake of simplicity.\n\n\nFormally, the overall loss can be written as\n\\begin{equation}\n\\mathcal{L} = \\mathcal{L}_{\\text{main}} + \\lambda \\cdot \\mathcal{L}_{\\text{aux}}\n\\end{equation}\nwhere $\\lambda$ is the weight for loss balancing and is empirically set to 1 in our experiments.\n\n\\section{Experiments}\n\\label{sect_exp}\nWe perform the following experiments to demonstrate and verify the proposed method: 1) we compare the results of our unsupervised approach and a supervised baseline, qualitatively and quantitatively; 2) we perform an ablation study of our approach using different maximum number nodes; and 3) we also study the effect of different image reconstruction losses.\n\n\\paragraph{Datasets:}\nWe use a synthetic dataset, called Simple Shape dataset, to demonstrate the proposed idea. The dataset contains 50k images with each presenting one of the three simple shapes including line, triangle, and rectangle. Each shape is processed with random affine transformation (including scale, rotation, shear, and translation) and is drawn on an empty black canvas with the size being $128\\times128$ pixels. Please see Fig~\\ref{fig_qualitative} for some visualized samples. Thus, for each image, the number of nodes varies from 2 to 4, which is why the maximum number of nodes is set as 4 in the main experiments.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{imgs\/imgs\/46500}\\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/imgs\/46506}\\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/imgs\/46509}\\\\\n\t\\caption{\\centering input image}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46500_gt-cropped}\\\\\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46506_gt-cropped}\\\\\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46509_gt-cropped}\\\\\n\t\\caption{\\centering ground truth graph}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_c_imgs\/46500_simple_shape_nodes4of4_ours} \\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_c_imgs\/46506_simple_shape_nodes4of4_ours} \\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_c_imgs\/46509_simple_shape_nodes4of4_ours} \\\\\n\t\\caption{\\centering coarse image}\n\t\\end{subfigure}    \n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs\/46500_simple_shape_nodes4of4_ours} \\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs\/46506_simple_shape_nodes4of4_ours} \\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs\/46509_simple_shape_nodes4of4_ours} \\\\\n\t\\caption{\\centering refined image}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs_from_g\/46500_simple_shape_nodes4of4_ours}\\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs_from_g\/46506_simple_shape_nodes4of4_ours}\\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs_from_g\/46509_simple_shape_nodes4of4_ours}\\\\\n\t\\caption{\\centering ours prediction (image)}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46500_ours-cropped}\\\\\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46506_ours-cropped}\\\\\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46509_ours-cropped}\\\\\n\t\\caption{\\centering ours prediction (graph)}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs_from_g\/46500_simple_shape_nodes4of4_baseline}\\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs_from_g\/46506_simple_shape_nodes4of4_baseline}\\\\\n\t\\includegraphics[width=\\linewidth]{imgs\/pred_imgs_from_g\/46509_simple_shape_nodes4of4_baseline}\\\\\n\t\\caption{\\centering baseline prediction (image)}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{.11\\textwidth}\n\t\\centering\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46500_baseline-cropped}\\\\\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46506_baseline-cropped}\\\\\n\t\\includegraphics[height=\\linewidth]{imgs\/graphs\/46509_baseline-cropped}\\\\\n\t\\caption{\\centering baseline prediction (graph)}\n\t\\end{subfigure}\n\t\\caption{The qualitative results of our approach. (a)-(b) are the inputs image and the corresponding graphs. (c)-(f) are the predictions from our approach, in which (c)-(d) are the output images from the network and (e)-(f) are the graph predictions in image and matrix formats. (g)-(h) are the graph predictions from the baseline, in image and matrix formats, respectively. }\n\t\\label{fig_qualitative}\n\\end{figure}\n\n\\paragraph{Baseline:} \n\nSince there is no existing previous method that performs the same task, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot generating graphs from images without any external supervision, we provide a baseline that shares a similar design of the network, but has access to the ground truth graph information. To have a fair comparison, the capacity of the baseline network, is aligned with the design of the proposed network as much as possible, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot the overall structure, number of layers, and channels are set as similar as possible. Since the ground truth is provided, the decoder is no longer needed for the baseline, and we directly apply two supervisions on the graph outputs of the encoder. First, the supervision for the node attention channels is provided by creating a ground truth heat map with Gaussian kernels at the node ground truth locations. As the node attentions can have random orders, we perform summation across the channels and compute the mean squire pixel error between the single channel attention and the ground truth heat map. Second, for each step, we perform the node matching between the prediction and the ground truth, to re-generate the temporary ground truth adjacency matrix aligned with the correct temporary node order. This allows for a cross-entropy classification loss to be computed. With the aforementioned two supervisions, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, pixel-wise heat map error and cross-entropy classification error, the graph can be learned in a fully-supervised manner.\n\nNote that the baseline is intended to formulate a relatively fair comparison with our approach. Thus, it is preferred that the two models have similar capacities, which is why we apply minimal modifications to our model. One could carefully re-design the nodes detection and relation classification mechanisms for the fully-supervised training scenario, and potentially achieve better performance. However, for investigating the fundamental difference coming from either using supervised and self-supervised training strategies, it is best to use the same network architectures as much as possible.\n\n\\paragraph{Metric:}\n\nTo evaluate the quality of scene graph, the image-wise SGGen metric is often used \\cite{lu_visual_2016, xu_scene_2017, yang_graph_2018}. The idea is to organize the prediction and ground truth graphs as two sets of triplets with the format of \\textit{<object, relationship, object>}. Then triplet matching is performed to compute the conventional Recall@k metric. The main reason only the recall is used and not also precision in previous work is that the manual annotations are sparse and it is not possible to annotate all existing relations between objects in images. However in our dataset, since the images are created from a given graph, we are able to capture the complete graph information. Thus, the precision metric is also applied. In our evaluation protocol, like previous work \\cite{lu_visual_2016, xu_scene_2017, yang_graph_2018}, we also consider the task as detection of triplets, and use F1-score with underlying metrics being precision and recall. \n\nSince our model might have multiple predictions for the same triplet, we pre-process the raw prediction and delete the redundant triplets before evaluation. The recall is computed without sorting the confidence and all the predicted triplets are used for the computation of precision. In this sense, our F-score metric containing precision and recall covers the previously used SGGen, which is essentially recall, and is more challenging in terms of triplet redundancy evaluation, which is essentially the precision.\n\n\\paragraph{Implementation details:} \n\nWe train our approach and the baseline using PyTorch \\cite{paszke_automatic_2017}, and we use Adam \\cite{kingma_adam:_2014} optimizer with batch size $= 128$, $\\beta_1 = 0.6$, and $\\beta_2 = 0.9$ for the training of both models. We also train the network in each setting with different random seeds for 10 times, and calculate the mean and standard deviation for each metric. For the \\textit{proposed} approach, we train with the initial learning rate 0.0005 for 30 epochs and reduce the learning rate by the factor of 10 for the last 10 epochs. As for the \\textit{baseline} approach, we notice that providing the full ground truth will significantly simplify the task, which is expected. Thus, we train the baseline model with the initial learning rate 0.0003 for 15 epochs and reduce the learning rate by the factor of 10 for the last 5 epochs. We find that the sub-task for the node attention prediction is particularly easy to train and over-fits quickly. Also, the supervision quality for the adjacency matrix is dependent on the quality of the predicted nodes, which is used for generating the temporary ground truth on the fly. To solve these issues for the fully-supervised baseline, for the first two epochs we train the node attention module and later we fix it and only train the relation classification module.\n\n\\subsection{Results}\\label{sect_results}\n\n\\paragraph{Self-reconstruction \\vs full supervision:}\nWe compare the performances of the proposed self-supervised approach and the fully-supervised baseline. The results are presented in Table~\\ref{tab_main_qiantitative} and Figure~\\ref{fig_qualitative}. \n\nOur unsupervised approach achieves a comparable performance as the baseline, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot 67.9 and 61.3 respectively, in terms of F1-score. This shows that, even without any external explicit ground truth as a training target, the self-reconstruction is sufficient to provide supervision for the task of graph prediction. Unlike many other encoder-decoder approaches that process the data in the same format, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot image-feature-image translation, in our case, the information in the bottleneck is a conventional graph with random node order and the corresponding adjacency matrix. By creating differentiable transformation modules, as introduced in Section~\\ref{sect_method}, an image-graph-image auto-encoding framework is able to regress the graph information in the canonical formation automatically. As discussed before, note that we do not claim that our approach is absolutely better than the baseline in terms of the performance. One can optimize the design of the baseline in many aspects and achieve performance improvements. However, we opt to keep the network design the same to gain more meaningful insights into our self-supervised approach.\n\nIt must be clarified that the unsupervised approach contains several limitations that the supervised counterpart does not share. First, even though we use a simple dataset, it is less efficient for the network to learn under the self-supervised setting, which can be noticed by the number of training epochs used by two approaches. Second, if the task is more complicated and challenging, the self-supervised learning would be less effective or even fail, compared to the fully-supervised approach. Third, during the experiments we notice that the randomness has a larger impact on our self-supervised approach: although F1-scores are with similar standard deviation, the recalls have noticeable larger variance. With a very small probability, the network fails to reconstruct the full image, which is not likely to happen in a fully-supervised training scenario. \n\n\\begin{table}\n\t\\caption{Quantitative results of our approach and supervised baseline}\n\t\\label{tab_main_qiantitative}\n\t\\centering\n\t\\begin{tabular}{lllll}\n\t\t\\toprule\n\t\tMethod     & Supervision & Precision & Recall & F1-score \\\\\n\t\t\\midrule\n\t\tOurs & Image reconstruction  & \\textbf{57.7}$\\pm$2.7  & \\textbf{84.1}$\\pm$12.6 & \\textbf{67.9}$\\pm$5.4\\\\\n\t\tBaseline & Full (nodes \\& edges) &  54.0$\\pm$5.9 & 70.9$\\pm$5.9 & 61.3$\\pm$5.9 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\n\n\\paragraph{Effect of maximum number of nodes:}\n\nIn the previous main experiment we set the pre-defined maximum number of nodes to 4, according to the property of the dataset. In this experiment, we study the performance change when the defined maximum number of nodes is redundant even for the most complicated rectangle shape, which is the typical setting for real tasks. Table~\\ref{tab_num_of_nodes} presents the quantitative results of different values for the maximum number of nodes. \n\nFrom the table, it can be concluded that the redundancy of maximum number of nodes will result in the performance drop of the triplets matching: 54.6, 49.9, and 41.7 F1-score when the number of nodes is 5, 6, and 8, respectively. This is mainly because, when the network has extra chances to reconstruct the image, the adjacency prediction tends to be conservative. For example, to reconstruct a triplet in an image, if there are two extra triplets for the network to predict, the network will tend to generate three triplets with the confidence of connectivity being 1\/3. This will result in a reconstructed image that aligns with the input image due to triplets overlaying, but none of these triplets is correctly classified as connected. \n\nHowever, note that when the redundancy is limited, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot 4 and 5,  the performance can remain at the acceptable level. Thus, in the real case that the size of the graph is unknown in a dataset without any graphs labels, as long as the maximum number of nodes is not extremely large, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot 100\\% extra nodes or more, one can expect to still have a decent graph quality.\n\n\\begin{table}\n\t\\caption{Quantitative results of our approach with different defined maximum number of nodes}\n\t\\label{tab_num_of_nodes}\n\t\\centering\n\t\\begin{tabular}{clll}\n\t\t\\toprule\n\t\tNumber of nodes & Precision & Recall & F1-score \\\\\n\t\t\\midrule\t\n\t\t4   & \\textbf{57.7}$\\pm$2.7  & \\textbf{84.1}$\\pm$12.6 & \\textbf{67.9}$\\pm$5.4\\\\\n\t\t5   & 46.1$\\pm$3.5 & 68.9$\\pm$14.9 & 54.6$\\pm$6.9 \\\\\n\t\t6   & 43.2$\\pm$6.7 & 62.0$\\pm$12.9 & 49.9$\\pm$5.9 \\\\\n\t\t8   & 31.4$\\pm$2.9 & 62.3$\\pm$5.5 & 41.7$\\pm$2.8 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\n\\paragraph{Loss ablations:}\n\\label{sect_result_loss_ablations}\n\n\\begin{table}\n\t\\caption{The results of our approaches trained with different losses.}\n\t\\label{tab_loss_ablation}\n\t\\centering\n\t\\begin{tabular}{ccc|l}\n\t\t\\toprule\n\t\t& \\multicolumn{2}{c|}{$\\mathcal{L}_{\\text{main}}$ on } & \\\\\n\t\t$\\mathcal{L}_{\\text{aux}}$ & Refined image & Coarse image &  F1-score \\\\\n\t\t\\midrule\n\t\tYes & MS-SSIM & - &\\textbf{67.9}$\\pm$5.4 \\\\\n\t\tNo & MS-SSIM & - &63.2$\\pm$8.2 \\\\\n\t\tYes & SSIM & - & 50.3$\\pm$17.7 \\\\\n\t\tYes & -   & MS-SSIM & 51.7$\\pm$7.1 \\\\\n\t\tYes & -   & SSIM & 12.1$\\pm$9.7 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\nWe also perform the ablation study of different loss settings, to verify the design choice of the training loss. Five loss settings are tested, with their performances listed in Table~\\ref{tab_loss_ablation}. For the sake of simplicity we only show F1-score in this experiment, since the precision and recall are highly correlated to it.\n\nOur default setting (first row in Table~\\ref{tab_loss_ablation}) exhibits the best performance, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot 67.9 F1-score, and significantly outperforms the alternatives. If the auxiliary loss is disabled, one can observe a performance drop by 4.7 F1-score and a increase of standard deviation by 2.8. This validates the contribution of the auxiliary loss: it stabilizes the training procedure and improves the node attention quality without requiring human annotations.\n\nWhen replacing the multi-scale structural similarity index measure (MS-SSIM) with regular SSIM loss (third row), a performance drop can be observed with a margin larger than 17 in F1-score. This is mainly because the structural similarity at multiple scales can better measure the graph difference represented by images and ignore the pixel-level image differences at higher resolutions. Instead of applying the losses after the refinement sub-network, we also apply them on the coarse reconstructed image directly provided by the differentiable CG module (fourth and fifth row). The results are 51.7 and 12.1 F1-score when using MS-SSIM and SSIM, respectively. The performance degradation is mainly due to the domain gap between the training images and online generated images from the differentiable CG module. This domain gap can also be observed in Figure \\ref{fig_qualitative}. In this case, the supervision is not passed through the refinement network, thus the domain transformation is not performed. Since the (MS-)SSIM measures pixel-level similarity between two images, the domain gap will result in additional noise during the loss computation and thus inhibit the image reconstruction task which should be domain-invariant and only focus on the graph structure itself. This experiment also shows the importance of the refinement network and verifies that it can perform the domain adaption task via self-supervised training.\n\n\\section{Conclusion}\n\nIn this work, we propose a novel neural network that can learn to estimate graph information, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot node and adjacency matrices, from image content without the need for manually annotated ground truth. At its core, the node and adjacency matrices are self-learned by properly designing the network architecture of the encoder, aligning it with a decoder based on differentiable image drawing techniques, and training this end-to-end differentiable system on basis of image reconstruction loss. In terms of the commonly used triplet matching metric, our approach achieves a performance comparable to the fully-supervised baseline. Although the current unsupervised approach is limited to line drawings of simple shapes and has certain limitations related to its task complexity and training stability, we believe this approach is an important stepping stone towards self-supervised learning of the image to graph translation task for more complex imagery.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nPhysics and applications of the whispering gallery microresonators have been attracting a lot of interest over the past decade.\nIn particular, the frequency comb generation in the ring microresonators \nhas revealed a plethora of complex nonlinear effects that included  \nthe formation of the so-called {\\em soliton combs}, see, e.g., \\cite{soliton1}. Microresonator combs and solitons are gradually finding their applications in the high precession spectroscopy \\cite{spectr} and information processing \\cite{koos}. The soliton combs in the microring resonators are the dissipative soliton pulses primarily relying on the balance between the Kerr nonlinearity and dispersion, that can be modeled using the Lugiato-Lefever (LL) equation \\cite{soliton1}. The LL model was originally introduced to explain spatial patterns and localized states in the wide-aperture nonlinear resonators \\cite{lug}. Apart from the microring resonators, the  bottle \\cite{bot,pol1,snap,prl,dvo}, microbubble \\cite{Farnesi2015,Lu2016}, spheroidal \\cite{mat} and microsphere \\cite{coen,suh}   resonators have also been developed for  sensing and frequency conversion applications. Four-wave mixing, nonlinear switching, Brillouin and Raman effects have been observed in the spheroidal \\cite{mat}, bottle \\cite{pol1,yang,Asano2016}, microbubble \\cite{Farnesi2015,Lu2016} and microsphere \\cite{coen,suh} resonators. \n\nIn this work, we are proposing a generalization of the LL model to study nonlinear effects and frequency comb generation in the bottle microresonators. These resonators are made of silica or semiconductor strands\/fibers (radius $r$) and operate  close to the cut-off frequency of a high order whispering gallery type mode. The side surface of the fiber is curved (bubbled) with the radius of curvature $R$, $R\\gg r$, see Fig. 1(c). This curvature shapes the 'bottle' and provides the axial confinement, that transforms the slowly propagating waveguide mode into a discrete set of the resonator modes \\cite{bot,snap,prl,dvo}. Dispersion near the waveguide cut-off is typically anomalous \\cite{yulin,josab1,josab}, which together with the positive Kerr nonlinearity  creates conditions for the phase-matched cascaded four-wave mixing and soliton formation. Light can be coupled into the resonator using the evanescent tails of  a tapered fiber mode aligned perpendicular to the bottle axis. \n\nAn advantage of the bottle resonators is that they can be integral parts of the surface nano-scale photonic circuits \\cite{snap}. If used as the Kerr comb generators, the bottle resonators can cover a very wide range of the comb repetition rates, which equals the cavity free spectral range (FSR), from THz to MHz \\cite{dvo}. This flexibility of the FSR design in microresonators has been first reported for microspheroidal resonators \\cite{mat}, which have geometry and hence modal properties similar to the bottle ones. Importantly, bottle resonators  provide the GHz to MHz FSR values simultaneously with the on-chip integration option \\cite{dvo}, while the ring resonators with comparable  FSRs are becoming too large for on-chip integration. Studying nonlinear effects in the low FSR devices is particularly interesting since the nonlinear shifts of the resonances start to compete and can exceed the FSR. Studies of these regimes  have started to emerge only recently in the context of the fiber loop resonators  \\cite{wab}, where 100's of meters of fiber required to achieve the few MHz FSRs.\nBelow, we introduce the bottle resonator LL model and demonstrate that the modes of the  nonlinear bottle resonators can form multiple coexisting nonlinear resonances and their instabilities can lead to the generation of the low repetition rate frequency combs.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.32\\textwidth]{f1a}\n\\includegraphics[width=0.32\\textwidth]{f1b}\n\\includegraphics[width=0.32\\textwidth]{f1c}\n\\caption{(a) Density plot showing the spatial profiles vs the eigen-frequency $\\epsilon$ of the $50$ modes in the linear bottle resonator.\n(b) Spatial profiles, $|\\tilde\\psi(z)|$, of the upper branches of the nonlinear modes numerically computed using Eq. (\\ref{eq3}) with $\\partial_t=0$: $z_p=0$, $P_0=1$, $\\kappa=10^{-3}$; $\\delta=0$ ($n=0,2$),  $\\delta=-10$ ($n=10,12$). (c) Geometry of a bottle resonator.}\n\\label{f1}\n\\end{figure*}\n\\section{Lugiato-Lefever equation for bottle resonators}\nThe modal structure of  bottle resonators has been studied theoretically and experimentally \\cite{snap,prl,dvo}. It was demonstrated that the linear Schr\\\"odinger equation with the parabolic potential and positive effective mass, corresponding to the anomalous group velocity dispersion, describes  the axial modal family belonging to one whispering gallery (azimuthal) resonance \\cite{snap,prl}. The same equation, but including nonlinearity has been derived from the Maxwell equations in the context of the slow-light mode in optical fibers with longitudinal index variations \\cite{josab1,josab}, which is exactly the technology used to fabricate bottle resonators \\cite{bot,pol1,snap,prl}. For our purposes, we also need to account the pump and loss terms, which are well established in the optical resonator context, starting from the classical LL paper \\cite{lug} and all the way to the recent  publications on the microresonator frequency combs, see, e.g., \\cite{soliton1,spectr}. Thus the generalized LL equation for bottle microresonators is\n\\begin{equation}\ni\\partial_T\\Psi = - \\frac{1}{2}d \\partial_Z^2\\Psi \n     + \\left(\\omega_0+\\frac{f^2}{2d} U(Z)\\right)\\Psi-i\\kappa_0\\Psi-f|\\Psi|^2\\Psi-fP(Z)e^{-i\\omega_p T}.\\label{eq1}\n\\end{equation}\nHere $T$ is the physical time and $Z$ is the coordinate along the resonator axis. It is appropriate to term Eq. (\\ref{eq1}) as the {\\em generalized LL} equation, because it differs from the classical one \\cite{soliton1,spectr,lug} by the potential term $U(Z)$  and the spatially varying pump $P(Z)=P_0e^{-(Z-Z_p)^2\/w^2}$. Here $P_0$ is the dimensionless pump amplitude,  $Z_p$ is the position of the input nano-fiber running across the bottle and $w$ is the characteristic width of the evanescently coupled light \\cite{snap,prl}.\n$\\omega_0$ is the reference frequency chosen close to the eigenfrequency of the ground states of the potential $U(Z)$, see below, and $\\omega_p$ is the pump frequency.  $\\kappa_0$ is the photon decay rate. $d>0$ is the group velocity dispersion coefficient \\cite{yulin,josab1,josab}. $f$ is the FSR parameter,  which enters Eq. (\\ref{eq1}) through an appropriate scaling of the dimensionless electric field envelope $\\Psi$ and of the pump and potential terms. \n\nLinear spectrum of the pump and loss free resonator is found  assuming\n$\\Psi(Z,T)=\\phi_n(Z)e^{-i\\omega_0T+i\\varepsilon_nT}$, which results in the eigenvalue problem\n$-\\varepsilon_n\\phi_n=-\\frac{d}{2}\\partial_Z^2\\phi_n+\\frac{f^2}{2d}U(Z)\\phi_n$.\nAssuming $U(Z)=Z^2$, we recover the spectrum of the quantum harmonic oscillator: $\\varepsilon_n=-f (n+ 1\/2)$. Thus $f$ is indeed the FSR, $\\omega_0+f\/2$ is the physical frequency of the fundamental cavity mode and $q=\\sqrt{d\/f}$ is the width of this mode.  One can assume as an estimate $f=60$~MHz, which physically  corresponds, e.g., to the silica cylinder with the  radius $r=300\\mu$m and with the bottle curvature   $R=1$km: $f\\simeq c\/(3\\pi)\/\\sqrt{rR}$, where $c=3\\cdot 10^8$m\/s~\\cite{dvo}. $R\\simeq 1$km has been demonstrated in  \\cite{prl}. The number $N$ of modes in a bottle resonator of the length $L$ is estimated from $f^2 L^2\/(2d)=f N$. Thus, $d=0.05Hz~m^2$ and $L=0.3$mm give the  fundamental mode width $q\\simeq 30\\mu$m and  $N\\simeq 50$. Meaning of all geometrical parameters introduced above is clarified in Fig. 1(c). In what follows we restrict our simulations to the  $N=50$ case, however,  one can increase $N$ to thousands by increasing $L$ just to few mm's, $N\\sim L^2$. $f|\\Psi|^2$ represents the nonlinear shift of the resonance frequencies. We are primarily interested here in the case, when nonlinear effects are relatively large, so that $f|\\Psi|^2$ is comparable to the FSR, $f$.\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.33\\textwidth]{f2a}\n  \\includegraphics[width=0.33\\textwidth]{f2b}\n  \\includegraphics[width=0.33\\textwidth]{f2c}\n  \\includegraphics[width=0.33\\textwidth]{f2d}\n  \\caption{(a-c) Structure of the multiple nonlinear resonances for the different pump positions and the resonator losses. Plots show $I=\\int_{-\\infty}^{\\infty}|\\tilde\\psi(z)|^2dz$ vs $\\delta$. (a) $z_p=0$ (maximum of the ground state mode), $\\kappa=10^{-3}$, (b) $z_p=0.79$ (maximum of the $n=1$ mode), $\\kappa=10^{-3}$, (c) $\\kappa=10^{-1}$, $z_p=0$.  The bold points show the onsets of the instabilities of the upper branches of the nonlinear resonances, which are stable below these points. The encircled numbers correspond to the modal index $n$. (d) The instability growth rates $\\mathrm{Re}\\, \\lambda$ vs $\\delta$ of the upper branch of the ground state resonance. Parameters as in (a) - solid line, and (c) - dotted line. The inset shows $\\mathrm{Re}\\, u(z)$ of the unstable perturbation.}\n \\label{f2}\n\\end{figure*}\n\nFor our numerical studies we have dimensionalized Eq. (\\ref{eq1}) to the form\n\\begin{equation}\ni\\partial_t\\psi=-(1\/2)\\partial_z^2\\psi+\\left(\\delta+U(z)\/2\\right) \\psi-i\\kappa\\psi-|\\psi|^2\\psi-P(z),\\label{eq3}\n\\end{equation} \nwhere $\\psi=\\Psi e^{i\\omega_pT}$,  scaled time is $t=fT$,  distance is  $z=Z\/q$,  $\\kappa=\\kappa_0\/f$ is the normalized loss rate and $\\delta=(\\omega_0-\\omega_p)\/f$ is the normalized detuning.\n$U(z)=z^2$ for $z<l$ and $U(z)=l^2$ for $z>l$, where  $l=L\/q$ is the scaled resonator length, $l^2\/2=N$ and $N=50$.\nDimensionless pump width $w\/q$ is $0.2$ in the numerical examples shown. The density plot in Fig. \\ref{f1}(a) shows  $|\\psi(z)|$ for the linear modes, calculated from $-2\\epsilon~\\psi=-\\partial_z^2\\psi+U(z) \\psi$, vs their eigen-frequencies, $\\epsilon\\simeq -(n+1\/2)$.\n\\section{Multistability, instabilities and comb generation}\nFor $\\kappa=P=0$, the nonlinear modes split from the linear spectrum at points $\\delta=-(n+1\/2)$ and  their amplitudes increase with $\\delta$ through the positive Kerr nonlinearity, see dashed lines in Figs. \\ref{f2}(a)-\\ref{f2}(c).\nAs soon as the pump and loss are introduced, the dashed lines split into pairs. If $P(z)=P(-z)$, as for $z_p=0$, then the odd modes are not excited, see Figs. \\ref{f2}(a) and \\ref{f2}(c). Otherwise, the even and odd modes show up together, see Fig. \\ref{f2}(b). In the  $\\kappa=0$ limit, the nonlinear modes extend to $\\delta\\to+\\infty$, so that for any $\\delta>0$ we have $N$ co-existing solutions: {\\em multistability}. As $\\kappa$ increases the intervals of $\\delta$, where the tilted nonlinear resonances  exist, become narrower and eventually shrink below the FSR, at what point  the {\\em multistability} is replaced by the more usual {\\em bistability}, cf. Figs. \\ref{f2}(a,b) and \\ref{f2}(c). Data for  Figs. \\ref{f2}(a)-\\ref{f2}(c) were obtained by numerically solving Eq. (\\ref{eq3}) assuming $\\psi(z,t)=\\tilde\\psi(z)$.  Some examples of the nonlinear mode profiles $\\tilde\\psi$  are shown in Fig. \\ref{f1}(b).\n\nIn order to study stability of the nonlinear modes $\\tilde\\psi$ with respect to  noise, we have linearized Eq. (\\ref{eq3})  using the substitution $\\psi(z,t)=\\tilde\\psi(z)+u(z)e^{\\lambda t}+v^*(z)e^{\\lambda^* t}$ with $|\\tilde\\psi|\\gg |u|,|v|$ and solved the resulting eigenvalue problem,\n\\begin{eqnarray}\n&& \\lambda u=-i\\left(\\delta+U(z)\/2\\right) u+(i\/2)\\partial^2_zu-\\kappa u+2i|\\tilde\\psi|^2u+i\\tilde\\psi^2v, \\label{eq4}\\\\ \\nonumber &&\n\\lambda v=i\\left(\\delta+U(z)\/2\\right) v-(i\/2)\\partial^2_zv-\\kappa v-2i|\\tilde\\psi|^2v-i\\tilde\\psi^{*2}u,\n\\end{eqnarray}\nnumerically.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{f3a}\n\\includegraphics[width=0.3\\textwidth]{f3b}\n\\includegraphics[width=0.3\\textwidth]{f3c}\n\\includegraphics[width=0.3\\textwidth]{f3d}\n\\caption{(a) Instability development of the ground state mode. Parameters as in Fig. \\ref{f1}(a), $\\delta=3$. Spectra of the field across the resonator at the initial (b) and the advanced  (c) stages of the comb generation calculated as $S(\\varepsilon,z)=|\\int_{t_0}^{t_0+\\tau}\\psi(z,t)e^{-i\\epsilon t}dt|$: $t_0=0$ in (b),\n  $t_0=1450$ in (c) and $\\tau=50$.   (d) The spectra at $z=0$ and $\\tau=50$:  $t_0=0$ (blue); $t_0=400$ (orange); $t_0=900$ (green); $t_0=1400$ (red). $\\epsilon$ in (b)-(d) corresponds to the physical frequency $\\omega_p-f\\varepsilon$.}\n  \\label{f3}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{f4a}\n\\includegraphics[width=0.3\\textwidth]{f4b}\n\\includegraphics[width=0.3\\textwidth]{f4c}  \n\\caption{\n(a) Instability development of the $n=8$ mode. Parameters as in Fig. \\ref{f1}(a), $\\delta=-5$. (b) Spectrum $S(\\varepsilon,z)$, see Fig. \\ref{f3},  at the advanced   stage of the comb generation: $t_0=1450$, $\\tau=50$.  (c) The spectra at $z=0$. The colors and the time intervals same as in Fig. \\ref{f3}(d).}\n\\label{f4}\n\\end{figure*}\nIn the LL equation without the potential term and with the homogeneous pump, the upper branch of the homogeneous solution is always unstable in the case of focusing nonlinearity \\cite{lug}. The harmonic potential spatially confines the modes,  and therefore the instability range shrinks, which is particularly pronounced for the modes with the low to moderate $n$'s, see Fig. \\ref{f2}. In particular, we have found the coexistence of the stable upper branches of the $n=0$ and $n=2$ resonances, Fig. \\ref{f2}(a), and of the \n$n=0,1$ and $3$ ones, see Fig. \\ref{f2}(b). Of course, the lowest amplitude solution is also stable at the same time. Thus  the true multistability is realized in our system, which is a relative rare situation in nonlinear optical devices. Increasing the losses leads to broadening of the resonances and either reduction or complete suppression of all the instabilities,  Fig. \\ref{f2}(c).\n\nThe inset in Fig. \\ref{f2}(d) shows the eigensolution of Eqs. (\\ref{eq4}) driving the instability of the ground state $n=0$ and the figure itself shows the corresponding growth rate as a function of $\\delta$. Thus the ground state becomes  unstable above some critical detuning with respect to the perturbations that are shaped like $n=1$ mode. The noise driven excitation of this instability leads to the development of the periodic in time and space oscillations of the localized wavepacket in the harmonic potential, see Fig. \\ref{f3}(a). These oscillations lose their regularity with time, while more of the higher order modes are getting excited, so that the spectral content of the field is broadening and the frequency comb is generated. Figs. \\ref{f3}(b) and \\ref{f3}(c) show the spectra of the intracavity field calculated at every point inside the resonator at the initial stage of the evolution and when the frequency comb has already fully developed. Fig. \\ref{f3}(d) shows how the  spectrum at  $z=0$ changes with time and  acquires the comb structure.  Since, practically, the signal is going to be collected at a point, this is the type of spectrum which is expected to be seen in the experiments. By looking at the space-time evolution of the field in Fig. \\ref{f3}(a), one can say that this comb corresponds to the quasi-soliton pulse oscillating in the harmonic potential and immersed into the sea of the weakly nonlinear modes represented by the discrete part of the spectrum for $\\epsilon<-1\/2$.\nThe  dynamics generally gets more complex as the pump frequency is tuned into a resonance with the higher order modes. Fig. \\ref{f4} shows the space-time and spectral evolution observed when we initialized Eq. (\\ref{eq3}) with the unstable $n=8$ mode. In this case,  several localized wavepackets emerge and interact  in the potential between themselves and with the extended nonlinear modes. This produces the combs, see Figs. \n\\ref{f4}(b) and \\ref{f4}(c), which are both broader and more intense than the ones in Fig. \\ref{f3}. \n\\section{Summary}\nWe have introduced a generalization of the LL model applicable to the bottle resonators and demonstrated  multistability and  generation of the low repetition rate combs in these devices.  \n\\section*{Funding}\nThe Leverhulme Trust (RPG-2015-456); H2020 (691011, Soliring); ITMO University (Grant 074-U01); RFBR (17-02-00081); RSCF (17-12-01413).\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nBosonic atoms in optical lattices,\n described by the Bose-Hubbard model~\\cite{jaksch:olatt,fisher:bhubb},\n display a non-trivial quantum phase transition between a  superfluid and Mott insulator.\n The latter is an incompressible state with an integer number of atoms per site.\n  In a\n trap the phase diagram is revealed by the spatial structure of the gas: one has concentric superfluid and insulating shells.\n This structure has been elegantly explored by\n radio frequency (RF) spectroscopy~\\cite{campbell:ketterle-clock-shift}, a technique which has also given insight into strongly interacting Fermi gases across the BEC-BCS crossover~\\cite{bloch:many-body-cold-atoms-review}.\nHere we use a Random Phase Approximation (RPA) that treats fluctuations around the strong coupling Gutzwiller mean field theory to explore the radio-frequency spectrum of lattice bosons.\n\nWe find two key results: (1) Our previous sum-rule based analysis \\cite{hazzard:rf-spectra-sum-rule} of experiments at MIT~\\cite{campbell:ketterle-clock-shift} stands up to more rigorous analysis: in the limit of small spectral shifts, the RPA calculation reduces to that simpler theory.  (2)  In a gas with more disparate initial and final state interactions  (such as Cesium), the spectrum becomes more complex, with a bimodal spectrum appearing even in a homogeneous gas.  The bimodality reveals key features of the many-body state.  For example, in the limit considered by Sun, Lannert, and Vishveshwara~\\cite{sun:rf-spectra-condensate-probe}, the spectral features are related to the nearest-neighbor phase coherence.  In the Gutzwiller approximation, the phase coherence directly maps onto the condensate density.  In this paper we provide a physical picture of this result and explain how this bimodality can be  observed in a spatially resolved experiment.\n\n\\subsection{RF Spectroscopy}\nIn RF spectroscopy, a radio wave is used to flip the hyperfine spin of an atom from  $\\ket{a}$ to $\\ket{b}$.  The rate of excitation reveals details about the many-body state because the $\\ket{a}$ and $\\ket{b}$ atoms have slightly different interactions.  Generically the interaction Hamiltonian is $H_{\\rm int}=\\sum_{j} U_{aa} n_a (n_a-1)\/2+U_{bb} n_b (n_b-1)\/2+U_{ab}n_a n_b$, with $U_{aa}\\neq U_{ab}\\neq U_{bb}$, where $n_{\\sigma}$ is the number of $\\sigma$-state atoms on site $j$.  In the simplest mean-field picture, the energy needed to flip an atom on site $j$ from state $a$ to state $b$ is shifted by an energy $\\delta\\omega= U_{bb} n_b+(U_{ab}-U_{aa}) n_a$.  Applying this  picture to an inhomogeneous gas suggests that the absorption spectrum  reveals a histogram of the atomic density.  Such a density probe is quite valuable: in addition to the aforementioned examples, it was the primary means of identifying Bose-Einstein condensation in atomic hydrogen \\cite{fried:h}.\n\n\n\\begin{figure}[hbtp]\n\\setlength{\\unitlength}{1.0in}\n\\begin{picture}(3.10,2.8)\n\\put(0.,-0.05){\n\\includegraphics[width=1.5in,angle=0]{TwoCorrelnCartoonMottExc}\n}\n\\put(1.55,-0.05){\n\\includegraphics[width=1.5in,angle=0]{TwoCorrelnCartoonSFExc}\n}\n\\put(0.72,1.2){\n\\includegraphics[width=1.6in,angle=0]{TwoCorrelnCartoonNoExc}\n}\n\\put(1.48,1.29){(a)}\n\\put(0.71,0.0){(b)}\n\\put(2.26,0.){(c)}\n\\end{picture}\n\\caption{(Color online) Illustration of two types of RF-active excitations of the lattice superfluid near the Mott transition.  Open (blue) circles are atoms in the $\\ket{a}$ state, filled (red) circles are atoms in the $\\ket{b}$ state, and the arrows indicate a delocalized particle while other particles are localized.\n(a) Illustrates the initial superfluid state, consisting of a dilute gas of atoms moving in a Mott background. Final states in (b) and (c), show the excitation of a core or delocalized atom.\n}\n\\label{fig:two-correlations}\n\\end{figure}\n\nRecently Sun, Lannert, and Vishveshwara~\\cite{sun:rf-spectra-condensate-probe} found a bimodal spectrum in a special limit of this problem, as did Ohashi, Kitaura, and Matsumoto~\\cite{ohashi:rf-spectra-dual-character} in a separate limit, calling into question this simple picture. We give a simple physical interpretation of the bimodality.\nAs illustrated in Fig.~\\ref{fig:two-correlations}, the superfluid state near the Mott insulator can be caricatured as a dilute gas of atoms\/holes moving in a Mott background.  An RF photon can either flip the spin of one of the core atoms, or flip the spin of one of the mobile atoms.  The energy of these two excitations will be very different, implying that the RF spectrum should be bimodal.  Through our RPA calculation, we verify this feature, calculating the frequencies of the two peaks and their spectral weights.  Interestingly, this calculation reveals that the two excitations in our cartoon model are strongly hybridized.\n\nWe find that that for parameters relevant to experiments on $^{87}$Rb, that the degree of bimodality is vanishingly small and our previous  sum  rule arguments \\cite{hazzard:rf-spectra-sum-rule} accurately describe such experiments.  On the other hand, there are opportunities to study other atoms (for example, Na, Cs, Yb) for which the bimodality may be more pronounced.  Moreover, if the interactions or tunneling rates can be tuned via a spin-dependent lattice or a Feshbach resonance then this spectral feature will appear in a dramatic fashion.\n\nThis bimodal spectrum, with one peak produced by the ``Mott\" component and another by the ``superfluid\" component, is reminiscent of the spectrum of a finite temperature Bose gas in the absence of a lattice.  As described by Oktel and Levitov~\\cite{oktel:cs-ref}, in that situation one sees one peak from the condensate, and one from the incoherent thermal atoms.  We would expect that at finite temperature our ``Mott\" peak continuously evolves into their ``thermal\" peak.\n\n\n\n\\section{Bose-Hubbard Model}\n\\subsection{Model and RF spectra}\nIn the rf spectra experiments we consider, initially all atoms are in the $a$-internal state  and the rf pulse drives them to the $b$-state. Consequently,\nwe consider two-component bosons trapped in the periodic potential formed by interfering laser beams, described by a Bose-Hubbard model~\\cite{jaksch:olatt},\n\\begin{eqnarray}\nH &=& -\\!\\!\\!\\!\\!\\!\\sum_{\\begin{array}{c}{\\scriptstyle \\langle i,j\\rangle}\\\\{\\scriptstyle \\sigma=\\{a,b\\}}\\end{array}}  t_\\sigma c^\\dagger_{i,\\sigma} c_{j,\\sigma}\n+ \\sum_{\\sigma,j}  (V_{j,\\sigma}-\\mu_\\sigma) c^\\dagger_{j,\\sigma} c_{j,\\sigma}\n\\nonumber \\\\\n&&{}+\\sum_{j} \\left (  \\sum_{\\alpha,\\beta}\\frac{U_{\\alpha\\beta}}{2} c^\\dagger_{j,\\alpha} c^\\dagger_{j,\\beta} c_{j,\\beta} c_{j,\\alpha}\\right ) ,\\label{eq:bh-ham-defn}\n\\end{eqnarray}\nwhere $c_\\sigma$ and $c^\\dagger_\\sigma$ are the annihilation and creation operators for states in the internal state $\\sigma$,\n$\\mu_\\sigma$ is the chemical potential,  $V_{j,\\sigma}$ is the external potential with $\\delta$, the vacuum $a$-$b$ splitting, absorbed into it, $U_{\\alpha\\beta}$ is the $\\alpha$ state-$\\beta$ state on-site interaction strength, and $t_\\sigma$ is the hopping matrix element.\nThe interactions are tunable via Feshbach resonances and spin-dependent lattices are also available~\\cite{deutsch}. For this latter setup, the hopping matrix elements may be tuned by the intensity of the lattices, and introducing small displacements of the lattice will reduce the overlap between the Wannier states of $a$ and $b$ atoms, and therefore may also be an efficient way to  control the relative size of $U_{aa}$ and $U_{ab}$.\nThe interaction $U_{bb}$ will be irrelevant: we will only consider the case where there is a vanishingly small concentration  of  $b$-state particles.\nIn calculating the response to RF photons we will take $V_j=\\text{constant}$.   Trap effects will later be included through a local density approximation~\\cite{hazzard:rf-spectra-sum-rule} which is valid for slowly varying traps~\\cite{pollet:mi,bergkvist:mi,wessel:mi,batrouni:mi,demarco:stability,dupuis:mi-sf-review,sengupta:bhubb-rpa,konabe:out-coupling-single-ptcl-spec,\nmenotti:trivedi-single-ptcl-spectral-weight,ohashi:rf-spectra-dual-character}.\n\nExperimentally the RF spectrum is measured by counting the number of atoms transferred from state $a$ to $b$ when the system is illuminated by a RF pulse.  These dynamics are driven by a perturbation\n\\begin{eqnarray}\nH_{\\text{rf}} &=& \\sum_j \\gamma(t) c^\\dagger_{j,b} c_{j,a} + \\text{H.c.}.\\label{eq:rf-pert}\n\\end{eqnarray}\nwhere $\\gamma(t)$ is proportional to the time-dependent amplitude of the applied RF field multiplied by the dipole matrix element between states $a$ and $b$:  typically $\\gamma$ is a sinusoidal pulse with frequency $\\omega$ with  a  slowly varying envelope ensuring a small bandwidth.  Due to the small wave-number of RF photons, recoil can be neglected.\n\nFor a purely sinusoidal drive, the number of atoms transferred per unit time for short times is\n\\begin{eqnarray}\n\\Gamma(\\omega) &=& \\frac{2\\pi}{\\hbar} \\sum_{i,f}p_i \\delta(\\omega-(E_f-E_i)) \\left|\\opsandwich{f}{H_{\\text{rf}}}{i}\\right|^2\\label{eq:rf-spectra-fgr-1}\n\\end{eqnarray}\nwhere  the sum is over the initial states (occupied with probability $p_i=e^{-\\beta E_i}$) and the final states, all of which are eigenstates of  $H$ with energies $E_i$ and $E_f$.  We will restrict ourselves to $T=0$ and the physically relevant case where the initial states contain no $b$-atoms.\n\n\\subsection{Sum Rules}\nTaking moments of Eq.~\\eqref{eq:rf-spectra-fgr-1}~\\cite{oktel:cs-ref,oktel:cs-ref2,oktel:rf-spectra},\nthe mean absorbed photon frequency is\n\\begin{eqnarray}\n\\expec{\\omega} &=&\n=\\frac{\\int \\! d\\omega \\,\\omega \\Gamma(\\omega)}{\\int \\! d\\omega \\, \\Gamma(\\omega)}=\n\\frac{\\expec{[H_{\\text{rf}},H]H_{\\text{rf}}}}{\\expec{H_{\\text{rf}}^2}} \\label{eq:average-shift-given-commutators}\\\\\n&=& \\delta -z(t_b-t_a) f_c\n+ \\left (  U_{ab}-U_{aa}\\right )  g_2 \\expec{n}.\\label{eq:sum-rule-general}\n\\end{eqnarray}\nWe defined $\\delta$ to be the vacuum $a$-$b$ splitting,  the local phase coherence factor is\n\\begin{eqnarray}\nf_c &=&\\frac{\\expec{c^\\dagger_{i,a} c_{j,a} }}{\\expec{n}},\\label{eq:cond-dens}\n\\end{eqnarray}\nwith $i$ and $j$ nearest neighbors,\nthe site filling is $n\\equiv c^\\dagger_a c_a$,\nand the lattice coordination is $z$.  The  zero-distance density-density correlation function is\n\\begin{eqnarray}\ng_2 &=& \\frac{\\expec{c_a^\\dagger c_a^\\dagger c_a c_a}}{\\expec{n}^2}.\n\\end{eqnarray}\n  The second term in Eq.~\\eqref{eq:sum-rule-general} may be interpreted as the mean shift in the kinetic energy when the spin of an atom is flipped.  In particular, within a strong-coupling mean-field picture $\\langle c^\\dagger_{i,a} c_{j,a} \\rangle=\\langle c^\\dagger_{i,a}\\rangle\\langle c_{j,a} \\rangle$ is the condensate density, which can therefore be measured with this technique.  The second term in  Eq.~\\eqref{eq:sum-rule-general} is the shift in the interaction energy.\n\nOur subsequent approximations will satisfy this sum rule.\nThis is non-trivial: for example, even in simultaneous limits of  $t_b=0$, $U_{ab}=U_{aa}$, and $t_a\\rightarrow 0$ considered in Ref.~\\cite{sun:rf-spectra-condensate-probe}, their results violate this sum rule by a factor of $\\sim 3$.\n\nSince it plays no role in the remainder of the discussion, we will set to zero the vacuum level splitting: $\\delta=0$.  This amounts to working in a ``rotating frame\".\n\n\n\\section{Random phase approximation\\label{sec:rpa}}\n\n\\subsection{General setup and solution}\n\n\nTo calculate the RF spectrum we employ a time-dependent strong-coupling mean-field theory which\n includes $k=0$ fluctuations around the static strong-coupling Gutzwiller mean field theory~\\cite{fisher:bhubb}.\nThis mean field theory is exact in the deep Mott limit and in the deep superfluid when $t_a=t_b$, and it\nyields fairly accurate ground states in the intermediate regime~\\cite{pollet:mi,bergkvist:mi,wessel:mi,batrouni:mi,demarco:stability}.\nRefs.~\\cite{menotti:trivedi-single-ptcl-spectral-weight,ohashi:rf-spectra-dual-character} previously used analogous RPA's to calculate the Bose-Hubbard model's quasiparticle spectra and RF spectra with $U_{ab}=0$, which reduces to the $k=0$ single particle spectra.\n\n\n\\begin{figure}[hbtp]\n\\setlength{\\unitlength}{1.0in}\n\\subfigure[]{\n\\hspace{-0.1in}\\includegraphics[width=1.625in,angle=0]{HomSystemLargeUSametMu198}\n}\n\\subfigure[]{\n\\hspace{-0.05in}\\includegraphics[width=1.625in,angle=0]{HomSystemLargeUSametMu202}\n}\n\\subfigure[]{\n\\hspace{-0.1in}\\includegraphics[width=1.675in,angle=0]{HomSystemRbSametMu2000}\n}\n\\subfigure[]{\n\\hspace{-0.15in}\n\\includegraphics[width=1.675in,angle=0]{HomSystemRbSametMu2004}\n}\n\\caption{(Color online) Homogeneous system's spectral density as a function of $\\omega\/U_{aa}$ and $t_a\/U_{aa}$ (whiter indicates larger spectral density) compared with sum rule prediction (red, single line).  Delta functions are broadened to Lorentzians for visualization purposes.\n(a,b)  We take $U_{ba}=1.2 U_{aa}$ and $t_b=t_a$, with (a) $\\mu = 1.98$ and (b) $\\mu=2.02$. (c,d) We take parameters corresponding to typical $^{87}$Rb experiments: $U_{ba}=1.025 U_{aa}$ and $t_b=t_a$, and take (c) $\\mu = 1.999$ and (d) $\\mu=2.004$.   In both cases, a double peak structure is visible, but the region of the phase diagram in which it is important is much smaller for $^{87}$Rb parameters than for Fig~(a,b)'s parameters.\n\\label{fig:homog-rf-spec}\n}\n\\end{figure}\n\n\nWe use the homogeneous time-dependent Gutzwiller variational ansatz\n\\begin{eqnarray}\n\\hspace{-0.1in}\\ket{\\psi(t)} \\!&=& \\!\\bigotimes_i\\! \\left [  \\sum_n \\left (  f_n(t) \\ket{n,0}_i + g_n(t) \\ket{n-1,1}_i\\right ) \\right ] \\label{eq:gutz-variational}\n\\end{eqnarray}\nwhere $\\ket{n_a,n_b}_i$ is the state at site $i$ with $n_a$ particles in the $a$ state and $n_b$ in the $b$ state.  The equation of motion for $f_n(t)$ and $g_n(t)$ are derived by minimizing the action $S=\\int dt {\\cal L}$, with Lagrangian\n\\begin{eqnarray}\n{\\cal L}=  \\langle \\psi |i\\partial_t |\\psi\\rangle - \\langle\\psi| H|\\psi \\rangle-\\lambda \\langle\\psi|\\psi\\rangle,\n\\end{eqnarray}\nwhere $\\lambda$ is a Lagrange multiplier which enforces conservation of probablility.  At time $t=-\\infty$, where $\\gamma(t)=0,$ we take $g_n=0$, and choose $f_n$ to minimize $ \\langle\\psi| H|\\psi \\rangle$,\n\\begin{eqnarray}\n\\lambda f_n &=& -t_a z \\left ( \\sqrt{n} \\alpha^* f_{n-1} + \\sqrt{n+1}\\alpha f_{n+1}\\right )  \\nonumber \\\\\n    &&{}+\\left (  \\frac{U_{aa}}{2} n(n-1)-\\mu n\\right )  f_n,\\label{eq:gutz-fns}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\alpha &=& \\sum_n \\sqrt{n} f_n^* f_{n-1}.\\label{eq:mean-field}\n\\end{eqnarray}\n Solving the subsequent dynamics to quadratic order in $\\gamma$, one finds\n\\begin{eqnarray}\n\\Gamma(t) &=& N_s \\int \\! dt' \\,\\gamma(t)\\gamma(t')\\chi^{(R)}(t-t'),\n\\end{eqnarray}\nwhere the retarded response function is\n\\begin{eqnarray}\n\\chi^{(R)}(t) &=& \\frac{1}{i} \\sum_n \\sqrt{n} \\left (    G_n^*(t) f_n -  G_n(t) f_n^* \\right ) .\\label{eq:rf-spectra-in-terms-of-Gns}\n\\end{eqnarray}\nThe Green's functions $G_n(t)$ satisfy the equations of motion for the $g_n$'s in the absence of an RF field, but in the presence of a delta function source, and boundary condition $G_n(t)=0$ for $t<0$.\nThe relevant equations are simplest in Fourier space, where\n$G_n(\\omega) = \\int \\! dt\\, e^{i\\omega t} G_n(t)$ obeys\n\\begin{eqnarray}\n\\sqrt{n} f_n &=&-\\omega G_n+\\sum_m \\Lambda_{nm} G_m\\label{eq:Gn-eqn}\n\\end{eqnarray}\nwhere $\\Lambda=\\bar \\Lambda+\\Theta$ is a Hermitian matrix.  The tridagonal part $\\bar\\Lambda$ is\n\\begin{eqnarray}\n\\bar\\Lambda_{n,n+1}&=&-zt_a\\alpha \\sqrt{n}\\\\\n\\bar\\Lambda_{n,n-1}&=&-zt_a\\alpha^* \\sqrt{n-1}\\\\\n   \\bar\\Lambda_{nn}&=&  -\\mu n -\\lambda +\\frac{U_{aa}}{2}(n-1)(n-2) \\\\\\nonumber&&+ U_{ab}(n-1).\n\\end{eqnarray}\nThe remaining contribution, $\\Theta$, is\n\\begin{eqnarray}\n \\Theta_{nm}= -z t_b f_{n-1} f^*_{m-1}.\n\\end{eqnarray}\nSpecializing to the case where $\\alpha(t)=\\alpha e^{i\\omega t}$, the response is given\nin terms of normalized eigenvectors $v_m$, with $\\sum_{m} \\Lambda_{nm} v^{(j)}_m=\\epsilon_j v_n^{(j)}.$  It takes the form of\na sum of delta-functions,\n\\begin{eqnarray}\nI(\\omega) &=& \\sum_j \\left(\\sum_m \\sqrt{m} f_m  v_m^{(j)}\\right)^2 \\delta(\\omega-\\epsilon_j)\\label{eq:spectra-general}.\n\\end{eqnarray}\n\nThe $f_n$'s are found at each point in the phase diagram by starting with a trial $\\alpha$, solving  Eq.~\\eqref{eq:gutz-fns}, then updating $\\alpha$ via Eq.~\\eqref{eq:mean-field} and iterating.  We find that almost all spectral weight typically lies in only one or two peaks.  Fig.~\\ref{fig:homog-rf-spec} shows sample spectra.\n The superfluid near the Mott state displays a multi-modal spectrum, but in the weakly interacting limit only a single peak is seen.  An avoided crossing is clearly visible in these plots.\n Fig.~\\ref{fig:double-branch-3d} shows the manifold of  spectral peaks in the  $t_a\/U_{aa}$ and  $\\mu\/U_{aa}$ plane, using height to denote frequency and opacity to denote spectral weight.  Taking moments of $\\chi^R(\\omega)$, we see that Eq.~\\eqref{eq:sum-rule-general} is satisfied.\n\n \\begin{figure}[hbtp]\n\\setlength{\\unitlength}{1.0in}\n\\begin{picture}(3.50,3.70)\n\\put(-0.4,0){\n\\includegraphics[width=4.in,angle=0]{spectrum-3d-white-background}\n}\n\\put(0.25,1.25){$\\mu\/U_{aa}$}\n\\put(3.0,1.0){$t\/U_{aa}$}\n\\put(1.99,1.45)\n{\n$\\omega\/U_{aa}$\n}\n\\end{picture}\n\\vspace{-0.6in}\n\\caption{(Color online)  Three-dimensional plot of RF spectral frequencies versus rescaled hopping $t_{a}\/U_{aa}$ and rescaled chemical potential $\\mu\/U_{aa}$ for  $U_{ab}\/U_{aa}=1.2$.\nLarger opacity indicates larger spectral weight.\nWhite lines represent contours of fixed $\\mu$, $U$ and $\\omega$.  The main branch is colored so that the progression from green to red to blue corresponds to increasing $\\omega$.\nThe double peaked spectrum is apparent from the ``double-valuedness\" of the surface.\n To avoid clutter, numerical values are omitted from the axes: the Mott plateaus occur at frequencies $\\omega=0,0.2U_{aa}$ and $0.4U_{aa}$, are each $U_{aa}$ wide and the first lobe's critical $t$ is around $0.029U_{aa}$ in 3D.\n}\n\\label{fig:double-branch-3d}\n\\end{figure}\n\n\n\n\n\n\\subsection{Limiting Cases}\nAlthough finding the spectrum in Eq.~\\eqref{eq:spectra-general} is a trivial numerical task, one can gain further insight by considering limiting cases.  First, when $U_{ab}=U_{aa}$ and $t_a=t_b$ the system possesses an $SU(2)$ symmetry.  In this limit we find that $G_n(t)=-i\\sqrt{n} f_n \\theta(t)$ is constant for $t>0$.  Thus our approximation gives a spectrum $I(\\omega)$ which is proportional to $\\delta(\\omega)$.  This result coincides with the exact behavior of the system: the operator $X=\\sum_j b_j^\\dagger a_j$ is a ladder operator, $[H,X]=\\delta X$, and can only generate excitations with energy $\\delta$ (set equal to zero in our calculation).    The fact that our approximations correctly capture this behavior is nontrivial: in a field theoretic language one would say that our equation of motion approach includes the vertex corrections necessary for satisfying the relevant ``Ward identities\"~\\cite{pethick:pseudopot-breakdown,baym:self-consistent-approx,zinn-justin:qft}.\n\nThe current $^{87}$Rb experiments are slightly perturbed from this limit, with $\\eta\\equiv (U_{ab}-U_{aa})\/U_{aa}\\approx-0.025$ and $t_b=t_a$.  We find that the $\\delta$-function is shifted by a frequency proportional to $\\eta$, but that the total spectral weight remains concentrated on that one frequency: the sum of the spectral weights at all other frequencies scale as $\\eta^2$.  Consequently it is an excellent approximation to treat the spectrum as a delta-function, and our RPA calculation reduces to the results in \\cite{hazzard:rf-spectra-sum-rule}.  We emphasize however that other atoms, such as Cesium, can be in a regime where $\\eta$ is large.\n\n\n\n\n\n\n\nWe gain further insight by considering the superfluid near the Mott phase with $t_a\/U_a\\ll1$.  Here one can truncate the basis to two states with total particle number $n$ and $n+1$ on each site.  Then the $f_n$'s and $G_n$'s can be found analytically: one only needs to solve  $2\\times2$ linear algebra problems.  In the $t_b=0$, $U_{ab}=U_{aa}$ limit, this is similar  to Ref.~\\cite{sun:rf-spectra-condensate-probe}'s approach, but includes the hopping self consistently, allowing us to satisfy the sum rule Eq.~\\eqref{eq:sum-rule-general}.\nThis\ntruncation\nis exact\nin the small $t_a$ limit, and yields\n\\begin{eqnarray}\n\\chi^{(R)}(\\omega) &=& A_+ \\delta(\\omega-\\omega_+) + A_- \\delta(\\omega-\\omega_-)\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\omega_{\\pm} =& \\frac{\\epsilon_1 + \\epsilon_2}{2} \\pm \\sqrt{\\Delta^2 + \\left ( \\frac{\\epsilon_1-\\epsilon_2}{2}\\right ) ^2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\epsilon_1 &\\equiv& (U_{ab}-U_{aa}) (n-1) +  z t_a f_{n+1}^2 (n+1) \\nonumber\\\\\n\\epsilon_2 &\\equiv& (U_{ab}-U_{aa})n + z\\left [  t_a (n+1) -  t_b\\right ]  f_n^2\n\\nonumber\\\\\n\\Delta &\\equiv& - \\sqrt{n(n+1)} t_a z f_n f_{n+1}.\n\\end{eqnarray}\nif $n\\ge 1$ and\n\\begin{eqnarray}\n\\epsilon_1 &\\equiv& z t_a f_1^2\\nonumber\\\\\n\\epsilon_2 &\\equiv&  z (t_a-t_b) f_0^2\n\\nonumber\\\\\n\\Delta &\\equiv& 0.\n\\end{eqnarray}\nif $n= 0$ (here, only the $\\epsilon_2$ peak has non-zero spectral weight).\nWe omit the cumbersome analytic expressions for the spectral weights $A_\\pm$. The spectrum consist of two peaks -- hybridized versions of the excitations caricatured in Fig.~\\ref{fig:two-correlations}.  One can identify $\\epsilon_1$ and $\\epsilon_2$ as the energies of those caricature processes, recognizing that the hybridization term, $\\Delta$, grows with $t_a$.  The avoided crossing between these modes is evident in Fig.~\\ref{fig:homog-rf-spec}.\n\n\n\n\n\n\\subsection{Inhomogeneous spectrum}\n\nWe model the trapped spectrum through a local density approximation.  We assume that a given point in the trap has the properties of a homogeneous gas with chemical potential $\\mu(r)=\\mu_0-V(r)$.  In Fig.~\\ref{fig:trap-rf-spec} we show the density profile and the spectrum corresponding to each point in space.  Also shown is the trap averaged spectrum.\nThe bimodality of the homogeneous spectrum is quite effectively washed out by the inhomogeneous broadening of the trap.  On the other hand, if one images the atoms flipped into the $b$ state as in Ref.~\\cite{campbell:ketterle-clock-shift}, there is a clear qualitative signature of the bimodality.  If one excites the system with an RF pulse whose frequency lies between the resonant frequencies of two Mott plateaus, one will excite two ``shells\" of atoms.  These shells should be clearly visible, even in column integrated data.\n\n\n\\begin{figure}[hbtp]\n\\setlength{\\unitlength}{1.0in}\n\\begin{picture}(3.5, 7.55)\n\\put(0.18,0.)\n{\\includegraphics[width=4.4in,angle=0]{trap-ave-compare-2}}\n\\end{picture}\n\\vspace{-0.68in}\n\\caption{\n(Color online) (a) Density $n$ as a function of distance to trap center rescaled by the lattice spacing, $r\/d$, in a local density approximation.  For all subfigures, we take $t_a\/U_{aa}=0.004$, which is moderately smaller than the tip of the first Mott lobe.\n(b-e) Left: spectrum  of a homogeneous gas with density $n(r)$, representing the spatially resolved spectrum observed in an experiment on a trapped gas.  Horizontal axis is position, vertical is frequency, color from dark to light represents increasing spectral density.  Continuous (red) curve denotes sum rule result for $\\langle \\omega \\rangle$.\nWe round the $\\delta$-functions to Lorentzians for visualization.  Right: trap-averaged spectrum for a 3D trap within our RPA (black, solid line) compared with sum rule (red, dashed line).  (b) $U_{ab}=1.2 U_{aa}, t_b=t_a$ (c) $U_{ab}= U_{aa}, t_b=t_a+0.1U_{aa}$ (d) $U_{ab}=1.2 U_{aa}, t_b=t_a+0.1U_{aa}$ (e) $^{87}$Rb parameters: $U_{ab}=1.025 U_{aa}, t_b=t_a$.\n}\n\\label{fig:trap-rf-spec}\n\\end{figure}\n\n\\section{Conclusions and discussion\\label{sec:discussion}}\nIn this paper we have shown that the RF spectra of a homogeneous Bose gas in an optical lattice will have two (or more) peaks in the superfluid state when the parameters are tuned close to the superfluid-Mott insulator phase transition. Physically, this bimodality is a result of the strong correlations in the system.  These correlations result in two distinct forms of excitations (which are strongly hybridized): those involving ``core\" atoms, and those involving delocalized atoms.  When $\\eta=(U_{ab}-U_{aa})\/U_{aa}$ is small, such as in the experiments on $^{87}$Rb, this bimodality is absent.\n\nOur approach, based upon applying linear response to a time dependent Gutzwiller mean field theory, is both simple and quite general.   It allows arbitrary interactions between both spin states, and it allows arbitrary spin-dependent hopping rates.  The major weakness of the theory is that it fails to fully account for short range-correlations: the atoms are in a quantum superposition of being completely delocalized, and being confined to a single site.  The physical significance of this approximation is most clearly seen when one considers the case where the final-state atoms have no interactions, $U_{ab}=0$, and see no trap or lattice.  Imaging the $b$-atoms after a time-of-flight is analogous to momentum resolved photoemission \\cite{jin:photoemission}, and would reveal the dispersion relationship of the single-particle excitations.  The fact that the spectrum consists of two sharp peaks means that all of the non-condensed atoms are approximated to have the same energy.  One will also see that their momentum is uniformly distributed throughout the first Brillioun zone.  In the strong lattice limit, where  the bandwidth is small, this approximation is not severe.\n\n\\section{Acknowledgements}\nWe thank Sourish Basu, Stefan Baur, Stefan Natu, Kuei Sun, Smitha Vishveshwara, Henk Stoof, Ian Spielman, and Mukund Vengalattore for useful discussions.  This material is based upon work supported by the National Science Foundation through grant No. PHY-0758104, and partially performed at the Aspen Center for Physics.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\n\n\n\n\n\nIn this article, we study the asymptotic\nerror rates for integration by quasi-Monte Carlo (QMC) as $n\\to\\infty$\nwhile $f$ is fixed.\nMost of the error upper bounds in QMC are based on fooling functions\n$f_n$ that, given $n$ integration points, are poorly integrated.\nBy contrast, most of the\npublished empirical results follow the integration\nerror for a single integrand $f$ as $n$ increases.\nThe upper bounds have us play against an adaptive adversary choosing\nan unfavorable $f_n$ at each sample size $n$\ninstead of keeping $f$  fixed as $n\\to\\infty$.\nThe error bounds, that we describe in more detail below,\nare typically $O(\\log(n)^r\/n)$ where $r$ can be as large as the\ndimension of the integrand's domain.\nThese bounds can be enormous and, to our knowledge, there has never\nbeen an integrand exhibited where a standard QMC point set\nis shown to need $r>1$.\nThat raises the question of whether $r>1$ is simply\na consequence of the adversarial formulation.\nThe alternative is that some function $f$ is a `persistent fooling function'\ncausing large errors for infinitely many $n$.\nIn an earlier version of this article we posed a question about whether\n{\\sl any} integrand of bounded variation in the sense of Hardy and Krause (BVHK)\non $[0,1]^d$ for any $d\\geqslant1$\nhas an integration error above $c\\log(n)^r\/n$ for infinitely many $n$\nwith $r>1$ and $c>0$ using a digital sequence such as Sobol's for the integration points.\nFor background on bounded variation in the sense of Hardy and Krause\nor in the sense of Vitali, we refer the reader to \\cite{variation}.\n\nWe owe a great debt to Erich Novak who pointed us to some unpublished\nwork of Trojan described in detail in Chapter 10 of\nthe information based complexity monograph of Traub, Wasilkowski\nand Wo\\'zniakowsk \\cite{trau:wasi:wozn:1988}.\nTrojan's work is about very general problems of computing\nlinear operators on Banach spaces based on the values of $n\\to\\infty$ linear functionals.\nHe shows that the adversarial worst case convergence rate is also very nearly the attained\nrate for some specific problem instances.\nIn the QMC context, that work pertains to a single integrand $f$\nas the number $n$ of evaluation points diverges to infinity.\nA consequence of that work is that for any infinite\nsequence of integration points, there are indeed integrands\nin BVHK$[0,1]^d$ with an absolute error larger than $c\\log(n)^{(d-1)\/2}\/(n\\log\\log(n))$\ninfinitely often, for any $c>0$.\nFurthermore, those integrands are present within a reproducing kernel Hilbert space (RKHS)\non a certain unanchored space.  They are dense in that space, though\nthis does not mean that the usual Gaussian processes on such spaces\ngive them positive measure.\nWe only get $r=(d-1)\/2$ logarithmic factors instead of $d$ or $d-1$\nof them.  The explanation is that we use an $L^2$ bound\njust like Roth \\cite{roth:1954} used in getting a lower bound on star discrepancy.\nA different analysis might yield larger $r$.\nThe $\\log\\log(n)$ factor in the denominator\ncan be replaced by a sequence that diverges more slowly.\n\nWe have not been able to construct a function in BVHK$[0,1]^d$\nthat provably needs $r>1$ powers of $\\log(n)$\nfor a Sobol' sequence \\cite{sobo:1967:tran}\nor the Halton \\cite{halt:1960}\nsequence, even when exploiting known weaknesses of commonly used QMC sequences.\nSo, we are left to wonder: where are the logs?\n\nAn outline of this paper is as follows.\nSection~\\ref{sec:back} presents some results from the QMC literature\nand introduces notation on some QMC sequences.\nSection~\\ref{sec:proofofbound} proves our main result\ndescribed above on existence of persistent fooling functions.\nSection~\\ref{sec:d=1} looks at the case $d=1$\nto exhibit some example functions requiring $r=1$\nfor the van der Corput sequence: $f(x)=1\\{x<2\/3\\}$\nand $f(x)=x$.\nSection~\\ref{sec:d=2} computes error for some $d=2$\ndimensional problems. The Halton and Sobol'\npoints there are closely related to van der Corput\npoints, yet two dimensional generalizations of the\nproblematic integrands from Section~\\ref{sec:d=1}\nfail to show a need for $r>1$.\nIn fact some of the empirical results are more consistent\nwith an $O(1\/n)$ error.\nSection~\\ref{sec:bigm} computes a local discrepancy\n$\\delta({\\boldsymbol{z}}$)\nfor  Sobol' nets with $d=2,3$ and $1\\leqslant m\\leqslant100$ where all components of ${\\boldsymbol{z}}$ equal $2\/3$\nchosen because $2\/3$ is difficult to approximate by dyadic rationals.\nIt also includes $d=4$ for $1\\leqslant m\\leqslant 50$.\nThe cases with $d>2$ are the closest we have found to needing $r>1$\nbut are inconclusive.\nSection~\\ref{sec:discussion} discusses these results.\n\n\\section{Background}\\label{sec:back}\n\nFrom the Koksma-Hlawka inequality \\cite{hick:2014}\ncombined with convergence rates for the star discrepancy \\cite{nied:1992},\nwe get the widely quoted convergence rates\nfor the error in quasi-Monte Carlo integration\nof a function $f:[0,1]^d\\to{\\mathbb{R}}$.\nAn integrand $f$ of bounded variation in the\nsense of Hardy and Krause, written $f\\in\\mathrm{BVHK}[0,1]^d$,\ncan be integrated with error $O(n^{-1}(\\log n)^{d-1})$\nusing $n$ function evaluations.  If we must use the first $n$ points\nof an infinite sequence, then the rate\n$O(n^{-1}(\\log n)^{d})$ is attainable.\nThis article is mostly about the infinite sequence version.\nBoth of these rates are often written $O(n^{-1+\\epsilon})$\nwhere $\\epsilon$ can be any positive constant\nbut $\\log(n)^d\\gg n^\\epsilon$ for many use cases of interest.\n\nFor high dimensional problems, such powers of\n$\\log(n)$ are enormous and then\nthere is genuine uncertainty about whether\n$O(n^{-1}(\\log n)^{d})$ is better than the\nroot mean squared error (RMSE) of $O(n^{-1\/2})$ from plain\nMonte Carlo (MC) at practically relevant  $n$.\nThese rates omit three implied constants: one in\nthe star discrepancy (see \\cite{faur:lemi:2014} for information),\none in the total variation of $f$\nand the third one is the standard deviation of $f$.\nThese unknown constants contribute to\nuncertainty about the $n$ at which QMC would\noutperform MC.\nA further complication is that the Koksma-Hlawka\nbound is for a worst case integrand.\nThe situation is quite different in Monte Carlo\n(MC) where the rate $\\sigma n^{-1\/2}$ holds for all finite $n$ making\nit simultaneously a guide to how accuracy progresses\nfor a single integrand of variance $\\sigma^2$ and\nthe RMSE formula (upper and lower bound)\nfor all integrands of variance~$\\sigma^2$.\n\n\nThat observed errors for realistic $n$ and large $d$\ndo not follow a trend like $\\log(n)^d\/n$\nwas reported by Schlier \\cite{schl:2004} among others.\nThat work also found that the variance of $f({\\boldsymbol{x}})$\nwas more useful than its total variation in explaining the empirical accuracy\nof QMC integration on test functions, despite the fact\nthat proved theoretical bounds for QMC error use total\nvariation and variance does not require any of the smoothness that\nQMC relies on.\nMany papers include empirically estimated convergence\nrates for individual $f$ found by fitting a regression model\nfor log error versus $\\log(n)$. See for instance L'Ecuyer \\cite{lecu:2018}.\nWe do not see results that look like a large power of $\\log(n)$\nis present.\n\nThis mismatch between empirical results and theoretical ones\nis troubling.  Empirical results alone don't give enough confidence\nthat they will apply to future problems.  Similarly, bounds that are\nfavorable (but asymptotic) or unfavorable (but worst case) could\nalso fail to provide a reliable guide to attained accuracy.\nThis mismatch has brought practical difficulties.\nFor instance, the logarithmic powers\nin the Koksma-Hlawka bound led Bratley, Fox and Niederreiter \\cite{brat:fox:nied:1992}\nto limit their software to $d\\leqslant 12$.\n\nFor some randomizations of digital nets\nthe RMSE is $O(n^{-1\/2})$ whenever $f\\in L^2[0,1]^d$ \\cite{snxs}\nand is also $O(\\log(n)^{(d-1)\/2}\/n)$ under further smoothness\nconditions \\cite{smoovar,localanti,yue:mao:1999}.\nIn such cases the large powers of $\\log(n)$ are\nsubject to a simultaneous $O(n^{-1\/2})$ bound that\nlimits how much worse randomized QMC can be compared to MC\nfor finite $n$.\nIt would be interesting to know whether something like that also\nholds for plain QMC. Perhaps the coefficient of $\\log(n)^r\/n$ is ordinarily\nvery small, or the effect is only relevant for impractically large $n$ or\nperhaps not even present for most commonly investigated integrands.\nFor a survey of randomized QMC see L'Ecuyer and Lemieux \\cite{lecu:lemi:2002}.\n\n\n\n\n\n\n\n\n\n\nWe conclude this section by describing $(t,m,d)$-nets and $(t,d)$-sequences\nusing the formulation from Niederreiter \\cite{nied:1987}.\nLet $b\\geqslant2$ be an integer.\nFor ${\\boldsymbol{k}} = (k_1,\\dots,k_d)\\in{\\mathbb{N}}} % natural numbers {1, 2, ..._0^d$ and\n${\\boldsymbol{c}} = (c_1,\\dots,c_d)\\in\\Z_0^d$\nwith $0\\leqslant c_j<b^{k_j}$ the half open hyper-rectangle\n\\begin{align}\\label{eq:elemint}\nE({\\boldsymbol{k}},{\\boldsymbol{c}}) = \\prod_{j=1}^d\\Bigl[\\frac{c_j}{b^{k_j}}, \\frac{c_j+1}{b^{k_j}}\\Bigr)\n\\end{align}\nis called an elementary interval in base $b$.\nIt has volume $b^{-|{\\boldsymbol{k}}|}$ where $|{\\boldsymbol{k}}|=\\sum_{j=1}^dk_j$.\nWe define its indicator function as\n$$\nI_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{x}})\n=I_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{x}};b)\n= 1\\{{\\boldsymbol{x}}\\in E({\\boldsymbol{k}},{\\boldsymbol{c}})\\}.\n$$\n\nFor integers $m\\geqslant t\\geqslant0$, $b\\geqslant2$, $n=b^m$ and $d\\geqslant1$\nthe points ${\\boldsymbol{x}}_0,\\dots,{\\boldsymbol{x}}_{n-1}\\in[0,1]^d$ are\na $(t,m,d)$-net in base $b$ if\n$$\n\\frac1n\\sum_{i=0}^{n-1} I_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{x}}_i) =\n\\int_{[0,1]^d}I_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{x}}){\\mathrm{d}}{\\boldsymbol{x}}=b^{-|{\\boldsymbol{k}}|}\n$$\nholds for all elementary intervals with $|{\\boldsymbol{k}}|\\leqslant m-t$.\nOther things being equal, smaller $t$ are better and\n$t=0$ is best, but the choices of $d$, $b$, and $m$\nimpose a lower bound on the possible $t$ which\nmay rule out $t=0$.\nThe minT project of \\cite{schu:schm:2009}\ntracks the best known values of $t$ as well as some\nlower bounds on $t$.\n\nIn this paper we emphasize infinite sequences.\nThe sequence ${\\boldsymbol{x}}_i\\in[0,1]^d$ for integers $i\\geqslant0$\nis a $(t,d)$-sequence in base $b$ if\n${\\boldsymbol{x}}_{rb^m},\\dots,{\\boldsymbol{x}}_{(r+1)b^m-1}$ is a $(t,m,d)$-net\nin base $b$ for all $m\\geqslant t$ and all integers $r\\geqslant0$.\nThese are extensible $(t,m,d)$-nets in that\nthe first $b^\\ell$ points of a $(t,d)$-sequence form a $(t,m+\\ell,d)$-net\nfor any integer $\\ell\\geqslant1$.\nThe most used $(t,d)$-sequences are the\n$(0,d)$-sequences in prime bases $p\\geqslant d$\nof Faure \\cite{faur:1982} and the $(t,d)$-sequences\nin base $2$ of Sobol' \\cite{sobo:1967:tran}.\n\nWe will make special use of the\nvan der Corput sequences in base $b\\geqslant2$.\nThese are $(0,1)$-sequences in base $b$.\nIf we write the natural number $i=\\sum_{k=1}^\\infty i_k b^{k-1}$\nwith digits $i_k\\in\\{0,1,\\dots,b-1\\}$, then the sum only has $K(i)<\\infty$\nnonzero terms and we then set\n$x_i = \\sum_{k=1}^{K(i)}b^{-k}i_k$.\nThis sequence has star discrepancy $D^*_n=O(\\log(n)\/n)$.\nFor $n=b^m$ it is a left endpoint rule containing points\n$i\/n$ for $0\\leqslant i< n$ and so it has $D^*_n = 1\/n$\nby \\cite[Theorem 2.6]{nied:1992}.\nThe original van der Corput sequence in base $b=2$\nis from \\cite{vand:1935:I}.\n\n\n\n\n\\section{Proof of the lower bound}\\label{sec:proofofbound}\n\nWe begin with a general theorem on worst-case errors.\nLater we specialize it to the QMC setting.\n\n\\begin{theorem}\\label{thm:asymptoticrate}\nLet $(F,\\Vert\\cdot\\Vert)$ be a Banach space and $S$ be a linear functional on $F$. For a sequence of continuous linear functionals $L_n$ on $F$, define\n\\begin{align*}\nN_n(f)&=(L_1(f),\\ldots,L_n(f)),\\quad\\text{and}\\\\\nr_n&=\\sup\\bigl\\{|S(f)|\\bigm| f\\in F, N_n(f)=\\bszero, \\Vert f\\Vert\\leqslant 1\\bigr\\}.\n\\end{align*}\nThen for any sequence of mappings $\\phi_n$ from $\\mathbb{R}^n$ to $\\mathbb{R}$, there exists $f\\in F$ such that\n\\begin{equation}\\label{eqn:target}\n    \\limsup_{n\\to \\infty}\\frac{ |S(f)-\\phi_n(N_n(f))|}{(\\log\\log n)^{-1}r_n}=+\\infty.\n\\end{equation}\n\\end{theorem}\n\\begin{proof} This follows from Theorem 2.1.1 in Chapter 10 of\n\\cite{trau:wasi:wozn:1988}\nwho cite unpublished work by Trojan.\n\\end{proof}\n\nWe will set $F$ to be some reproducing kernel Hilbert space (RKHS) contained in BVHK$[0,1]^d$, $S(f)$ to be $\\int_{[0,1]^d}f({\\boldsymbol{y}})\\,{\\mathrm{d}}{\\boldsymbol{y}}$ and $N_n(f)$ to be $(f({\\boldsymbol{x}}_0),\\ldots,f({\\boldsymbol{x}}_{n-1}))$. We note that evaluation at ${\\boldsymbol{x}}_i$ is a continuous linear functional in an RKHS. As the theorem suggests, we can use $r_n\/\\log\\log n$ as a lower bound on the asymptotic convergence rate achievable by all functions in BVHK$[0,1]^d$, so it remains to determine a lower bound on $r_n$.\n\nTo derive such a lower bound, we will apply the proof techniques used in Roth's lower bound on the $L^2$ discrepancy.\nSee Chen and Travaglini \\cite{chen:trav:2009} for a nice summary.\nDick, Hinrichs and Pillichshammer \\cite{dick:hinr:pill:2015}\nuse this strategy to prove that the worst-case error of any equal-weight quadrature rule is\nlower bounded by $\\Omega(n^{-1}(\\log n)^{(d-1)\/2})$ when $F$ is the RKHS with kernel $K({\\boldsymbol{x}},{\\boldsymbol{y}})=\\prod_{j=1}^d (1+\\min(x_j,y_j))$. Wozniakowski \\cite{wozn:1991} points out that the same strategy still works if the equal weight requirement is removed.\nBelow we illustrate Roth's technique by showing that $r_n=\\Omega(n^{-1}(\\log n)^{(d-1)\/2})$ if $F$ is chosen to be the  RKHS with kernel\n \\begin{align}\\label{eqn:kernel}\n     K({\\boldsymbol{x}},{\\boldsymbol{y}})=\\prod_{j=1}^d \\frac{4}{3}+\\frac{1}{2}\\Bigl(x_j^2+y_j^2-x_j-y_j-|x_j-y_j|\\Bigr).\n \\end{align}\nThis is the unanchored space introduced in\n\\cite{dick:sloa:wang:wozn:2004}.\nIt has the inner product\n\\begin{align}\\label{eqn:innerproduct}\n    (f,g)=\\sum_{u\\subseteq\\{1,\\ldots,d\\}}\\int_{[0,1]^{|u|}}\\bigg(\\int_{[0,1]^{d-|u|}}\\frac{\\partial^{|u|}f}{\\partial {\\boldsymbol{y}}_u}({\\boldsymbol{y}})\\,{\\mathrm{d}}{\\boldsymbol{y}}_{-u}\\bigg) \\bigg(\\int_{[0,1]^{d-|u|}}\\frac{\\partial^{|u|}g}{\\partial {\\boldsymbol{y}}_u}({\\boldsymbol{y}})\\,{\\mathrm{d}}{\\boldsymbol{y}}_{-u}\\bigg)\\, {\\mathrm{d}}{\\boldsymbol{y}}_u\n\\end{align}\nwhere ${\\partial^{|u|}f}\/{\\partial {\\boldsymbol{y}}_u}$ is the partial derivative of $f$ taken once with respect to\neach $y_j$ with $j\\in u$.\nAny $f$ belonging to this RKHS has mixed partial derivative\n${\\partial^{|u|}f}\/{\\partial {\\boldsymbol{y}}_u}\\in L^2$ for any $u\\subseteq\\{1,\\ldots,d\\}$, so $f$ belongs to BVHK$[0,1]^d$\nusing equation (5) and Proposition~13 of \\cite{variation}.\n\nLetting $\\bsone\\in F$ be the function equal to $1$ for all ${\\boldsymbol{y}}\\in[0,1]^d$,\nit is straightforward to verify that\n$$(f,\\bsone)=\\int_{[0,1]^{d}}f({\\boldsymbol{y}}) \\,{\\mathrm{d}}{\\boldsymbol{y}}=S(f).$$\nIn other words, the function $\\bsone$ is the Riesz representation of integration over $[0,1]^d$. Moreover, by the reproducing property\n$(f,K({\\boldsymbol{x}}_i,\\cdot))=f({\\boldsymbol{x}}_i).$\nTherefore\n\\begin{align*}\n    r_n&=\\sup\\bigl\\{|S(f)|\\mid f\\in F, f({\\boldsymbol{x}}_0)=\\cdots=f({\\boldsymbol{x}}_{n-1})=0, \\Vert f\\Vert\\leqslant 1\\bigr\\}\\\\\n    &=\\sup\\bigl\\{|(f,\\bsone)|\\mid f\\in F, (f,K({\\boldsymbol{x}}_0,\\cdot))=\\cdots=(f,K({\\boldsymbol{x}}_{n-1},\\cdot))=0, \\Vert f\\Vert\\leqslant 1\\bigr\\}.\n\\end{align*}\nThis is the well-known least squares projection problem. The maximizer is proportional to the projection of $\\bsone$ into the orthogonal complement of the linear span of $\\{K({\\boldsymbol{x}}_0,\\cdot),\\ldots,K({\\boldsymbol{x}}_{n-1},\\cdot)\\}$.\nTherefore\n\\begin{align*}\n    r_n=\\min_{a_0,\\ldots,a_{n-1}} \\Bigl\\Vert\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert.\n\\end{align*}\nNow we prove that $\\Vert\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Vert=\\Omega(n^{-1}(\\log n)^{(d-1)\/2})$\nfor any choice of $a_0,\\dots,a_{n-1}$, including  $a_i=1\/n$ as used in QMC.\n\\begin{theorem}\\label{thm:Roth}\nLet $K$ be the kernel~\\eqref{eqn:kernel} for the unanchored RKHS.\nFor any points  ${\\boldsymbol{x}}_0,\\ldots,{\\boldsymbol{x}}_{n-1}\\in[0,1]^d$ and any  weights $a_0,\\ldots,a_{n-1}\\in{\\mathbb{R}}$\n\\begin{align}\\label{eq:Roth}\n\\Bigl\\Vert1-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert\\geqslant A_d\\frac{(\\log n)^{(d-1)\/2}}{n}\n\\end{align}\nholds for some positive number $A_d$ independent of $n$.\n\\end{theorem}\n\\begin{proof}\nBecause the function $\\bsone$ is the Riesz representation of integration over $[0,1]^d$,\n$(\\bsone,\\bsone)=\\int_{[0,1]^{d}}\\bsone \\,{\\mathrm{d}}{\\boldsymbol{y}}=1,$ and\n$$(K({\\boldsymbol{x}},\\cdot),1)=\\prod_{j=1}^d \\int_0^1 \\frac{4}{3}+\\frac{1}{2}\\Bigl(x_j^2+y_{j}^2-x_j-y_{j}-|x_j-y_{j}|\\Bigr)\\,{\\mathrm{d}} y_j=1.$$\nTherefore\n$$\\Bigl(\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot),\\bsone\\Bigr)\n=(\\bsone,\\bsone)-\\sum_{i=0}^{n-1} a_i (K({\\boldsymbol{x}}_i,\\cdot),\\bsone)=1-\\sum_{i=0}^{n-1} a_i.$$\nOn the other hand,\n$$\\Bigl|\\Bigl(\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot),\\bsone\\Bigr)\\Bigr|\n\\leqslant \\Vert\\bsone\\Vert\\times \\Bigl\\Vert\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert=\\Bigl\\Vert\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert,$$\nso we get the first lower bound\n\\begin{equation}\\label{eqn:bound1}\n    \\Bigl\\Vert1-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert\\geqslant \\Bigl|1-\\sum_{i=0}^{n-1} a_i\\Bigr|.\n\\end{equation}\nNext we evaluate $\\Vert1-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Vert^2$ directly.\nIf we ignore all summands except for $u=\\{1,\\ldots,d\\}$ in \\eqref{eqn:innerproduct}, we get\n\\begin{align}\\label{eqn:ainorm}\n\\Bigl\\Vert1-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert^2&\\geqslant \\int_{[0,1]^d} \\bigg(\\sum_{i=0}^{n-1} a_i\\frac{\\partial^{d}K({\\boldsymbol{x}}_i,{\\boldsymbol{y}})}{\\partial y_1\\cdots\\partial y_d}\\bigg)^2 \\,{\\mathrm{d}}{\\boldsymbol{y}} \\nonumber\\\\\n   &= \\int_{[0,1]^d} \\bigg(\\sum_{i=0}^{n-1} a_i\\prod_{j=1}^d (y_j-1\\{y_j>x_{ij}\\})\\bigg)^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}\n\\end{align}\nwhere $x_{ij}$ is the $j$th component of ${\\boldsymbol{x}}_i$.\n\nThe left hand side of~\\eqref{eqn:ainorm} is a squared norm in the RKHS\nwhile the right hand side is a plain $L^2$ squared norm.\nTo provide a lower bound, we construct a function $h$ satisfying\n\\begin{align*}\n&\\int_{[0,1]^d} h({\\boldsymbol{y}})^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}=O((\\log n)^{d-1}),\\quad\\text{and}\\\\\n&\\int_{[0,1]^d} h({\\boldsymbol{y}})\\bigg(\\sum_{i=0}^{n-1} a_i\\prod_{j=1}^d (y_j-1\\{y_j>x_{ij}\\})\\bigg) \\,{\\mathrm{d}}{\\boldsymbol{y}}=\\Omega\\Bigl(\\frac{(\\log n)^{d-1}}{n}\\Bigr),\n\\end{align*}\nneither of which involve the RKHS inner product, so the function $h$ does not have to be in the RKHS.\n\nDefine $E({\\boldsymbol{k}},{\\boldsymbol{c}})$ to be the $d$-dimensional interval\nfrom~\\eqref{eq:elemint} with $b=2$, that is\n$$\nE({\\boldsymbol{k}},{\\boldsymbol{c}})=\\prod_{j=1}^d\\Bigl[\n\\frac{c_j}{2^{k_j}},\n\\frac{c_j+1}{2^{k_j}}\n\\Bigr)\n$$\nwhere ${\\boldsymbol{k}} = (k_1,\\dots,k_d)\\in{\\mathbb{Z}}^d$\nand ${\\boldsymbol{c}} = (c_1,\\dots,c_d)\\in{\\mathbb{Z}}^d$\nsatisfy $k_j\\geqslant0$ and $0\\leqslant c_j <2^{k_j}$. Given ${\\boldsymbol{k}}$, we define $|{\\boldsymbol{k}}|=\\sum_{j=1}^dk_j$.\nFor a given vector ${\\boldsymbol{k}}$,\nthe $2^{|{\\boldsymbol{k}}|}$ elementary intervals $E({\\boldsymbol{k}},{\\boldsymbol{c}})$\npartition $[0,1)^d$ into congruent sub-intervals.\n\nFor each $E({\\boldsymbol{k}},{\\boldsymbol{c}})$, define $U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})$ by\n$$\nU_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})=\n\\begin{cases}\n(-1)^{\\sum_{j=1}^d 1\\{2^{k_j}y_j-c_j<1\/2\\}}, & {\\boldsymbol{y}}\\in E({\\boldsymbol{k}},{\\boldsymbol{c}})\\\\\n0, &\\text{else}.\n\\end{cases}\n$$\nIf we divide $E({\\boldsymbol{k}},{\\boldsymbol{c}})$ into $2^d$ sub-intervals, the value of $U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})$ is constant on each sub-interval and it alternates between $1$ and $-1$.\n\nIt is straightforward to verify that $\\int_{[0,1]^d}U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})U_{{\\boldsymbol{k}}',{\\boldsymbol{c}}'}({\\boldsymbol{y}})\\,{\\mathrm{d}}{\\boldsymbol{y}}=0$ if ${\\boldsymbol{k}}\\neq {\\boldsymbol{k}}'$ or ${\\boldsymbol{c}}\\neq {\\boldsymbol{c}}'$. It is trivially true if $E({\\boldsymbol{k}},{\\boldsymbol{c}})\\cap E({\\boldsymbol{k}}',{\\boldsymbol{c}}')=\\varnothing$.\nIf instead $E({\\boldsymbol{k}},{\\boldsymbol{c}})\\cap E({\\boldsymbol{k}}',{\\boldsymbol{c}}')\\neq \\varnothing$, then there must be some $j$ with $k_j<k'_j$ and observe that as a function of $y_j$, $U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})U_{{\\boldsymbol{k}}',{\\boldsymbol{c}}'}({\\boldsymbol{y}})$ equals $1$ on a $1\/2^{k_j+1}$-length interval and equals $-1$ on an adjacent $1\/2^{k_j+1}$-length interval, so the integration over variable $j$ always returns $0$. Then for any set $P$ of $({\\boldsymbol{k}},{\\boldsymbol{c}})$ pairs,\n\\begin{equation}\\label{eqn:Unorm}\n   \\int_{[0,1]^d}\\bigg(\\sum_{({\\boldsymbol{k}},{\\boldsymbol{c}})\\in P}U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})\\bigg)^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}=\\sum_{({\\boldsymbol{k}},{\\boldsymbol{c}})\\in P}\\int_{[0,1]^d}U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}=\\sum_{({\\boldsymbol{k}},{\\boldsymbol{c}})\\in P} 2^{-|{\\boldsymbol{k}}|}.\n\\end{equation}\n\nNow let $\\mathcal{P}=\\{{\\boldsymbol{x}}_0,\\ldots,{\\boldsymbol{x}}_{n-1}\\}$ and choose $m$ so that $2n\\leqslant 2^m<4n$. For any ${\\boldsymbol{k}}$ with $|{\\boldsymbol{k}}|=m$, define the set $P_{\\boldsymbol{k}}$ to be\n$$P_{{\\boldsymbol{k}}} =\\{{\\boldsymbol{c}}\\mid E({\\boldsymbol{k}},{\\boldsymbol{c}})\\cap \\mathcal{P}=\\varnothing\\}.$$\nFor each ${\\boldsymbol{c}}\\in P_{\\boldsymbol{k}}$,\n$$\\int_{[0,1]^d}U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})\\prod_{j=1}^d (y_j-1\\{y_j>x_{ij}\\}) \\,{\\mathrm{d}}{\\boldsymbol{y}}=\\frac{1}{4^{m+d}}.$$\nBecause there are $2^m$ intervals associated with ${\\boldsymbol{k}}$, the cardinality of $P_{{\\boldsymbol{k}}}$ is at least $2^m-n\\geqslant n$. Hence\n\\begin{equation}\\label{eqn:Uprod}\n    \\int_{[0,1]^d} \\bigg( \\sum_{{\\boldsymbol{c}}\\in P_{{\\boldsymbol{k}}}}  U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}})\\bigg)\\prod_{j=1}^d (y_j-1\\{y_j>x_{ij}\\})\\,{\\mathrm{d}}{\\boldsymbol{y}}\\geqslant \\frac{n}{4^{m+d}}.\n\\end{equation}\nNow we define\n$$h({\\boldsymbol{y}})=\\sum_{{\\boldsymbol{k}}:|{\\boldsymbol{k}}|=m} \\sum_{{\\boldsymbol{c}}\\in P_{{\\boldsymbol{k}}}} U_{{\\boldsymbol{k}},{\\boldsymbol{c}}}({\\boldsymbol{y}}).$$\nThe number of ${\\boldsymbol{k}}$ with $|{\\boldsymbol{k}}|=m$ is the number of ways to partition $m$ into $d$ nonnegative ordered integers,\nwhich equals ${m+d-1\\choose d-1}$. Equation~\\eqref{eqn:Unorm} and $2n\\leqslant 2^m<4n$ imply that\n$$\\int_{[0,1]^d} h({\\boldsymbol{y}})^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}=\\sum_{{\\boldsymbol{k}}:|{\\boldsymbol{k}}|=m} \\sum_{{\\boldsymbol{c}}\\in P_{{\\boldsymbol{k}}}} 2^{-m}\\leqslant \\sum_{{\\boldsymbol{k}}:|{\\boldsymbol{k}}|=m}1={m+d-1\\choose d-1}\\leqslant C_d (\\log n)^{d-1}$$\nfor some positive number $C_d$ independent of $n$. On the other hand, equation~\\eqref{eqn:Uprod} implies that\n$$    \\int_{[0,1]^d} h({\\boldsymbol{y}})\\prod_{j=1}^d (y_j-1\\{y_j>x_{ij}\\})\\,{\\mathrm{d}}{\\boldsymbol{y}}\\geqslant {m+d-1\\choose d-1}\\frac{n}{4^{m+d}}\\geqslant  \\frac{c_d(\\log n)^{d-1}}{n}$$\nfor another positive number $c_d$ independent of $n$.\n\nBy the Cauchy-Schwarz inequality and equation~\\eqref{eqn:ainorm}\n\\begin{align*}\n   &\\int_{[0,1]^d} h({\\boldsymbol{y}})\\bigg(\\sum_{i=0}^{n-1} a_i\\prod_{j=1}^d (y_j-1\\{y_j>x_{ij}\\})\\bigg) \\,{\\mathrm{d}}{\\boldsymbol{y}}\\\\\n   &\\leqslant \\bigg(\\int_{[0,1]^d} h({\\boldsymbol{y}})^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}\\bigg)^{\\frac{1}{2}}\\bigg(\\int_{[0,1]^d} \\bigg(\\sum_{i=0}^{n-1} a_i\\prod_{j=1}^d (y_j-1\\{y_j>x_{ij}\\})\\bigg)^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}\\bigg)^{\\frac{1}{2}}\\\\\n   &\\leqslant \\bigg(\\int_{[0,1]^d} h({\\boldsymbol{y}})^2 \\,{\\mathrm{d}}{\\boldsymbol{y}}\\bigg)^{\\frac{1}{2}}\\Bigl\\Vert\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert\n\\end{align*}\nwhich provides the lower bound\n\\begin{align*}\n    \\Bigl\\Vert1-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert\n&\\geqslant (C_d (\\log n)^{d-1})^{-\\frac{1}{2}} \\frac{c_d(\\log n)^{d-1}}{n} \\sum_{i=0}^{n-1} a_i=\\bigg(\\frac{c_d}{C_d^{1\/2}}\\sum_{i=0}^{n-1} a_i\\bigg)\\frac{(\\log n)^{(d-1)\/2}}{n} .\n\\end{align*}\n\nCombining the above lower bound with equation~\\eqref{eqn:bound1} we get\n$$\\Bigl\\Vert\\bsone-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert\n\\geqslant \\max\\Biggl(\\Bigl|1-\\sum_{i=0}^{n-1} a_i\\Bigr|,\\bigg(\\frac{c_d}{C_d^{1\/2}}\\sum_{i=0}^{n-1} a_i\\bigg)\\frac{(\\log n)^{(d-1)\/2}}{n} \\Biggr).$$\nFor $\\lambda>0$,\n$$\n\\min_{a\\in{\\mathbb{R}}}\\max( |1-a|,\\lambda a) = \\frac{\\lambda}{\\lambda+1}\n$$\nso that\n\\begin{align*}\n    \\Bigl\\Vert1-\\sum_{i=0}^{n-1} a_i K({\\boldsymbol{x}}_i,\\cdot)\\Bigr\\Vert\n&\\geqslant\n\\frac{{c_d(\\log n)^{(d-1)\/2}}\/{(nC_d^{1\/2})}}{1+{c_d(\\log n)^{(d-1)\/2}}\/{(nC_d^{1\/2})}}\n\\end{align*}\nand we let $A_d =(c_d\/C_d^{1\/2})\/(1+c_dM_d\/C_d^{1\/2})$\nfor $M_d = \\sup_{n\\in{\\mathbb{N}}}\\log(n)^{(d-1)\/2}\/n$.\n\\end{proof}\n\\begin{corollary}\nFor any sequence of points $({\\boldsymbol{x}}_i)_{i\\geqslant0}$ in $[0,1]^d$, there exists a function $f$ in the RKHS with kernel defined by \\eqref{eqn:kernel} such that\n$$\\limsup_{n\\to \\infty}\\frac{\\big|\\int_{[0,1]^{d}}f({\\boldsymbol{x}}) \\,{\\mathrm{d}}{\\boldsymbol{x}}-\\frac{1}{n}\\sum_{i=0}^{n-1} f({\\boldsymbol{x}}_i)\\big|}{(n\\log\\log n)^{-1}(\\log n)^{(d-1)\/2}}=+\\infty$$\n\\end{corollary}\n\\begin{proof}\nApply Theorem~\\ref{thm:asymptoticrate} with the lower bound on $r_n$ from Theorem~\\ref{thm:Roth}.\n\\end{proof}\n\n\n\n\\section{Discrepancy and the case of $d=1$}\\label{sec:d=1}\n\nLet $x_0,x_1,\\dots,x_{n-1}\\in[0,1]$.\nThe local discrepancy of these points at $\\alpha\\in[0,1]$ is\n$$\n\\delta_n(\\alpha) = \\frac1n\\sum_{i=0}^{n-1}1\\{x_i<\\alpha\\}-\\alpha\n$$\nand the star discrepancy is $D_n^*=\\sup_{0\\leqslant\\alpha\\leqslant1}|\\delta_n(\\alpha)|$.\nNo infinite sequence $x_i$ can have $D_n^*=o(\\log(n)\/n)$.\nUsing results from discrepancy theory we see below that there are specific\nvalues of $\\alpha$ for which $D_n^*=\\Omega(\\log(n)\/n)$.\nFor those values $1\\{x<\\alpha\\}-\\alpha$ is a persistent\nfooling function.  We show below that $f(x)=x$ is also a persistent\nfooling function for the van der Corput sequence.\n\n\nThe set of $\\alpha\\in[0,1]$\nwith $|\\delta_n(\\alpha)|=o(\\log(n)\/n)$ has Hausdorff dimension 0\nfor any sequence $(x_i)_{i\\geqslant0}\\subset[0,1]$. See \\cite{hala:1981}.\nSo $r=1$ is not just available for $d=1$ it is the usual\nrate for functions of the form $f(x) = 1\\{x<\\alpha\\}$.\n\nFor $x_i$ taken from the van der Corput sequence,\nand $\\alpha = \\sum_{k=1}^\\infty a_k\/2^k$ for bits $a_k\\in\\{0,1\\}$,\nDrmota, Larcher and Pillichshammer \\cite{drmo:larc:pill:2005} note\nthat $n|\\delta_n(\\alpha)|$\nis bounded as $n\\to\\infty$, if and only if $\\alpha$ has a representation\nwith only finitely many nonzero $a_k$.  Further, letting\n$$h_\\alpha(m) = \\#\\{ k<m\\mid a_k\\ne a_{k+1}\\}$$\ntheir Corollary 1 in our notation has\n\\begin{align}\\label{eq:theircor1}\n\\lim_{m\\to\\infty}\\frac1{2^m}\\#\\bigl\\{\n1\\leqslant n \\leqslant 2^m \\mid n\\delta_n > (1-\\epsilon) h_\\alpha(m)\n\\bigr\\}=1\n\\end{align}\nfor any $\\epsilon >0$.\nThe base $2$ representation of $2\/3$ is $0.10101\\cdots$\nand so $h_{2\/3}(m)=m$.\nIt follows that $f(x) = 1\\{x<2\/3\\}$ has\n$|\\hat\\mu_n-\\mu|>c\\log(n)\/n$ infinitely often for some $c>0$.\nEven more, the fraction of such $n$ among the first $N=2^m$\nsample sizes becomes ever closer to $1$ as $m\\to\\infty$.\n\nIf we average the local discrepancy over $\\alpha$ we get\n$$\n\\int_0^1\\delta_n(\\alpha)\\,{\\mathrm{d}}\\alpha = \\frac1n\\sum_{i=0}^nx_i-\\frac12\n$$\nwhich is the integration error for the function $f(x)=x$\nthat we study next.\nIn our study of $f(x)=x$, we use sample sizes $n$\nwith base $2$ expansion $10101\\cdots101$.\nThat is, for some $L\\geqslant1$\n$$n=n_L = \\sum_{\\ell=0}^L4^{\\ell}.$$\nThe first few values of $n_L$ are\n$1$, $5$, $21$, $85$, $341$, $1365$, and $5461$.\n\n\\begin{proposition}\\label{prop:sumx}\nFor integers $0\\leqslant i<n$, let $x_i$ be the $i$'th van der Corput\npoint in base $2$.\nThen\n\\begin{align}\\label{eq:sumx}\n\\sum_{i=0}^{n-1}x_i =\n\\sum_{k=1}^\\infty\n\\lfloor 2^{-k-1}n\\rfloor+(2^{-k}n-2\\lfloor 2^{-k-1}n\\rfloor-1)_+.\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nAs $i$ increases from $0$, the $k$'th digit of $x_i$\ncomes in alternating blocks of $2^k$ zeros and $2^k$ ones,\nstarting with zeros.  For $0\\leqslant i<r2^{k+1}$ with\nan integer $r\\geqslant0$, the $k$'th digits sum to $r2^k$\nbecause there are $r$ complete blocks of $2^k$ ones.\nThe number of ones among $i_k$ for $0\\leqslant i<n$ is then\n$$\n2^k\\lfloor 2^{-k-1}n\\rfloor+(n-2^{k+1}\\lfloor 2^{-k-1}n\\rfloor-2^k)_+\n$$\nwhere $z_+=\\max(z,0)$.  The first term counts ones\nfrom $\\lfloor 2^{-k-1}n\\rfloor$ complete blocks of $2^{k+1}$\nindices.  That leaves an incomplete block of\n$n-2^{k+1}\\lfloor 2^{-k-1}n\\rfloor$ indices $i$\nof which the first $2^k$, should there be that many,\nmust be $0$s.  Any indices past the first $2^k$ are\nones providing the second term above.\nTo complete the proof, the $k$'th digits have a coefficient\nof $2^{-k}$ in $x_i$ and summing over digits yields~\\eqref{eq:sumx}.\n\\end{proof}\n\nThe sum over $k$ in~\\eqref{eq:sumx} only needs to go\nas far as the number of nonzero  binary digits\nin $n-1$.\nFor larger $k$, both parts of the $k$'th term are zero.\n\n\n\n\\begin{proposition}\nFor $L\\geqslant0$,  let $n_L = \\sum_{\\ell=0}^L4^\\ell$.\nFor integers $i\\geqslant0$ let $x_i$ be the van der Corput points in base $2$\nand for $n\\geqslant1$ let $\\hat\\mu_{n}=(1\/n)\\sum_{i=0}^{n-1}x_i$.\nThen\n$$\n\\limsup_{L\\to\\infty} \\frac{n_L}{\\log(n_L)} |\\hat\\mu_{n_L}-1\/2|\n>c\n$$\nif $0<c<1\/(8\\log(2))$.\n\\end{proposition}\n\\begin{proof}\nWe need $K(n_L)=2L-1$ base $2$ digits to represent $n_L$.\nWe can write $n_L = \\sum_{\\ell=0}^L2^{2\\ell}$.\nThen for $1\\leqslant k\\leqslant 2L-1$,\n\\begin{align*}\n\\lfloor 2^{-k-1}n_L\\rfloor\n&\n=\\Bigl\\lfloor2^{-k-1}\\sum_{\\ell=0}^L2^{2\\ell}\\Bigr\\rfloor\n=2^{-k-1}n_L-\\sum_{\\ell=0}^{\\lfloor (k-1)\/2\\rfloor}2^{2\\ell-k-1}.\n\\end{align*}\nNext let\n\\begin{align*}\n\\theta_k\\equiv\n\\sum_{\\ell=0}^{\\lfloor (k-1)\/2\\rfloor}2^{2\\ell-k-1}\n=2^{-k-1}\\sum_{\\ell=0}^{\\lfloor (k-1)\/2\\rfloor}2^{2\\ell}\n= 2^{-k-1}\\frac{4^{\\lfloor (k+1)\/2\\rfloor}-1}3\n\\end{align*}\nand then\n\\begin{align*}\n(2^{-k}n_L-2\\lfloor 2^{-k-1}n_L\\rfloor-1)_+\n&=\n(2^{-k}n_L-2^{-k}n_L+2\\theta_k-1)_+\n=(2\\theta_k-1)_+\n\\end{align*}\nBecause $\\theta_k<1\/3$ we have\n$(2\\theta_k-1)_+=0$.\nUsing Proposition~\\ref{prop:sumx},\n\\begin{align*}\n\\frac1{n_L}\\sum_{i=0}^{n_L-1}x_i\n&=\\frac1{n_L}\\sum_{k=1}^{K(n_L)}\\lfloor 2^{-k-1}n_L\\rfloor\n=\\frac1{n_L}\\sum_{k=1}^{K(n_L)}\n2^{-k-1}n_L-\\theta_k\\\\\n&=\n\\frac12-2^{-K(n_L)-1}-\\frac1{n_L}\\sum_{k=1}^{K(n_L)}\\theta_k.\n\\end{align*}\nNow because $\\theta_k\\geqslant\\theta_2=1\/8$,\n$|\\hat\\mu_{n_L}-1\/2| \\geqslant %\nK(n_L)\/(8n_L)$\nand so\n\\begin{align*}\n\\frac{n_L}{\\log(n_L)}|\\hat\\mu_{n_L}-1\/2|\n&\\geqslant \\frac{K(n_L)}{8\\log(n_L)}\n> \\frac18\\frac{2L-1}{(L+1)\\log(4)}\\to\\frac1{4\\log(4)},\n\\end{align*}\ncompleting the proof. \n\\end{proof}\n\n\nWe can see why the sample sizes $n_L$ give unusually innaccurate estimates\nof the integral of $x$ in the van der Corput sequence.\nThose values of $n$ consistently miss getting into the `ones block'\nfor digit $k$.\n\n\n\nFigure~\\ref{fig:vdcempiricals} shows some empirical\nbehavior of the scaled errors for the two integrands\nwe considered in this section.\nThe scaled error there is essentially the number of observations\nby which the count of points in $[0,2\/3)$ differs from\n$2n\/3$.  It is $n|\\delta_n(2\/3)|$ which is the customary\nscaling in the discrepancy literature.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.9\\hsize]{figvdcempiricals}\n\\caption{\\label{fig:vdcempiricals}\nBoth panels show the scaled error $n|\\hat\\mu_n-\\mu|$ versus $n$\nfor the first $2^{14}$ points of the van der Corput sequence.\nThe top panel has the integrand $f(x)=x$ and the values\nfor $n=n_L$ are indicated with open circles.  The bottom panel has the integrand $f(x) = 1\\{x<2\/3\\}$\nand the values $n=\\tilde n_L$ are indicated.\n}\n\\end{figure}\n\nThe integrands $x$ and $1\\{x<2\/3\\}$ both have\ntotal variation one.  It would be interesting to know\nwhat integrand of total variation one has the largest\nvalue for $\\limsup_{n\\to\\infty} n|\\hat\\mu_n-\\mu|\/\\log(n)$\nin the van der Corput sequence.\n\nFor integration, it is advisable to use $n=b^m$ for $m\\geqslant0$\nin the van der Corput sequence. Then the Koksma-Hlawka inequality\ngives us $|\\hat\\mu_{b^m}-\\mu|=O(1\/b^m)$ as $m\\to\\infty$\nfor any $f$ of bounded variation on $[0,1]$ because\nthe van der Corput sequence has $D^*_{b^m} = b^{-m}=1\/n$.\nAny $(t,d)$-sequence for $d=1$ has $D^*_{b^m}=O(1\/n)$.\nFor $d=1$, bounded variation in the sense of Hardy\nand Krause reduces to the familiar one dimensional notion\nof bounded variation.\nAs a result a log power $r>0$ can apply to\nthe limit as $n\\to\\infty$ through positive\nintegers but not as $n=b^m$ for $m\\to\\infty$.\n\n\n\n\n\\section{Empirical investigations for $d=2$}\n\\label{sec:d=2}\n\nThis section reports on an empirical search for\nan integrand with errors that are $\\Omega( \\log(n)^r\/n)$\nfor some $r>1$ when using a sequence of points\nwith $D_n^*=O(\\log(n)^2\/n)$.  We know from Section~\\ref{sec:proofofbound}\nthat this is attainable for $1<r<3\/2$\nand not ruled out for $3\/2\\leqslant r<2$.\nWe search using some knowledge of the\nweaknesses of the van der Corput points for the functions\nfrom Section~\\ref{sec:d=1}.\n\nThe search must take place with $d\\geqslant2$ and so $d=2$\nis the natural first place to look for a problematic integrand.\nIt is clear that the integrand should not be additive because\nthen integrating it involves a sum of two one dimensional\nintegration problems where we know that $r>1$ is not needed.\nWe look at two generalizations of the van der Corput\nsequence.  The first is the Halton sequence for $d=2$.\nIn that sequence, ${\\boldsymbol{x}}_{i1}$ is the van der Corput sequence\nin base $2$ and ${\\boldsymbol{x}}_{i2}$ is the van der Corput sequence\nin base $3$.\nThe second sequence is the Sobol' sequence in base $2$.\nIt has $t=0$ and like the Halton points, the first\ncomponent ${\\boldsymbol{x}}_{i1}$ is the van der Corput sequence\nin base $2$.  For $n=2^m$, the second component\n${\\boldsymbol{x}}_{i2}$ of the Sobol' sequence is a permutation\nof the first $n$ elements of the van der Corput sequence.\n\n\nWe look first at the scaled error $n|\\hat\\mu_n-\\mu|$\nfor $f({\\boldsymbol{x}}_i) = (x_{i1}-1\/2) (x_{i2}-1\/2)$.\nThis integrand has integral zero and two dimensional Vitali variation of one.\nIt integrates to zero over each component of ${\\boldsymbol{x}}$ when the other\ncomponent is fixed at any value.  It has no nonzero ANOVA component\nof dimension smaller than $2$.  Each of the factors is a persistent\nfooling function for the van der Corput points.\n\nFor the Halton sequence, the scaled error equals $1\/4$ when $n=1$\nand it remains below $1\/4$ for $2\\leqslant n\\leqslant 2^{20}$.  It repeatedly approaches $1\/4$\nfrom below.\nFigure~\\ref{fig:d2prodempirical} shows the first $2^{19}$ scaled errors\nfor this product integrand for both Halton and Sobol' points.\nFor both constructions we see no evidence that the error is\n$\\Omega(\\log(n)^r\/n)$ for $r>1$.  In fact, what is quite surprising\nthere is the apparent $O(1\/n)$ error rate, which is then\n{\\sl better} than what the van der Corput sequence attains for $f(x)=x$.\n\nBy plotting only $2^{19}$ points\ninstead of all $2^{20}$ it becomes possible to see something of\nthe structure within the triangular peaks for the Sobol' points.\nThe linear scale for $n$ (versus a logarithmic one) shows\na clear repeating pattern consistent with an $O(1\/n)$ error rate.\nWhether or not the errors are $O(1\/n)$, this integrand is\nnot a promising one to investigate further in the\nsearch for an integrand needing $r>1$.\n\n\\begin{figure}[t]\\centering\n\\includegraphics[width=.9\\hsize]{figd2prodempirical.png}\n\\caption{\\label{fig:d2prodempirical}\nThe panels show scaled errors $n|\\hat\\mu_n-\\mu|$ versus $n$\nfor the integrand $f({\\boldsymbol{x}})=(x_1-1\/2) (x_2-1\/2)$ on $[0,1]^2$.\nThe top panel is for Halton points and the bottom one is\nfor Sobol' points.\n}\n\\end{figure}\n\n\nAnother challenging integrand for van der Corput\npoints was $1\\{x<2\/3\\}$.\nFor the Sobol' points we then take\n$f({\\boldsymbol{x}})=\\prod_{j=1}^2( 1\\{x_j<2\/3\\}-2\/3)$\nfor $d=2$.  Once again we have removed\nthe additive contribution to the integrand\nthat we know is integrated at the $\\log(n)\/n$\nrate making it easier to discern the effect\nof the bivariate component.\nFor Halton points, we do not use $2\/3$\nas that has a terminating expansion in base $3$\nthat defines the second component of the ${\\boldsymbol{x}}_i$.\nWe use\n$f({\\boldsymbol{x}})=(1\\{x_1<2\/3\\}-2\/3) (1\\{x_2<3\/5\\}-3\/5)$\nfor Halton points because $3\/5$ does not have a terminating\nexpansion in base $3$ that could make the problem\nartificially easy.  Both of these functions have\ntwo dimensional Vitali variation equal to one, just\nlike $\\prod_{j=1}^2(x_j-1\/2)$.\nFigure~\\ref{fig:d2rectempirical} shows the scaled errors\n$n|\\hat\\mu-\\mu|\/\\log(n)$. The logarithmic scaling\nin the denominator is there so that\na value of $r>1$ would give an infinite series of new records.\nWe don't see many such new records for $n\\leqslant 2^{22}$\nfor either of the two test cases. These integrands were designed\nto exploit weaknesses in the Sobol' and Halton sequences\nand they did not produce the appearance of errors growing\nfaster than $\\log(n)\/n$.  It remains possible that errors\ngrow like $\\log(n)^r\/n$ for $r>1$ but with the records being set very sparsely.\nThe situation is quite different from Figure~\\ref{fig:vdcempiricals}\nin the one dimensional case where we see a steady sequence of\nrecord values with a fractal pattern in the empirical errors.\n\n\n\\begin{figure}[t]\\centering\n\\includegraphics[width=.9\\hsize]{figd2rectempirical.png}\n\\caption{\\label{fig:d2rectempirical}\nThe panels show scaled errors $n|\\hat\\mu_n-\\mu|\/\\log(n)$ versus $n>1$.\nThe top panel has $f({\\boldsymbol{x}})=(1\\{x_1<2\/3\\}-2\/3)\\times(1\\{x_2<2\/3\\}-2\/3)$\nfor Sobol' points.\nThe bottom panel has\n$f({\\boldsymbol{x}})=(1\\{x_1<2\/3\\}-2\/3)\\times(1\\{x_2<3\/5\\}-3\/5)$\nfor Halton points.\n}\n\\end{figure}\n\nThere are some other integrands that could be difficult for\nQMC.  A badly approximable irrational number $\\theta$\nis one where the distance between $n\\theta$\nand $\\Z$ is above $c\/n$ for some $c>0$ and all integers $n\\geqslant1$.\nFor instance, $\\theta=\\sqrt{2}-1$ is badly approximable.\nAn integrand like $\\prod_{j=1}^d (1_{x_j<\\theta}-\\theta)$\ncould pose a challenge to QMC methods, but some exploration\nwith Halton and Sobol' points was inconclusive: there was\nno empirical indication that $r>1$ would be required.\nAnother potentially difficult integrand is $\\prod_{j=1}^d (x_j^\\theta-1\/(1+\\theta))$\nfor $0<\\theta<1$. Partial derivatives of this integrand with respect to a subset\nof components $x_j$ are unbounded.  That would make them fail the sufficient\nconditions for scrambled nets to attain $O(n^{-3\/2+\\epsilon})$ RMSE\nin \\cite{smoovar,localanti,yue:mao:1999}.\nThese integrands did not show a need for $r>1$.\n\nNext, we consider the function $f({\\boldsymbol{x}}) = 1\\{x_1+x_2<1\\}$.\nThis integrand has infinite Vitali variation\nover $[0,1]^2$. Therefore it also has infinite Hardy-Krause\nvariation  and so the Koksma-Hlawka bound degenerates to $+\\infty$.\nBecause $f$ is Riemann integrable we know that $\\hat\\mu_n\\to\\mu_n$\nif $D_n^*\\to0$.\nFinding that $r>1$ for this $f$ would not provide\nan example of a BVHK function needing that $r>1$.\nIt remains interesting to see what happens because the case is\nnot covered by QMC theory without randomization.\n\nFigure~\\ref{fig:d2notbvempirical} shows scaled errors\nfor this integrand, using an exponent of $r=1$  for $\\log(n)$.\nThere is a striking difference between the results for Sobol'\npoints versus Halton points.  We see a few approximate doublings of the\nscaled error for Sobol' points even when using $r=1$.\nThis does not prove that\nwe need $r>1$ here; perhaps we saw the last doubling\nor perhaps similar jumps for larger $n$ arise at extremely sparse intervals.\nIt does serve to raise some additional\nquestions.  For instance, why are Halton points so much more\neffective on this integrand, and to which other integrands\nof unbounded variation might that apply?\nFor smooth integrands, Sobol' points commonly perform\nmuch better when $n$ is a power of two than otherwise.\nHere, those sample sizes did not bring much better outcomes.\n\n\n\\begin{figure}[t]\\centering\n\\includegraphics[width=.9\\hsize]{figd2notbvempirical.png}\n\\caption{\\label{fig:d2notbvempirical}\nThis shows\nscaled errors $n|\\hat\\mu_n-\\mu|\/\\log(n)$ versus $n>1$.\nThe integrand is $1\\{x_1+x_2<1\\}$ which has infinite\nVitali variation on $[0,1]^2$. The upper curve is for Sobol' points.\nThe lower curve is for Halton points. There is a reference\nline at scaled error of zero.\n}\n\\end{figure}\n\n\\section{Very large $m$ for Sobol' nets}\\label{sec:bigm}\n\nIf the need for $r>1$ is only evident for very large $n$\nthen we might fail to detect it by computing $\\hat\\mu_n$.\nIf we restrict to sample sizes $n=2^m$  for $m\\geqslant0$ then\nwe can use properties of the generating matrices of Sobol'\nsequences to compute the scaled error $n|\\hat\\mu-\\mu|$\nfor very large $n$ when $f$ is the indicator function\nof a set $[\\bszero,{\\boldsymbol{a}})$.\nThe first $n=2^m$ Sobol' points form a $(t,m,s)$-net in\nbase $2$ for which the Koksma-Hlawka bound gives\na rate of $O(n^{-1}\\log(n)^{d-1})$.  As a result we must\nlook to $d=3$ or larger for a problem that needs $r>1$\nfor these values of $n$.\nWe choose ${\\boldsymbol{a}} = (2\/3,2\/3,\\dots,2\/3)$ of length $d$\nas $2\/3$ is difficult to approximate in base $2$.\n\nWe can partition $[0,2\/3)^d$ into a countable number\nof elementary intervals in base $2$.\nThe number of points of the digital net\n${\\boldsymbol{x}}_0,\\dots,{\\boldsymbol{x}}_{b^m-1}$\nthat are in $E({\\boldsymbol{k}},{\\boldsymbol{c}})$ equals the number of solutions\n$\\vec{i}\\in \\{0,1\\}^m$ to a linear equation\n$C\\vec{i}=\\vec{a} \\mathrm{\\,mod\\,} 2$\nfor a matrix $C\\in\\{0,1\\}^{|{\\boldsymbol{k}}|\\times m}$\nthat takes the first $k_j$ rows from the $j$'th\ngenerating matrix of the digital net, for $j=1,\\dots,d$\nand some $\\vec{a}\\in\\{0,1\\}^m$.\nSee \\cite{pan:owen:2021:tr} for a description and\na discussion of how to find the number of such points.\nFor our Sobol' points we use  the direction\nnumbers from Joe and Kuo \\cite{joe:kuo:2008}.\n\nTo make the computation finite,\nwe replace $[0,2\/3)^d$ by $[0,a_m)^d$\nwhere $a_m=\\lfloor 2^m (2\/3)\\rfloor\/2^m$.\nFor $a\\in(0,1)$, let $\\mu(a) = a^d$\nand $\\hat\\mu(a) =(1\/n)\\sum_{i=0}^{n-1}1_{{\\boldsymbol{x}}_i\\in[0,a)^d}$.\nThen\n\\begin{align*}\n0&\\leqslant \\mu\\Bigl(\\frac23\\Bigr)-\\mu(a_m)\n= \\Bigl(\\frac23-a_m\\Bigr)\\sum_{j=0}^{d-1}\n\\Bigl(\\frac23\\Bigr)^ja_m^{d-j-1}\n\\leqslant \\Bigl(\\frac23\\Bigr)^{d-1}\\frac{d}n.\n\\end{align*}\nThe number of the first $n$ Sobol'\npoints that belong to $[0,2\/3)^d\\setminus[0,a_m)^d$\nis at most $d$. Therefore\n\\begin{align*}\n0&\\leqslant\n\\hat\\mu(2\/3)-\\hat\\mu(a_m)\\leqslant d\/n\n\\end{align*}\nNow\n\\begin{align*}\n&\\quad n(\\hat\\mu(2\/3)-\\mu(2\/3))\n-n(\\hat\\mu(a_m)-\\mu(a_m))\\\\\n&=\nn(\\hat\\mu(2\/3)-\\hat\\mu(a_m)\n-n(\\mu(2\/3)-\\mu(a_m))\\\\\n&\\in [-(2\/3)^{d-1}d,d]\\subseteq [-4\/3,d]\n\\end{align*}\nso the absolute error in using $n((\\hat\\mu(a_m)-\\mu(a_m))$\ninstead of $n((\\hat\\mu(2\/3)-\\mu(2\/3))$\nis at most $d$.\n\n\nFigure~\\ref{fig:errorvsmd} shows the results for $d=2,3,4$.\nFor $d=2$ we see strong linear trend\nlines in the scaled error consistent with\nneeding $r=1$.  For $d=3,4$ the trend is not\nobviously linear but it does not have a compelling\nand repeating $r>1$ pattern even at $n=2^{50}$ for $d=4$\nor $n=2^{100}$ for $d=3$.\n\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{0.34\\hsize}\n\\includegraphics[width=\\textwidth]{errorvsm,d=2}\n\\end{minipage}\n\\hspace*{-.35cm}\n\\begin{minipage}{0.34\\hsize}\n\\includegraphics[width=\\textwidth]{errorvsm,d=3}\n\\end{minipage}\n\\hspace*{-.35cm}\n\\begin{minipage}{0.34\\hsize}\n\\includegraphics[width=\\textwidth]{errorvsm,d=4}\n\\end{minipage}\n\\caption{\\label{fig:errorvsmd}\nThe panels show the signed scaled error\n$n(\\hat\\mu-\\mu)$ for a Sobol' sequence\nwhen $f=\\prod_{j=1}^d1_{x_j<2\/3}$.\nThe panels have $d=2,3,4$.\nThe sample sizes go to $2^{100}$ for\n$d=2,3$ and to $2^{50}$ for $d=4$.\nThe computed values differ from the true\nones by at most $d$ for all $m$.\n}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\\label{sec:discussion}\n\nFor any $\\epsilon>0$\nand any $d\\geqslant1$ and any infinite sequence of points ${\\boldsymbol{x}}_i\\in[0,1]^d$\nand any $c>0$ that there are functions in BVHK$[0,1]^d$ where the QMC error\nexceeds $c(\\log n)^{r}\/n$ infinitely often when $r<(d-1)\/2$.\nFor $d=1$ we can easily construct such functions for $r=1$\nusing results from discrepancy theory.  We don't know of\nany specific examples with $r>1$ and simple multidimensional\ngeneralizations of the functions needing $r=1$ for $d=1$\ndid not show an apparent need for $r>1$ when $d=2$.\nThe best candidates we saw for the Sobol' sequence\nare $f({\\boldsymbol{x}}) =\\prod_{j=1}^d1\\{x_j<2\/3\\}$ for $d=3$ or $4$\nbut we have no proof that they require $r>1$.\n\n\n\nOne surprise is that comparing\nFigures~\\ref{fig:vdcempiricals} and~\\ref{fig:d2prodempirical} we see an error for $f({\\boldsymbol{x}})=(x_1-1\/2)(x_2-1\/2)$ that appears\nto be $O(1\/n)$ while the one dimensional\nfunction $f(x)=x$ has error $\\Omega(\\log(n)\/n)$ (theoretically\nand also empirically over small $n$).\nA second surprise is that for a two dimensional function\nof unbounded variation we see very different behavior\nfor Sobol' and Halton points in\nFigure~\\ref{fig:d2notbvempirical}.  The error for Sobol'\npoints appears to grow faster than $\\log(n)\/n$\nwhile that for Halton points is far lower and might\neven grow at a different rate.  Neither of these surprises\ncontradict known theory but it is odd to see the two\ndimensional problem apparently easier to solve than\na corresponding one dimensional one and it is also odd\nto see Halton points appear to be so much more robust\nto unbounded variation than Sobol' points.\n\nSo, where could the logs be?  It is possible that they are only\npractically important for enormous $n$, or what is almost\nthe same thing, that they are present with a very tiny implied constant.\nIt is also possible that the commonly investigated integrands\nhave error $O(\\log(n)\/n)$ even for $d\\geqslant2$.\n\n\\section*{Acknowledgments}\nThis work was supported by the U.S.\\ NSF under\ngrant IIS-1837931.  Thanks to Fred Hickernell and\nErich Novak for discussions related to this problem.\nWe are also grateful to Traub, Wasilkowski and Wo\\'zniakowski\nfor ensuring that Trojan's work was not lost.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum systems with classically chaotic counterparts possess\nunique characteristic features as summarized, e.g., in\n\\cite{rev,haake}. Following the semiclassical approach, one\noften relates quantum properties of a system to its classical\nmotion, using for example a direct comparison of\n phase space portraits of the classical dynamics and\n wave function quasiprobability representations in the phase\nspace (via Husimi or Wigner functions) \\cite{rev,rvj,holt}.\nEven in the case of globally\nchaotic dynamics, individual unstable classical trajectories\n can  be retraced\nby stationary quantum eigenfunctions which are ``scarred\" by the\nclassical\nsolution \\cite{Hell84}. When the classical phase space is mixed -\npartially chaotic and partially regular - a similar separation\ninto regular and irregular wavefunctions is possible in the\nquantum world \\cite{Perc73}.\n Stable regions of phase space (tori) lend\nthemselves to semiclassical EBK\n (Einstein-Brillouin-Keller)\nquantization, yielding both the approximate eigenenergies and\nthe corresponding wavefunctions \\cite{MF65}. Similarly,\n there are ``irregular\" wavefunctions living in the region of\nchaotic classical motion. Some of them can be associated\n with residual structures\nof classically regular motion such as cantori while the other\nare practically structureless.\nIn low-dimensional systems, KAM\n(Kolmogorov-Arnold-Moser)\ntori provide impenetrable\nborders;\n the only way regular and\nirregular wave functions may communicate with each other is by\nquantum mechanical\ntunneling processes. In higher-dimensional systems, classical Arnold\ndiffusion provides another mechanism of transport, a process\nwhich is, however, typically quite slow \\cite{LL81}. On the other\nhand, quantum mechanical tunneling through impenetrable borders\nof classical mechanics may be quite effective. Once the particle\ntunnels from, say, a stable island into the surrounding chaotic\nphase space, it can visit distant regions following the classically\nchaotic transport mechanism. In particular, it can tunnel into some other\nstable island thus providing the coupling between two\nwavefunctions localized on distinct and separated islands or\nit can wander very far away (possibly leading, e.g., to ionization\nas in atoms driven by external radiation).\n\nInterestingly, this \n``chaos assisted'' tunneling mechanism posseses\nunique features typically absent in the standard ``barrier''\ntunneling of quantum mechanics, such as a great sensitivity to\nthe variation of external parameters manifesting itself in\nfluctuations of observable quantities. Previous work\nconsidered mainly model one-dimensional time dependent\nsystems \\cite{LB90,GDJH91,PGL92} or model two-dimensional\nautonomous systems \\cite{BTU93,BBEM93,TU94,LU96}. A similar\nproblem in the scattering case has also been discussed on a\nkicked model system \\cite{GS94}. We shall consider here a\nrealistic, experimentally accessible (although simplified, see\nbelow) system - namely the hydrogen atom illuminated by\nmicrowave radiation. Instead\nof considering tunneling between two regions (tori) mediated by the\nchaotic transport between them, we shall rather consider the\nsingle tunneling process out of the stable island. Then the\nchaotic diffusion process will lead to ionization. While in the\nformer case, the probability may flow periodically between two\nregions linked by the tunneling coupling, in our problem the\nprocess is irreversible and constitutes the mechanism of the\ndecay.\n\nThe paper is organized as follows. Section II contains the description\nof the systems studied - the hydrogen atom in the field of a microwave\nradiation of either circular or linear polarization - and a general\npresentation of the ionization via chaos assisted tunneling.\nSection III presents a simple\nmodel for the description of the fluctuations present in the\ndecay, catalized by chaos assisted tunneling.\nWe present there the distribution of resonance widths\nand consider also the distribution of energy shifts of the\nsingle, initially localized state due to the coupling to other\n``delocalized'' states, and, via these states, to the continuum.\nThis theory is confronted with\nthe numerical data obtained for the hydrogen atom in the field\nof microwave radiation of either circular or linear polarization\nin Section IV. Finally we present conclusions in Section V\nwhile the Appendix contains the details of the derivation of\nformulae presented in Section III.\n\n\\section{Nondispersive electronic wave packets and their ionization}\n\nIn order to obtain the simplest study of quantum transport\nthrough a chaotic sea, one should use an initial state as localized\nas possible in phase space as, for example, a minimum wave packet\nlocalized on a classical stable equilibrium point.\nUnfortunately, in atomic systems, no stable equilibrium point of the electron\noutside the nucleus exists.\n\nA simple alternative is to use a nonlinear resonance between the\ninternal motion of the electron and an external driving force.\nRecently, interesting new objects have been proposed in the studies of\nhydrogen atoms illuminated by microwave radiation of either linear \\cite{ab1}\nor circular \\cite{bb} polarization: the so called non-dispersive\nwave packets. The corresponding classical dynamics picture\ncorresponds to the stable  resonance island\nembedded in a chaotic sea. For the motion contained within\nthe principal 1:1 resonance between the Kepler frequency of the\nunperturbed Rydberg electron and the frequency of the driving field,\nthe frequency of the electronic motion is locked on the external\nmicrowave frequency. Semiclassically, a wave packet localized on such a\nregular island will be confined to it modulo \nthe exponentially\ndecaying\ntails of the wavefunction which may extend into the chaotic region.\nIn a quantum treatment, one finds\nwave packets which are really single eigenstates of the atom dressed\nby the microwave field, i.e. single eigenstates of the\ncorresponding Floquet \\cite{sfloq} Hamiltonian \\cite{ab1,dzb95}.\nThey are localized in all spatial dimensions and propagate along the classical\ntrajectory in the same way a classical particle would do.\nFor a generic case (e.g.\nlinear polarization microwaves, or more generally, any time-periodically\nperturbed system \\cite{holt}), it undergoes periodic deformations\nwhich faithfully follow the change of shape of the resonance island\nover one period, repeating the same shape every period.\nOnly in the case of circular microwave polarization,\nthe shape of the wave packet eigenstate does not change. This is\ndue to the fact that the time-dependence may be removed from the\nHamiltonian of the problem by a transformation to the frame\nrotating with the field \\cite{proh,groz,zgd96}.\n\nAs mentioned above, a finite\n$\\hbar$ value leads to quantum mechanical tunneling from\nthe island to the chaotic sea surrounding it. Then the electron\ngains energy \nfrom the driving field \nand eventually becomes ionized by a process\nclassically known as chaotic diffusive ionization. Since many different\npaths link the initial wave packet with the continuum, its\nionization time (or its reciprocal - the ionization rate or resonance width)\nfluctuates strongly with the parameters of the problem, the\nmicrowave frequency, $\\omega$, or its amplitude, $F$ \n\\cite{zdb95}.\nTherefore, these wave packets are ideally suited for a quantitative\nstudy of the ionization promoted by chaos assisted tunneling.\n\n\\subsection{Circularly polarized microwaves}\n\nLet us consider first the conceptually simpler case of hydrogen atoms\nilluminated by a circularly polarized microwave (CPM) \\cite{bb,dzb95,zdb95}.\nThe problem is fully three dimensional; however, as\nit has been shown elsewhere \\cite{dzb95,zdb95}, one can consider\nthe quantum dynamics\nin the space  restricted to the polarization plane of the microwave field.\nWhile\nthis excludes possible excitations in the direction\nperpendicular to the polarization plane, the dynamics of\nthe wave packets and their properties are qualitatively\nnot affected by the reduced dimensionality \\cite{dzb95,zdb95}. In the\nfollowing, we shall present results obtained within\nsuch a reduced two-dimensional (2D) model.\n\nThe time-dependence of the\nHamiltonian describing the CPM ionization of H atoms is explicitely\nremoved by transforming to the coordinate frame which\ncorotates with the microwave field \\cite{proh,groz},\nwhere it reads (in atomic units):\n\\begin{equation}\nH=\\frac{{\\bf p}^2}{2}-\\frac{1}{r}+Fx-\\omega\\ell_z,\n\\label{hcirc}\n\\end{equation}\nwith $\\ell_z$ the angular momentum operator.\n\nAt the center of the \nprincipal resonance island between the Kepler\nand the microwave frequency, a periodic orbit exists whose period\nexactly matches\nthe period of the microwave. In the laboratory frame, this\nis a circular orbit with radius $x$ such that:\n\\begin{equation}\n\\frac{1}{\\omega^2x^2}+\\frac{F}{\\omega^2}=x.\n\\label{pos}\n\\end{equation}\nWe introduce the effective\nprincipal quantum number $n_0$ (not necessarily an integer) corresponding\nto this main resonance:\n\\begin{equation}\nn_0 = \\omega^{-1\/3}.\n\\label{n0}\n\\end{equation}\nDue to the classical scaling of the Coulomb problem \\cite{del},\namong the two parameters\n$\\omega$ and $F$\nonly one is necessary to \ntune the dynamics classically.\nThus, we define quantities (position and microwave electric field) scaled\nwith respect to $n_0$:\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\nx_0 = x n_0^{-2} = x \\omega^{2\/3}\\\\\nF_0 = F n_0^{4} = F \\omega^{-4\/3}\n\\end{array}\n\\right.\n\\end{equation}\n$F_0$ represents the ratio of the external microwave\nfield to the Coulomb field of the nucleus on the\nunpertubed resonant circular orbit. The classical dynamics\ndepends only on this parameter.\nThe scaled radius $x_0$ of the resonant circular orbit is the solution\nof the scaled equation:\n\\begin{equation}\n\\frac{1}{x_0^2}+F_0=x_0\n\\label{pos0}\n\\end{equation}\n\nIn the corotating frame, the resonant orbit corresponds to an equilibrium point.\nThis point is stable if the\ndimensionless stability parameter\n\\begin{equation}\nq=\\frac{1}{\\omega^2x^3}=\\frac{1}{x_0^3}\n\\label{stab}\n\\end{equation}\nis chosen in the interval $8\/9<q<1$ \\cite{bb}. Then, the existence of a\nwave packet localized in the vicinity of the fixed point is assured in\nthe semiclassical limit. It appears in the rotating frame as a\nstationary eigenstate of the Hamiltonian, Eq.~(\\ref{hcirc}), localized\naround the equilibrium point and in the laboratory frame as a localized\nwave packet following a circular trajectory without spreading.\n\nThe energies of the wave packet eigenstates in the\nrotating frame are given (in \nharmonic approximation around the stable fixed point) by \\cite{dzb95}\n\\begin{equation}\nE_0=\\frac{1-4q}{2q^{2\/3}}\\omega^{2\/3}+(N_++\\frac{1}{2})\\omega_+-(N_-\n+\\frac{1}{2})\\omega_-,\n\\label{semicl}\n\\end{equation}\nwith\n\\begin{equation}\n\\omega_{\\pm}=\\sqrt{\\frac{2-q\\pm\\sqrt{q(9q-8)}}{2}}\\omega\n\\label{modes}\n\\end{equation}\nthe normal mode frequencies of the locally harmonic Hamiltonian which confines\nthe wave packet, and $N_{\\pm}$ the number of quanta in these modes.\nIn the following, we shall consider the ionization of the ``ground\nstate'' wave packet corresponding to $N_+=N_-=0$.\nSuch a wave packet can be expanded over the usual atomic eigenstates.\nIt is a coherent superposition of mainly circular\nstates \\cite{dzb95}. The frequency  $\\omega $ is\nclose to the resonance between atomic circular states with\nprincipal quantum numbers\n$n\\rightarrow n\\pm 1$ with $n\\simeq n_0$.\nThus these states are strongly coupled by the\nmicrowave field. It can be shown that, for such a frequency, the overlap\nof the wave packet state with circular states is, for a sufficiently high\n$n_0,$ Gaussian distributed with a maximum at $n_0$ and the width\nof the order of $\\sqrt{n_0}$.\n\nTo find the wave packets numerically, we diagonalize the time-independent\nHamiltonian in the rotating frame,\nEq.(\\ref{hcirc}), in a Sturmian basis \\cite{zgd96}.\nThe so-called complex rotation method\n\\cite{rein}\nallows us to take exactly into account the coupling to the continuum.\nWe refer the reader to Ref.~\\cite{zgd96}\nfor the description of the technical details. Let us mention here only\nthat a diagonalization yields complex energies, $E_i-i\\Gamma_i\/2$, where\nthe real parts\n$E_i$ are the positions of the resonances while the $\\Gamma_i$ correspond\nto their widths, i.e. their ionization rates.\nIn this approach, spontaneous emission from the wave packet eigenstates\nto lower lying states is neglected. This is a reasonable approximation\nas they are composed of mainly circular states which have very long\nspontaneous lifetimes, typically of the order of several millions periods.\nIn all calculations discussed hereafter, the decay of wave packet\neigenstates is dominated by field induced ionization (via chaos assisted\ntunneling) and {\\em not} by spontaneous emission \\cite{ab97,ibb97}.\n\nThe present results are obtained from the\ndiagonalization of matrices of size up to $200000$. We\nuse the Lanczos algorithm \\cite{gd91} to extract few eigenvalues and\nthe corresponding eigenvectors in the vicinity of the energy\npredicted by the semiclassical expression, Eq.~(\\ref{semicl}). The\nwave packet eigenstate is then identified by its large\noverlap with the circular state with principal quantum number \nclose to $n_0=\\omega^{-1\/3}$, and by its large dipole moment. Typically, due\nto the accuracy of the semiclassical prediction, it is enough\nto extract \nfew eigenvalues of the matrix. For \nour present purpose, it is the\n{\\it deviation} of the exact resonance position from the semiclassical\nprediction, and the {\\it ionization rate}\nwhich are of great interest for us.\n\nFor our statistical analysis, it is\nreasonable to collect the data,\nfor a {\\it  fixed} classical dynamics, i.e. at {\\it fixed} $F_0$\nvalue varying simultaneously\n$\\omega$ and $F$ around some mean values.\nOn the other hand, quantum mechanics does {\\em not} preserve the classical\nscaling, the finite $\\hbar$ value introduces another scale into the\nproblem - say the energy of the state or\n the number of photons necessary for the ionization. We shall see\nthat this has important consequences for the characteristics of the\nionization process -- the statistical properties of the widths\ndepend on $n_0$.\n\nWe shall present the numerical data for both circular and\nlinear polarization \nof the driving field simultaneously. Thus before presenting the data,\nwe review the wave packet properties in linearly polarized\nmicrowaves (LPM)\n\\cite{ab1}.\n\n\\subsection{Linearly polarized microwaves}\n\nIf the atom is irradiated by an electromagnetic field of linear\npolarization defining the $z$-axis, the angular momentum projection\n$m$ on the field axis is a good quantum number, due to the rotational\nsymmetry of the Hamiltonian:\n\\begin{equation}\nH=\\frac{{\\bf p}^2}{2}-\\frac{1}{r}+Fz\\cos(\\omega t).\n\\label{hlp}\n\\end{equation}\nOne is therefore left with two spatial degrees of freedom and the\nexplicit, time periodic dependence of the Hamiltonian which cannot be\neliminated. However, the temporal periocicity of the problem (for\nconstant driving field amplitude $F$) allows\nfor the application of the\nFloquet theorem and the identification of the eigenfunctions\n$|\\psi_i \\rangle$ of the\natom {\\em in} the field as solutions of the stationary Floquet\nequation \\cite{sfloq}\n\\begin{equation}\n(H-i\\partial _t)|\\psi_i \\rangle=\\epsilon|\\psi_i \\rangle,\n\\label{floqeq}\n\\end{equation}\nwhere spatial coordinates and time are treated now on an equal\nfooting. The Floquet theorem guarantees that the eigenfunctions\n$|\\psi_i \\rangle$ are periodic with the period $T=2\\pi\/\\omega$ of the\ndriving field, and form a complete basis of the problem.\nThe Floquet states are nothing but the ``dressed\nstates'' of the atom in the field \\cite{cct}.\n\nAgain, when Coulomb attraction and driving field amplitude become\ncomparable, the classical dynamics of the Rydberg electron\nturns chaotic and phase space is divided into\nregions of regular and irregular motion. At sufficiently large field\namplitudes, only the principal resonance between the driving frequency\nand the Kepler motion is left as an (elliptic) island of regular motion in the\nchaotic sea \\cite{ab1}. Unlike the CPM case, the stable periodic orbit at the\ncenter of the elliptic island is not characterized by a set of\nsimple analytical expressions. However, the local oscillatory motion\ncan be plugged into the form of a Mathieu equation \\cite{bz},\n the numerical\nsolution of which provides good estimates of the energy of the quantum\nmechanical ground state and of the first excited states\nof the local hamiltonian \\cite{holt,sk}.\nThe oscillator ground state is the\nwave packet eigenstate of the atom in the field and\nfollows the classical, periodic evolution of the principal resonance\nisland \\cite{hh,ab1}. Depending on the value of the classical angular\nmomentum and\nits projection on the symmetry axis, the wave packet may probe\nthe Coulomb singularity and consequently displays some transient\ndispersion which mimicks the acceleration of the classical particle at\nthe aphelion \\cite{ab1}.\n\nSince, in the LPM case, the \nnumerical detection of the wave packet is less\nstraightforward than for CPM, we restrict ourselves to the\ninvestigation of the hydrogen atom confined to one spatial dimension,\nalong the field polarization axis. As the driving field amplitude is\nincreased, it is this direction along which chaos is born in the full\ndynamics of the 3D atom \\cite{ab1,i3e,ab2},\nand therefore this approximation will be\nsufficient for our present purposes.\n\nThe LPM  case has the same scaling property than the CPM case:\nthe classical dynamics depends only on the scaled field\n$F_0=F\\omega^{-4\/3}$.\nNumerically, the wave packets are\nidentified by their\nlarge (as compared to other nearby Floquet states)\noverlap with the state $\\mid n_0\\rangle$ \\cite{comment}, which is\nresonantly coupled to its nearest neighbour $\\mid n_0+1\\rangle$ by the\ndriving frequency.\nSimilarly to the CPM case, the quasienergies $E_i$ and\nresonance widths $\\Gamma_i$ of the wave packets\nare used for the statistical analysis.\nAll data samples are characterized by a fixed value of the\nparameter $F_0$, hence they correspond to the same\n structure of classical phase space. For a given value of\n$F_0$, $\\omega$ (and, accordingly, $F$) has been scanned to give\n$1000$ eigenvalues per $F_0$ value.\n\n\\subsection{Numerical results}\n\nThe typical behaviour of the wave packet ionization width versus\nthe microwave frequency has already been presented  in \\cite{zdb95}\nfor a fixed microwave amplitude. It displays very strong fluctuations\nover several orders of magnitude for small changes of the frequency\n(typically of the order of 1 part in 1000).\nThese fluctuations -- although perfectly deterministic --\nlook completely random and are strongly reminiscent of the\nuniversal conductance fluctuations observed in mesoscopic systems \\cite{meso}.\nIndeed, the ionization width measures the rate at which an electron\ninitially localized close to the stable resonant trajectory ionizes,\ni.e. escapes to infinity. In other words, the ionization width\ndirectly measures the \nconductance of the atomic system from the initial\npoint to infinity.\nIn the quantum language, the ionization width is due to the coupling\n(via tunnel effect) between the localized wave packet and states\nlying in the chaotic sea surrounding it. While the energy (or quasi-energy\nin the LPM case) of the wave packet is a smooth function of the\nparameters $F$ and $\\omega,$ (see Eq.~(\\ref{semicl})), the energies\nof the chaotic states  display a complicated behaviour characterized\nby level repulsion and strong avoided crossings. By chance, it may happen\nthat -- for specific values of the parameters -- there is a quasi-degeneracy\nbetween the wave packet eigenstate and a chaotic state. There, the two\nstates are more efficiently coupled by tunneling and the ionization\nwidth of the wave packet eigenstate increases. \nThis is the very origin\nof the observed fluctuations. Simultaneously, the repulsion between\nthe two states should slightly modify the energy (real part of the complex\neigenvalue) of the wave packet state. A simple way of measuring this\neffect is to compute also the shift of the real part of the\nenergy level with respect to its \nsemiclassical position (which does not exhibit the repulsion from a \nnear-degenerate state).\n\nAs mentioned above, we study the\nfluctuations for a fixed value of the classically\nscaling parameter $F_0$ versus $n_0=\\omega^{-1\/3}.$\nExemplary ionization\nwidth and {\\it level shift} fluctuations are presented in Fig.~\\ref{fig1}\non a logarithmic scale for the CPM case.\nNote that both quantities fluctuate over\nseveral orders of magnitude and that the widths are more sensitive\nto changes of $n_0$. Since the shifts can take both positive\nand negative values, the absolute value of shifts is plotted.\n\nImportantly we must mention that the shifts plotted in Fig.~\\ref{fig1}\nand used later for the statistical analysis are not obtained directly\nfrom the difference between the resonance energy and the semiclassical\nprediction, Eq.~(\\ref{semicl}), as anticipated before. \nThese differences show\na bias -- the average shift is non zero. It indicates that although\nEq.~(\\ref{semicl}) well predicts the wave packet energy (to within a\nfraction of the mean level spacing), the remaining difference is\nnot solely due to the fluctuations. There is a slowly varying\npart in it which most probably originates from the unharmonic\ncorrections. The latter could be estimated; however, since in the LPM case,\nwe do not have any good semiclassical prediction, we find the fluctuating\npart of the shift in both cases by \nsubtracting from the exact quantum\nenergies the smooth background. The\nlatter is obtained by a low order polynomial fit of the wave packet\neigenenergies as a function of the parameter $n_0.$ \n\nTo describe the fluctuations {\\it quantitatively}, which is the main\naim of this study, we calculate the statistical distributions\nof the ionization widths $P(w)$ and of the energy shifts\n$P(s).$ Typical distributions are displayed as histograms\non a double logarithmic scale in Fig.~\\ref{fig2}.\nThe data are those of Fig.~\\ref{fig1}.\nThe use of logarithmic scales is useful to show quantitatively\nthe fluctuations over several orders of magnitudes.\n>From this figure, we immediately obtain the following qualitative conclusions:\n\\begin{itemize}\n\\item\nThe distribution of energy shifts $P(s)$ tends to a constant as\n$s \\rightarrow 0.$\n\\item Above some critical value, the distribution drops and decreases\nroughly algebraically with $s.$ The slope is close to $-2$, indicating\na $P(s) \\equiv 1\/s^2$ behaviour.\n\\item Finally, $P(s)$ falls off abruptly above some value.\n\\item The distribution of ionization widths $P(w)$ behaves algebraically\nwith $w$ at small width, with a slope close to $-1\/2.$\nHence, the small widths are the most probable ones.\n\\item Above some critical value, the distribution drops faster,\nroughly with a $1\/w^{3\/2}$ behaviour.\n\\item Finally, similarly to the shift distribution, there is a\nfinal sharp cutoff.\n\\end{itemize}\n\nThese conclusions are similar to the ones obtained in a slightly\ndifferent physical situation, when two symmetric regular islands are\ncoupled to a single chaotic sea (see Ref.~\\cite{TU94} for a complete\ndiscussion of this physical situation). There, states lying on the\nregular islands appear in doublets with even\/odd parities with respect\nto the discrete symmetry exchanging the two regular islands. The splitting\nof the doublet is a direct measure of the chaos assisted process where\nthe particle tunnels from one regular island, then diffuses in the chaotic sea\nand finally tunnels to the other regular island. This process is very similar\nto the one studied in the present paper where the particle tunnels and\nthen diffuses towards infinity.\nThe splitting distribution observed in \\cite{TU94,LU96} is indeed very\nsimilar to our shift distribution as explained below.\n\nIn the following section, we propose a simple model - mainly based on \nphysical ideas similar to those used in \\cite{TU94} - which allows\nus to understand\nthe properties of our shift and width distributions.\n\n\\section{Statistical Model}\n\n\nWe shall consider a simple statistical model amenable to an\nanalytical treatment. Despite its simplicity, we shall see\nthat it is capable of\ndescribing quantitatively the fluctuations of resonance widths and\nshifts observed in the numerical analysis of the microwave ionization of\nhydrogen atoms.\n\nThe model system is directly based on the understanding\nwe have of the origin of the fluctuations and follows the idea\nalready used in~\\cite{TU94}.\nThe idea is to consider the wave packet eigenstate as coupled\nrandomly to a set of chaotic states (described by Random Matrix Theory)\nwhich are \nthemselves randomly coupled to the atomic continuum.\nMore precisely, our model consists of a single state $|0\\rangle$ (representing a\nstate localized on the elliptic island) coupled (weakly) to\nthe space of $N$ ``chaotic'', irregular levels, which are close to \n$|0\\rangle$ in energy.\nThe latter\nstates are considered to be, strictly speaking, resonances\nrather than bound states due to the mechanism responsible\nfor decay (e.g., the ionization of an atom).\nWhile we consider\na single localized state in the model, there may be several other\nstates localized on the same island. They will have typically\nmuch different energies and their mutual coupling via the\nchaotic states is negligible.\n\nThe Hamiltonian\nmatrix may be represented as a $(N+1)\\times (N+1)$ matrix:\n\\begin{equation}\n{\\cal H}=\\left(\n\\begin{array}{cc}\n0 & \\sigma \\langle V| \\\\\n\\sigma |V\\rangle & H_0-i {\\cal W}\n\\end{array}\n\\right),\n\\label{mat1}\n\\end{equation}\nwhere we have assumed that the energy of the localized state sets\nthe zero of\nthe energy scale.\nThe first vector in the basis represents the localized state,\nthe following $N$ ones the chaotic states.\nIn the chaotic subspace, the statistical properties of the Hamiltonian\nare well represented by a $N\\times N$ random matrix $H_0$.\nWe shall deal with\nproblems with a preserved (generalized) time-reversal invariance --\n$H_0$ should belong, therefore, to\nthe Gaussian Orthogonal Ensemble (GOE) of real symmetric matrices~\\cite{haake,bohigas}.\nThe matrix elements of $H_0$ are independent random Gaussian variables:\n\\begin{equation}\nP((H_0)_{ij}) = \\frac{\\exp \\left(-\\frac{(H_0)_{ij}^2}{2\\sigma_{ij}^2} \\right)}\n{\\sqrt{2\\pi \\sigma_{ij}^2}},\n\\end{equation}\nwith\nthe variance satisfying\n\\begin{equation}\n\\sigma_{ij}^2 = (1+\\delta_{ij}) \\frac{\\pi^2 \\Delta^2}{N}.\n\\end{equation}\nHere, $\\Delta $ is the mean level spacing between consecutive\nchaotic states close to energy 0.\n\nFollowing a commonly accepted approach \\cite{HILSS}\nthe coupling of the chaotic state\nto the continuum is introduced by the decay matrix:\n\\begin{equation}\n{\\cal W} = \\frac{\\gamma}{2} |W\\rangle \\langle W|\n\\label{w}\n\\end{equation}\nwhere the $N$ component real vector $|W\\rangle $ describes the coupling of\nthe chaotic states to the continuum.\nAs in Ref.~\\cite{HILSS}, we take this\nvector to be composed of Gaussian\ndistributed random numbers with vanishing mean and unit variance.\nThe real coefficient $\\gamma$ measures the strength of the\ndecay to the continuum.\nSuch a form implies that there is only one significant\ndecay channel for chaotic states. This is far from obvious and,\nas discussed below, probably true only at relatively low field\nstrength.\nWhen there are several open ionization channels, a convenient form\nof the decay matrix is~\\cite{HILSS}:\n\\begin{equation}\n{\\cal W}=\\sum_{k=1}^{M} \\frac{\\gamma_k}{2} \\ |W^{[k]}\\rangle \\langle W^{[k]}|,\n\\label{Gam}\n\\end{equation}\nwhere $M$ denotes the number of open channels\n(degeneracy of the continuum) and the $N$ component\nreal\nvectors $|W^{[k]}\\rangle $ describe the coupling of the chaotic states\nto channels $k=1,..,M$. Again, we take these\nvectors to be composed of Gaussian\ndistributed random numbers with vanishing mean and unit variance.\nThe $\\gamma_k$ real coefficients measure the strength of the\ndecay to continuum $k$.\n\nIn a similar way, the (real) $N$-component vector $|V\\rangle $ in\nEq.~(\\ref{mat1}) describes the\ncoupling of the localized state $|0\\rangle$ to the chaotic states.\nEach component of $|V\\rangle $ is taken as a Gaussian distributed random\nnumber of zero mean and unit variance. The coefficient $\\sigma$ in\nEq.~(\\ref{mat1}) is a\nmeasure of the strength of the coupling between the localized state\nand the chaotic subspace.\n\nIf the coupling to the continuum is neglected $(\\gamma=0)$, the model\ndescribes a single bound state randomly coupled to $N$ chaotic states.\nThis is exactly the model succesfully used in~\\cite{TU94} to describe the \nsplitting of doublets induced by chaos assisted tunneling.\n\nThe model has several free parameters: the mean level\nspacing between chaotic states, $\\Delta$, the strength of the\ncoupling with the ionization channel, $\\gamma$,\nand the strength of the coupling to the localized state, $\\sigma$.\nThere is a trivial scaling law of the Hamiltonian ${\\cal H}$, Eq.~(\\ref{mat1}),\nwhich implies that, except for a global multiplicative factor, there\nare only two relevant dimensionless parameters,\n$\\sigma\/\\Delta$ and $\\gamma\/\\Delta$ in the model.\nFor several open channels, the relevant parameters are $\\sigma\/\\Delta$ and\n$\\gamma_k\/\\Delta,\\ k=1..M.$\n\n\nDue to the interaction with ``chaotic\" resonances, the state $|0\\rangle$\nis not an eigenstate of the full Hamiltonian ${\\cal H}$, Eq.~(\\ref{mat1}).\nHowever, in most cases, the coupling to chaotic resonances is weak\nand the true eigenstate does not differ very much from $|0\\rangle :$ this\nis the perturbative regime that we shall consider in the rest of this paper.\nThis regime is obtained when the coupling is much smaller than the mean\nlevel spacing between chaotic states, i.e.:\n\\begin{equation}\n\\sigma \\ll \\Delta .\n\\label{pert1}\n\\end{equation}\n\nWe will see below that this condition is always satisfied for the\nreal physical system (hydrogen atom + microwave field) for the microwave\nfields studied in this paper. At very high field, when the nondispersive\nwave packet is destroyed, this pertubative approximation should break down.\nThe physical interpretation is clear: if Eq.~(\\ref{pert1}) is not satisfied,\nthe localized state is spread over several eigenstates of ${\\cal H}$ and\ncompletely looses its identity. It has been shown elsewhere \\cite{zbd96} that,\nin the perturbative regime, the localized state can be interpreted as a soliton\nweakly interacting with the background of chaotic states, but\nessentially keeping its shape when a parameter (for example the microwave\nfield strength) is varied.\n\nIn the following, we will also assume that the ionization rates (widths)\nof the chaotic states are small compared to their mean level spacing,\ni.e. that the decay matrix $W$, Eq.~(\\ref{Gam}), can be considered as\na pertubation. This implies:\n\\begin{equation}\n\\gamma \\ll \\Delta .\n\\label{pert2}\n\\end{equation}\nSuch a condition is not {\\em strictly} necessary, but it makes calculations\nsimpler. Physically, it means that the various chaotic\nresonances are isolated. This is a typical situation in our system -\nthe ionization of an atom in\nthe presence of microwave driving for not too strong microwave amplitudes.\n\nWith the above assumptions -- motivated by the physics of the process\nstudied --\nthe shift and width of the localized state may be obtained\nusing the lowest nonvanishing order of perturbation theory.\nSuch an approach is justified unless an accidental degeneracy\nbetween the localized state and one of the eigenstates of the matrix\n$H_0$ occurs. Neglecting such degeneracies (which only affect the tail of\nthe distribution, see Sec.~IV.A below) and performing\nan average over the random matrix ensemble defined by Eq.~(\\ref{mat1})\nmakes it possible to extract\nthe analytic expressions both for the distribution of shifts\nand for the distribution of widths. The details\nof the derivation are given in the Appendix.\nWe give here only the important results.\n\nThe shift distribution  is obtained\nalong a similar way to Leyvraz and Ullmo\n\\cite{LU96} and takes the form of a Cauchy law\n\\begin{equation}\nP(s)=\\frac{1}{\\pi}\\frac{s_0}{s_0^2+s^2},\n\\label{eqs}\n\\end{equation}\nwith\n\\begin{equation}\ns_0=\\frac{\\pi \\sigma^2}{\\Delta}.\n\\label{s0ab}\n\\end{equation}\nImportantly, this result is independent of the\ndegree of correlation among the eigenvalues of the $H_0$ matrix: the\nsame result is obtained for an uncorrelated spectrum (Poisson-like\ndistributed eigenvalues, physically corresponding to coupling\nof the localized state with a set of ``regular\" states, instead of\nchaotic ones as in the model described above),\na GOE or a picket fence (harmonic ocsillator)\nspectrum \\cite{LU96}. This Cauchy distribution is the same than the one\nobtained \\cite{TU94,LU96} in the absence of ionization.\n\nThe situation is a bit more complicated for the width distribution.\nFor Poissonian distributed\neigenvalues of $H_0$ (uncorrelated spectrum), a similar Cauchy distribution\nis obtained for the {\\em square root}\nof the width,\ncorresponding to the\nfollowing distribution of widths (see Appendix):\n\\begin{equation}\nP_{\\rm Poisson}(w)=\\frac{1}{\\pi}\\frac{\\sqrt{w_0}}{\\sqrt{w}(w+w_0)},\n\\label{eqx}\n\\end{equation}\nwith\n\\begin{equation}\nw_0=\\frac{4\\sigma^2\\gamma}{\\Delta^2}.\n\\end{equation}\nFor a matrix $H_0$ belonging to the GOE, the distribution\nis slightly more complicated (see Appendix):\n\\begin{equation}\nP_{\\rm GOE}(w) = \\frac{2}{\\pi^2} \\frac{{w'_0}^{3\/2} \\ln (w+\\sqrt{1+w\/w'_0})\n+ \\sqrt{ww'_0(w+w'_0)})}{w(w+w'_0)^{3\/2}}\n\\label{eqx2}\n\\end{equation}\nwith\n\\begin{equation}\nw'_0= \\frac{\\pi^2 \\sigma^2 \\gamma}{\\Delta^2} = \\frac{\\pi^2}{4}\\ w_0.\n\\end{equation}\nAlthough it is distinct from Eq.~(\\ref{eqx}), it has in fact quite a similar\nshape, see Fig.~\\ref{fig2.2}.\nIn particular, if $P_{\\rm GOE}$ is rescaled by a global multiplicative\nfactor of about 20\\%, it is almost indistinguishable from $P_{\\rm Poisson}.$\nQuite a large statistics is necessary to determine which of\nthe two distributions is a correct one for a given data set (see\nalso next Section).\n\nWe can also obtain the distribution of widths for the chaotic states in\nthe perturbative regime. This is the well-known ``non-overlaping resonances\"\nregime~\\cite{brody,bg:prl,dupret} where the widths are distributed according\nto a Porter-Thomas distribution:\n\\begin{equation}\nP_{PT}(w)=\\frac{1}{\\sqrt{w\\overline{w}}}\n\\exp(-w\/2\\overline{w}),\n\\label{PT}\n\\end{equation}\nwhere $\\overline{w}$ denotes the average width.\nFor small $w$, $P_{PT}$ diverges as $w^{-1\/2}$ while\nfor large $w$ it decays exponentially.\n\n\n\n\\section{Analysis of the data}\n\n\\subsection{Quantitative analysis of the fluctuations with the statistical\nmodel}\n\nThe exact expressions, Eqs.(\\ref{eqs}-\\ref{eqx2}), obtained\nin the perturbative regime, reproduce qualitatively most of the\nstatistical distributions numerically observed for\nthe shift and width of the nondispersive wave packet\nof the hydrogen atom in a microwave field, see Fig.~\\ref{fig2}.\nFor the shift distribution $P(s),$ the distribution is constant\nnear $0$ and decays like $1\/s^2$ at large $s$. The only difference\nis the absence of the sharp cut-off in the perturbative expression.\nThis can be easily understood: the large energy shifts correspond\nto quasi-degeneracies between the localized state and one specific\nchaotic state, i.e. to the immediate vicinity of an avoided crossing.\nThere, the simple pertubative scheme breaks down and the actual shift\nremains finite as the perturbative expression diverges as the inverse\nof the unperturbed spacing between the chaotic and localized states\nHence,\nthe actual distribution has to fall faster than the perturbative one\nat large shifts, as numerically observed.\n\nThe width (ionization rate) distribution behaves similarly.\nBoth the initial $1\/w^{1\/2}$ regime and the following $1\/w^{3\/2}$ regime\nobserved for the \nwave packet\nare well reproduced by the simple statistical model. Again, the difference\nis the absence of the cut-off for very large widths. \nThe reason is identical,\nnamely the breakdown of the perturbative approximation.\n\nNevertheless, we can go beyond the perturbative scheme, using\nthe statistical model described in the previous section, but calculating\nnumerically the shift and width distribution.\nThis has been done by numerical diagonalization of the complex Hamiltonian,\ni.e. of\nrandom matrices\ncorresponding to Eq.~(\\ref{mat1}), generated according to the rules\ngiven above. A diagonalization of a matrix of size\n$N=80$ yields a single shift and width for chosen values of\n$\\sigma\/\\Delta$ and $\\gamma\/\\Delta$.\nUsing different \nrandom matrix realizations\nwe accumulate up to 50~000 data for comparison. We have verified\nthat the distributions obtained do not depend on the matrix size $N$.\n\n\nIn Fig.~\\ref{fig2.1}, we show the numerical results\nfor the shift distribution obtained for \nthe hydrogen atom in a circularly polarized\nmicrowave field with the\nperturbative analytical expression for our random matrix model\n(pure Cauchy distribution),\nEq.~(\\ref{eqs}), and with the\nfull non-perturbative result using our statistical model,\nboth on a linear scale \n-- well suited for small and moderate\nshifts -- and on a double logarithmic scale -- well suited for the\ntail of the distribution at large shifts.\nAs expected, the perturbative analytical expression reproduces\nthe numerically observed distribution, except for the exponentially\nsmall tail at large shift. The full non-perturbative distribution\nis found to be in excellent agreement with the numerical data\nfor the real system -- the hydrogen atom in a circularly polarized\nmicrowave field, which proves that our simple statistical model\nactually catches the physics of the chaos assisted tunneling\nphenomenon. A similar conclusion has been reached \\cite{TU94} for\ndoublet splittings induced by chaos assisted tunneling.\n\nIn Fig.~\\ref{fig2.2}, a similar analysis is done for the\ndistribution of widths, both on linear and\ndouble logarithmic scale. On the linear scale, instead\nof the width distribution,\nEq.~(\\ref{eqx2}), itself which diverges at zero\n(see Appendix), we plotted\nthe distribution for the square root of the width,\nwhich tends to a constant value at zero. As can be seen,\nthe agreement is again excellent over the full range,\nthe perturbative expression being inaccurate in the tail only,\nas expected. In addition to the\nperturbative analytical expression, Eq.(\\ref{eqx2}),\nwe have also drawn the distribution\nexpected when the states in the chaotic sea have eigenergies\ndescribed by a Poisson distribution rather than a GOE one, Eq.~(\\ref{eqx}).\nBoth distributions are similar far from the origin and differ by\nabout 20\\% at $w=0.$ At first glance, it seems that the Poisson\ncurve agrees slightly better than the GOE curve, which is somewhat\nsurprising and not understood\nas chaotic motion surrounding the stable island\nsuggests the choice of the GOE. However, the deviation is at the border\nof statistical significance.\nIn the double-logarithmic plot, we have also added the Porter-Thomas\ndistribution, Eq.~(\\ref{PT}), which reproduces correctly the tail\nat large widths.\n\nIt is remarkable that both the shift and the width\ndistributions are so well reproduced by the random matrix\nmodel.\nThis proves that our simple statistical model carries the essential part\nof the physics. The data presented are the first, as far as we know, manifestation\nof the chaos assisted tunneling process in a realistic, experimentally\naccessible systems.\n\nEven if the perturbative expression, Eq.~(\\ref{eqx}),\nincorrectly describes the\nnon-perturbative large width tail,\nit is\nclear that the initial $w^{-1\/2}$ decrease, similar to\n$P_{PT}$, Eq.~(\\ref{PT}), is followed by a regime of $w^{-3\/2}$ behaviour.\nThe length of this $w^{-3\/2}$ behaviour provides an interesting\ninformation about the system. For strong coupling, $\\sigma \\ge \\Delta,$\nthe localized state $|0\\rangle$ is strongly mixed with the chaotic state.\nThus, its width\ndistribution would be the same as that of other resonance states, i.e.\na Porter-Thomas distribution. Hence, the $w^{-3\/2}$ part\nthere shrinks to zero and the power $-1\/2$ law is followed immediately by\nthe exponential tail.\nThe relative importance of the $w^{-3\/2}$ part in the\nwidth distribution indicates, therefore, the presence of the\nweak coupling, perturbative regime.\nCompared to the pure chaotic state where fluctuations\nof the widths are already known to be large, the effect of additional tunneling\nis to shift some part of the width distribution towards small\nwidths while keeping the exponential cut-off, that is\nto increase the fluctuations. In the perturbative limit, the fluctuations\nbecome so large that the average width \n{\\em diverges}.\n\nThe success of the statistical model allows to give a complete\nphysical interpretation of the observed data:\n\\begin{itemize}\n\\item The smallest shifts and widths, observed for\n\\begin{eqnarray}\ns < s_0\\\\\nw < w_0\n\\end{eqnarray}\nwith probabilities behaving, respectively, as $s^0$ \nand $w^{-1\/2}$,\ncorrespond to the localized state lying far from quasi-degeneracies\nwith one of the chaotic states. Then, the localized state is weakly\ncontaminated by the various surrouding chaotic levels. For example, its shift\nis the sum of the effect of level repulsion by the various chaotic states.\nChaotic states with higher (resp. lower) energy push the localized state down\n(resp. up) in energy, which globally results in a small shift with a random \nsign.\nThe width results from the interference between the elementary ionization \namplitudes\ncontributed by \nthe various chaotic states. \nAs there is only one open decay channel, the\namplitudes -- not the probabilities -- have to be added. \nAs they are essentially random\nuncorrelated variables, the interference is mainly destructive, producing\nsmall widths with a large probability.\n\n\\item The intermediate shifts and widths, observed for\n\\begin{eqnarray}\ns_0 < s << \\sigma\\\\\nw_0 < w << \\gamma\n\\end{eqnarray}\nwith probabilities behaving respectively as $s^{-2}$ and $w^{-3\/2}$,\ncorrespond to one chaotic state being much closer in energy\nto the localized state than to the other chaotic states, but\nnevertheless sufficiently far to be \ncoupled only perturbatively. Then,\nthe single coupling to this particular chaotic state dominates both\nthe shift and the width of the localized state.\n\n\\item The largest shifts and widths, observed for\n\\begin{eqnarray}\ns \\simeq \\sigma \\\\\nw \\simeq \\gamma\n\\end{eqnarray}\ncorrespond to the quasi-degeneracy between the localized state and one chaotic\nstate, with strong non-perturbative mixing between the two.\nThis is where the exponentially decreasing tails are observed.\n\\end{itemize}\n\nIn the latter two cases, as one chaotic state is dominant, approximate\nexpressions for the distributions can be obtained by a simple\ntwo-level model. Although the expression of the shift and the width are\nthen easy to obtain (diagonalization of a $2\\times 2$ complex matrix),\nwe have not been able to perform analytically the averaging\nover the ensemble of random matrices.\n\n\\subsection{Extraction of the relevant physical parameters from the statistical data}\n\n\nThe simplest possible measures of the fluctuations of a quantity are typically\nits average value and the variance. Inspection of\nthe perturbative distributions for both the shifts, Eq.~(\\ref{eqs}), and\nthe widths, Eq.~(\\ref{eqx}), suggests, however, some caution.\nIndeed, the average value of the width is not defined (the corresponding\nintegral diverges) and the same is true for the variance of the shift\n(the average value is zero due to the symmetry of the distribution).\nThis is because of the existence of long algebraic tails, $1\/s^2$ and\n$1\/w^{3\/2}$, in the distributions. The variance of the shift and the\naverage value of the width are infinite because of the diverging contributions\nof these tails. This is an example of unusual random processes such as Levy\nflights~\\cite{levy} where rare events play the dominant role.\nUltimately,\nit is because, in perturbation theory, the contamination of a state by its\nneighbors is proportional to the ratio of the coupling matrix element\nto the energy difference and consequently decreases slowly: even a far distant\nlevel can have a large effect.\n\nFor the full non-perturbative \ndistributions, the exponential cutoff\nat large values kills the divergence of the various integrals and\naverage values and variances -- as well as higher moments of the\ndistributions -- are well defined and could be calculated.\nTheir precise values, however, depend crucially on the position of the cutoffs.\nHence, distributions identical for small widths (shifts) and only\ndiffering at large widths (shifts) may have completely different\naverage values and variances.\nCalculating such quantities requires a very accurate knowledge\nof the tails of the distributions. The average values and the\nvariances are thus fragile and difficult to calculate on a real\nsystem like the hydrogen atom in a microwave field; they do not provide\nus with the most interesting physical information.\n\nRather than the average values, we prefer to define typical values.\nThe typical width $\\tilde{w}$ lies at the middle of the distribution, \nsuch that half of the widths are larger and half of them smaller, i.e.\n\\begin{equation}\n\\int_0^{\\tilde{w}} P(w)\\ dw = \\frac{1}{2}\n\\end{equation}\nWith such a definition, the typical width is robust, only weakly sensitive\nto the tail of the distribution.\n\nA similar definition holds for the typical shift, slightly modified\nto take into account that $s$ can be either positive or negative:\n\\begin{equation}\n\\int_0^{\\tilde{s}} P(|s|)\\ d|s| = \\frac{1}{2}.\n\\end{equation}\n\nFor the perturbative distributions in the statistical random matrix model,\nthe typical widths and shifts can be calculated from\nEqs.~(\\ref{eqs}),(\\ref{eqx}),(\\ref{eqx2}):\n\\begin{equation}\n\\left\\{\n\\begin{array}{lll}\n\\displaystyle \\tilde{s}&=&\\displaystyle  \\frac{\\pi \\sigma^2}{\\Delta}\\\\\n\\displaystyle \\tilde{w}&=& \\displaystyle \\frac{4 \\sigma^2 \\gamma}{\\Delta^2}\\ \\ \\ \\ {\\rm for\\ a\\ Poisson\n\\ spectrum}\\\\\n\\displaystyle \\tilde{w}&=& \\displaystyle \\frac{5.48 \\sigma^2 \\gamma}{\\Delta^2}\\ \\ \\ \\ {\\rm for\\ a\\ GOE\n\\ spectrum}.\n\\end{array}\n\\right.\n\\label{typical}\n\\end{equation}\nFor the full non-perturbative \nrandom model distributions -- as long as the basic hypothesis\nof small coupling, Eqs.~(\\ref{pert1}) and (\\ref{pert2}), are true --\nwe carefully checked that\nthe typical widths and shifts are not significantly \n(within few tenths of percent)\ndifferent from the previous analytic expressions.\n\nFor a real physical system like the wave packets \nin the\nhydrogen atom exposed to a microwave\nfield, we can reliably extract from the statistical data the typical\nwidth and shift. We have also compared the numerically obtained\ndistributions with non-perturbative distributions of our statistical\nmodel and performed best fits of the former by the latter. Again, we checked\nthat the typical width and shift for the best fit do not differ\nby more than \nfew tenths of percent from the direct measures. This implies\nthat the typical shift and width can be safely used to extract\nfrom the statistical distributions the relevant \nphysical parameters\n$\\sigma$ (coupling between the localized\nstate and the chaotic states) and $\\gamma$ (ionization widths of the\nchaotic states). Slightly different values are obtained\nif the Poisson or GOE expressions are used.\nIn the following, we have used the GOE expression.\n\nIn fact, only $\\sigma^2\/\\Delta$ and\nthe dimensionless parameter $\\gamma\/\\Delta$\n(from the ratio $\\tilde{w}\/\\tilde{s}$) can be easily extracted.\nObtaining the other dimensionless parameter $\\sigma\/\\Delta$ or\n$\\sigma$ and $\\gamma$ themselves requires to know the mean spacing $\\Delta$ \nbetween\nchaotic resonances.\nSurprisingly, this is not straightforward.\nTo understand the problem, consider\nthe diagonalization of the Floquet Hamiltonian for the LPM case. The\nnumber of states present in a single Floquet zone depends on the\nnumber of photon blocks included in the diagonalization. When\nit is increased, new states appear in the vicinity of the wave packet state\ncorresponding to either low lying atomic states (with very different\nenergy but shifted upwards by an integer times the photon frequency) or\nhighly excited states or resonances (shifted downwards).\nThese states should not \ncontribute to the determination\nof $\\Delta$ since they have a vanishing overlap with atomic states\nbuilding the wave packet.\nHence, the mean level spacing $\\Delta$ between \nchaotic states\nis a somewhat ambiguous quantity. However -- as will be seen \nat the beginning of the next section --\nall our results are obtained either in the perturbative regime\n$\\sigma ,\\gamma \\ll \\Delta$ or close to it, and the mean spacing\n$\\Delta$ is just a scaling parameter. A rough estimate of $\\Delta$\nis obtained assuming that only states with similar principal quantum number,\nlet us say differing by less than a factor 2, are efficiently coupled.\nExperimental results on hydrogen atoms in a linearly polarized microwave field\n\\cite{koch}\nsuggest that the physics of ionization is entirely dominated\nby states with principal quantum number less than $2n_0$\n(when the experimental cut-off is changed from infinity to $2n_0$,\nno important change is observed). At low microwave field,\nthe number of efficiently coupled states is smaller,\nbut this is not the regime of chaotic diffusion we are interested in.\nChaotic motion requires overlap between classical resonance islands,\ni.e. efficient coupling between states of largely different principal\nquantum numbers.\nThis gives the following approximate mean level spacing, used later in\nthis paper:\n\n\\begin{equation}\n\\Delta\\approx \\frac{1}{n_0^4}.\n\\end{equation}\n\n\n\\subsection{Tunneling and chaotic ionization rates}\n\\label{sectyp}\n\nFig.~\\ref{fig3} shows the typical width and shift for the non-spreading\nwave packet of the hydrogen atom in a circularly polarized microwave field\nat frequency $\\omega=1\/(40)^3,$\nas a function of the scaled electric field $F_0=F(40)^4.$\nEach point in this curve results from the numerical diagonalization\nof several hundred matrices, each of typical size several tens\nof thousands, for neighbouring values of the microwave\nfield strength and frequency.\nThe statistical model described above makes it possible to separate\nthe intrinsic huge fluctuations of the ionization rate\nand extract values of the various couplings.\nThis is very clear in Fig.~\\ref{fig3} where both the typical width and the\ntypical shift are relatively smooth functions of the field strength,\nwith short range fluctuations smaller than a factor 2, whereas the\nraw shift and width display fluctuations over \nat least 3 orders of magnitude,\ncompare Fig.~\\ref{fig3} with Fig.~\\ref{fig1}.\n\nIn Fig.~\\ref{fig3}, one can easily check that the typical width\nis always smaller than the typical shift by at least \none order\nof magnitude. As the ratio of the two is $\\gamma\/\\Delta$, see\nEq.~(\\ref{typical}), this implies that inequality (\\ref{pert2})\nis verified.\nAlso, the typical shift is smaller than the mean level spacing\n(represented by the dashed line) which, using \nEq.~(\\ref{typical}),\nshows that inequality (\\ref{pert1}) is also verified.\nAltogether, this proves that our data are effectively obtained\nin the perturbative regime.\n\nThe third observation in Fig.~\\ref{fig3} is that neither the typical\nwidth, nor the typical shift are monotously increasing functions of the\nmicrowave field strength, but display various bumps. These bumps\nare obviously strongly correlated, which indicates that they\nare due to variations of the tunneling rate $\\sigma$ rather than\nvariations of $\\gamma.$\nIndeed, in Fig.~\\ref{fig4},\nwe plot the dimensionless parameters $\\sigma\/\\Delta$ and $\\gamma\/\\Delta$\ndeduced from the typical shift and width.\nIt confirms that the tunneling rate $\\sigma\/\\Delta$ is a slowly increasing\nfunction of the\nfield strength with various structures. The bumps occur\nprecisely at values of the field strength where there is\na resonance between the\neigenfrequencies $\\omega_+$ and $\\omega_-$, see Eq.~(\\ref{modes}),\nof the motion in the vicinity of the stable fixed point supporting\nthe non spreading wave packet. This has been analyzed in reference~\\cite{zbd96}\nwhere it is shown that the bump around $F_0=0.023$ corresponds to the\n1:4 resonance and the bump just below $F_0=0.04$ to the 1:3 resonance.\nIn the vicinity of such a resonance, the classical dynamics is strongly\nperturbed, some secondary resonant tori and islands appear. The bumps\nin Fig.~\\ref{fig4} are just quantum manifestations of an increased\ntransport rate induced by these classical resonances.\nNot surprisingly, $\\gamma\/\\Delta$, which represents the ionization rate\nof states surrounding the resonance island, is practically\nnot affected by these resonances (only small residual\noscillations are visible around $F_0=0.04$). On the other hand, it increases\nvery fast up to scaled field $F_0\\approx 0.04$\nwhere it saturates to a roughly constant value. This has a simple\nsemiclassical explanation. Below $F_0\\approx 0.04$, chaos is not established\naround the principal resonance island, there still exist some regular\ntori further in phase space which strongly slow down the classical\nchaotic diffusion. Above 0.04, only the principal resonance island survives,\nthe chaotic ionization rate is quite large ($\\gamma\/\\Delta$ is of the order\nof 0.1) and only slowly increases with the\nfield strength.\n\nStrictly similar observations can be made for the hydrogen atom exposed\nto a linearly polarized microwave field, which proves that they\nare not specific to one system under study, but rather general\nproperties of chaos assisted tunneling followed by chaotic\ndiffusion. Fig.~\\ref{fig5} displays the typical width and typical shift\nof the non-spreading wave packet for $n_0=40$, as a function\nof the scaled microwave field $F_0=Fn_0^4.$ Again, these data are\nobtained in the perturbative regime, see Eqs.~(\\ref{pert1})-(\\ref{pert2}),\nand display obviously correlated bumps. The dimensionless tunneling\nrate $\\sigma\/\\Delta$ and chaotic ionization rate $\\gamma\/\\Delta,$\nshown in Fig.~\\ref{fig6}, indicate that the bumps are due to\nsecondary resonances\ninside the primary resonance island between the external microwave frequency\nand the internal Kepler motion. Comparison between Fig.~\\ref{fig4}\nand Fig.~\\ref{fig6} shows that both the tunneling rate\n and chaotic ionization rate\nare of the same order of magnitude in linear and circular polarization,\nwith similar changes versus $F_0$, up to possibly a roughly\nconstant multiplicative factor.\nThis is a confirmation of the experimental observation that\nvery similar ionization threshold frequency dependences\n are observed in the two cases provided $F_0$ is appropriately\nrescaled\n\\cite{koch:circ}. As in \\cite{koch:circ} we observe that larger\nvalues of $F_0$ are necessary in LPM to result in the similar\nbehaviour as for CPM.\n\nTo make the study complete, we have also studied how the\ntypical width and the typical shift change when the principal\nquantum number $n_0$ -- or equivalently, the microwave\nfrequency $\\omega=1\/n_0^3$ -- is changed.\nThe result is shown in Fig.~\\ref{fig7} for the circular polarization,\nfor a fixed scaled microwave field $F_0=0.0426.$ In this plot,\nthe classical dynamics of the system is absolutely fixed, the only\nvarying parameter being the effective Planck's constant\n$\\hbar_{\\rm eff}=1\/n_0.$ The striking phenomenon is the fast decrease\nof both the typical width and typical shift with $n_0.$ In the logarithmic\nscale of Fig.~\\ref{fig7}, is appears as a straight line indicating\nan exponential decrease with $n_0.$ Also, the two quantities\ndecrease along parallel lines, which -- according to \nEq.~(\\ref{typical}) --\nindicates that the tunneling rate $\\sigma$ is responsible\nfor this decrease.\nIn Fig.~\\ref{fig8}, we plotted the dimensionless tunneling\nrate $\\sigma\/\\Delta$ and chaotic ionization rate $\\gamma\/\\Delta$ as\na function of $n_0.$ Note that $\\sigma\/\\Delta$ is plotted using a logarithmic\nscale and $\\gamma\/\\Delta$ on a linear scale!\nThe exponential decrease is of the form:\n\\begin{equation}\n\\sigma\/\\Delta \\simeq \\exp (-n_0S) = \\exp \\left( - \\frac{S}{\\hbar_{\\rm\neff}}\\right)\n\\end{equation}\nwith $S\\approx 0.06 \\pm 0.01$\n(extracted from the plot).\n\nSuch a dependence is typical for a tunneling process. $S$ then\nrepresents the tunneling action from the stable fixed point\nwhere the wave packet sits to the chaotic region surrounding the\nresonance island. If complex orbits are used, \n$S$ can be thought\nas the imaginary part of the action of a complex tunneling\norbit \\cite{whelan,PROF.DR.AMAURY-MOUCHET}. In our realistic system, finding such complex orbits\nis much more difficult than in the few\nmodel systems where this analysis has been done. We have not been able to find\nthe complex path associated with the tunneling process, but\nour purely quantum results may provide a guide in this search, as\nthey show that the imaginary action has to be of the order of 0.06\nfor $F_0=0.0426.$\n\nOn a linear scale, see Fig.~\\ref{fig9}, the dimensionless chaotic ionization\nrate\n$\\gamma\/\\Delta$ is a slowly increasing function of $n_0.$ A simple\nclassical analysis using the so-called Kepler map \\cite{i3e}, which is known\nto produce relatively good predictions for the ionization\nthreshold of Rydberg states by a microwave field \\cite{zgd96,bd93}\npredicts a linear dependence versus $n_0,$ while the numerical result\nseems rather a quadratic function. \nThis discrepancy\ncould be due either\nto the approximations done to obtain the Kepler map\nor to the fact that, for high $n_0,$ the statistical model\nused to extract $\\gamma\/\\Delta,$ see \nEqs.~(\\ref{eqs}),(\\ref{eqx}),(\\ref{eqx2}),(\\ref{typical}), is no\nlonger valid because several ionization channels are open (see next section).\n\nLet us note that to get 800 data points\nfor $n_0=100$ (enough to determine the typical shift\nand width) requires about 40 hours of Cray J98 single\nprocessor CPU time. The presented results are, in this sense,\nquite costly (the size of diagonalized matrices exceeded 200 000\nin this case).\n\n\n\\subsection{Limitations of the model}\n\n\nAlthough the simple statistical model well describes the fluctuations\nof the width and shift of the non-spreading wave packet\nfor a range of scaled microwave field ($F_0\\in [0.04,0.08]$ for\nLPM and $F_0\\in [0.02,0.06]$ for CPM, both for $n_0$ around 40) --\nthe region\nwhere experiments are usually done -- additional difficulties\nappear for lower and higher field values.\n\nFor lower $F_0$ values, a statistically\nsignificant part of the data show very small widths, at the limits\n of the numerical precision. \nAt the same time, \nplots of the wave packets similar to those shown \n in \\cite{dzb95} suggest that the states are more extended,\nextending far from the stable island. The \nsituation then may not correspond\nto a clear-cut case of the chaos assisted tunneling process. \nOur LPM data \nindicate \nthat in such cases the singularity \nfor small widths is much\nstronger than $\\Gamma^{-1\/2}$.\nSimilarly, in the CPM case,\nwe did not present the random matrix fit for $F_0=0.038,$ as a\nsignificant part of the data is \naffected there by a strong classical\n$1:3$ resonance \\cite{zbd96}. Thus we do not \nface a clear case\nof a single localized state  but\nrather two strongly coupled localized states\ndecaying via a chaos assisted tunneling process. Since such a case\nis quite rare, we prefer to exclude it from the analysis and not to \nconstruct the extension of the random matrix theory. It could not\nbe tested convincingly on a single case, anyway.\n\nMost importantly, we could not extend the random model fits to\nhigher $F_0$ values for a very simple\nreason. There, we observed indications of the \nopening of other ionization channels\n(see Eq.~(\\ref{Gam})). A typical signature of such\na behaviour is the disappearance of the\nsingularity $\\Gamma^{-1\/2}$ in the distribution of the widths.\nTo understand this, note\nthat the typical Porter-Thomas distribution\nbehaves, for small widths, as \n$\\Gamma^{M\/2-1}$ with $M$ being\nthe number of open channels \\cite{brody,dupret}. \nIn the chaos assisted tunneling\nprocess leading to ionization, while the full distribution\ndiffers from the Porter-Thomas distribution, as exemplified\nearlier for the single channel case, the small width functional behaviour is\nsimilar in both cases.\n\nThe study of the available data reveals\nthat the opening of the second, and possibly third ionization\nchannel appears gradually with the increasing microwave amplitude,\n$F_0$. Thus the different possible ionization channels are not\nof equal importance, i.e., they are not equivalent (in the language\nof the random matrix theory \\cite{HILSS}). To build the random\nmatrix model of the process one then needs to introduce\nadditional free parameters describing the strength of the coupling\nto the additional ionization channels, i.e. various values\nfor the $\\gamma_k$ in Eq.~(\\ref{Gam}). Although such a procedure\nis quite straightforward, it is clear that fitting these parameters\nto two data sets (shifts and widths) provides little information\nand must be ambiguous. Typical distributions of the square root of\nthe width obtained for large microwave amplitudes are presented in\nFig.~\\ref{fig9} for LPM and CPM wave packets.\nNote the presence of the hole for small widths.\n\n\nOn the other hand, since in the perturbative limit, the level shifts\ndepend only on the real coupling between the localized state and\nthe remaining chaotic subspace, \nEqs.~(\\ref{s0ab}), (\\ref{typical}), \none can expect that the shifts will\nbe still well described by the Cauchy law, Eq.~(\\ref{eqs}), independently\nof the opening of additional ionization channels.\nIndeed, it is the case as exemplified in\nFig.~\\ref{fig9} for LPM and CPM wave packets.\n\nSimilarly, the opening of additional ionization channels is expected\nin the semiclassical limit. The limit is realized by decreasing\nthe microwave frequency, then the wave packet is composed of\ncircular states of higher $n_0$. While the data corresponding to\nthe single channel decay have been obtained for $n_0=40$ at\n$F_0=0.0426,$\nwe observed the opening of the second channel for the same $F_0$\nstarting at $n_0=60$. Panel (e) in Fig.~\\ref{fig9}\npresents the histogram of the square root of the width for $n_0=90$\nand shows the existence of at least two open channels.\nAgain the corresponding shift distribution is not affected and\nis well described by the Cauchy distribution [panel (b)].\n\n\n\\section{Physical Interpretation and Conclusions}\n\nWe have presented a statistical theory of ionization\ncatalyzed by chaos assisted tunneling. The corresponding\nphysical picture is built of a single state, localized on a stable\nisland and coupled (quantum mechanically, due to a finite value of\n$\\hbar$) to the surrounding chaotic sea. Once the tunneling\ninto the sea takes place, the diffusive \nchaotic excitation leads finally to\nionization.\n\nA random matrix theory model allows us to determine analytically\nthe distribution of the energy shifts \n(induced by the interaction with the chaotic sea)\nof the localized state, as well as\nthe distribution of its widths (ionization rates), in the\nperturbative limit. Non-perturbative corrections may also be\nunderstood and estimated. We concentrated on the simplest case of\nsingle channel ionization -- the model then is characterized \nby few parameters only. In that case, both the distributions of shifts\nand widths have long algebraic tails\nexplaining the large scale fluctuations of both quantities.\nThese fluctuations are a characteristic feature of chaos assisted\ntunneling processes. Fluctuations (and universal properties of\nfluctuations) are well established properties of chaotic systems.\nIn the ionization brought about by chaos assisted tunneling,\nthe combination of a weak tunneling process with chaotic\ncoupling to the continuum increases dramatically the range\nof the fluctuations, by extending the distribution considerably \ntowards extremely small widths, i.e. metastable states.\n\nThe developed theory has been confronted with numerical data\nobtained for the shifts and widths of non-spreading wave packets --\nstates localized on a stable 1:1 resonance island between\nthe Kepler frequency of a Rydberg electron and the\nfrequency of an externally applied microwave field of either\nlinear or circular polarization\n-- a system accessible to present experiments. The numerical\ndata have been obtained for simplified models of the atom -\na one-dimensional atom in LPM and a two-dimensional atom in\nCPM. This allowed us to study the frequency range well in the\nexperimental region - the important atomic states building\nthe wave packet correspond to the principal quantum numbers\nused in the experiments. The principal reason for the simplification\nis that fully three-dimensional numerical calculations -- although\npossible for a single set of parameters\nas exemplified by us before \\cite{zdb95} -- are\nstill prohibitive for present day computers.\nMore importantly, however, the statistical\nproperties of non-spreading wave packet states are not affected\nby the \nreduced dimensionality of the atom as tested by us for the\nthree-dimensional CPM case.\n\nIt turns out that the developed statistical theory very well\ndescribes the numerical data in the range of the single channel decay, \nwhich makes us confident that it contains\nthe essential ingredients of the physical process. The quality of the fits\nallows to extract order from chaos, that is to extract from\nstrongly fluctuating quantities, see Fig.~\\ref{fig1}, the physical\nparameters describing the coupling of the non-spreading wave packet\nto the chaotic states\n(tunneling rate) and the \nionization rate of chaotic states. \nThese parameters exhibit reasonably smooth behaviour.\nFor example, we have shown\nthat secondary \nclassical resonances inside the regular island increase the tunneling rate.\nAs an unambiguous signature of a\ntunneling process\nwe also could demonstrate the exponential decrease of the tunneling rate\nwith the principal quantum number.\n\n\nLet us emphasize the importance of the fluctuations of the ionization\nrate (width) of non-spreading wave packets in the hydrogen atom.\nIn a real experiment, it is likely that the atoms will experience\nvarious values of the microwave field strength -- either\nbecause of spatial inhomogeneities or because they are prepared\nby a slow increase of the microwave strength as explained in\n\\cite{zd:jpb97} -- and more or less\naverage the short range fluctuations of the ionization rate.\nIn the total, the residual ionization of the atom will be given\nby the average ionization rate, a quantity dominated by the\nfluctuations towards large ionization rates and which can be\n significantly larger than the typical ionization rate.\nFor example, for the data in Fig.~\\ref{fig1} discussed\nin this paper, the average ionization rate is about 6.4 times\nlarger than the typical ionization rate. In the limit\nof the perturbative regime, the ratio of the two even diverges!\nThis is an example of physical processes as Levy flights where the\nphysics of a fluctuating system is dominated by rare events.\n\n>From a practical point of view, the present study also tells us that\nthe lifetimes of the non-spreading wave packets either in CPM or in LPM\nare rather long. Indeed, for $n_0=60$\nand $F_0=0.0426$  -- these values are representative\nof what could be used in a real experiment, microwave frequency\naround 30 GHz and microwave field \namplitude of the order of 10V\/cm --\nthe typical lifetime of the non-spreading wave packet in CPM,\ndue to ionization catalyzed\nby chaos assisted tunneling, is of the order of several micro-seconds, that is\nabout 100 000 Kepler periods. However, fluctuations\nby one or two orders of magnitude are expected around this typical value.\nEven the longest lifetimes should be shorter\nthan the natural lifetime, due to spontaneous emission, of the\norder of a fraction of a second.  At higher $n_0\\simeq 100,$\nthe typical ionization lifetime is of the order of several milliseconds,\ni.e. 10 million Kepler periods, but still shorter than\nthe lifetime induced by spontaneous emission \\cite{ibb97}. \nHence, for practical\nexperiments in CPM, spontaneous emission should not be a problem. In LPM,\nspontaneous emission is a slightly stronger effect, but \nlargely dominated by chaos assisted tunneling ionization \nfor $n_0\\leq 100$\n\\cite{ab97}.\n\nFinally, the physical situation and the model described\nhere are not restricted to atomic non-spreading wave packets.\nIt should describe physical systems where a given state is weakly\ncoupled to a dense family of completely different other states\nwhich can decay on a rather long time scale. Then, the effective\ndecay rate of the initial state, induced by the coupling with\nthe family of decaying states, should present\nhuge fluctuations. An example is given in nuclear physics by the so-called\nsuper-deformed nuclei \\cite{superdeformed} where the ground state of a super-deformed\nnucleus can only decay by coupling to highly excited\n(hence chaotic) states of the non-deformed nucleus.\nOur model then predicts the distribution of lifetimes\nof super-deformed nuclei.\n\n\\acknowledgments\nCPU time on a Cray C98 computer has been\nprovided by IDRIS and RZG.\nLaboratoire Kastler Brossel de\nl'Universit\\'e Pierre\net Marie Curie et de l'Ecole Normale Sup\\'erieure is\nunit\\'e associ\\'ee 18 du CNRS. J.Z. acknowledges support of KBN\nunder project No.~2P03B~03810.\nThe additional support under the bilateral collaboration scheme\n(J.Z. and D.D.) of the French Embassy in Poland, no.76209 \nand the Programme International de Coop\\'eration Scientifique\n(CNRS) no.408 is appreciated.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}