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arXiv:2110.06606 · Scientific Reports

Bayesian Optimization of Bose-Einstein Condensates

Tamil Arasan Bakthavatchalam, Suriyadeepan Ramamoorthy, Malaikannan Sankarasubbu, Ramaswamy Radha, Vijayalakshmi Sethuraman

Published Mar 3, 2021• 12 citations
arXivHTMLPDF

Abstract

Machine Learning methods are emerging as faster and efficient alternatives to numerical simulation techniques. The field of Scientific Computing has started adopting these data-driven approaches to faithfully model physical phenomena using scattered, noisy observations from coarse-grained grid-based simulations. In this paper, we investigate data-driven modelling of Bose-Einstein Condensates (BECs). In particular, we use Gaussian Processes (GPs) to model the ground state wave function of BECs as a function of scattering parameters from the dimensionless Gross Pitaveskii Equation (GPE). Experimental results illustrate the ability of GPs to accurately reproduce ground state wave functions using a limited number of data points from simulations. Consistent performance across different configurations of BECs, namely Scalar and Vectorial BECs generated under different potentials, including harmonic, double well and optical lattice potentials pronounces the versatility of our method. Comparison with existing data-driven models indicates that our model achieves similar accuracy with only a small fraction ([Formula: see text]th) of data points used by existing methods, in addition to modelling uncertainty from data. When used as a simulator post-training, our model generates ground state wave functions [Formula: see text] faster than Trotter Suzuki, a numerical approximation technique that uses Imaginary time evolution. Our method is quite general; with minor changes it can be applied to similar quantum many-body problems.

DOI
10.1038/s41598-021-84336-0

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Machine Learning methods are emerging as faster and efficient alternatives to numerical simulation techniques. The field of Scientific Computing has started adopting these data-driven approaches to faithfully model physical phenomena using scattered, noisy observations from coarse-grained grid-based simulations. In this paper, we investigate data-driven modelling of Bose-Einstein Condensates (BECs). In particular, we use Gaussian Processes (GPs) to model the ground state wave function of BECs as a function of scattering parameters from the dimensionless Gross Pitaveskii Equation (GPE). Experimental results illustrate the ability of GPs to accurately reproduce ground state wave functions using a limited number of data points from simulations. Consistent performance across different configurations of BECs, namely Scalar and Vectorial BECs generated under different potentials, including harmonic, double well and optical lattice potentials pronounces the versatility of our method. Comparison with existing data-driven models indicates that our model achieves similar accuracy with only a small fraction ([Formula: see text]th) of data points used by existing methods, in addition to modelling uncertainty from data. When used as a simulator post-training, our model generates ground state wave functions [Formula: see text] faster than Trotter Suzuki, a numerical approximation technique that uses Imaginary time evolution. Our method is quite general; with minor changes it can be applied to similar quantum many-body problems.
Abstract We show that both single-component and two-component Bose-Einstein condensates’ (BECs) ground states can be simulated by a deep convolutional neural network. We trained the neural network via inputting the parameters in the dimensionless Gross-Pitaevskii equation (GPE) and outputting the ground-state wave function. After the training, the neural network generates ground-state wave functions with high precision. We benchmark the neural network for either inputting different coupling strength in the GPE or inputting an arbitrary potential under the infinite double walls trapping potential, and it is found that the ground state wave function generated by the neural network gives the relative chemical potential error magnitude below 10−3. Furthermore, the neural network trained with random potentials shows prediction ability on other types of potentials. Therefore, the BEC ground states, which are continuous wave functions, can be represented by deep convolutional neural networks.
In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.
The data-driven approach shows great potential for designing new materials with unprecedented properties by using machine learning and optimization. Recently, a data-driven framework was successfully applied to design a unit cell of metamaterial achieving super-compressibility, despite being built out of brittle base material. The key element of the framework is the algorithm called Gaussian processes regression (GPs) – a unique machine learning method that provides with the uncertainty of the prediction, which can be used during the design process to account for inherent material imperfections, ensuring the robustness of the design. Despite their superior predictive performance, however, GPs suffer from scalability issues, which limit their application to relatively small design problems. In the future, those limitations could be surpassed by quantum computing. This research aims at demonstrating how quantum computing could enhance the computational design of materials, specifically by replacing the expensive machine learning step of the data-driven design framework with an exponentially faster quantum algorithm for Gaussian processes (QGP). This objective is achieved in two steps. First, the QGP algorithm was Implementation and simulated within a quantum computing framework (Qiskit), which allowed to understand and control its performance (in particular its accuracy), proving the feasibility for practical applications. Furthermore, the numerical tests exposed a mechanism for inducing a low-rank approximation, which allows for additional speed-up, making the QGP algorithm similar to classical sparse Gaussian processes, which rely on low-rank approximations to improve the scalability of full GPs. In the second part of this research, the implemented QGP algorithm was integrated within a computational framework for the data-driven design of materials and applied to two example design problems of optimization a unit cell for a super-compressible metamaterial. The results obtained with the QGP were comparable to those obtained with classical methods (also from literature), which proved the feasibility of the concept.
Bose–Einstein condensates (BECs), a state of matter formed when a low-density gas of bosons is cooled to near absolute zero, continue to motivate novel work in theoretical and experimental physics. Although BECs are most commonly studied in stationary ground states, time-varying BECs arise when some aspect of the physics governing the condensate varies as a function of time. We study the evolution of time-varying BECs under non-autonomous Gross–Pitaevskii equations (GPEs) through a mix of theory and numerical experiments. We separately derive a perturbation theory (in the small-parameter limit) and a variational approximation for non-autonomous GPEs on generic bounded space domains. We then explore various routes to obtain time-varying BECs, starting with the more standard techniques of varying the potential, scattering length, or dispersion, and then moving on to more advanced control mechanisms such as moving the external potential well over time to move or even split the BEC cloud. We also describe how to modify a BEC cloud through evolution of the size or curvature of the space domain. Our results highlight a variety of interesting theoretical routes for studying and controlling time-varying BECs, lending a stronger theoretical formulation for existing experiments and suggesting new directions for future investigation.
© Q.-C. Chen et al. We develop a tensor network framework based on the quantic tensor train (QTT) format to efficiently solve the Gross-Pitaevskii equation (GPE), which governs Bose-Einstein condensates under mean-field theory. For sufficiently regular functions, the QTT approach enables exponentially fine grids with computational cost that grows only linearly. By adapting the time-dependent variational principle (TDVP) and gradient descent methods, we accurately handle the GPE’s nonlinearities within the QTT structure. Our approach enables high-resolution simulations with drastically reduced computational cost on highly refined grids. We benchmark the ground states and dynamics of BECs, including vortex lattice formation and breathing modes, demonstrating superior performance over regular-grid-based methods and stable long-time evolution due to saturating bond dimensions. This establishes QTT as a powerful tool for nonlinear quantum simulations.
We develop a tensor network framework based on the quantic tensor train (QTT) format to efficiently solve the Gross-Pitaevskii equation (GPE), which governs Bose-Einstein condensates under mean-field theory. For sufficiently regular functions, the QTT approach enables exponentially fine grids with computational cost that grows only linearly. By adapting the time-dependent variational principle (TDVP) and gradient descent methods, we accurately handle the GPE’s nonlinearities within the QTT structure. Our approach enables high-resolution simulations with drastically reduced computational cost on highly refined grids. We benchmark the ground states and dynamics of BECs, including vortex lattice formation and breathing modes, demonstrating superior performance over regular-grid-based methods and stable long-time evolution due to saturating bond dimensions. This establishes QTT as a powerful tool for nonlinear quantum simulations.