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Physical Review A

Learning ground states of spin-orbit-coupled Bose-Einstein condensates by a theory-guided neural network

Xiaodong Bai, Dongxiao Zhang

Published Dec 27, 2021• 8 citations
Publisher

Abstract

The ground state of spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) is an intriguing subject that has not yet been fully explored. In particular, how to quickly obtain ground states for discovering their new structures is currently under active debate. In this work, we construct a theory-guided neural network (TgNN) to explore the ground states of one-dimensional BECs with and without SOC. We find that such method is markedly superior to the ordinary deep neural network due to theory guidance of the underlying problems. The former can replace the traditional method to directly give any ground state within a valid parameter plane using only imaginary-time evolution data of nine solutions as the training data, without the tedious step-by-step iterative calculation process. Furthermore, we show that the method of TgNN exhibits higher reliability and a better ability to generalize beyond regimes covered with the training data. This method provides a promising technique for discovering novel structures of ground states in parameterized spatiotemporal systems.

DOI
10.1103/physreva.104.063316

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The ground state of spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) is an intriguing subject that has not yet been fully explored. In particular, how to quickly obtain ground states for discovering their new structures is currently under active debate. In this work, we construct a theory-guided neural network (TgNN) to explore the ground states of one-dimensional BECs with and without SOC. We find that such method is markedly superior to the ordinary deep neural network due to theory guidance of the underlying problems. The former can replace the traditional method to directly give any ground state within a valid parameter plane using only imaginary-time evolution data of nine solutions as the training data, without the tedious step-by-step iterative calculation process. Furthermore, we show that the method of TgNN exhibits higher reliability and a better ability to generalize beyond regimes covered with the training data. This method provides a promising technique for discovering novel structures of ground states in parameterized spatiotemporal systems.
In this work, a coupled theory-guided neural network (coupled-TgNN) is constructed to explore the ground states of one-dimensional binary Bose-Einstein condensates with spin-orbit coupling and a Lee-Huang-Yang term. We find that this method is markedly superior to the ordinary deep neural network due to both the theoretical guidance for the underlying problem and the coupling of neural networks. The former shows better accuracy and more strong robustness and can rapidly construct any ground state using only imaginary-time evolution data of solutions as the training data, without the tedious step-by-step iterative calculation process. In addition, based on the coupled-TgNN approach, a phase transition boundary is also discovered, which clearly distinguishes the single-peak ground state phase from the striped phase. The results not only greatly reduce computational time for exploring the properties of the ground states but also provide a promising technique for discovering phase transitions in other coupled nonlinear systems.
Machine learning has been a powerful tool to study various models in many fields, which requires plenty of data to ensure accuracy. It is a challenge to reduce the dependence on the amount of data and ensure that the predictions are both accurate and physical. We develop a theory-guided neural network (TgNN) to explore the ground states of two-component spin–orbit coupled Bose–Einstein condensation in two-dimension. Only randomly selected few data from the twelve ground states as training data, TgNN performs higher accuracy, broader generalization ability and stronger robustness than deep neural network (DNN). Moreover, in the neighborhood of the phase transition where the predictions of DNN have obvious deviations, while TgNN still keeps relative accuracy. TgNN is less dependent on data and exhibits potential for high-dimensional and complicated models. • TgNN accurately predicts all ground states of 2D SOC BEC, which uses only one-tenth of the total data. • TgNN provides higher accuracy, broader generality ability and stronger robustness than DNN. • TgNN has more advantages than DNN in solving critical problems.
An important and incompletely answered question is whether machine learning methods can be used to discover the excitation of rogue waves (RWs) in nonlinear systems, especially their dynamic properties and phase transitions. In this work, a theory-guided neural network (TgNN) is constructed to explore the RWs of one-dimensional Bose-Einstein condensates. We find that such method is superior to the ordinary deep neural network due to theory guidance of underlying problems. The former can directly give any excited location, timing, and structure of RWs using only a small amount of dynamic evolution data as the training data, without the tedious step-by-step iterative calculation process. In addition, based on the TgNN approach, a phase transition boundary is also discovered, which clearly distinguishes the first-order RW phase from the non-RW phase. The results not only greatly reduce computational time for exploring RWs, but also provide a promising technique for discovering phase transitions in parameterized nonlinear systems.
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.
Properties of the ground-state solitons, which exist in the spin–orbit coupling (SOC) Bose–Einstein condensates (BEC) in the presence of optical lattices, are presented. Results show that several system parameters, such as SOC strength, lattice depth, and lattice frequency, have important influences on properties of ground state solitons in SOC BEC. By controlling these parameters, structure and spin polarization of the ground-state solitons can be effectively tuned, so manipulation of atoms may be realized.
Binary mixtures of Bose-Einstein condensates (BECs) trapped in deep optical lattices and subjected to equal contributions of Rashba and Dresselhaus spin-orbit coupling (SOC) are investigated in the presence of a periodic time modulation of the Zeeman field. SOC tunability is explicitly demonstrated by adopting a mean-field tight-binding model for the BEC mixture and by performing an averaging approach in the strong modulation limit. In this case, the system can be reduced to an unmodulated vector discrete nonlinear Schr\"odinger equation with a rescaled SOC tuning parameter $\ensuremath{\alpha}$, which depends only on the ratio between amplitude and frequency of the applied Zeeman field. We consider the attractive interaction case and focus on the effect of the SOC tuning on the localized ground states. The dependence of the spectrum of the linear system on $\ensuremath{\alpha}$ has been analytically characterized. In particular, we show that extremal curves (ground and highest excited states) of the linear spectrum are continuous piecewise functions (together with their derivatives) of $\ensuremath{\alpha}$, which consist of a finite number of decreasing band lobes joined by constant lines. This structure also remains in the presence of inter- and intra-species interactions, the nonlinearity mainly introducing a number of localized states in the band gaps. The stability of ground states in the presence of the modulating field has been demonstrated by real-time evolutions of the original (unaveraged) system. Localization properties of the ground state induced by the SOC tuning, and a parameter design for possible experimental observation, have also been discussed.