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Unitary Dynamics of Strongly Interacting Bose Gases with the Time-Dependent Variational Monte Carlo Method in Continuous Space

Giuseppe Carleo, Lorenzo Cevolani, Laurent Sanchez-Palencia, Markus Holzmann

Published Sep 1, 2017• 26 citations
Publisher

Abstract

We introduce the time-dependent variational Monte Carlo method for continuous-space Bose gases. Our approach is based on the systematic expansion of the many-body wave function in terms of multibody correlations and is essentially exact up to adaptive truncation. The method is benchmarked by comparison to an exact Bethe ansatz or existing numerical results for the integrable Lieb-Liniger model. We first show that the many-body wave function achieves high precision for ground-state properties, including energy and first-order as well as second-order correlation functions. Then, we study the out-of-equilibrium, unitary dynamics induced by a quantum quench in the interaction strength. Our time-dependent variational Monte Carlo results are benchmarked by comparison to exact Bethe ansatz results available for a small number of particles, and are also compared to quench action results available for noninteracting initial states. Moreover, our approach allows us to study large particle numbers and general quench protocols, previously inaccessible beyond the mean-field level. Our results suggest that it is possible to find correlated initial states for which the long-term dynamics of local density fluctuations is close to the predictions of a simple Boltzmann ensemble.

DOI
10.3929/ethz-b-000191675

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We introduce the time-dependent variational Monte Carlo method for continuous-space Bose gases. Our approach is based on the systematic expansion of the many-body wave function in terms of multibody correlations and is essentially exact up to adaptive truncation. The method is benchmarked by comparison to an exact Bethe ansatz or existing numerical results for the integrable Lieb-Liniger model. We first show that the many-body wave function achieves high precision for ground-state properties, including energy and first-order as well as second-order correlation functions. Then, we study the out-of-equilibrium, unitary dynamics induced by a quantum quench in the interaction strength. Our time-dependent variational Monte Carlo results are benchmarked by comparison to exact Bethe ansatz results available for a small number of particles, and are also compared to quench action results available for noninteracting initial states. Moreover, our approach allows us to study large particle numbers and general quench protocols, previously inaccessible beyond the mean-field level. Our results suggest that it is possible to find correlated initial states for which the long-term dynamics of local density fluctuations is close to the predictions of a simple Boltzmann ensemble.
This thesis is concerned with non-equilibrium phenomena in interacting quantum many-body systems. Specifically, we investigate the time-evolution and relaxation dynamics of Bose gases in a highly restricted geometry, which constrains the dynamics to one spatial dimension. This leads to a description of the system in terms of a simple model Hamiltonian which permits exact many-body quantum mechanical solutions due to its integrability. In the first part of this thesis, a computational method is developed to obtain experimentally relevant correlation functions in the framework of the coordinate Bethe ansatz. We employ this method to compute exact ground-state correlation functions of the Lieb-Liniger gas for up to seven particles covering the whole regime of repulsive interactions. We also investigate the dynamics of the system after an instantaneous change of the interaction strength. This quantum quench deposits large amounts of energy that cannot be dissipated due to the system being closed, and so the dynamics far from equilibrium are probed. We prepare the system in two different initial states, and quench to the same final interaction strength. The latter is determined in such a way that the added energy due to the quench is the same for both scenarios. Conventional statistical mechanics predicts the same relaxed state, but due to the integrability of the system all considered correlation functions of the relaxed states differ from each other and also from the thermal ones. We then investigate the dynamics and relaxed state for a quench from zero to repulsive interactions in more detail, focussing on the mechanism of relaxation and the involved time-scales. We find that local correlation functions relax on time-scales determined by the interaction strength, in contrast to non-local correlation functions, whose relaxation time-scale is proportional to the system size. Next, we employ the same methodology to study the one-dimensional Bose gas with attractive interactions. In this case many-body bound states are permissible solutions of the Lieb-Liniger model. We compare exact ground-state correlation functions of up to seven particles to their corresponding mean-field solution. The latter displays a quantum phase transition at a critical interaction strength, marking the transition from a uniform-density state to a localized bright soliton. Our exact results agree remarkably well with the corresponding mean-field solution past the critical point. We also investigate the dynamics following an interaction strength quench, starting again from the non-interacting ground state. Bound states strongly influence correlation functions for all post-quench interaction strengths, and local correlation functions are largely increased compared to their initial value. In the last part of this thesis, we investigate the behavior of the one-dimensional Bose gas under periodic driving of the interaction strength. To this end, we extend the coordinate Bethe-ansatz formalism by employing Floquet theory to obtain solutions for the full time- dependent Hamiltonian. This realizes an extension to non-integrable systems in a way that allows controlled breaking of integrability. We compare the dynamics of the system following an interaction quench to a fixed final interaction strength to that of a system with periodically modulated post-quench interaction strength. For fast driving, the system evolves according to the time-averaged Hamiltonian and correlation functions are nearly identical to those of the undriven system. The response of the system qualitatively changes at resonances of the time-averaged Hamiltonian, where we observe energy absorption and a marked change in correlation functions. However, the system does not absorb energy indefinitely.
We present a new variational method to study the dynamics of a closed bosonic many-body system, the time-dependent hypernetted-chain Euler-Lagrange method, tHNC. Based on the Jastrow ansatz, it accounts for quantum fluctuations in a non-perturbative way. The tHNC method scales well with the number of dimensions, as demonstrated by our results on one-, two-, and three-dimensional systems. We apply the tHNC method to interaction quenches, i.e. sudden changes of the interaction strength, in homogeneous Bose gases. When the quench is strong enough that the final state has roton excitations (as found and predicted for dipolar and Rydberg-dressed Bose-Einstein condensates, respectively), the pair distribution function exhibits stable oscillations. For validation, we compare tHNC results with time-dependent variational Monte Carlo results in one and two dimensions.
Determining the dynamics of interacting integrable many-particle quantum systems at finite times after homogeneous quantum quenches is a long-standing challenge. We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths. We discuss the emergence of a "sign problem" for more general dynamical correlators and setups.
We consider the non-equilibrium dynamics of the interacting Lieb–Liniger gas after instantaneously switching the interactions off. The subsequent time evolution of the space- and time-dependent correlation functions is computed exactly. Different relaxation behavior is observed for different correlation functions. The long time average is compared with the predictions of several statistical ensembles. The generalized Gibbs ensemble restricted to a fixed number of particles is shown to give correct results at large times for all length scales.
The description of strongly correlated quantum many-body systems far from equilibrium presents a fundamental challenge due to the vast amount of information it requires. We introduce a generalization of the Jastrow ansatz for time-dependent wave functions that offers an efficient and exact description of the time evolution of various strongly correlated systems. Previously known exact solutions are characterized by scale invariance, enforcing self-similar evolution of local correlations, such as the spatial density. However, we demonstrate that a complex-valued time-dependent Jastrow ansatz (TDJA) is not restricted to scale invariance and can describe a broader class of dynamical processes lacking this symmetry. The associated time evolution is equivalent to the implementation of a shortcut to adiabaticity (STA) via counterdiabatic driving along a continuous manifold of quantum states described by a real-valued TDJA, providing a framework for engineering exact STA in strongly correlated many-body quantum systems. We illustrate our findings in systems with inverse-square interactions, such as the Calogero-Sutherland and hyperbolic models, supplemented with pairwise logarithmic interactions, as well as in the long-range Lieb-Liniger model, where bosons experience both contact and Coulomb interactions in one dimension. Our results enable the study of quench dynamics in all these models and serve as a benchmark for numerical and quantum simulations of nonequilibrium strongly correlated systems with continuous variables.