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

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.

Popular Summary

Everyday objects move pretty predictably. When an apple falls from a tree, its motion is fully and easily described by its trajectory. Quantum particles such as electrons and protons are much more complex. Describing their motion requires a complex mathematical tool called a wave function to determine the probability of where the quantum “apple” is at a given time. This becomes even harder when many quantum particles are involved. Predicting their exact motion in those circumstances is far beyond reach, even on the most powerful supercomputer. In our work, we have developed a new theoretical approach to accurately study the dynamics of a gas of quantum particles that can help solve the most puzzling problems concerning the collective behavior of quantum objects over long times.

A fundamental open question we address is whether a strongly interacting quantum gas, well isolated from the external environment, still obeys the laws of thermodynamics after being driven out of equilibrium. Despite impressive theoretical and experimental progress over the last years, this and other fundamental questions about quantum thermalization remain open. Our new stochastic method (time-dependent variational Monte Carlo) allows us to accurately study the dynamics of one-dimensional strongly interacting bosons. We show that the absence and occurrence of thermalization of the potential energy can be observed in ultracold atom experiments in a controlled way using a switch from noninteracting to interacting initial states followed by a sudden quench of the interacting strength.

Our methods pave the way for predicting the out-of-equilibrium dynamics of two- and three-dimensional quantum gases and fluids more accurately than current approximation methods.

Figure 1

Ground-state properties of the Lieb-Liniger model as obtained from different variational approaches: (a) relative accuracy of the ground-state energy, (b) one-body density matrix, (c) density-density pair correlation functions. $U^{(2)}$ and $U^{(3)}$ denote results for the two- and three-body expansion (present work), $U_{\mathrm{AG}}^{(2)}$ is the parametrized two-body Jastrow state of Ref. [52], ${\mathrm{cMPS}}_1$ results are from Ref. [23], and ${\mathrm{cMPS}}_2$ are those very recently reported in Ref. [24]. Distances $r$ are in units of the inverse density $1/n$. Our variational results are obtained for $N=100$ particles. Finite-size corrections on local quantities are negligible, and very mildly affect the reported large-distance correlations. Overall statistical errors are of the order of symbol sizes for (a) and line widths for (b) and (c).

Figure 2

Time-dependent expectation value of local two-body correlations after a quantum quench from a noninteracting state, ${\gamma }_i=0$, to ${\gamma }_f$. (a) tVMC results are compared with BA results obtained for a small number of particles [56, 59]. The correlation function is rescaled to have $g_2(0,0)=1$. (b) tVMC results for $N=100$ particles and ${\gamma }_f=1$, 2, 4, 8 (from top to bottom) compared to the quench action (QA) predictions from Ref. [57] (dashed lines), to the Boltzmann thermal averages at the effective temperature $T^{\star }$ (dot-dashed lines), and to the GGE thermal averages prediction (rightmost dashed lines). Statistical error bars on tVMC data are of the order of the line widths.

Figure 3

Time-dependent expectation value of local two-body correlations after a quantum quench from the interacting ground state at ${\gamma }_i=1$ (a) and ${\gamma }_i=4$ (b). Long-term dynamical averages (red continuous lines) are compared to thermal averages at the effective temperature set by energy conservation (black dashed lines). Uncertainties on the thermal averages are of the order of the line width, and are larger for small values of ${\gamma }_f$.

Figure 4

Time-dependent expectation value of the two-body correlations after a quantum quench from a noninteracting state, ${\gamma }_i=0$, to ${\gamma }_f=4$ at three different distances, $|x_1-x_2|=0$, $L/10$, $L/4$. Here, the system is on a lattice with $L/a=40$ lattice sites. The full line is obtained by exact diagonalization (ED) of the Hamiltonian; the other curves are from tVMC results truncated at the level of $U^{(2)}$. In (a), we show the convergence with different time-step discretization. In (b), we show the approach to the continuum for tVMC simulations using $U^{(2)}$ for discretizations $L/a=40$, 80, and 160.

Figure 5

(a) Time-dependent expectation value of the two-body correlations $g_2(r,t)$ after a quantum quench from a noninteracting state, ${\gamma }_i=0$, to ${\gamma }_f=8$ at three different distances, $|x_1-x_2|=0$, $L/10$, $L/4$. Here, the system is on a lattice with $L/a=40$ lattice sites. The full line is obtained by exact diagonalization of the Hamiltonian; the other curves are from tVMC results truncated at the level of $U^{(2)}$ or $U^{(3)}$. In contrast to ${\gamma }_f=4$ shown in Fig. 4, systematic differences of $U^{(2)}$ compared to the exact results are more visible here; the exact dynamics is recovered by inclusion of three-body terms $U^{(3)}$ into the tVMC wave function. (b) Off-diagonal single-particle density matrix $g_1(r,t)$ at three different distances, $r=L/10$, $L/4$, and $L/2$. The exact results (ED–black dashed line) are for all purposes indistinguishable from the $U^{(3)}$ curves.