Simulation methods for open quantum many-body systems

Abstract

Coupling a quantum many-body system to an external environment dramatically changes its dynamics and offers novel possibilities not found in closed systems. Of special interest are the properties of the steady state of such open quantum many-body systems, as well as the relaxation dynamics towards the steady state. However, new computational tools are required to simulate open quantum many-body systems, as methods developed for closed systems cannot be readily applied. We review several approaches to simulate open many-body systems and point out the advances made in recent years towards the simulation of large system sizes. CONTENTSI. Introduction 1 A. The open quantum many-body problem 1 B. The Markovian quantum master equation 2 C. Steady state solution versus time evolution 2 D. Differences to equilibrium problems 3 E. Paradigmatic models 3 II. Stochastic methods 4 III. Tensor network methods 5 A. One spatial dimension 5 B. Extensions to higher dimensions 8 IV. Variational methods 11 A. The variational principle for open quantum systems 11 B. Comparison with mean-field methods 12 C. Variational tensor network methods 13 D. Variational quantum Monte-Carlo methods 14 V. Phase space and related methods 15 A. Truncated Wigner approximation 15 B. BBGKY hierarchy equations

Non-Hermitian Physics

“…It is our hope that this review article will be a useful guide for researchers who are interested in this burgeoning field. Going beyond the works included in this review, interested readers may refer to other excellent reviews that are more specialized in particular areas/topics, such as optics [32][33][34], acoustics [35][36][37], parity-timesymmetric systems [38][39][40], mesoscopic physics and quantum resonances [23,[41][42][43], ultracold atoms [44,45], driven-dissipative systems [46][47][48][49][50], optomechanics [51], exciton-polariton condensates [20,52], nonlinear phenomena [53], biological transport [54], active matter [55], and random matrices and disordered systems [56][57][58].…”

“…We make several important remarks on the role of dynamics conditioned on measurement outcomes, which are often overlooked in literature. Such conditional dynamics is in general distinct from the unconditional dynamics obeying the master equation; the latter has been well explored over the last couple of decades especially in the context of driven-dissipative systems, and we refer to excellent review papers [20,[46][47][48][49][50] for developments in that direction. In contrast, as detailed in Sec.…”

Constructing k-local parent Lindbladians for matrix product density operators

“…See e.g. [67] for a review of methods based on MPDOs and a discussion of alternatives. There are open-source software packages that can be used for such simulations, e.g.…”

Determination of the critical exponents in dissipative phase transitions: Coherent anomaly approach

“…In particular, it would be very important to have reliable numerical methods to determine them. Various powerful methods have been developed to simulate quantum many-body systems [21,30,[37][38][39][40][41][42][43][44][45][46][47]. It is however important to stress that, from the numerics perspective, directly accessing the critical properties of the steady-state may be rather demanding.…”

“…CAM also provides a route for systematically improving the accuracy of computed results, by further enlarging the maximum cluster size. To achieve this goal, there are several promising approaches worth to pursue in future work, involving a combination of corner-space renormalization [21], neural-networks [40][41][42][43] and/or tensor-networks [46,47]. On one side, the corner-space renormalization method and the recently introduced neural-networks quantum ansatz were shown to simulate the steady states of open quantum manybody systems, benchmarked with a few examples including small 2D clusters.…”