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arXiv:1610.07633

Comment on: `Single-shot simulations of dynamic quantum many-body systems' [arXiv:1501.03224]

P. D. Drummond, Joachim Brand

Published Oct 19, 2016• 2 citations
arXivHTMLPDF

Abstract

In their recent paper [Nature Physics 15, 451 (2006)], Sakmann and Kasevich study the formation of fringe patterns in ultra-cold Bose gases and claim: `Here, we show how single shots can be simulated from numerical solutions of the time-dependent many-body Schrödinger equation.' It would be remarkable if they had solved this exponentially complex equation. Instead they solve nonlinear equations with the aim to approximate the solution of the Schrödinger equation. The authors proceed to criticize phase-space approaches to simulating quantum dynamics and claim the impossibility of interpreting single trajectories of the truncated Wigner (tW) method as single-shot experimental outcomes. Here we aim to provide relevant context and elaborate why we disagree with the authors' claims.

DOI
10.48550/arxiv.1610.07633

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In their recent paper [Nature Physics 15, 451 (2006)], Sakmann and Kasevich study the formation of fringe patterns in ultra-cold Bose gases and claim: `Here, we show how single shots can be simulated from numerical solutions of the time-dependent many-body Schrödinger equation.' It would be remarkable if they had solved this exponentially complex equation. Instead they solve nonlinear equations with the aim to approximate the solution of the Schrödinger equation. The authors proceed to criticize phase-space approaches to simulating quantum dynamics and claim the impossibility of interpreting single trajectories of the truncated Wigner (tW) method as single-shot experimental outcomes. Here we aim to provide relevant context and elaborate why we disagree with the authors' claims.
Understanding the collective behaviour of many-body quantum systems is an important subject in many areas of physics. With advances in ultra-cold gas experiments, the dynamics of strongly-interacting systems can now be studied in the lab. However, there is a paucity of theoretical techniques available to simulate such systems. One technique is phase-space methods, often known as the Truncated Wigner Approximation; however, its applicability in its naive form is limited. In this work, we expound on techniques to expand the regimes in which it can be effective. This involves creating a novel phase-space that is tailored to the problem at hand, and associated classical equations of motion. We show techniques for lattice systems with local finite Hilbert spaces, for fermionic systems, and for many-body localized systems. In all cases, we benchmark the accuracy of the approximation against exact results.
Ultracold Bose gases can be used to experimentally test many-body theory predictions. Here we point out that both exact conservation laws and dynamical invariants exist in the topical case of the one-dimensional Bose gas, and these provide an important validation of methods. We show that the first four quantum conservation laws are exactly conserved in the approximate truncated Wigner approach to many-body quantum dynamics. Center-of-mass position variance is also exactly calculable. This is nearly exact in the truncated Wigner approximation, apart from small terms that vanish as ${N}^{\ensuremath{-}3/2}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ with fixed momentum cutoff. Examples of this are calculated in experimentally relevant, mesoscopic cases.
We introduce a method to solve nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating two steps. First, we transform the Wigner function into a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. Next, we simulate the dynamics in this frame using a numerical method that implements this time-dependent nonlinear change of variables. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics. Published by the American Physical Society 2024
In this thesis, the behavior of ultracold bosonic systems in external potentials is investigated. Such systems are typically realized as Bose-Einstein condensates, where dilute atomic gases cooled to extremely low temperatures exhibit collective quantum behavior. Because of the large degree of control they allow over the external potential and over the interaction parameters of the many-body system, Bose-Einstein condensates constitute a testing ground for the dedicated study of coherence, correlations, and quantum phase transitions in interacting bosonic systems. This thesis presents an investigation of the many-body physics of ultracold bosons, focusing specifically on how the many-body properties of these ultracold bosonic systems change with changing external potential, how they manifest themselves in experimentally accessible observables, and how these properties can be analyzed using both numerical many-body methods and data-driven approaches. The first part of this thesis focuses on the theoretical description of ultracold bosonic systems in external trapping potentials. Using the multiconfigurational time-dependent Hartree method for bosons (MCTDH-X), numerically exact simulations were performed that allow the calculation of reduced density matrices, natural orbitals, and correlation functions. These quantities were used to analyze the coherence properties and fragmentation of trapped bosonic systems. In particular, double-well systems were investigated as model scenarios in which the gradual formation of a central barrier leads to a redistribution of natural occupations and a transition from a condensed to a fragmented many-body state. In this context, we demonstrate the usefulness of single-shot images, which represent projective measurements of the many-body wave function, for the experimental investigation of the many-body physics of fragmenting system. Single-shot images can be simulated numerically using MCTDH-X, but they are more commonly obtained through experimental measurements. The possibility of extracting many-body information in an experimental context, where said many-body information is not otherwise accessible, is discussed. Supervised machine learning models were trained on simulated single-shot data sets to investigate whether information about the underlying many-body state can be inferred directly from measurement outcomes. The results show that machine learning models are capable of learning patterns in the single-shot images that correlate with properties of the underlying many-body state and can therefore predict density-related quantities with good accuracy. The second part of this thesis focuses on a joint theoretical and experimental investigation of a system of ultracold bosons coupled to an optical cavity. In this system, the atoms interact with a dynamically generated optical potential produced by the cavity field. We use a new theoretical framework that combines a mean-field treatment of the cavity field with a many-body description of the atomic degrees of freedom to be able to capture the emergence of a superradiant self-organized phase and the subsequent transition to a self-organized Mott-insulating state. Our simulations show good agreement with the experimental results, and we discuss where the use of a beyond mean-field treatment in the simulations becomes essential in describing the physics of the different involved phases and quantum states. In summary, the results presented in this thesis demonstrate how numerical many-body simulations and machine learning methods can be used to analyze experimentally accessible observables and to gain insight into the many-body physics of ultracold bosonic systems.