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On Bose-Einstein condensates in disordered media

Marius Lemm, Simone Rademacher, Jingxuan Zhang

TL;DR

The paper addresses the question of whether disorder can slow the dynamics of a quantum many-body system in any dimension. It studies interacting bosons on a lattice in the mean-field regime with a disordered potential and proves that fluctuations around a localized condensate propagate with an arbitrarily small effective velocity $\varepsilon$, tunable by the disorder strength $\lambda$. The main technical innovation is an interaction-picture analysis around the Anderson-localized one-body dynamics, yielding two key results: (i) a bound on the local number of fluctuations showing slow propagation, and (ii) a locally enhanced mean-field approximation for local observables, with a light-cone $|x| \le \varepsilon t$ outside which dynamics are exponentially suppressed. The results are substantiated by propositions verifying the localization hypotheses for quasi-periodic and random potentials, linking Anderson localization to slow transport in disordered many-body Bose systems and suggesting robustness across dimensionality.

Abstract

We consider the quantum dynamics of interacting bosons in the mean-field regime when they are subjected to a disordered potential, which is either random or quasi-periodic. Starting from a spatially localized Bose-Einstein condensate, we prove that fluctuations around the condensate propagate with a small velocity due to the disorder. This provides an example of a disordered many-body system with provably slow transport behavior in any spatial dimension. The main technical novelty is an interaction picture analysis relative to the Anderson-localized one-body dynamics.

On Bose-Einstein condensates in disordered media

TL;DR

The paper addresses the question of whether disorder can slow the dynamics of a quantum many-body system in any dimension. It studies interacting bosons on a lattice in the mean-field regime with a disordered potential and proves that fluctuations around a localized condensate propagate with an arbitrarily small effective velocity , tunable by the disorder strength . The main technical innovation is an interaction-picture analysis around the Anderson-localized one-body dynamics, yielding two key results: (i) a bound on the local number of fluctuations showing slow propagation, and (ii) a locally enhanced mean-field approximation for local observables, with a light-cone outside which dynamics are exponentially suppressed. The results are substantiated by propositions verifying the localization hypotheses for quasi-periodic and random potentials, linking Anderson localization to slow transport in disordered many-body Bose systems and suggesting robustness across dimensionality.

Abstract

We consider the quantum dynamics of interacting bosons in the mean-field regime when they are subjected to a disordered potential, which is either random or quasi-periodic. Starting from a spatially localized Bose-Einstein condensate, we prove that fluctuations around the condensate propagate with a small velocity due to the disorder. This provides an example of a disordered many-body system with provably slow transport behavior in any spatial dimension. The main technical novelty is an interaction picture analysis relative to the Anderson-localized one-body dynamics.

Paper Structure

This paper contains 20 sections, 12 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of setup and main result
  3. Organization of the paper
  4. Setup and main results
  5. Notation.
  6. Assumptions
  7. Main results
  8. Examples
  9. Quasi-periodic potential
  10. Random potential
  11. Discussion
  12. Fluctuations around the Hartree evolution
  13. Proof of Theorem \ref{['thm1']}
  14. Estimate for $\tau_t^{(0)}$
  15. Estimate for $\tau_t^{\mathrm{int}}$
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $r>0$, $R\ge 2r$, and $\rho=R-r.$ Assume that Conditions C1 and C2 hold and assume further that Then, given any $\epsilon>0$, there exist $\lambda_*=\lambda_*(\epsilon, \rho,d)>0$, $C = C(r,d)>0$, and $K=K(U)>0$ such that for all $\lambda\ge\lambda_*$, all $0\le t \le \rho/\epsilon$, and any bounded local operator $O$ acting on $\ell^2$ with kernel satisfying it holds that

Figures (3)

  • Figure 1: The geometry in the main result (see Theorem \ref{['thm2']}). We probe the system at time $t$ with a local observable $O$ acting on a small ball $B_r$ around the origin. We assume that the initial condensate is supported outside of the larger ball $B_R$. In LRZa, we prove that for any external potential, the mean-field approximation is enhanced when $\rho> vt$ for an $\mathcal{O}(1)$ constant $v$ independent of the potential. Here we prove that for a disordered potential, the mean-field approximation is enhanced when $\rho> \epsilon t$ for a small constant $\epsilon>0$ determined by the disorder strength $\lambda$.
  • Figure 2: Our main result establishes the effective space time light cone $\abs{x}\le \epsilon t$ with $\epsilon>0$ arbitrarily small for sufficiently strong disorder (darker shaded region), outside of which propagation of the condensate and quantum fluctuations around it are exponentially suppressed.
  • Figure 3: Schematic diagram for the geometric splitting.

Theorems & Definitions (19)

  • Theorem 2.1: Main result --- slow spreading of mean-field error
  • Corollary 2.2: Many-body propagation bound
  • proof
  • Theorem 2.3: Slow propagation of fluctuations
  • Remark 2.1
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1: LRZa, Lem. 4.2
  • Lemma 4.1
  • proof
  • ...and 9 more