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Out-of-time-ordered correlators of mean-field bosons via Bogoliubov theory

Marius Lemm, Simone Rademacher

TL;DR

This work provides a rigorous bridge between quantum information scrambling and nonlinear dispersive PDE dynamics for a system of mean-field bosons. By leveraging refined Bogoliubov theory around a Bose–Einstein condensate obeying the nonlinear Hartree equation, the authors show that the N→∞ limit of OTOCs is governed by a nonlinear symplectic Bogoliubov flow encoded in a map $\Theta(t;0)$ on $L^2(\mathbb{R}^3)\oplus L^2(\mathbb{R}^3)$. They establish a multivariate Gaussian limit for time-indexed observables (MCLT) and a higher-order out-of-time-ordered Wick rule, with precise error bounds of the form $O(N^{-1/2}e^{e^{Ct}})$, connecting quantum chaos diagnostics to a tractable nonlinear PDE problem. The results yield an explicitly computable initial scrambling velocity and an exponential bound on the butterfly velocity, highlighting a new PDE-driven route to study quantum chaos and potential many-body localization phenomena in mean-field bosonic systems. Overall, the paper demonstrates that mean-field bosons exhibit scrambling that is captured by an effective nonlinear Bogoliubov dynamics, providing both theoretical insight and practical computational pathways for OTOCs in large quantum systems.

Abstract

Quantum many-body chaos concerns the scrambling of quantum information among large numbers of degrees of freedom. It rests on the prediction that out-of-time-ordered correlators (OTOCs) of the form $\langle [A(t),B]^2\rangle$ can be connected to classical symplectic dynamics. We rigorously prove a variant of this correspondence principle for mean-field bosons. We show that the $N\to\infty$ limit of the OTOC $\langle [A(t),B]^2\rangle$ is explicitly given by a suitable symplectic Bogoliubov dynamics. In practical terms, we describe the dynamical build-up of many-body entanglement between a particle and the whole system by an explicit nonlinear PDE on $L^2(\mathbb{R}^3) \oplus L^2(\mathbb{R}^3)$. For higher-order correlators, we obtain an out-of-time-ordered analog of the Wick rule. The proof uses Bogoliubov theory. Our finding spotlights a new problem in nonlinear dispersive PDE with implications for quantum many-body chaos.

Out-of-time-ordered correlators of mean-field bosons via Bogoliubov theory

TL;DR

This work provides a rigorous bridge between quantum information scrambling and nonlinear dispersive PDE dynamics for a system of mean-field bosons. By leveraging refined Bogoliubov theory around a Bose–Einstein condensate obeying the nonlinear Hartree equation, the authors show that the N→∞ limit of OTOCs is governed by a nonlinear symplectic Bogoliubov flow encoded in a map on . They establish a multivariate Gaussian limit for time-indexed observables (MCLT) and a higher-order out-of-time-ordered Wick rule, with precise error bounds of the form , connecting quantum chaos diagnostics to a tractable nonlinear PDE problem. The results yield an explicitly computable initial scrambling velocity and an exponential bound on the butterfly velocity, highlighting a new PDE-driven route to study quantum chaos and potential many-body localization phenomena in mean-field bosonic systems. Overall, the paper demonstrates that mean-field bosons exhibit scrambling that is captured by an effective nonlinear Bogoliubov dynamics, providing both theoretical insight and practical computational pathways for OTOCs in large quantum systems.

Abstract

Quantum many-body chaos concerns the scrambling of quantum information among large numbers of degrees of freedom. It rests on the prediction that out-of-time-ordered correlators (OTOCs) of the form can be connected to classical symplectic dynamics. We rigorously prove a variant of this correspondence principle for mean-field bosons. We show that the limit of the OTOC is explicitly given by a suitable symplectic Bogoliubov dynamics. In practical terms, we describe the dynamical build-up of many-body entanglement between a particle and the whole system by an explicit nonlinear PDE on . For higher-order correlators, we obtain an out-of-time-ordered analog of the Wick rule. The proof uses Bogoliubov theory. Our finding spotlights a new problem in nonlinear dispersive PDE with implications for quantum many-body chaos.
Paper Structure (15 sections, 7 theorems, 196 equations)

This paper contains 15 sections, 7 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian.
  3. Initial state.
  4. Description of the condensate through the nonlinear Hartree equation
  5. Observables and summary of main results
  6. Setup and main results
  7. Properties of the nonlinear Bogoliubov dynamics $\mathcal{L}_{(t;s)}$
  8. Many-body time evolution
  9. Fluctuation dynamics
  10. Connection between many-body and nonlinear Bogoliubov dynamics
  11. Multivariate Gaussian: Proof of Theorem \ref{['thm:MCLT']}
  12. Proof of Theorem \ref{['thm:corrfct']}
  13. Proof of Theorem \ref{['thm:OTOC']}
  14. Proof of Corollary \ref{['cor:OTOC']}
  15. Conclusions and future directions

Key Result

Theorem 2.1

Let $v$ satisfy ass:v and $\varphi_0 \in H^{4}( \mathbb{R}^3)$ with $\| \varphi_0 \| =1$. Let $t>0$ and $\varphi_t$ denote the solution of the Hartree equation def:hartree with initial data $\varphi_0$. Fix two self-adjoint, real operators $A,B$ satisfying the regularity assumption Let $\Psi_N$ be given by eq:psidefn, $A_t^{(j)}$ by def:Atj and $\mathcal{B}_0$ by def:B0. Then we have where the s

Theorems & Definitions (16)

  • Remark 1.1: Connection to Lieb-Robinson bounds
  • Theorem 2.1: OTOC asymptotics
  • Corollary 2.2
  • Theorem 2.3: Higher OTOCs with regularization
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: Out-of-time-ordered Wick rule
  • Lemma 3.1
  • proof
  • Lemma 3.2: Lemma 3.2 RS
  • ...and 6 more