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

Marius Lemm; Simone Rademacher

doi:10.22331/q-2025-01-13-1587

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

Marius Lemm¹ and Simone Rademacher²

¹Department of Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
²Department of Mathematics, Ludwig-Maximilians-Universität München, Theresienstr. 39, 80333 Munich, Germany

Published:	2025-01-13, volume 9, page 1587
Eprint:	arXiv:2312.01736v2
Doi:	https://doi.org/10.22331/q-2025-01-13-1587
Citation:	Quantum 9, 1587 (2025).

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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.

Popular summary

We study the scrambling of quantum information that occurs over time as many quantum particles interact. We work with bosonic particles whose interactions are weak in a certain sense, the so-called bosonic mean-field regime. We derive an asymptotic description of the scrambling of quantum information, which becomes increasingly precise as the total number of particles becomes large. The description itself is given in terms of a nonlinear PDE in 3 dimensions that no longer depends on the total number of particles. It is therefore in principle computable, e.g., numerically. The result is mathematical, in the sense that we give a rigorous proof. The heuristic reason why we are able to control a quantum systems with many interacting particles is that in the mean-field regime that we study, most of the bosons exhibit Bose-Einstein condensation. This means most of the particles are approximately described by the same quantum state (called the "condensate") and this significantly reduces the complexity of the collective quantum state.

► BibTeX data

@article{Lemm2025outoftimeordered,
  doi = {10.22331/q-2025-01-13-1587},
  url = {https://doi.org/10.22331/q-2025-01-13-1587},
  title = {Out-of-time-ordered correlators of mean-field bosons via {B}ogoliubov theory},
  author = {Lemm, Marius and Rademacher, Simone},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1587},
  month = jan,
  year = {2025}
}

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