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arXiv:2303.00021

Quantum equilibration and measurements -- bounds on speeds, Lyapunov exponents, and transport coefficients obtained from the uncertainty relations and their comparison with experimental data

Saurish Chakrabarty, Zohar Nussinov

Published Feb 28, 2023• 0 citations

Abstract

We discuss our recent study of local quantum mechanical uncertainty relations in quantum many body systems. These lead to fundamental bounds for quantities such as the speed, acceleration, relaxation times, spatial gradients and the Lyapunov exponents. We additionally obtain bounds on various transport coefficients like the viscosity, the diffusion constant, and the thermal conductivity. Some of these bounds are related to earlier conjectures, such as the bound on chaos by Maldacena, Shenker and Stanford while others are new. Our approach is a direct way of obtaining exact bounds in fairly general settings. We employ uncertainty relations for local quantities from which we strip off irrelevant terms as much as possible, thereby removing non-local terms. To gauge the utility of our bounds, we briefly compare their numerical values with typical values available from experimental data. In various cases, approximate simplified variants of the bounds that we obtain can become fairly tight, i.e., comparable to experimental values. These considerations lead to a minimal time for thermal equilibrium to be achieved. Building on a conjectured relation between quantum measurements and equilibration, our bounds, far more speculatively, suggest a minimal time scale for measurements to stabilize to equilibrium values.

DOI: 10.48550/arxiv.2303.00021

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We introduce “ local uncertainty relations ” in thermal quantum many-body systems from which fundamental bounds can be derived. These inequalities imply rigorous universal non-relativistic speed limits (independent of interaction range) and, as we focus on in the current work, bounds associated with transport (e.g., the diffusion constant and shear viscosity). Using the chaos bounds and other considerations, we introduce simplified non-rigorous universal transport coefficient inequalities and compare our resulting bounds against experimental data.

Using local uncertainty relations in thermal many body systems, we illustrate that there exist universal speed limit (regardless of interaction range), bounds on acceleration or force/stress, acceleration or material stress rates, diffusion constant, viscosity, other transport coefficients, electromagnetic or other gauge field strengths, correlation functions of arbitrary spatio-temporal derivatives, Lyapunov exponents, thermalization times, and transition times between orthogonal states in non-relativistic quantum thermal systems, and derive analogs of the Ioffe-Regel limit. In the

\hbar \to 0

limit, all of our bounds either diverge (e.g., our speed and acceleration limit) or vanish (as in, e.g., our viscosity and diffusion constant bounds). Our inequalities hold at all temperatures and, as corollaries, imply general power law bounds on response functions at both asymptotically high and low temperatures. Our results shed light on measurements and on how apparent nearly instantaneous effective collapse to an eigenstate may arise in macroscopic interacting many body quantum systems.

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