Quantum Effect on Classical Chaos, and Localization, Smoothing, and Irreversibility of Quantum System

Giulio Casati

K. Shiokawa, B. L. Hu

Urban Duh, Marko Žnidarič

We describe a chaos scenario valid for chaotic many-body quantum systems, the simulation of which is one of the prime candidates for demonstrating quantum advantage, and clarify its importance for classical truncation-based simulations. By studying the spectral properties of the truncated propagator, we, for the first time, identify fractal structures responsible for many-body quantum chaos, analogous to the fractality in classical chaos emerging from the time evolution of initially smooth long-wavelength densities into ever finer short-wavelength densities. Namely, we show that the leading slowest-decaying Ruelle-Pollicott eigenoperators have a non-trivial fractal dimension quantifying their non-locality, visible also in the divergence of their condition numbers. Furthermore, we find that unitarity imposes an asymptotic equality between the temporal decay rate of local correlations and this many-body fractal dimension. With this insight, the scenario for many-body quantum chaos becomes clear: over time, local operators evolve toward increasingly non-local ones with a quantifiable fractal structure, thereby naturally leading to effective non-unitary relaxation on the subspace of local operators -- a kind of many-body Kolmogorov cascade in the space of operators -- which enables accurate truncation-based simulations. Our predictions are demonstrated numerically in various one- and two-dimensional quantum circuits, with and without conservation laws, as well as by exact results in dual-unitary circuits.

I. Puha

K. Shiokawa, B. L. Hu

Decoherence in quantum systems which are classically chaotic is studied. The Arnold cat map and the quantum kicked rotor are chosen as examples of linear and nonlinear chaotic systems. The Feynman-Vernon influence functional formalism is used to study the effect of the environment on the system. It is well known that quantum coherence can obliterate chaotic behavior in the corresponding classical system. But interaction with an environment can under general circumstances quickly diminish quantum coherence and reenact classical chaotic behavior. How effectively decoherence works to sustain chaos, and how the resultant behavior qualitatively differs from the quantum picture, depending on the coupling of the system with the environment and the spectral density and temperature of the environment. We show how recurrence in the quantum cat map is lost and classical ergodicity is recovered due to the effect of the environment. Quantum coherence and diffusion suppression are instrumental to dynamical localization for the kicked rotor. We show how environment-induced effects can destroy this localization. Such effects can also be understood as resulting from external noises driving the system. Peculiar to decohering chaotic systems is the apparent transition from reversible to irreversible dynamics. We show such transitions in the quantum cat map and the kicked rotor and distinguish it from apparent irreversibility originating from dynamical instability and imprecise measurements. By performing a time reversal on and following the quantum kicked rotor dynamics numerically, we show how the otherwise reversible quantum dynamics acquires an arrow of time upon the introduction of noise or interaction with an environment.

Michael Berry

Is there a connection between the Riemann zeros and the quantum physics of classical chaos? Can the relation between spin and statistics be understood within elementary quantum mechanics? How are the phase singularities in classical optics smoothed by quantum effects?