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Chaos Solitons & Fractals

The classical limit of non-integrable quantum systems, a route to quantum chaos

Mario Castagnino, Olímpia Lombardi

Published Oct 18, 2005• 22 citations
Publisher
DOI
10.1016/j.chaos.2005.08.149

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We show that cavity quantum electrodynamics (QED) devices can realize dissipative Sachdev-Ye-Kitaev (SYK) universality, a paradigmatic setting for quantum chaos in open many-body systems. Ultracold fermions with disordered, all-to-all cavity-mediated interactions provide two complementary routes: atomic spontaneous emission in a single-mode cavity and photon leakage from a multimode cavity. Strikingly, both converge to the same non-Hermitian random-matrix universality despite originating from integrable and chaotic closed-system limits, respectively. In the single-mode cavity, dissipation creates quantum chaos from an integrable Hamiltonian. We trace this convergence to a tunable growth in dissipative rank, controlled, respectively, by the Lamb-Dicke parameter and the cavity-mode spacing. The integrability-to-chaos crossover is accompanied by a dynamical crossover from long-lived prethermal memory to rapid thermalization, visible in single-atom-resolved densities.
A novel concept for quantum chaos in spin systems is proposed which differs distinctively from concepts currently in use. It is argued (a) that the definition of quantum chaos cannot be based on the correspondence principle and (b) that quantum chaos is not equivalent to statistical behavior. Quantum chaos in spin systems implies the existence of a new type of spectrum consisting of excitations which do not form regular patterns with a multiparameter continuum structure in the thermodynamic limit, thus removing the constraint imposed by van Hove singularities that time-dependent correlation functions for pure quantum states cannot decay more rapidly than as powers of t. The irregular spectrum is caused by strong level repulsion resulting from the lack of a sufficient number of conservation laws in nonintegrable quantum many-body systems. The similarities and differences between quantum and classical chaos and, more generally, between quantum (non)integrability and classical (non)integrability are discussed in detail.
Quantum chaos remains a challenging concept because classical signatures of chaos do not directly transfer to quantum systems. This thesis investigates which indicators of chaos are shared between classical and quantum mechanics and how they can be used to characterise non-integrable many-body systems. We examine ergodicity and thermalisation by means of the Eigenstate Thermalisation Hypothesis (ETH) as signatures of non-integrability, and study the Out-of-Time-Ordered Correlator (OTOC) as a probe of information scrambling and as a measure of sensitivity to initial conditions. We briefly discuss the appearance of chaotic wavefunctions and observations such as quantum scars and quantum ergodicity in individual eigenmodes of the Bunimovich stadium potential. Using the OTOC, we discuss extensions of the Lyapunov exponent to quantum systems and apply these ideas to the Sachdev–Ye–Kitaev (SYK) model. In the large-N limit, and using the AdS/CFT correspondence, we find that the model might potentially be the dual of a black hole and that it is maximally chaotic. We conclude by discussing additional proposed signatures of quantum chaos and prospects for experimental realisations of the SYK model.