Quantum and Classical Lyapunov Exponents in Atom-Field Interaction Systems

Abstract

The exponential growth of the out-of-time-ordered correlator (OTOC) has been proposed as a quantum signature of classical chaos. The growth rate is expected to coincide with the classical Lyapunov exponent. This quantum-classical correspondence has been corroborated for the kicked rotor and the stadium billiard, which are one-body chaotic systems. The conjecture has not yet been validated for realistic systems with interactions. We make progress in this direction by studying the OTOC in the Dicke model, where two-level atoms cooperatively interact with a quantized radiation field. For parameters where the model is chaotic in the classical limit, the OTOC increases exponentially in time with a rate that closely follows the classical Lyapunov exponent.Quantum chaos tries to bridge quantum and classical mechanics. The search for quantum signatures of classical chaos has ranged from level statistics [1,2] and the structure of the eigenstates [3,4] to the exponential increase of complexity [5,6] and the exponential decay of the overlap of two wave packets [7][8][9][10]. Recently, the pursuit of exponential instabilities in the quantum domain has been revived by the conjecture of a bound on the rate growth of the out-of-time-ordered correlator (OTOC) [11,12]. First introduced in the context of superconductivity [13], the OTOC is now presented as a measure of quantum chaos, with its growth rate being associated with the classical Lyapunov exponent. The OTOC is not only a theoretical quantity, but has also been measured experimentally via nuclear magnetic resonance techniques [14][15][16][17].The correspondence between the OTOC growth rate and the classical Lyapunov exponent has been explicitly shown in two cases of one-body chaotic systems, the kicked-rotor [18] and, after a first unsuccessful attempt [19], the stadium billiard [20]. It was also achieved for chaotic maps [21]. For interacting many-body systems, while exponential behaviors for the OTOC have been found for the Sachdev-Ye-Kitaev model [11,22] and for the Bose-Hubbard model [23,24], a direct demonstration of the quantum-classical correspondence has not yet been made. Studies in this direction include [6,[25][26][27][28][29] and [30].Here, we investigate the OTOC for the Dicke model [31,32]. Comparing with one-body systems, the model is a step up toward an explicit quantum-classical correspondence for interacting many-body systems, since it contains N atoms interacting with a quantized field.The Dicke model was originally proposed to explain the collective phenomenon of superradiance: the field mediates interatomic interactions, which causes the atoms to act collectively [31,33]. Superradiance has been experimentally studied with ultracold atoms in optical cavities [34][35][36][37][38][39]. The model has also found applications beyond superradiance in various different fields. It has been employed, for instance, in studies of ground-state and excited-state quantum phase tran-sitions [33,[40][41][42][43][44], entanglement creation [45], nonequilibrium dynamic...

Does Scrambling Equal Chaos?

“…This differs from the exponential growth rate of an OTOC, λ OTOC , which is rather the log of the phase space average of sensitivity squared. Since the log of the average is larger than the average of the log, we have [22][23][24][25][26][27][28]:…”

“…Originally, it was conceived to describe N two-level atoms coupled to a single optical cavity mode. Several recent works [25,27,28] considered the OTOCs in this model. In the classical limit, its Hamiltonian H = 1 2 (q 2 + p 2 ) + x + γzq (A1) describes an SU (2) (pseudo)-spin (x, y, z) and a harmonic oscillator (p, q), interacting with a coupling constant γ > 0.…”

“…5(b) and also Refs. [25,27]. This suggests that λ OTOC ≥ ω 1 is a tight bound, if the OTOC is computed in an ensemble that overlaps with the saddle point.…”

Paraxial wave function and Gouy phase for a relativistic electron in a uniform magnetic field

“…The obtained connection between relativistic QM in the FW representation and the paraxial equations can be applied to a wide class of problems connected with relativistic electron (positron, muon) beams in different external fields [51][52][53][54][55][56][57] (e.g., crossed magnetic and electric fields).…”

Dynamical Detection of Level Repulsion in the One-Particle Aubry-André Model

“…The more traditional approach to quantum chaos, however, focuses on the properties of the spectrum and uses level statistics as in random matrix theory (RMT) as its main signature [10][11][12][13]. There are several examples of cases where a correspondence between the exponential growth of the OTOC and level repulsion as in RMT has been found [14][15][16][17][18], but exceptions also exist [19][20][21]. In the present work, we propose a way to directly detect the effects of level repulsion in the evolution of a quantum system.…”