Chaos and Thermalization in the Spin-Boson Dicke Model

Abstract

We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called “efficient basis” over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.

Classical route to ergodicity and scarring in collective quantum systems

“…In this case, the semiclassical phase space density described by the Q-distribution evolves following a Fokker-Planck type diffusion equation, which facilitates thermalization to a microcanonical form in the chaotic regime. More recently, the validity of ETH has been tested for the DM [246,247], which also confirms thermalization in the chaotic regime. Additionally, the accuracy of the microcanonical description increases for higher energy states, which suggests an energy-dependent ergodic behavior (see also the discussion in section 8.2).…”

“…Although the validity of ETH has been tested for many interacting quantum systems [10,25,[236][237][238][239][240][241][242][243][244][245][246][247], a rigorous proof is still lacking. In fact, whether or not ETH is a necessary condition for thermalization is still a debatable issue [26,[248][249][250].…”

Dynamical transition from localized to uniform scrambling in locally hyperbolic systems

“…Recently it was shown [18,19] that an initial exponential growth of OTOCs does not necessarily imply chaotic dynamics of the system's classical counterpart, i.e., such OTOC behavior alone cannot serve as clear-cut probe of quantum chaos. These works [18,19] and further ones picking up the same idea [20][21][22][23] show that for quantum (many-body) systems with a classical limit and a semiclassical regime, it is sufficient to have local instabilities in a (possibly integrable) phase space to generate exponentially growing OTOCs. Several examples of this situation have been numerically studied [24,25], including basic models such as an inverted harmonic oscillator [26].…”

Quantum chaos in the Dicke model and its variants

“…The superradiant phase transition of the Dicke model has been studied in recent experiments [74][75][76][77][78]. The presence of quantum chaos and quantum phase transition in the Dicke model has long been an area of interest [79][80][81][82][83][84][85][86]. A demonstration of quantum-classical correspondence, using OTOC for the Dicke model, was made in the unitary case [87].…”