Quantum chaos and thermalization in the two-mode Dicke model

Abstract

We discuss the onset of quantum chaos and thermalization in the two-mode Dicke model, which describes the dipolar interaction between an ensemble of spins and two bosonic modes. The two-mode Dicke model exhibits normal to superradiant quantum phase transition with spontaneous breaking either of a discrete or continuous symmetry. We study the behaviour of the fidelity out-of-time-order correlator (FOTOC) derived from the Loschmidt echo signal in the quantum phases of the model. We show that the exponential growth of the FOTOC in the beginning of the time evolution cannot be related to a classical unstable point in the general case. Furthermore, we find that the collective spin observable in the two-mode Dicke model quickly saturates to its long-time average value, and shows very good agreement between its diagonal ensemble average and microcanonical average even for a small number of spins. We show that the temporal fluctuations of the expectation value of the collective spin observable around its average are small and decrease with the effective system size, which leads to thermalization of the spin system.

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].…”