Adiabatic invariants for the regular region of the Dicke model

Abstract

Adiabatic invariants are introduced and shown to provide an approximate second integral of motion for the non-integrable Dicke model, in the energy region where the system exhibits a regular dynamics. This low-energy region is always present and has been described both in a semiclassical and a full quantum analysis. Its Peres lattices exhibit that many observables vary smoothly with energy, along lines which beg for a formal description. It is shown how the adiabatic invariants provide a rationale to their presence in many cases. They are built employing the Born-Oppenheimer approximation, valid when a fast system is coupled to a much slower one. As the Dicke model has a one bosonic and one fermionic degree of freedom, two versions of the approximation are used, depending on which one is the faster. In both cases a noticeably accord with exact numerical results is obtained. The employment of the adiabatic invariants provides a simple and clear theoretical framework to study the physical phenomenology associated to this energy regime, far beyond the energies where the quadratic approximation can be employed. Adiabatic invariants in the Dicke model.For the Dicke model it was first simulated by means of a Bose-Einstein condensate in an optical cavity [21,22]. Also, an open version of the model was realized employing Raman transitions [23], and dynamical non-equilibrium superradiant phase transition has been also observed [24]. Moreover, by means of superconducting QED the Rabi model [25,26] and few-atoms Dicke-like models [27,28] have been explored in this direction.More than ten years ago, C. Emary and T. Brandes built approximate solutions of the Dicke model by means of the Holstein-Primakoff realization of the SU(2) algebra in the thermodynamic limit [29]. This approach describes the behavior of the model for the ground-and low energy-states, exhibiting the superradiant QPT easily and showing the presence of chaos in the spectrum of the Hamiltonian. Besides, the truncated Holstein-Primakoff approach makes it possible to extract the critical exponents for the groundstate energy per particle, the fraction of excited atoms, the number of photons per atom, their fluctuations, and the concurrence [30,31].Enhanced by these experimental and theoretical results, and thanks to its algebraic properties, the Dicke model has become an excellent tool from the theoretical point of view for exploring several features of quantum many-body systems with collective degrees of freedom, e.g., the QPT, the Excited-State Quantum Phase Transitions [32,33,34,35] and their relation with the thermal phase transition [36,37], the onset of quantum chaos and its correspondence to the classical limit [38,39,40,41], quantum quenches [42,43] and the problem of equilibrium and thermalization in isolated manybody quantum systems [44,45], to name a few. This is the motivation for looking for as many analytical or semi-analytical descriptions of the spectra and observables of the model as it is possible. The truncated Holstein-Primakoff approximatio...

Ubiquitous quantum scarring does not prevent ergodicity

“…This plot is equivalent to a Peres lattice 52 for expectation values of observables, as used in studies of chaos and thermalization. In the low-energy regular regime, is organized along lines that can be classified with quasi-integrals of motion linked with classical periodic orbits 53 . Conversely, as the system enters the chaotic region at higher energies, the distribution of becomes dense and looses any order.…”

Constant of Motion Identifying Excited-State Quantum Phases

“…This constant of motion signals to which part of the semiclassical phase space a given quantum state is attached and assigns it a conserved quantum number, with important thermodynamic consequences. We present general arguments and illustrate our findings with the paradigmatic Rabi (RM) [40,41] and the Dicke (DM) [42][43][44][45][46] models, discussing dynamical and thermodynamic consequences.…”

“…These models fulfill the condition for the existence of the operator Ĉ. From their semiclassical energy surfaces [15,[43][44][45][46]53] [Figs. 2(a) and 2(d)], we propose that the relevant dynamical function is fðxÞ ¼ q − q c ðα; λÞ, where q c ðα; λÞ is the canonical coordinate corresponding to the ESQPT critical energy.…”

Constant of motion identifying excited-state quantum phases

“…This means that dynamics and thermodynamics are qualitatively different in both phases, independently of the number of degrees of freedom of the classical analogue. We present general arguments, and illustrate our findings with the paradigmatic Rabi (RM) [38,39] and the Dicke (DM) [40][41][42][43][44] models.…”