Long-time evolution in the dicke model

Reinhard Honegger

Aleksandrina V. Kirkova, Peter A. Ivanov

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.

Caspar Groiseau, Stuart J. Masson, A. S. Parkins

We study trajectories of collective spin states of an ensemble of spinors. The spinors considered here are either trapped ions in free space or atoms confined in a cavity, both systems of which are engineered through their interactions with light fields to obey an effective Dicke model. In an appropriate limit of the Dicke model, one obtains one-axis twisting dynamics of the collective spin and evolution after a finite time to a spin cat state or, in the long-time limit, the Dicke state ${|S,0\ensuremath{\rangle}}_{x}$, conditioned upon there being no photon emissions from the system (i.e., no quantum jumps). If there is a jump, however, the system evolves probabilistically into one of a finite number of entangled-state cycles, where the system then undergoes a persistent sequence of jumps between two Dicke-state superpositions in a rotated basis. The different cycles can be distinguished by the frequency at which jumps occur.

Feng Pan, Tao Wang, Jing Pan +2 more

B M Rodríguez-Lara, H M Moya-Cessa

We show a right unitary transformation approach based on Susskind–Glogower operators that diagonalizes a generalized Dicke Hamiltonian in the field basis and delivers a tridiagonal Hamiltonian in the Dicke basis. This tridiagonal Hamiltonian is diagonalized by a set of orthogonal polynomials satisfying a three-term recurrence relation. Our result is used to deliver a closed form, analytic time evolution for the case of a Jaynes–Cummings–Kerr model and to study the time evolution of the population inversion, reduced field entropy and Husimi's Q -function of the field for ensembles of interacting two-level systems under a Dicke–Kerr model.

Richard M. Friedberg