Ground state of the gauge invariant Dicke model: condensation of the photons in non-classical states

TL;DR

This paper explores the ground states of modified Dicke models, revealing distinct phase transition behaviors and photon condensation phenomena, including non-classical squeezed states, in systems with two-level particles and gauge invariance.

Contribution

It introduces and analyzes two modifications of the Dicke model, highlighting their qualitative differences in phase transition behavior and photon state properties.

Findings

Phase transition into a classical field state is possible in the first model.

In the second model, the order parameter is zero despite non-zero photon number.

The system exhibits non-classical squeezed states of the electromagnetic field.

Abstract

We investigate the ground state of two physically motivated modifications of the Dicke model. The first modification corresponds to particles whose phase space contains only two states, for example, particles with spin 1/2 or artificially created qubits. The second modification describes two-level systems that arise as a result of truncating the full Hilbert space of atoms to two levels that are in resonance with the electromagnetic field and are described by the gauge-invariant Dicke model. We demonstrate that the behavior of these systems is qualitatively distinct in both cases. In particular, in the first scenario, a phase transition into the state with a non-zero amplitude of the classical field is possible, while in the second case, the so-called order parameter $\eta = \braket{\hat{a}}$ of the field's phase transition into a coherent state with photon condensation is zero. At the same time, the average number of photons $\bar{n} = \braket{\hat{a}^\dagger \hat{a}} \neq 0$, and the collective excitation in the system manifests a non-classical "squeezed" state of the field. We analyze the observable characteristics of both systems in a wide range of variation of their parameters.