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

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, 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 superradiant phase transition (SPT) into a state with a non-zero amplitude of the classical field is possible, while in the second case, the so-called order parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>η</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="false">〈</mml:mo> <mml:mover accent="true"> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> <mml:mo stretchy="false">〉</mml:mo> </mml:mrow> </mml:math> which would indicate condensation into a coherent state remains zero, meaning that the SPT is forbidden. It resolves the long-standing discussion about the SPT in the Dicke model. At the same time, the average number of photons <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo stretchy="false">〈</mml:mo> <mml:msup> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mo>†</mml:mo> </mml:mrow> </mml:msup> <mml:mover accent="true"> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>ˆ</mml:mo> </mml:mrow> </mml:mover> <mml:mo stretchy="false">〉</mml:mo> </mml:mrow> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:math> , and the collective excitation in the system manifests as 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.