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Physical Review A

Are super-radiant phase transitions possible?

James M. Knight, Yakir Aharonov, G. T. C. Hsieh

Published Apr 1, 1978• 70 citations
Publisher

Abstract

An argument is given which rules out super-radiant phase transitions in which the atomic states involved are connected by electric-dipole matrix elements. The argument is based on gauge invariance, and is valid even in the presence of direct interatomic interactions. Its relevance is demonstrated by presenting a number of two-level models exhibiting super-radiant phase transitions which are not forbidden by previous sum-rule arguments. As a preliminary to the argument, the nature of the super-radiant phase transition is discussed, and a simple model is given which reproduces its main features. It is argued, finally, that in at least one case in which our argument fails a super-radiant phase transition is possible in principle.

DOI
10.1103/physreva.17.1454

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An argument is given which rules out super-radiant phase transitions in which the atomic states involved are connected by electric-dipole matrix elements. The argument is based on gauge invariance, and is valid even in the presence of direct interatomic interactions. Its relevance is demonstrated by presenting a number of two-level models exhibiting super-radiant phase transitions which are not forbidden by previous sum-rule arguments. As a preliminary to the argument, the nature of the super-radiant phase transition is discussed, and a simple model is given which reproduces its main features. It is argued, finally, that in at least one case in which our argument fails a super-radiant phase transition is possible in principle.
In a gauge invariant formulation of the molecular electric dipole-photon interaction, the rigorous coupling is strictly linear in the photon creation and photon annihilation operators. The linear coupling allows for a super-radiant phase transition as in the Hepp-Lieb formulation. A previous notion of a quadratic-coupling ``no-go theorem'' for super-radiance is incorrect. Also incorrect is a previous assertion that the dipole-photon coupling has absolutely no effect on the thermal equations of state. These dubious assertions were based on incorrect canonical transformations which eliminated the electric field (and thereby eliminated the dipole-photon interaction) which is neither mathematically nor physically consistent. The correct form of the canonical transformations are given in this work which allows for the physical reality of super-radiant condensed matter phases.
The transition to the super-radiant regime in the Dicke model of two-level atoms interacting with a one-mode radiation field has been argued to be impossible for many years. Recently, however, experimental results show that this transition is possible in an open system [1, 2]. Here we exhibit the power of the use of coherent states with restored symmetry to describe the superradiant regime of the Dicke Hamiltonian. These symmetry-adapted states allow for an analytic description of both the ground and first excited states. From these, analytical expressions for all expectation values of interest are obtained, valid for any number of atoms. All these results are exact in the thermodynamic limit.
The thermodynamic equilibrium properties of the Dicke Hamiltonian for a collection of spin- 1 / 2 particles interacting via magnetic dipole interaction with a single mode of the radiation field and with each other by spin-spin interaction are analysed and the conditions for the existence of a super-radiant phase transition are derived. It is shown, using the approximate long-range spin-spin interaction (MI), in the Dicke Hamiltonian, that the conditions for the existence of the super-radiant phase transition are invariant with respect to isotropic MI whether or not the A 2 term is included.
The possibility of the superradiant phase transition in polarizable materials described by the minimal-coupling Hamiltonian with the longitudinal dipole-dipole interaction is examined. We try to reduce the Hamiltonian into the Dicke one in the homogeneous and infinite case, and discuss the stability of normal ground state by the formalism of the Green function in the spatially inhomogeneous case. The presence of the longitudinal dipole-dipole interaction does not enable the superradiant phase transition, if the transverse and longitudinal fields are decoupled. Although the full dipole-dipole interaction can be eliminated in the electric-dipole gauge in the absence of overlap between individual atomic dipoles, we cannot reduce the Hamiltonian to the Dicke one, because the elimination is justified only if all the transverse and longitudinal fields remain. Even if the transverse and longitudinal fields are mixed in spatially inhomogeneous systems, the normal ground state is still stable if the system does not show the superradiant phase transition in the homogeneous case.