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Journal of Physics A Mathematical and General

Phase transition of a system of two-level atoms

Miguel Orszag

Published Nov 1, 1977• 13 citations
Publisher

Abstract

A very simple treatment is presented to describe a phase transition present in Dicke's model for super-radiance, in the off-resonance case and for a single radiation mode. The results agree with previous authors, although the use of mod n( states provides a transparent physical interpretation. The results show that if the A 2 term is kept in the Hamiltonian, the phase transition persists, thus clarifying some recent controversy on the subject. The thermodynamic variables are calculated above and below T c . The phase transition is shown to be a consequence of an instability of the ground state. The behaviour of (a Dagger a) and (S Z ) is explored as a function of temperature. (a Dagger a) goes to zero as (T c -T) near the critical temperature.

DOI
10.1088/0305-4470/10/11/025

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A very simple treatment is presented to describe a phase transition present in Dicke's model for super-radiance, in the off-resonance case and for a single radiation mode. The results agree with previous authors, although the use of mod n( states provides a transparent physical interpretation. The results show that if the A 2 term is kept in the Hamiltonian, the phase transition persists, thus clarifying some recent controversy on the subject. The thermodynamic variables are calculated above and below T c . The phase transition is shown to be a consequence of an instability of the ground state. The behaviour of (a Dagger a) and (S Z ) is explored as a function of temperature. (a Dagger a) goes to zero as (T c -T) near the critical temperature.
The authors construct the classical analogue of the phenomena of super-radiance in the zero temperature limit and show that a simple geometrical interpretation can be given in the integrable case. The non-integrable case is also studied and in both cases they find bifurcation of equilibrium for the same parameter values where this phase transition is known to occur in the thermodynamical context. The ground state of the system is also studied in the framework of a mean field approximation and a simple analytical expression obtained. A connection between the classical and quantum points of view is presented.
A system of NNN two-level atoms interacting with a quantized field, the so-called Dicke model of superradiance, is studied. By making use of a set of Glauber's coherent states for the field, the free energy of the system is calculated exactly in the thermodynamic limit. The results agree precisely with those obtained by Hepp and Lieb, who studied the same model using a different method. The exhibition of a phase transition of the system is presented mathematically in an elementary manner in our approach. The generalization to the case of finitely many radiation modes is also presented.
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.
By using the thermodynamic Green's function method, the authors study the phase transition of a collection of two- as well as three-level atoms interacting with the electromagnetic field and contained within a volume much smaller than the smallest resonance wavelength (Dicke model). They show for the case of three-level atoms the existence of two critical temperatures where a second-order phase transition takes place. The upper critical temperature is determined by the largest of the three coupling constants in the three-level model, whereas the lower critical temperature is a function of the remaining two coupling constants. It is also shown that for a collection of two- as well as three-level atoms, the critical temperature depends upon the number of atoms contained within lambda r 3 ( lambda r is a resonance wavelength) rather than the density as has previously been suggested. Also, using the Green's function method, we show the formal equivalence of the Dicke model in thermodynamic equilibrium with the BCS model of superconductivity.
It is shown that the rotating wave approximation in the minimal coupling Hamiltonian with the A 2 term leads to an incorrect energy spectrum of the polariton mode: in particular, to a spurious polariton instability which should not occur for a photon-matter system consisting of radiation and atoms without the Coulomb interaction. Thus the counter-rotating term in the presence of the A 2 term plays a crucial role in the instability property of the system. The super-radiant phase transition in the Dicke model is shown to be accompanied by polariton instability, and therefore by instability of the ground state of the photon-matter system. The rotating wave approximation, as well as the neglect of the A 2 term, should therefore not be made for the study of the thermodynamic phase transition and instability properties of the ground state of the system.
The Dicke Hamiltonian describes the simplest quantum system with atoms interacting with photons: N two level atoms inside a perfectly reflecting cavity which allows only one electromagnetic mode. It has also been successfully employed to describe superconducting circuits which behave as artificial atoms coupled to a resonator. The system exhibits a transition to a superradiant phase at zero temperature. When the interaction strength reaches its critical value, both the number of photons and of atoms in excited states in the cavity, together with their fluctuations, exhibit a sudden increase from zero. Employing symmetry-adapted coherent states it is shown that these properties scale with the number of atoms, that their reported divergences at the critical point represent the limit when this number goes to infinity, and that in this limit they remain divergent in the superradiant phase. Analytical expressions are presented for all observables of interest, for any number of atoms. Comparisons with exact numerical solutions strongly support the results.