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Journal of Physics A Mathematical and General

Phase transition in the multimode two- and three-level Dicke model (Green's function method)

Chi C. Sung, C. M. Bowden

Published Nov 1, 1979• 16 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/12/11/035

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Phase transition of a system of two-level atoms

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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.

Journal of Physics A Mathematical and General•Nov 1, 1977•13 citations
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.
The authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.
The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir. In the thermodynamic limit, the fermion Dicke model can be analized using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superrandiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations. There is quantum phantum critical behavior when the coupling constants satisfy an especific condition. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations in the case using the rotating-wave approximation, recovering the well know results. Second, the case only considering virtual processes. In the last case, it is possible to have a superradiance phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at some critical coupling, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.
We consider an ensemble of three-level particles in Lambda configuration interacting with two bosonic modes. The Hamiltonian has the form of a generalized Dicke model. We show that in the thermodynamic limit this model supports a superradiant quantum phase transition. Remarkably, this can be both a first- and a second-order phase transition. A connection of the phase diagram to the symmetries of the Hamiltonian is also given. In addition, we show that this model can describe atoms interacting with an electromagnetic field in which the microscopic Hamiltonian includes a diamagnetic contribution. Even though the parameters of the atomic system respect the Thomas-Reiche-Kuhn sum rule, the system still shows a superradiant phase transition.
The Dicke model describes an ensemble of N identical two-level atoms (qubits) coupled to a single quantized mode of a bosonic field. The fermion Dicke model should be obtained by changing the atomic pseudo-spin operators by a linear combination of Fermi operators. The generalized fermion Dicke model is defined introducing different coupling constants between the single mode of the bosonic field and the reservoir, g1 and g2 for rotating and counter-rotating terms respectively. In the limit N → ∞, the thermodynamic of the fermion Dicke model can be analyzed using the path integral approach with functional method. The system exhibits a second order phase transition from normal to superradiance at some critical temperature with the presence of a condensate. We evaluate the critical transition temperature and present the spectrum of the collective bosonic excitations for the general case (g1 ̸ = 0 and g2 ̸ = 0). There is quantum critical behavior when the coupling constants g1 and g2 satisfy g1 + g2 = (ω0 Ω) 1 2, where ω0 is the frequency of the mode of the field and Ω is the energy gap between energy eigenstates of the qubits. Two particular situations are analyzed. First, we present the spectrum of the collective bosonic excitations, in the case g1 ̸ = 0 and g2 = 0, recovering the well known results. Second, the case g1 = 0 and g2 ̸ = 0 is studied. In this last case, it is possible to have a superradiant phase when only virtual processes are introduced in the interaction Hamiltonian. Here also appears a quantum phase transition at the critical coupling g2 = (ω0 Ω) 1 2, and for larger values for the critical coupling, the system enter in this superradiant phase with a Goldstone mode.
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.