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Journal of Physics A Mathematical and General

Effect of spin-spin interaction on the super-radiant phase transition for the Dicke Hamiltonian

C. M. Bowden, Chi C. Sung

Published Jan 1, 1978• 7 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/11/1/016

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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.
We consider the full Dicke spin-boson model composed by a single bosonic mode and an ensemble of N identical two-level atoms with different couplings for the resonant and anti-resonant interaction terms, and incorporate a dipole–dipole interaction between the atoms. Assuming that the system is in thermal equilibrium with a reservoir at temperature β −1 , we compute the free energy in the thermodynamic limit N → ∞ in the saddle-point approximation to the path integral and determine the critical temperature for the super-radiant phase transition. In the zero temperature limit, we recover the critical coupling of the quantum phase transition, presented in the literature.
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.
We study the optical signatures of the quantum critical behaviour associated with the quantum phase transition exhibited by the Dicke model in the thermodynamic limit. We obtain an effective Hamiltonian for the radiation field, which resembles a degenerate parametric amplifier and which reproduces the Dicke's model critical behaviour. We identify the state of the radiation field in the sub-radiant and super-radiant phases, and show that the optical squeezing and photon statistics present striking behaviour in the vicinity of the phase transition. Our results are both of theoretical interest and of relevance to recent experimental proposals that hope to realize the quantum phase transition in the Dicke model.
We discuss the coherent behavior of a polarized, nuclear or electron, spin system for which the magnetic dipole radiation emitted in the radio-frequency region, has approximately quadratic dependence on the number of spins. An effective method of describing these phenomena is provided by computer simulation of a microscopic model of the spin system. Important aspects of this numeric simulation are described, together with a comparison with the theoretical predictions. The behavior of the transverse component of the magnetic moment, M+(t){M}^{+}(t)M+(t), in super-radiant conditions is studied. In addition, the role of dipole-dipole interactions in super-radiation phenomena is investigated in detail. It is shown that some important features of super-radiation cannot be described with the Bloch equations.
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.
We consider spin-boson models composed by a single bosonic mode and an ensemble of N identical two-level atoms. The situation where the coupling between the bosonic mode and the atoms generates resonant and nonresonant processes is studied, where the whole system is in thermal equilibrium with a reservoir at temperature β−1. Phase transitions from ordinary fluorescence to super-radiant phase in three different models are investigated. First a model where the coupling between the bosonic mode and the jth atom is via the pseudo-spin-operator σ(j)z is studied. Second, we investigate the generalized Dicke model, introducing different coupling constants between the single mode bosonic-field and the environment, g1 and g2, for rotating and counter-rotating terms, respectively. Finally a modified version of the generalized Dicke model with intensity-dependent coupling in the rotating terms is considered. In the first model, for an arbitrary value for the coupling constant, the zero mode contributes in rendering the canonical entropy a negative quantity for low temperatures. For small coupling constants, the partition function is analytic for all temperatures. The last two models present phase transitions, even when only Hamiltonian terms which generate virtual processes are considered.