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Quantum Optics Journal of the European Optical Society Part B

The classical analogue of the super-radiant phase transition in the Dicke model

Marcus A. M. de Aguiar, K. Furuya, Maria Carolina Nemes

Published Oct 1, 1991• 17 citations
Publisher

Abstract

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.

DOI
10.1088/0954-8998/3/5/005

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•Nov 27, 2017•0 citations
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 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 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 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 Dicke model exhibits a variety of phase transitions. The quantum phase transition from the normal phase to the super-radiant phase is marked by a dramatic change in the scaling of the participation ratio. We find that the ground state in the super-radiant phase exhibits multifractality manifest in the participation ratio scaling as the square root of the full Hilbert-space dimension. The thermal phase transition temperature, for which an exact analytical expression may be obtained using the partition function, is strikingly captured by the mutual information between two spins. In the excited-state quantum phase transition within the super-radiant phase, there are two cutoff energies; the central energy bands between the lower and upper cutoff energies show distinctly different behaviors. While the level statistics of the central band is Wigner-Dyson, the lower and upper bands show mixed behavior that is closer to Poisson than Wigner-Dyson. This finding is corroborated with the aid of several eigenvector properties: von Neumann entanglement entropy between spins and bosons, the mean photon number, concurrence between two spins, and participation ratio.
We prove that superradiant phase transitions (SPTs) of the Dicke model and its generalizations in the thermodynamic and classical oscillator limit are indeed of the same type. In this sense, we unify SPTs under both limits at zero and finite temperature. We show that the mean-field approximation for bosons is exact in both cases, and compute the structure and location of the phase transitions in parameter space using a concise analytic method. Moreover, we illustrate how SPT properties (first order, second order, or none) are related to symmetry. Finally, we uncover general features of the phase structure in the space of parameters of these models with dipolar couplings. There will be a region of normal phase in the neighborhood of the origin of the space of dipolar couplings $\stackrel{P\vec}{\ensuremath{\gamma}}$, and that generally one flows radially in this space to a superradiant phase.
The Hamiltonian of the two-photon Dicke model is diagonalized for the normal phase and super-radiant phase beyond the mean-field method respectively, giving the critical coupling strength. Besides a spectral collapse, the super-radiant phase transition is presented by the vanishing of the excitation energy, which is lower than the mean-field results. Finite-size scaling exponents for the ground-state energy and the atomic angular momentum are analytically derived from scaling hypothesis, belonging to the same scaling universality of the one-photon Dicke model.