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arXiv:1107.2945 · Journal of Physics A Mathematical and Theoretical

A path integral approach to the full Dicke model with dipole–dipole interaction

M. Aparicio Alcalde, Jorge Stephany, Nami Fux Svaiter

Published Nov 22, 2011• 5 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/50/505301

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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 full Dicke model is composed of a single bosonic mode and an ensemble of NNN identical two-level atoms. In the model, the coupling between the bosonic mode and the atoms generates resonant and nonresonant processes. We also consider a dipole-dipole interaction between the atoms, which is able to generate entangled states in the atomic system. By assuming thermal equilibrium with a reservoir at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$, the transition from fluorescent to superradiant phase and the quantum phase transition are investigated. It is shown that the critical behavior of the full Dicke model is not modified by the introduction of the dipole-dipole interaction.
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.
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 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.
We introduce and study the disordered Dicke model in which the spin-boson couplings are drawn from a random distribution with some finite width. Regarding the quantum phase transition we show that when the standard deviation σσσ of the coupling strength gradually increases, the critical value of the mean coupling strength μμμ gradually decreases and after a certain σσσ there is no quantum phase transition at all; the system always lies in the super-radiant phase. We derive an approximate expression for the quantum phase transition in the presence of disorder in terms of μμμ and σσσ, which we numerically verify. Studying the thermal phase transition in the disordered Dicke model, we obtain an analytical expression for the critical temperature in terms of the mean and standard deviation of the coupling strength. We observe that even when the mean of the coupling strength is zero, there is a finite temperature transition if the standard deviation of the coupling is sufficiently high. Disordered couplings in the Dicke model will exist in quantum dot superlattices, and we also sketch how they can be engineered and controlled with ultracold atoms or molecules in a cavity.
The Dicke model describing an ensemble of two-state atoms interacting with a single quantized mode of the electromagnetic field (with omission of the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{2}$ term) exhibits a zero-temperature phase transition at a critical value of the dipole coupling strength. We propose a scheme based on multilevel atoms and cavity-mediated Raman transitions to realize an effective Dicke model operating in the phase transition regime. Optical light from the cavity carries signatures of the critical behavior, which is analyzed for the thermodynamic limit where the number of atoms is very large.