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arXiv:quant-ph/9809051 · Physical Review A

Collective effects in the collapse-revival phenomenon and squeezing in the Dicke model

Guy Z. Ramon, Constantin Brif, A. Mann

Published Sep 1, 1998• 26 citations
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Abstract

Resonant interaction of a collection of two-level atoms with a single-mode coherent cavity field is considered in the framework of the Dicke model. We focus on the role of collective atomic effects in the phenomenon of collapses and revivals of the Rabi oscillations. It is shown that the behavior of the system strongly depends on the initial atomic state. In the case of the initial half-excited Dicke state we account for a number of interesting phenomena. The correlations between the atoms result in a suppression of the revival amplitude, and the revival time is halved, compared to the uncorrelated fully excited and ground states. The phenomenon of squeezing of the radiation field in the atom-field interaction is also discussed. For the initial fully excited and ground atomic states, the field is squeezed on the short-time scale, and squeezing can be enhanced by increasing the number of atoms. Some empirical formulas are found which describe the behavior of the system in excellent agreement with numerical results. For the half-excited Dicke state, the field can be strongly squeezed on the long-time scale in the case of two atoms. This kind of squeezing is enhanced by increasing the intensity of the initial coherent field and is of the same nature as revival-time squeezing in the Jaynes-Cummings model. The appearance of this long-time squeezing can be explained using the factorization approximation for semiclassical atomic states.

DOI
10.1103/physreva.58.2506

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Resonant interaction of a collection of two-level atoms with a single-mode coherent cavity field is considered in the framework of the Dicke model. We focus on the role of collective atomic effects in the phenomenon of collapses and revivals of the Rabi oscillations. It is shown that the behavior of the system strongly depends on the initial atomic state. In the case of the initial half-excited Dicke state we account for a number of interesting phenomena. The correlations between the atoms result in a suppression of the revival amplitude, and the revival time is halved, compared to the uncorrelated fully excited and ground states. The phenomenon of squeezing of the radiation field in the atom-field interaction is also discussed. For the initial fully excited and ground atomic states, the field is squeezed on the short-time scale, and squeezing can be enhanced by increasing the number of atoms. Some empirical formulas are found which describe the behavior of the system in excellent agreement with numerical results. For the half-excited Dicke state, the field can be strongly squeezed on the long-time scale in the case of two atoms. This kind of squeezing is enhanced by increasing the intensity of the initial coherent field and is of the same nature as revival-time squeezing in the Jaynes-Cummings model. The appearance of this long-time squeezing can be explained using the factorization approximation for semiclassical atomic states.
We report here results for collapses and revivals phenomenon for a collection of M two-level atoms contained in a volume having dimensions much smaller than the radiation wavelength and existing in the Dicke state |r,m〉. We use a perturbation approach and obtain in the lowest contributing order of perturbation second harmonic, besides the main Rabi frequency. The amplitude of the fundamental depends on m only, but the amplitude of the harmonic has dependence on r as well as m. If the atoms are in superradiant states, only the second harmonic is obtained. Revival time for the second harmonic is one half that for the fundamental. It is interesting to note that the number of atoms M does not matter in Rabi frequency, collapse times, revival times or in amplitudes, and only r and m matter. Values of r and m of atoms in a Dicke state can be determined, in principle, by observing Rabi oscillations and average atomic inversion.
The resonant interaction of A initially unexcited two-level atoms with different quantum fields in a high-Q single-mode cavity (Dicke model) is studied analytically in one of the limits in which the eigenvalues spectrum of the model may be assumed as equidistant. The time evolution of atomic inversion of a single two-level atom coupled to squeezed vacuum or thermal cavity fields (Jaynes-Cummings model) is irregular. Here we show that the dynamical response of the inversion of an assemblage of A two-level atoms to such fields may be regular, i.e., may reveal collapses and revivals and that the properties of the model with an initial squeezed vacuum field differ from those for coherent or thermal fields.
The atomic system on which we focus is a single four-level atom interacting with two-mode cavity fields. We obtain the wavefunction when the atom is initially in an excited state. The result presented in this context is employed to discuss the collapses–revivals phenomenon, the photon statistics and frequency sum squeezing phenomenon. The influence of the detuning on these phenomena are investigated for two four-level atomic systems. We found that the presence of the detuning leads to an increase of the collapse time while the amount of squeezing decreases. Also, the photon statistics is affected by the existence of the detuning parameters where the anti-bunching effect appears.
Based on the time-convolutionless master-equation approach, we investigate squeezing of light field in a dissipative Jaynes–Cummings model. The results show that squeezing light can be generated when the atom transits to a ground state from an excited state, and then a collapse-revival phenomenon will occur in the squeezing of light field due to atom-cavity coupling. Enhancing the atom-cavity coupling can increase the frequency of the collapse-revival of squeezing. The stronger the non-Markovian effect is, the more obvious the collapse-revival phenomenon is. The oscillatory frequency of the squeezing is dependent on the resonant frequency of the atom-cavity.
In the present paper, we consider the interaction of a single mode radiation initially in a coherent state with an assembly of two two-level atoms in some different states using the Hamiltonian, H = ω(a+a + Sz)+g(a S+ + a+ S-) in the natural units, where a+ and a are the creation and annihilation operators for radiation, Sz, S± are the collective Dicke operators, g is the coupling constant, ω is the energy of the photons and energy difference between the two atomic levels. We solve it exactly. We study ordinary squeezing and amplitude-squared squeezing of radiation for the general operators, [Formula: see text] and Yθ=1/2(a2e-iθ+a+2eiθ), when atoms are fully excited, super-radiant or in the ground state. We get a large ordinary squeezing (nearly 88% squeezing) for the super-radiant atomic state, at a time, which is 0.451 of the first revival time, and this should be easily observable. We also get a large amplitude-squared squeezing (nearly 20%) at the time, which is about 0.033 of the first revival time. Variation of variances near their minima with coupling time gt, square root of mean photon number |α| and phase are also discussed.
We consider the dynamics of atomic and field coherent states in the nonresonant Dicke model. At weak coupling an initial product state evolves into a superposition of multiple field coherent states that are correlated with the atomic configuration. This process is accompanied by the buildup and decay of atom-field entanglement and leads to the periodic collapse and revival of Rabi oscillations. We provide a perturbative derivation of the underlying dynamical mechanism that complements the rotating wave approximation at resonance. The identification of two different time scales explains how the dynamical signatures depend on the sign of detuning between the atomic and field frequency and predicts the generation of either atomic or field cat states in the two opposite cases. We finally discuss the restrictions that the buildup of atom-field entanglement during the collapse of Rabi oscillations impose on the validity of semiclassical approximations that neglect entanglement.