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International Journal of Modern Physics B

COLLAPSES AND REVIVALS IN M TWO-LEVEL ATOMS INTERACTING WITH A SINGLE MODE COHERENT RADIATION

Ranjana Prakash, Pramila Shukla

Published Jun 20, 2008• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s021797920803940x

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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.
Collapses and revivals phenomenon in system of a single two-level atom and two two-level atoms existing in some superposed states and interacting with a single mode superposed coherent radiation is studied. For superposed Dicke states |1/2, ±1/2> of a single two-level atom interacting with even or odd coherent state, only odd revivals occur. For two two-level atoms, it is found that the collapse and revival times for even and odd coherent states are equal to one half of the corresponding times for coherent state. For this system, Rabi oscillations occur with a main mean frequency and also some second harmonics are present, in general. However, if the two atoms are in the superradiant state, only the second harmonics with large amplitude are obtained. The appearance of weak double frequency revivals for superposed atomic and coherent states was studied and a condition for their disappearance is found.
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.
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.
Collapse and revival of the Doppler-Rabi oscillations of a two-level atom moving in a cavity electromagnetic field are analyzed. The coupled atom-field dynamics are predicted accurately by numerical calculation and approximately by using the stationary phase approximation combined with the Poisson summation formula. The collapse and revival patterns are shown to be qualitatively different in the cases of moving atom and atom at rest. In particular, quantum revivals of Doppler-Rabi oscillations occur with a period determined by the Doppler shift of the atomic transition frequency. This regime of Doppler-Rabi oscillations requires that the Rabi frequency and the Doppler shift satisfy the condition Ω R ≪ Ω D . Under the inverse relation, the collapse- revival phenomenon generally does not occur. It is shown that even a small amount of atom-cavity detuning eliminates collapse-revival behavior. The analysis is performed for both coherent and thermal cavity fields.
A two-level atom interacting with a cavity field initialized in a coherent state exhibits the collapse and revival features in its population inversion dynamics under the Jaynes–Cummings interaction. When the cavity field is prepared in a number-state filtered coherent state, obtained by filtering the number state | m 〉 from a coherent state | α 〉 , we observe micro-Rabi oscillations in the collapse region instead of a perfect collapse. We find that the Rabi frequency of these micro oscillations depends only on the filtered photo-number m, whereas the amplitude depends on both m and α . This dependence enables one to infer the filtered number state from the coherent-state distribution by analyzing the time evolution of the atomic population inversion. Conversely, the amplitude and frequency of the micro-Rabi oscillations can be tuned by filtering an appropriate number state. Furthermore, we investigate the effect of filtering multiple number states from a coherent state on the atomic population inversion dynamics.
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.