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Physical Review A

Dynamics of a three-level atom in a two-mode squeezed vacuum

W. K. Lai, V. Buek, P. L. Knight

Published Nov 1, 1991• 53 citations
Publisher

Abstract

In this paper we study the dynamics of a three-level atom interacting with a two-mode squeezed vacuum, a highly correlated state in which each mode contains the same number of quanta yet exhibits a thermal photon distribution. Under exact resonance, the Rabi oscillations exhibit an irregular behavior reminiscent of those in the thermal-state Jaynes-Cummings model. If the individual field modes are sufficiently detuned from the intermediate atomic level, the Rabi oscillations become quasiperiodic. The three-level system in this case reduces to an effective two-level system in which two-photon processes dominate. The times of revivals of the Rabi oscillations in this limit depend only on the atom-field coupling constants and the detuning. For comparison, we present a parallel study of the case in which the initial field is prepared in an uncorrelated two-mode state, such as a two-mode thermal state or a two-mode coherent state. Under the full resonance condition, the former leads to an irregular evolution of the Rabi oscillations, while the latter leads to the usual collapse and revival phenomena. If the individual modes are sufficiently detuned from the intermediate atomic level, we observe the same quasiperiodicity of the Rabi oscillations as in the case of the two-mode squeezed vacuum but with a doubling of revival times (for equal atom-field coupling constants). Whereas the two-mode squeezed vacuum and the two-mode thermal-state transition dynamics lead to correlation between the field modes, the two-mode coherent state leads to anticorrelation (when sufficiently detuned from the intermediate atomic level).

DOI
10.1103/physreva.44.6043

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Journal of the Optical Society of America B•Oct 1, 1989•18 citations
In this paper we study the dynamics of a three-level atom interacting with a two-mode squeezed vacuum, a highly correlated state in which each mode contains the same number of quanta yet exhibits a thermal photon distribution. Under exact resonance, the Rabi oscillations exhibit an irregular behavior reminiscent of those in the thermal-state Jaynes-Cummings model. If the individual field modes are sufficiently detuned from the intermediate atomic level, the Rabi oscillations become quasiperiodic. The three-level system in this case reduces to an effective two-level system in which two-photon processes dominate. The times of revivals of the Rabi oscillations in this limit depend only on the atom-field coupling constants and the detuning. For comparison, we present a parallel study of the case in which the initial field is prepared in an uncorrelated two-mode state, such as a two-mode thermal state or a two-mode coherent state. Under the full resonance condition, the former leads to an irregular evolution of the Rabi oscillations, while the latter leads to the usual collapse and revival phenomena. If the individual modes are sufficiently detuned from the intermediate atomic level, we observe the same quasiperiodicity of the Rabi oscillations as in the case of the two-mode squeezed vacuum but with a doubling of revival times (for equal atom-field coupling constants). Whereas the two-mode squeezed vacuum and the two-mode thermal-state transition dynamics lead to correlation between the field modes, the two-mode coherent state leads to anticorrelation (when sufficiently detuned from the intermediate atomic level).
In this paper we investigate the sensitivity of coupled atomic transitions to correlations between field modes. In particular, we show that a three-level atom interacting with a cavity field supporting two quantized modes can be highly sensitive to the nature of the correlations between the modes. Correlated two-mode states include the two-mode squeezed state, whereas the SU(2) coherent state is an example of an anticorrelated state. We study in detail the dynamics of a three-level atom in the \ensuremath{\lambda} configuration, interacting with the two modes of an SU(2) coherent state. We show that for certain choices of the atom-field coupling constants, the generalized two-photon Rabi frequency for the system becomes independent of the photon number, leading to a purely periodic behavior in time of the atomic population probabilities; for other choices of the atom-field coupling constants, we observe the usual collapse and revival of the Rabi oscillations. We study also the dynamics of the field, in particular the correlations between the modes. Finally, for comparison purposes, we present a parallel study of the case in which the field modes are initially uncorrelated.
We investigate the quantum dynamics for a molecular-state-model system interacting with a resonant two-mode squeezed coherent field, in which each mode is initially correlated. As an example, we consider a three-state molecular model, which mimics the electronic excited states of trans-octatetraene obtained by a full-configuration-interaction calculation using the Pariser-Parr-Pople Hamiltonian, interacting with a two-mode squeezed coherent field. For comparison, we also perform parallel studies, in which the initial fields are prepared in uncorrelated two-mode states, i.e., a two-mode coherent state and a two-mode thermal state. It is well known that the case of two-mode coherent field leads to the usual collapse-revival phenomena of the Rabi oscillations, while the case of a two-mode thermal state leads to an irregular evolution of them. Although the two-mode squeezed coherent field exhibits similar collapse-revival behavior to that for the two-mode coherent field, some differences are detected in the amplitudes and periods of collapse-revival oscillations. Further, the dynamics of off-diagonal molecular density matrices between the ground and second excited states for these two fields are found to be distinctively different from each other. Such attractive behavior is found to be closely associated with the two-mode photon-phase dynamics obtained by the Pegg-Barnett phase operator and the initial quasiprobability (Q function) distribution of photons. These differences in phase properties are also shown to significantly affect the dynamics of the information entropy for a molecule, i.e., molecular entropy, which represents the degree of entanglement between the molecule and photons.
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 dynamics of the Jaynes–Cummings (JCM) model with a nonlinear Kerr medium at initially thermal and squeezed vacuum fields is discussed. At a special choice of the detuning the Rabi frequency can have a minimum at the initial mean photon number [nbar]. Then the situation becomes especially interesting and instead of the first-order ‘usual’ revivals resembling those manifested by the standard coherent JCM, the nonlinear JCM exhibits second-order revivals, also for initially thermal and squeezed vacuum fields, at least for not too large [nbar]. The revival times are highly sensitive to the nature of [nbar]. In the first case the revival period of the oscillations is dependent on whether [nbar] is integer or non-integer (situation similar to an initially coherent field) while in the latter case it depends additionally on the parity of the integer [nbar]. Moreover, for the squeezed vacuum model appearance of the revivals at an integer [nbar] is accelerated when compared to coherent and thermal models.
Squeezed states of the Jaynes-Cummings (JC) model are constructed and the statistical properties of the field oscillator in these states are studied. The presence of interaction between the field oscillator and the atom results in the appearance of slow temporal variations in the average values of the dynamic quantities. Two cases essentially differ: When only one of two modes of the JC model is excited, the variations in time, in particular, of the average value of the coordinate are governed by the mode anharmonicity. In the case when both modes of the JC model with close frequencies are excited, the temporal variations represent collapse and revival of oscillations with the frequency 2kn+12k\sqrt {n + 1} 2kn+1​ , where k is the constant of the atom-field oscillator interaction and n is the number of photons. The constructed squeezed vacuum states are not two-photon states, because the probability of finding an odd number of photons is not equal to zero and varies with time.
With the help of a two-photon Jaynes-Cummings model, which is an exactly solvable model, we have dealt analytically with the dynamical behavior of the atom and the field for a two-photon process in squeezed and coherent fields under the appropriate approximation. The features of collapses and revivals of Rabi oscillations for the atom and the field in a two-photon process represent strict periodicity. The phase jump and the large and negative frequency for the combination Rabi oscillation are exhibited in the case of a squeezed field. The results show that squeezing in a field strikingly affects the evolution behavior of the atom and the field.