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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Population inversion and stationary lineshapes in a driven two-atom system in a squeezed vacuum

Z. Ficek, Mohamed Ridza Wahiddin

Published Feb 1, 1995• 1 citation
Publisher

Abstract

We study the effect of a broadband squeezed vacuum on the dynamics of a two-atom system driven by a near resonant laser field. We show that only in the absence of the driving field are the collective atomic levels selectively populated. Otherwise all collective atomic levels are populated independently of the initial conditions and time. Moreover, we show that the mode correlations, characteristics for the squeezed vacuum, induce a population inversion which, in contrast to the selective population of the collective levels, is not destroyed by the driving field. The stationary lineshapes for the two atomic transitions are also studied. We show that in the squeezed vacuum the population in the upper collective level can exhibit three distinct peaks centred at the frequencies omega 0 and omega 0 +or- Omega 12 , where omega 0 is the atomic transition frequency and Omega 12 is the dipole-dipole interaction potential. The central peak is induced by the mode correlations, whereas the sidebands are induced by the driving field and show a strong dependence on the relative phase between the squeezed vacuum and the driving field. The population in the intermediate level shows a distinct peak at the frequency omega 0 + Omega 12 , and holes at omega 0 and omega 0 - Omega 12 .

DOI
10.1088/1355-5111/7/1/003

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We study the effect of a broadband squeezed vacuum on the dynamics of a two-atom system driven by a near resonant laser field. We show that only in the absence of the driving field are the collective atomic levels selectively populated. Otherwise all collective atomic levels are populated independently of the initial conditions and time. Moreover, we show that the mode correlations, characteristics for the squeezed vacuum, induce a population inversion which, in contrast to the selective population of the collective levels, is not destroyed by the driving field. The stationary lineshapes for the two atomic transitions are also studied. We show that in the squeezed vacuum the population in the upper collective level can exhibit three distinct peaks centred at the frequencies omega 0 and omega 0 +or- Omega 12 , where omega 0 is the atomic transition frequency and Omega 12 is the dipole-dipole interaction potential. The central peak is induced by the mode correlations, whereas the sidebands are induced by the driving field and show a strong dependence on the relative phase between the squeezed vacuum and the driving field. The population in the intermediate level shows a distinct peak at the frequency omega 0 + Omega 12 , and holes at omega 0 and omega 0 - Omega 12 .
The resonance fluorescence system of a coherently driven two-level atom in the presence of a broadband squeezed vacuum field whose central frequency is not in resonance with either the driving field frequency or the atomic transition frequency is analyzed. Solutions of the Bloch equations of the system are presented in terms of continued fractions for arbitrary strength of the coherent driving field. Numerical results are presented for the first-harmonic quadrature components of the absorption spectrum and the fluorescent intensity. The effects of the squeezed vacuum field detuning on the absorption spectrum result in hole burning or a dip structure in the weak-field case and two absorption–amplification peaks in the strong-field case. Results are sensitive to the relative phase of the squeezed vacuum.
Master equation methods are used to investigate the effects of a broad-band squeezed vacuum on a three-level atom of the lambda configuration. The two-mode squeezed vacuum is treated as a Markovian reservoir in a non-stationary phase-dependent state. In addition to the squeezed vacuum the atom is driven by two coherent laser fields each of which, depending on the polarization, can couple to one or both of the atomic transitions. We show that in general the optical Bloch equations for the atomic density matrix elements have oscillatory coefficients, thereby necessitating the use of Floquet methods. For the case of equal laser frequencies, which are also equal to the carrier frequency of the squeezed vacuum, the coefficients of the Bloch equations become time independent and stationary solutions for the populations and coherences are obtained by standard matrix methods. For the ordinary vacuum the usual coherent population trapping effect at two-photon resonance is obtained, with the upper state population being zero. An unsqueezed thermal field partially destroys the trapping effect as the upper state population is no longer zero at two-photon resonance. The squeezed vacuum has the effect of improving the trapping in that the coherence hole becomes more pronounced for some values of the relative phase between the squeezed vacuum and the driving fields. The additional effects of a coherence transfer rate between the two optical coherences, which occurs for special choices of angular momentum quantum numbers are also studied. For the case of equal laser frequencies, the inclusion of this coherence transfer process destroys population trapping and reduces the lambda system to a two-level system. However, for the case of unequal laser frequencies, the coherence transfer process in combination with the squeezed vacuum can restore to some extent the population trapping. We show that other features that do not occur for two-level atoms, such as stationary population inversions between pairs of the atomic levels, also depend on the relative phase and can be enhanced in the squeezed vacuum. In the case of unequal frequencies of the driving fields the population in the upper state depends on the relative phase only when the carrier frequency of the squeezed vacuum is equal to one of the two frequencies of the driving fields. When the carrier frequency of the squeezed vacuum is slightly detuned from both frequencies of the driving fields, the population in the upper state is insensitive to the relative phase but is dependent on the degree of squeezing. For large detunings, the population does not show any dependence on the degree of squeezing and its distribution in function of the two-photon detuning is similar to that in the thermal vacuum field.
A comprehensive study is made of the absorption spectrum of a weak-probe beam irradiating a two-level atom damped by a broadband squeezed vacuum and driven by a strong laser field. We show that for a resonant driving field whose phase does not coincide with the principal directions of the squeezing ellipse, the absorption spectrum is asymmetric and exhibits absorption-emission profiles at the Rabi sidebands. We find that this feature arises from unequal populations of the dressed states of the system, which occur when all components of the Bloch vector are coupled to the driving field. A symmetric spectrum with dispersionlike profiles at the Rabi sidebands is revealed for an off-resonant driving field. Furthermore, the spectrum may exhibit an emission profile at the central frequency, signifying an amplification of the probe beam. We point out that the amplification at the central component is not attributed to population inversion in both the atomic bare states and in the dressed atomic states. We find that the amplification originates from the coherent population oscillations and is impossible in the absence of the squeezed vacuum. In addition, we examine how these features are affected by the detuning of the driving field and an imperfect squeezed vacuum.
The authors present analytical and numerical results for resonance fluorescence produced by a two-level atom damped by a broad-band squeezed vacuum. Previous theoretical analyses of this model have assumed that the laser frequency and the carrier frequency of the squeezed vacuum are equal. The authors introduce a non-zero detuning between the laser frequency and the carrier frequency while maintaining resonance between the laser frequency and atomic resonance. In the absence of the driving field there is a squeezing threshold: above threshold the decay rate of the atomic polarization can be reduced, and below threshold a squeezing-induced two peak spectrum and frequency shift, which is analogous to the Bloch-Siegert shift, are predicted. In the presence of the driving field, the spectrum is asymmetric for Rabi frequencies smaller or comparable to the detuning of the squeezed vacuum. A symmetric spectrum is obtained for large Rabi frequencies. An interpretation is predicted by means of a dressed-state analysis. The authors find that for not too large Rabi frequencies a remarkably large population inversion can be induced in the dressed states, even at exact resonance of the driving field, provided that the laser frequency and carrier frequency are detuned. Moreover, for detection times larger than the inverse detuning, the spectrum is insensitive to the phase between the driving field and the squeezed vacuum, in marked contrast to the case of no detuning.
Using the coupled-system approach we calculate the optical spectra of the fluorescence and transmitted fields of a two-level atom driven by a squeezed vacuum of bandwidths smaller than the natural atomic linewidth. We find that in this regime of squeezing bandwidths the spectra exhibit unique features, such as a hole burning and a three-peak structure, which do not appear for a broadband excitation. We show that the features are unique to the quantum nature of the driving squeezed vacuum field and do not appear when the atom is driven by a classically squeezed field. We find that a quantum squeezed-vacuum field produces squeezing in the emitted fluorescence field which appears only in the squeezing spectrum while there is no squeezing in the total field. We also discuss a nonresonant excitation and find that depending on the squeezing bandwidth there is a peak or a hole in the spectrum at a frequency corresponding to a three-wave-mixing process. The hole appears only for a broadband excitation and results from the strong correlations between squeezed-vacuum photons.
The influence of the dipole-dipole interaction on the final atomic state of a two-atom system driven by a coherent laser field and damped by a broad-band squeezed vacuum is discussed. It is shown that in the absence of a coherent laser field the final atomic state is independent of the dipole-dipole interaction and can be the pure two-atom squeezed state only if detuning of the squeezed vacuum field from atomic resonance is equal to zero. In the presence of a coherent laser field, the final atomic state depends on the dipole-dipole interaction. As the latter increases, the final atomic state shifts from a mixed state to the pure two-atom squeezed state. However, this happens only if the driving field and the squeezed field frequencies are tuned to the atomic transition frequency. For the driving field and the squeezed vacuum field frequencies detuned from atomic resonance the system decays into a mixed state irrespective of the dipole-dipole interaction between the atoms.