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Journal of Modern Optics

Generalized Bloch-Siegert Shift in a Squeezed Vacuum

Z. Ficek

Published Dec 1, 1993• 8 citations
Publisher

Abstract

The time evolution of the atomic dipole moment of a two-level atom damped by an off-resonance squeezed vacuum is studied. Our results are valid for arbitrary detuning between the carrier frequency of the squeezed vacuum and the atomic transition frequency and show that the atomic dipole moment can oscillate with two dominant frequencies. These frequencies are shifted from their central values by an amount which is proportional to the degree of squeezing. We show that this shift of the frequencies is the exact equivalent of the optical Bloch-Siegert shift in the case of the ordinary vacuum damping without the rotating-wave approximation.

DOI
10.1080/09500349314552361

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The resonance fluorescence of a two-level atom driven by a coherent laser field and damped by a finite bandwidth squeezed vacuum is analysed. We extend the Yeoman and Barnett technique to a non-zero detuning of the driving field from the atomic resonance and discuss the role of squeezing bandwidth and the detuning in the level shifts, widths and intensities of the spectral lines. The approach is valid for arbitrary values of the Rabi frequency and detuning but for the squeezing bandwidths larger than the natural line-width in order to satisfy the Markoff approximation. The narrowing of the spectral lines is interpreted in terms of the quadrature-noise spectrum. We find that, depending on the Rabi frequency, detuning and the squeezing phase, different factors contribute to the line narrowing. For a strong resonant driving field there is no squeezing in the emitted field and the fluorescence spectrum exactly reveals the noise spectrum. In this case the narrowing of the spectral lines arises from the noise reduction in the input squeezed vacuum. For a weak or detuned driving field the fluorescence exhibits a large squeezing and, as a consequence, the spectral lines have narrowed linewidths. Moreover, the fluorescence spectrum can be asymmetric about the central frequency despite the symmetrical distribution of the noise. The asymmetry arises from the absorption of photons by the squeezed vacuum which reduces the spontaneous emission. For an appropriate choice of the detuning some of the spectral lines can vanish despite that there is no population trapping. Again this process can be interpreted as arising from the absorption of photons by the squeezed vacuum. When the absorption is large it may compensate the spontaneous emission resulting in the vanishing of the fluorescence lines.

Journal of Modern Optics•Sep 1, 1998•22 citations
The time evolution of the atomic dipole moment of a two-level atom damped by an off-resonance squeezed vacuum is studied. Our results are valid for arbitrary detuning between the carrier frequency of the squeezed vacuum and the atomic transition frequency and show that the atomic dipole moment can oscillate with two dominant frequencies. These frequencies are shifted from their central values by an amount which is proportional to the degree of squeezing. We show that this shift of the frequencies is the exact equivalent of the optical Bloch-Siegert shift in the case of the ordinary vacuum damping without the rotating-wave approximation.
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.
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.
The master equation for a two-level atom interacting with a strong coherent field and damped into a reservoir formed by a finite bandwidth squeezed vacuum is derived. The master equation extends the Yeoman and Barnett approach to a non-zero detuning of the driving field from the atomic resonance and allows to discuss the role of squeezing bandwidth and the detuning in the level shifts, widths and intensities of spectral lines. The approach is valid for arbitrary values of the Rabi frequency and detuning but for the squeezing bandwidths larger than the natural linewidth in order to satisfy the Markov approximation.
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.
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.
The resonance fluorescence of a two-level atom driven by a coherent laser field and damped by a finite bandwidth squeezed vacuum is analysed. We extend the Yeoman and Barnett technique to a non-zero detuning of the driving field from the atomic resonance and discuss the role of squeezing bandwidth and the detuning in the level shifts, widths and intensities of the spectral lines. The approach is valid for arbitrary values of the Rabi frequency and detuning but for the squeezing bandwidths larger than the natural line-width in order to satisfy the Markoff approximation. The narrowing of the spectral lines is interpreted in terms of the quadrature-noise spectrum. We find that, depending on the Rabi frequency, detuning and the squeezing phase, different factors contribute to the line narrowing. For a strong resonant driving field there is no squeezing in the emitted field and the fluorescence spectrum exactly reveals the noise spectrum. In this case the narrowing of the spectral lines arises from the noise reduction in the input squeezed vacuum. For a weak or detuned driving field the fluorescence exhibits a large squeezing and, as a consequence, the spectral lines have narrowed linewidths. Moreover, the fluorescence spectrum can be asymmetric about the central frequency despite the symmetrical distribution of the noise. The asymmetry arises from the absorption of photons by the squeezed vacuum which reduces the spontaneous emission. For an appropriate choice of the detuning some of the spectral lines can vanish despite that there is no population trapping. Again this process can be interpreted as arising from the absorption of photons by the squeezed vacuum. When the absorption is large it may compensate the spontaneous emission resulting in the vanishing of the fluorescence lines.