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Journal of the Optical Society of America B

SU(2) squeezing in the optical Bloch equations

P. K. Aravind

Published Nov 1, 1987• 2 citations
Publisher

Abstract

It is pointed out that the steady-state solutions of the optical Bloch equations can exhibit SU(2) squeezing for suitable ratios of the transverse and longitudinal relaxation times. This augments my recent observation [ J. Opt. Soc. Am. B3, 1712 ( 1986)] of squeezing in the transient solutions and also establishes more closely the parallel with atomic squeezing in resonance fluorescence.

DOI
10.1364/josab.4.001847

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Le Journal de Physique Colloques•Jun 1, 1988•0 citations
It is pointed out that the steady-state solutions of the optical Bloch equations can exhibit SU(2) squeezing for suitable ratios of the transverse and longitudinal relaxation times. This augments my recent observation [ J. Opt. Soc. Am. B3, 1712 ( 1986)] of squeezing in the transient solutions and also establishes more closely the parallel with atomic squeezing in resonance fluorescence.
Resonance fluorescence from a two-level atom is first shown to permit squeezing for an arbitrary intensity of the exciting field provided that the resonance is time dependent. When the exciting field increases in intensity, the squeezing shifts toward shorter times and its maximum value becomes double, for strong exciting fields, its maximum value in the steady state. Moreover, it is shown that intervals of time exist in which the normally ordered variance of the fluorescence field is positive, despite the presence of squeezing in the atomic variables. Also, the signal-to-noise ratio in the transient regime attains high values, predictably making the observation of squeezed states experimentally feasible.
The atomic pseudospin operators describing a two-level atom, and obeying the SU(2) Lie algebra, can exhibit reduced quantum fluctuations (“squeezing”) in certain states of the atom. These states include both pure and mixed states of the atom. Parameterizing the atom by the spherical polar components S, θ, and ϕ of its pseudospin vector S, we determine the parameter ranges in which squeezing can occur and the magnitude of this squeezing. We illustrate our considerations by examining two models, namely, the coherent-state Jaynes–Cummings model and the optical Bloch equations, in which mixed-state SU(2) squeezing arises dynamically.
The nonclassical nature of the so-called squeezed states of electromagnetic fields has recently been discussed in connection with the phenomenon of resonance fluorescence from a coherently driven two-level atom.1 It has been shown that either the in-phase component E1 or the out-of-phase component E2 of the fluorescent field can become squeezed if the Rabi frequency Ω of the ex- citing field is sufficiently low. Squeezing is defined as a negative value of the normally ordered variance 〈: (ΔE1)2:〉 (or 〈:(ΔE2)2:〈). To satisfy this condition in the steady state, if the exciting field is tuned perfectly to the atomic transition, the Rabi frequency Ω has to be less than √2y, with 2y = A—the Einstein coefficient for spontaneous emission. Off-resonance excitation further lowers this critical value of 0. Walls and Zoller1 have indicated, without giving the details, that, in a transient regime, the squeezing effect can become twice as large as in the steady state.
We identify discrepancies between two published predictions of the resonance fluorescence from a two-level atom driven by a strong bichromatic field [J. Opt. Soc. Am. B 8,8,8, 1163 (1991) and Phys. Rev. A 48,48,48, 3092 (1993)]. These discrepancies are shown to arise from subtleties in extracting the time-averaged resonance fluorescence from the expansions used to solve the optical Bloch equations. We construct a solution to these equations that corrects the results of the latter reference and reconciles the two predictions.
Although the resonance fluorescence spectrum of coherent light scattering from a two-level atom has been discussed in many papers, the effects of longitudinal, transverse relaxation and optical pumping have not yet been considered simultaneously in the motion equations. By solving the optical Bloch equations including the longitudinal relaxation time T 1 , transverse relaxation time T 2 and optical pumping parameter Δ 0 , in the case T 2 /T 1 <<1, we obtain the spectrum with dip in the central peak besides the familiar spectrum of three peaks. In addition, we find that the parameter Δ 0 and detuning Δω have influence on the symmetry, height and width of the spectrum of the three peaks.