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Squeezed States in Transient Regime of Resonance Fluorescence

Z. Ficek, R. Tanaś, S. Kielich

Published Jun 18, 1984• 0 citations
OpenAlex

Abstract

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.

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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.

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Asymmetric two-level two-level resonance fluorescence spectra and dressed state populations in a squeezed vacuum

S Smart, S. Swain

The authors calculate the resonance fluorescence spectrum of a two-level atom interacting with a resonant classical applied field and a squeezed vacuum for general values of the squeezing phase. They find that the spectra are asymmetric under the following conditions: (i) the Rabi frequency must not be too large and (ii) the squeezing phase should be of athe order pi /2. Large asymmetries are obtainable, even if the three peaks of the spectrum are clearly resolved. An interpretation is provided by means of a dressed state analysis. Under conditions (i) and (ii), they also find that very large differences in the dressed state populations can occur.

Quantum Optics Journal of the European Optical Society Part B•Apr 1, 1993•27 citations
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.
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.
A theoretical study is made of the steady-state intensity and squeezing properties of the fluorescent light from a three-level-atom system in a ladder configuration, which is subject to spontaneous-emission decay to the electromagnetic-field vacuum. Two cases are examined: in the equispaced (ES) level case, the two atomic transition frequencies are nearly equal and two photon absorption processes occur from a single coherent laser field that couples both transitions and has a small detuning from each; in the non- equispaced (NES) level case, the atomic transition frequencies are rather different and two photon absorption processes occur from two coherent laser fields, each coupled to a single transition and in near resonance with it. In both cases, a situation of small two-photon detuning occurs. Optical Bloch equations for the atomic density matrix in rotating frames are given and matrix expressions for determining the steady-state populations and coherences are obtained. Analytic expressions are also given for special cases. The fluorescent intensity is obtained from the populations of the intermediate and upper states, with the normally ordered variance (NOV) for quadrature components of the fluorescent field involving, in addition, the atomic coherences. The fluorescent intensities are shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases.In both cases, the fluorescent intensities show the well known resonances from the upper state at zero two-photon detuning and from the intermediate state at zero one-photon detuning for the lower transition. However, further resonances are also found. The intermediate-state intensity displays a resonance at zero two-photon detuning. Also, the upper-state intensity shows a resonance at zero one-photon detuning for the upper transition, but only for the NES case; evidently, the transfer rate from upper to lower transition coherences destroys this resonance for the ES case. The time-averaged NOV is also shown graphically as a function of two-photon detuning for a variety of one-photon detunings and Rabi frequencies for the ES and NES cases. For the ES case, the quadrature frequency is chosen as the single laser frequency and for the NES case, it is chosen as either the average of the two laser frequencies or the lower transition laser frequency. Squeezing occurs near zero two-photon detuning for both ES and NES cases, though only for the quadrature frequency equal to the average laser frequency in the latter case. It also occurs near zero detuning for the lower transition for both cases, though only for the quadrature frequency equal to the lower transition laser frequency in the NES case. The squeezing minima show a splitting effect for large Rabi frequencies corresponding to two-photon or one-photon Rabi splitting of the dressed atom levels for the situation near two-photon or one-photon resonance, respectively. The large squeezing near two-photon resonance for moderate Rabi frequencies and large one-photon detuning corresponds to essentially pure three-level squeezing, while that near one-photon resonance for weak Rabi frequencies involves two-level squeezing.
The quadrature squeezing spectra produced in the resonance fluorescence of a two-level atom are investigated, including the effects of a finite laser bandwidth due to phase diffusion. For resonant excitation at low intensities, the fluorescence field exhibits narrow bandwidth squeezing in the out-of-phase quadrature, centered at the laser frequency. Otherwise, squeezing occurs only in the in-phase quadrature. Specifically, for slightly off-resonance and weak excitation, the squeezing still centers at the laser frequency, with a finite bandwidth of the order of $\ensuremath{\gamma}.$ More importantly, for far-off-resonance and strong excitation, the resonance fluorescence exhibits two-mode squeezing around the Rabi sideband frequencies. Thus we have a source of two-mode squeezing that is very easily frequency tuned simply by adjusting the generalized Rabi frequency. The presence of a laser linewidth substantially reduces the degree of squeezing obtainable. The fluorescence shows no squeezing at all when the laser linewidth is greater than the atomic spontaneous emission linewidth.
The quadrature squeezing spectra produced in the resonance fluorescence of a two-level atom driven by a strong coherent field and a weak amplitude-fluctuating field are investigated. For far-off-resonance and strong excitation, the resonance fluorescence exhibits two-mode squeezing around the Rabi sideband frequencies. Thus we have a source of two mode squeezing that is very easily frequency tuned simply by adjusting the generalized Rabi frequency. The presence of the strength of the stochastic process reduces the degree of squeezing obtainable. The quadrature squeezing spectra are also effected by the relative phase of the two fields.
Interaction of single-mode coherent light with dipole-forbidden two-level atoms via non-resonant virtual level is studied in this paper. It is shown that, in the case of two-photon resonance fluorescence , the atomic system generates three bands of correlated photon pairs with the carrier frequencies ω k 0 , ω k 0 ± Γ , where Γ=[Ω 2 +(Δ/2) 2 ] 1/2 , 2Ω is the two-photon Rabi frequency and Δ is the detuning from the two-photon resonance. It is established that amplitude-squared squeezing occurs from the sidebands of the two-photon resonance fluorescence spectrum . When the atom number increases, the amplitude-squared squeezing also increases.
The authors calculate the resonance fluorescence spectrum of a two-level atom interacting with a resonant classical applied field and a squeezed vacuum for general values of the squeezing phase. They find that the spectra are asymmetric under the following conditions: (i) the Rabi frequency must not be too large and (ii) the squeezing phase should be of athe order pi /2. Large asymmetries are obtainable, even if the three peaks of the spectrum are clearly resolved. An interpretation is provided by means of a dressed state analysis. Under conditions (i) and (ii), they also find that very large differences in the dressed state populations can occur.