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Journal de physique

Squeezed states in N-atom time-dependent resonance fluorescence

Z. Ficek, R. Tanaś, S. Kielich

Published Jan 1, 1987• 10 citations
Publisher

Abstract

The dependence of squeezing on the number of atoms is discussed in the time-dependent collective resonance fluorescence of N two-level atoms, coherently driven by a strong resonant laser field. It is shown that, for strong driving field, squeezing appears only in the transient regime of resonance fluorescence. The location of squeezing in time is independent of the number of atoms and the initial atomic populations. It is found that despite the reduction in the amount of squeezing per single atom, squeezing increases in the whole resonance fluorescence field as the number of atoms increases, and attains its maximal value for moderate N. Moreover, the maximal value of squeezing is dependent on the intensity of the driving field. As the latter increases, the maximum of squeezing increases and shifts towards larger numbers of atoms. For a very large number of atoms, however, squeezing is altogether removed.

DOI
10.1051/jphys:0198700480100169700

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The dependence of squeezing on the number of atoms is discussed in the time-dependent collective resonance fluorescence of N two-level atoms, coherently driven by a strong resonant laser field. It is shown that, for strong driving field, squeezing appears only in the transient regime of resonance fluorescence. The location of squeezing in time is independent of the number of atoms and the initial atomic populations. It is found that despite the reduction in the amount of squeezing per single atom, squeezing increases in the whole resonance fluorescence field as the number of atoms increases, and attains its maximal value for moderate N. Moreover, the maximal value of squeezing is dependent on the intensity of the driving field. As the latter increases, the maximum of squeezing increases and shifts towards larger numbers of atoms. For a very large number of atoms, however, squeezing is altogether removed.
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 photon-counting statistics of squeezed states in collective time-dependent resonance fluorescence is calculated for the homodyne detection technique. The secular approximation is applied to obtain analytical expressions for the photon-counting distribution produced by homodyning the fluorescent field and coherent radiation. Squeezing is shown to appear only in the transient regime of resonance fluorescence. The time location of squeezing is independent of the number of atoms, N, but squeezing increases with increasing N. Moreover, squeezing occurs only for a judicious choice of the phase of the local oscillator and occurs over long counting times T. The time interval T is strongly dependent on the initial atomic population. If initially all atoms are in their excited states, squeezing appears with a delay twice as long as that for initially nonexcited atoms.
The squeezing properties of the fluorescence field emitted by a two-level atom driven by a coherent laser field in a squeezed vacuum are calculated. We show that in the region of the anomalous resonance fluorescence the emitted field exhibits squeezing that is much larger than that in the input squeezed vacuum. The squeezing spectrum attains a minimum value that corresponds to 75% squeezing. We also find that, in the total fluorescence field, squeezing attains an optimum achievable value in the fluorescence field emitted by a two-level atom. The optimum squeezing is associated with the collapse of the system into a pure state.
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 single two-level atom driven by a beam of squeezed light is studied in the weak field limit. We consider the situation where the atom is coupled to the ordinary vacuum and only a few field modes corresponding to the driving field are squeezed. The field produced by the degenerate optical parametric oscillator is used as the driving field. This driving field can be described in terms of two real independent Gaussian processes. Heisenberg equations of motion are solved in the steady-state, and analytic expressions for the fluorescent light intensity and the spectrum of fluorescent light are derived. We also consider photon statistics of fluorescent light. In particular, squeezing, antibunching, and sub-Poissonian statistics of fluorescent photons are discussed and analytic expressions for the quadrature variance and the two-time intensity correlation function are presented. Contrary to the case of coherent excitation the second-order intensity correlation function does not factorize.