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Physical Review A

Multiple sidebands in cooperative resonance fluorescence: Exact semiclassical results

P. D. Drummond, Saad S. M. Hassan

Published Aug 1, 1980• 37 citations
Publisher

Abstract

We report on the semiclassical solution of the Bloch equations for cooperative resonance fluorescence with collective damping. These equations are shown to be equivalent to the relevant master equation, in the limit of large cooperation number and time scales faster than the one-atom lifetime (reciprocal Einstein $A$ coefficient). The resulting spectrum is derived for a coherent driving field of arbitrary intensity. Additional spectral peaks, as well as the central triplet, are predicted above threshold. A physical explanation of the extra peaks is proposed.

DOI
10.1103/physreva.22.662

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Collective Atomic Dynamics in Resonance Fluorescence

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Journal of Experimental and Theoretical Physics Letters•Jun 1, 2020•5 citations
We report on the semiclassical solution of the Bloch equations for cooperative resonance fluorescence with collective damping. These equations are shown to be equivalent to the relevant master equation, in the limit of large cooperation number and time scales faster than the one-atom lifetime (reciprocal Einstein AAA coefficient). The resulting spectrum is derived for a coherent driving field of arbitrary intensity. Additional spectral peaks, as well as the central triplet, are predicted above threshold. A physical explanation of the extra peaks is proposed.
We discuss a mean-field quantum-mechanical model which describes the dynamics of a homogeneously broadened system of two-level atoms contained in a pencil-shaped resonant cavity and driven by a coherent resonant field EI{E}_{I}EI​. The model is treated in the semiclassical approximation. The model is justified on the basis of Maxwell-Bloch equations with two coherently coupled directions of propagation, with boundary conditions for the fields taken into account. Above a suitable critical density of atoms the system exhibits a bistable behavior including both the stationary situation and the transient, both the light transmitted in the forward direction and the fluorescent light. Bistability is shown to be a consequence of atomic cooperation. We give a simple description of optical bistability leading to new predictions for the transient behavior of the transmitted light and for the spectrum of the fluorescent light. The damping constant which characterizes the rate of appraoch to the stationary situation exhibits a hysteresis cycle. In the low-transmission regime the approach is monotonic, whereas in the highly transmitting situation the approach is oscillatory. One finds a critical slowing down in correspondence with the values EI(+){{E}_{I}}^{(+)}EI​(+), ${{E}_{I}}^{(\ensuremath{-})}$ of the incident field where the transmitted field (as well as the total fluorescence intensity, the rate of approach to the stationary state, etc.) changes discontinuously. On the basis of the regression hypothesis we give a qualitative description of the spectrum of the fluorescent light. This is shown to undergo hysteresis and discontinuous changes at the same values EI(+){E}_{I}^{(+)}EI(+)​, ${E}_{I}^{(\ensuremath{-})}$ of the incident field. Below the critical density of atoms one recovers the usual picture of resonance fluorescence, with a continuous transition from a single-line spectrum to a three-peaked structure when the Rabi frequency becomes equal to the natural linewidth. Above the critical density the triplet appears only when the Rabi frequency becomes equal to the cooperative linewidth of pure super-fluorescence (i.e., for EI=EI(+){E}_{I}={E}_{I}^{(+)}EI​=EI(+)​). Moreover, it appears discontinuously: when EI{E}_{I}EI​ crosses the value EI(+){E}_{I}^{(+)}EI(+)​ the spectrum changes from a single narrow line to a triplet with well-separated sidebands.
For pt.I see ibid., vol.14, p.4171 (1981). Semiclassical and quantal time-dependent results are presented for a model of N identical two-level atoms occupying the same site (the Dicke model) and driven by a CW off-resonance laser field. The exact semiclassical time-dependent solutions are derived from an analysis of the Fokker-Planck equation for the system in the atomic coherent states representation. Steady-state behaviour predicted by these solutions is in agreement with that obtained from the exact quantal steady-state analysis in the thermodynamic limit N to infinity . The semiclassical theory predicts (in the steady state) one type of fluorescent spectrum, which is a single sharp line ( delta function) centered at the driving field frequency. This is like the case at exact resonance and below threshold. Quantum mechanically, analytical results are obtained in the strong-field limit and within the secular approximation of the master equation: the fluorescence spectrum is the usual dynamical Stark triplet, apart from a cooperative factor N 2 . The absorption spectrum and the second-order intensity autocorrelation function are also calculated. Numerical results are also presented for finite N(N<or approximately=5). The effect of detuning on the additional sidebands (occurring at higher harmonics) in both the fluorescent and adsorption spectrum are examined.
Cooperative effects on squeezing in resonance fluorescence from a finite number N of atoms are examined. Analytical expressions for the spectrum of squeezing and a quantity Q(T) related to the photon-counting statistics of the squeezed states are presented for N=1 and 2. These expressions are obtained by invoking the high-field approximation of the relevant master equation. The analytical results are supplemented by the numerical solutions of the exact master equation for N\ensuremath{\le}10. It is shown that for intense fields or large detunings, squeezing in resonance fluorescence is predominantly due to the quantum nature of the sidebands of the Mollow triplet. As a manifestation of cooperative effects, squeezing present in the total as well as the sideband radiation increases with the number of atoms. This is further confirmed from the behavior of the photon-statistics function Q(T).
We study cooperative properties of the emission spectrum of A two-level atoms interacting with a quantized field in a lossless cavity. We show, in a unified treatment, the system anharmonicity, the fine structure, and size of the spectral bands. Further, we show that for an initially inverted atomic system and a thermal cavity field the atomic cooperativity allows the merging of well-defined sidebands, absent in the single-atom case. Thus the cooperativity reduces the destructive effect of the large dispersion of the blackbody photon distribution. For an increasing number of atoms, the central band is split into a doublet. The dip in the doublet is filled when the cavity field is very strong. Also, the extra sidebands have smaller amplitudes than the time-dependent contributions to the spectrum. \textcopyright{} 1996 The American Physical Society.
The analytical solution for the spectrum of resonance fluorescence from a collective system of two atoms is presented. It is shown that the spectrum is a decreasing function of frequency in the neighborhood $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{L}=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$ ($\ensuremath{\Omega}$ is the Rabi frequency) for moderately large $\ensuremath{\Omega}$ and exhibits resonant structure only for extremely large $\ensuremath{\Omega}\ensuremath{\approx}100\ensuremath{\gamma}$. For moderately large $\ensuremath{\Omega}$ the resonant contributions at $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{L}=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$ are destroyed by the nonresonant contributions; hence the additional structures at $\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$ can be resolved only by filtering the nonresonant contributions.