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Physical Review Letters

Intensity Correlations of a Cooperative System

G. S. Agarwal, Lorenzo M. Narducci, D.H. Feng, Robert Gilmore

Published May 7, 1979• 104 citations
Publisher

Abstract

We suggest that the intensity correlation function could be an important tool in the study of atomic cooperative behavior because it contains predominantly components at 0, $\ifmmode\pm\else\textpm\fi{}4\ensuremath{\Omega}$ in the limit of large Rabi frequency ($2\ensuremath{\Omega}$) and large cooperation number. This is in contrast with the single-atom prediction where the intensity correlation function contains only frequency components at 0, $\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}$. The master equation for the collective system is solved analytically in the secular approximation.

DOI
10.1103/physrevlett.42.1260

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We suggest that the intensity correlation function could be an important tool in the study of atomic cooperative behavior because it contains predominantly components at 0, \ifmmode±\else\textpm\fi4\ensuremathΩ\ifmmode\pm\else\textpm\fi{}4\ensuremath{\Omega}\ifmmode±\else\textpm\fi4\ensuremathΩ in the limit of large Rabi frequency (2\ensuremathΩ2\ensuremath{\Omega}2\ensuremathΩ) and large cooperation number. This is in contrast with the single-atom prediction where the intensity correlation function contains only frequency components at 0, \ifmmode±\else\textpm\fi2\ensuremathΩ\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Omega}\ifmmode±\else\textpm\fi2\ensuremathΩ. The master equation for the collective system is solved analytically in the secular approximation.
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.
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).
The spectrum of single-atom fluctuations in cooperative situations such as in the context of optical bistability and tristability is calculated by applying the method of system size expansion. The correlation functions of the fluctuating forces in quantum Langevin equations for any pair of atoms are found to be of order 1N\frac{1}{N}N1​. The relaxation matrix for single-atom fluctuations is shown to have the same structure as in the noncooperative situation but with an effective field determined by the state equation for multistability.
The eigenstates of the Jaynes-Cummings Hamiltonian (the dressed atomic states) are widely used to describe the interaction between a single two-level atom and a single mode of the electromagnetic field. They provide a particularly useful basis when the secular approximation is justified. For an atom interacting fog with n photons in a cavity, the secular approximation requires that the n-photon Rabi frequency 2 n g be much larger than the atomic and cavity linewidths. Normally this condition is met when n is large and the coupling constant g is much smaller than the atomic and cavity linewidths. In this paper we allow g to be larger than the linewidths, and we allow the photon number to be small. We show that when the cavity is driven by a coherent field, the usual secular approximation cannot be used. The natural basis states become are now the eigenstates of the Jaynes-Cummings Hamiltonian plus the external field interaction. We solve this eigenvalue problem by separating the field and atomic parts of the Hamiltonian using a displacement and squeeze in the Hilbert-space of the field. We find that the Rabi frequencies are renormalized downwards as the strength of the driving field is increased. They eventually vanish at a critical field strength. At this critical field strength we observe a spontaneous symmetry breaking in the steady-state solution of the master equation for a single atom interacting with a driven cavity mode.