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Annual Meeting Optical Society of America

Photon antibunching in optical bistability

Robert J. Brecha, Min Xiao, Harry J. Kimble

Published Jan 1, 1988• 0 citations
Publisher

Abstract

While the problem of photon antibunching in optical bistability has been the subject of numerous theoretical investigations, there appears to be no general result in the literature that includes a complete specification of the dependence of antibunching on the system parameters. Motivated by the importance of such a result to experimental observation of this effect, we present a treatment that includes the parametric dependences on cavity and atomic detunings, cooperativity parameter, nonradiative (collisional) damping, and the ratio of cavity and atomic damping rates. We also attempt to analyze the contribution of fourth-order correlations of the fluctuating fields, which have been neglected in previous treatments. Although the antibunching initially increases with decreasing system size due to the approximate scaling of quantum fluctuations as the inverse of the saturation photon number ns, higher-order fluctuations begin playing an increasingly important role at values of ns ⪝ 1. We show that the linearized theory itself does not necessarily break down for arbitrarily small values of ns (as previously suggested), but that photon antibunching is lost in this limit, thus placing restrictions on the design of an experiment.

DOI
10.1364/oam.1988.mm4

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While the problem of photon antibunching in optical bistability has been the subject of numerous theoretical investigations, there appears to be no general result in the literature that includes a complete specification of the dependence of antibunching on the system parameters. Motivated by the importance of such a result to experimental observation of this effect, we present a treatment that includes the parametric dependences on cavity and atomic detunings, cooperativity parameter, nonradiative (collisional) damping, and the ratio of cavity and atomic damping rates. We also attempt to analyze the contribution of fourth-order correlations of the fluctuating fields, which have been neglected in previous treatments. Although the antibunching initially increases with decreasing system size due to the approximate scaling of quantum fluctuations as the inverse of the saturation photon number ns, higher-order fluctuations begin playing an increasingly important role at values of ns ⪝ 1. We show that the linearized theory itself does not necessarily break down for arbitrarily small values of ns (as previously suggested), but that photon antibunching is lost in this limit, thus placing restrictions on the design of an experiment.
Photon antibunching has been predicted at weak intensities along the lower branch in optical bistability. Existing theories assume large atomic numbers to justify their linearized analysis. From such a theory photon antibunching in absorptive bistability is given for weak intensities by where C is the bistability parameter, μ is the ratio of atomic and cavity decay times, and N is the number of atoms. Clearly this result becomes invalid as the system size is reduced to the limit of a single atom; then g(2) (0) is negative for moderate values of C. I calculate g(2) (0) for the transmitted light from an optical cavity containing a single atom, where the injected field, the atom, and the cavity are all resonant.
Photon antibunching in intracavity second-harmonic generation and intracavity atomic absorption inside a high-Q optical cavity is discussed. A linearized description of quantum dynamics in the weak-field limit is used because the system size, as characterized by the number of photons n0{\mathit{n}}_{0}n0​ needed to probe the nonlinearity of the system, is often quite large for realistic parameters. In both cases standard results are recalled that demonstrate that antibunching is a small effect given by g(2){\mathit{g}}^{(2)}g(2)(0)-1\ensuremath{\approxeq}-1/n0{\mathit{n}}_{0}n0​ for large values of n0{\mathit{n}}_{0}n0​. We show that when higher-order terms that are usually ignored in linearized treatments are retained, g(2){\mathit{g}}^{(2)}g(2)(0) satisfies the lower bound g(2){\mathit{g}}^{(2)}g(2)(0)\ensuremath{\ge}0 even for small values of n0{\mathit{n}}_{0}n0​. It is further shown that by employing a high-Q cavity to suppress the coherent part of the spectrum of field fluctuations, perfect antibunching can be achieved even for large systems. Conditions under which this is possible are derived and curves are presented to illustrate the behavior.
We specialize the quantum-statistical theory of optical bistability developed in "Optical Bistability. II" to a low-QQQ cavity. From a linearized Fokker-Planck equation derived in the positive PPP representation, the steady-state correlation matrix and spectrum of fluctuations are calculated. Expressions which normally assume a positive-definite diffusion matrix are shown to be valid for nonpositive-definite diffusion. For absorptive bistability we find agreement with results derived previously by quite different methods. More generally we extend previously published results by including atomic and cavity detunings. We compare the transmitted spectrum for dispersive bistability with that for absorptive bistability, and photon antibunching in dispersive bistability is discussed.
The quantum statistics of bistable systems have been studied since the beginnings of optical bistability theory. Predicted quantum-statistical effects include quantum induced transitions between states, evidences of atomic collectivity in the transmitted spectrum, photon antibunching, and squeezing.(1) The enthusiasm shown in theoretical analyses of these phenomena has not been matched by experiments, however; one reason being that the effects are very small, generally scaling with the size of the system (in this squeezing is an exception) as measured by saturation photon number nS or number of atoms N. The recent experiments of Rosenberger et al.,(2) which closely meet the idealized conditions called for by theory have ns ~ 103 and N ~ 105. While these numbers are too large, these experiments suggest that it might be feasible to measure quantum-statistical effects in a new generation of smaller systems. This paper addresses theoretical questions raised by a move to small systems. Existing theories are themselves restricted by assumptions of large system size, invoked to justify their various truncation, factorization and linearization procedures.
The radiative properties of atoms inside an optical cavity can be dramatically changed with respect to those in free space. In the strong-coupling limit, the reversible dipole interaction between an atom and the cavity field dominates the irreversible decay resulting from spontaneous emission and cavity damping. In this limit the atom and field can no longer be regarded as separate quantities but instead form a composite dynamic system. To enhance the dipole coupling constant and increase the photon storage time, our experiments are performed with a small cavity 1 mm long, with a finesse of 8 × 104. On average, 0.4 photons are sufficient to saturate the resonance transition at 852 nm for the cesium atoms used, while the cooperativity parameter for a single atom is 2. Optical bistability is studied in this system over the range 3 < N < 65 atoms inside the cavity, with the semiclassical theory predicting bistability for N > 5. We report observations of manifestly quantum effects in this regime, including photon antibunching and sub-Poissonian photon statistics, with the magnitude of these nonclassical effects observed to be largely independent of the number of intracavity atoms.
We study the spectrum of squeezing in the light transmitted by a system of two-level atoms in an optical cavity driven by a coherent field, as recently observed by Kimble and collaborators. Our analysis does not introduce any restriction on the detuning parameters or on the relative order of magnitude of the atomic and cavity relaxation rates. Using the dressed-mode formalism, we derive explicit analytic expressions for the spectrum of squeezing and for the second-order intensity correlation function. We describe how the spectrum depends on the various parameters of the system; this study provides general indications on how to optimize the conditions to observe squeezing in optical bistability. A special attention is devoted to the changes of the spectrum under variation of the detuning parameters or on the ratio of cavity to atomic damping rate. In all cases, the changes are continuous and smooth, with quantitative variations in the level of squeezing. This indicates the existence of a single avenue to squeezing in this system. The bad-cavity direction is identified as especially convenient to obtain a significant amount of squeezing.