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Physical Review A

Fluctuation theory in quantum-optical systems

L. A. Lugiato, Federico Casagrande, L. Pizzuto

Published Dec 1, 1982• 57 citations
Publisher

Abstract

We show that the most complete and accurate description of fluctuations in quantum-optical systems is that obtained using the generalized Wigner distribution for the macroscopic observables of the system. The smallness of the inverse of the saturation photon number entitles us to neglect the terms with derivatives of order higher than second order in the time-evolution equation for the quasiprobability distribution. This treatment allows us to also describe correctly nonclassical effects such as photon antibunching or "squeezing," which are maltreated or even destroyed if one neglects the atom-atom correlations. We derive and analyze the Fokker-Planck equation for the Wigner function in the case of the usual laser and compare the results with those following from other approaches. In the region very high above threshold we find a small antibunching effect, which was not discovered in previous treatments due to the neglect of atom-atom correlations.

DOI
10.1103/physreva.26.3438

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We show that the most complete and accurate description of fluctuations in quantum-optical systems is that obtained using the generalized Wigner distribution for the macroscopic observables of the system. The smallness of the inverse of the saturation photon number entitles us to neglect the terms with derivatives of order higher than second order in the time-evolution equation for the quasiprobability distribution. This treatment allows us to also describe correctly nonclassical effects such as photon antibunching or "squeezing," which are maltreated or even destroyed if one neglects the atom-atom correlations. We derive and analyze the Fokker-Planck equation for the Wigner function in the case of the usual laser and compare the results with those following from other approaches. In the region very high above threshold we find a small antibunching effect, which was not discovered in previous treatments due to the neglect of atom-atom correlations.
Quantum fluctuations for multiphoton transitions in driven optical systems are treated on the basis of a Fokker-Planck equation for the Wigner function. The equation is considered without adiabatic elimination of the atoms or the field. The linear stability is analyzed and time correlations are derived. A spectrum of transmitted light associated with vacuum Rabi splitting is obtained. The squeezing and antibunching effects in the atomic system and the field are discussed. This theory coincides with the adiabatic elimination theory for the case of both good- and bad-cavity limits.
An exact ccc-number generalized Fokker-Planck equation is derived for six noncommuting variables: the field variables, atomic populations, and atomic polarization of a laser model treated extensively by Lax. Langevin procedures yield the correct Fokker-Planck terms. Adiabatic eliminations are performed on the classical Markoff process given by the Fokker-Planck equation to compare with previous results. It is found that higher-order diffusion coefficients influence the second-order ones slightly during adiabatic elimination.
In Chapter 1 we consider the question of squeezing in a two-photon laser. A two-photon laser is modelled as N two-level atoms interacting via a two-photon transition with a single resonant cavity field mode. A Fokker-Planck equation containing all leading noise terms is derived. A linearised fluctuation analysis reveals that there is no squeezing in a two-photon laser with injected signal. A small amount of squeezing is present in two-photon optical bistability. In Chapter 2 a microscopic model of degenerate four wave mixing including the quantisation of the medium is presented. Time-dependent solutions show ideal squeezing to be obtained in the limits of large detuning, low saturation and zero loss. The effect of nonideal atomic parameters is analysed. Equations describing the spatial propagation of the field through the medium are also derived. In Chapters 3 and 4 we are concerned with quantum effects, in particular violation of Bell's Inequalities, present in optical systems. An optical field produced by an intracavity four wave mixing is shown to exhibit the nonclassical features of photon antibunching, squeezing and violation of Cauchy-Schwarz and Bell's Inequalities. These intrinsic quantum mechanical effects are shown to be associated with the nonexistence of a positive normalizable Glauber-Sudarshan P-function.
By employing the input-output theory of damped quantum systems developed by Gardiner and Collett,1 we discuss the propagation of nonclassical fields through optical systems containing frequency selective elements such as Fabry-Perot interferometers. Our results are of general applicability and are expressed in terms of the linearized drift and diffusion coefficients of the generalized Fokker-Planck equation describing the intracavity field of the nonlinear source. The two nonclassical effects of photon antibunching and squeezing are treated in detail with reference to the examples of intracavity harmonic conversion and optical bistability. Not surprisingly photon antibunching can be enhanced by the suppression of the coherent part of the spectrum of intensity fluctuations with a narrow bandwidth cavity. However the enhancement is ultimately limited and the antibunching lost altogether as the amplitude of the coherent carrier is reduced to a level comparable to that of the field fluctuations (which are set by the system size). Following the suggestion of Levenson et al., 2 we investigate the use of auxiliary cavities for the control of phase and amplitude in the propagation and detection of squeezed states of light. While for a small degree of squeezing such filter cavities operate in a straightforward fashion, a mixing of field quadratures can occur as the degree of squeezing increases leading to a loss of nonclassical behavior.
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.
Th author introduces an exact and simple technique to convert the atomic part of the master equation for the density operator of a quantum optical system into a partial differential equation for a quasiprobability distribution in an arbitrary representation. He focusses on the special cases of two-level atoms and normal and symmetrical orderings, corresponding respectively to the P-function and Wigner representations. As an example of his results he derives the equation for the Wigner function of single-mode optical bistability with two-level atoms, and shows how he recovers the well known Fokker-Planck approximation.