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Physical Review A

Homodyne photon statistics of the subthreshold degenerate parametric oscillator

Arlis Dodson, Reeta Vyas

Published Apr 1, 1993• 26 citations
Publisher

Abstract

Homodyne statistics when the light from a degenerate parametric oscillator (DPO) is mixed with coherent light from a local oscillator are discussed. By using dynamical models of these beams we discuss the photon sequences underlying the superposed beam in terms of photoelectric pulse sequences recorded by a detector. The field produced by the parametric oscillator is expressed in terms of two independent real Gaussian random variables. Using this property of the field variables, we derive the generating function for the photoelectron statistics analytically. From this generating function, expressions for the photoelectron-counting distribution, factorial moments, and the waiting-time distribution are derived. These quantities are directly measurable in photon-counting experiments. The results reported here are applicable to arbitrary strengths of the signal and the local-oscillator fields. We also show that, depending on the relative strength and relative phase between coherent light from the local oscillator and squeezed light from the DPO, the superposed light beam may exhibit many interesting nonclassical features. These and other effects are described and curves are presented to illustrate the behavior.

DOI
10.1103/physreva.47.3396

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Homodyne statistics when the light from a degenerate parametric oscillator (DPO) is mixed with coherent light from a local oscillator are discussed. By using dynamical models of these beams we discuss the photon sequences underlying the superposed beam in terms of photoelectric pulse sequences recorded by a detector. The field produced by the parametric oscillator is expressed in terms of two independent real Gaussian random variables. Using this property of the field variables, we derive the generating function for the photoelectron statistics analytically. From this generating function, expressions for the photoelectron-counting distribution, factorial moments, and the waiting-time distribution are derived. These quantities are directly measurable in photon-counting experiments. The results reported here are applicable to arbitrary strengths of the signal and the local-oscillator fields. We also show that, depending on the relative strength and relative phase between coherent light from the local oscillator and squeezed light from the DPO, the superposed light beam may exhibit many interesting nonclassical features. These and other effects are described and curves are presented to illustrate the behavior.
We discuss homodyne and heterodyne statistics when squeezed light from a parametric oscillator is mixed with coherent light from a local oscillator. We also consider the case where the fields or the intensities of the two nondegenerate beams from a parametric oscillator are combined. By using a dynamic model of these beams we discuss photon sequences underlying these beams in terms of photoelectric pulse sequences recorded by a detector. Analytic expressions for quantities directly measurable in photon counting experiments are derived, using the method developed in our earlier investigations.1
We discuss two- and three-fold photon statistics for the squeezed light generated by subthreshold degenerate (DPO) and nondegenerate (NDPO) parametric oscillators. In higher-order statistics details of coherence properties are more strongly reflected than in the single-fold statistics. Many experiments use two-fold statistics to extract spectral information about the incident beam. We use phase-space representation to express the operator field variables in terms of independent Gaussian random processes.1 Using the Gaussian property of these field variables and making a Karhunen-Loeve expansion in terms of orthogonal functions, analytical expressions for the two- and three-fold generating functions for the DPO and the NDPO are obtained. From this generating function various statistical quantities such as factorial moments and joint photon counting probability distributions are calculated.
Nonclassical effects exhibited by the homodyned field when light from an optical parametric oscillator (OPO) is mixed with coherent light from a local oscillator are discussed. Calculations are carried out exactly without making the strong local oscillator approximation. It is found that, depending upon the local oscillator phase and the parameters characterizing the light from the OPO, the second order intensity correlation function of the superposed field violates various classical inequalities g(2)(0) ≥ 1, g(2)(0) ≥g(2)(τ), and |g(2)(0) –1| ≥ | g(2)(τ)–1|. Thus we can see sub-Poissonian or super-Poissonian statistics and bunching or antibunching. We express the fields at the beam splitter in terms of independent real Gaussian stochastic processes with coherent component. Using these properties of the dynamical field variables, we derive analytic expressions for quantities directly measurable in photon counting experiments. It is found that the waiting time distribution shows two extrema at nonzero times. The second order intensity correlation function is found to vanish at nonzero time in violation of a classical inequality.
Nonclassical light beams generated by the degenerate optical parametric oscillator operating below threshold are analyzed in terms of photoelectron-counting sequences. The positive-P representation is used to calculate the generating function for photoelectron statistics in a closed form. This generating function is used to derive expressions for the photoelectron-counting and waiting-time distributions. The dependence of these distributions on mean photon number inside the cavity and efficiency of detection is studied. The relationship between photoelectron-counting sequence and the photon emission sequence is used to present a simple physical picture of light beams produced by the degenerate parametric oscillator.
Homodyne statistics of light generated by an atomic system exhibiting optical bistability are analyzed. Using the dynamical equations of motion for a single atom in a coherently driven cavity in the good cavity limit, we show that the homodyne field can be described in terms of two independent real Gaussian stochastic processes and a coherent component. By making a Karhunen-Lo\`eve expansion of the field variables we derive the generating function for the photoelectron statistics. From this generating function photoelectron-counting distribution, factorial moments, and waiting-time distribution are obtained analytically. These quantities are directly measurable in photon-counting experiments. We show that the homodyne field exhibits many interesting nonclassical features including nonclassical effects in higher-order factorial moments.