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Optics Express

Theory of photon statistics and squeezing in quantum interference of a sub-threshold parametric oscillator

Roger Andrews, Edward R. Pike, Sarben Sarkar

Published Jan 13, 2003• 7 citations
Publisher

Abstract

A multimode theory describing quantum interference of a subthreshold optical parametric oscillator (OPO) with a coherent local oscillator (LO) in a homodyne detection scheme is presented. Analytic expressions for the count rates in terms of the correlation time and relative phase difference between the LO and OPO have been derived. The spectrum of squeezing is also derived and the threshold for squeezing obtained in terms of the crystal nonlinearity and LO and OPO beam intensities.

DOI
10.1364/oe.11.000007

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Acta Physica Sinica•Jan 1, 1999•0 citations
A multimode theory describing quantum interference of a subthreshold optical parametric oscillator (OPO) with a coherent local oscillator (LO) in a homodyne detection scheme is presented. Analytic expressions for the count rates in terms of the correlation time and relative phase difference between the LO and OPO have been derived. The spectrum of squeezing is also derived and the threshold for squeezing obtained in terms of the crystal nonlinearity and LO and OPO beam intensities.
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
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.
The quantum features of ultrashort-pulse optical parametric oscillators (OPOs) are investigated theoretically in the nonlinear regime near and above threshold. Viewing the pulsed OPO as a multimode open quantum system, we rigorously derive a general input-output model that features nonlinear coupling among many cavity (i.e., system) signal modes and a broadband single-pass (i.e., reservoir) pump field. Under appropriate assumptions, our model produces a Lindblad master equation with multimode nonlinear Lindblad operators describing two-photon dissipation and a multimode four-wave-mixing Hamiltonian describing a broadband, dispersive optical cascade, which we show is required to preserve causality. To simplify the multimode complexity of the model, we employ a supermode decomposition to perform numerical simulations in the regime where the pulsed supermodes experience strong single-photon nonlinearity. We find that the quantum nonlinear dynamics induces pump depletion as well as corrections to the below-threshold squeezing spectrum predicted by linearized models. We also observe the formation of non-Gaussian states with Wigner-function negativity and show that the multimode interactions with the pump, both dissipative and dispersive, can act as effective decoherence channels. Finally, we briefly discuss some experimental considerations for potentially observing such quantum nonlinear phenomena with ultrashort-pulse OPOs on nonlinear nanophotonic platforms.
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.