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Physical Review A

Many-body quantum theory of the optical parametric oscillator

Christopher J. Mertens, T. A. B. Kennedy, S. Swain

Published Sep 1, 1993• 26 citations
Publisher

Abstract

A many-body theory of quantum fluctuations in the optical parametric oscillator is given. Dyson equations and diagrammatic representation of photon Green functions are derived from a combination of quantum Langevin and many-body techniques. Single-photon Green functions are used to compute dynamic spectra of the output using a self-consistent scheme which incorporates nonlinear, many-photon polarization effects. The latter scale with system size and their influence on the nonlinear quantum dynamics is addressed. We present results of dynamic output spectra and quantum-noise properties beyond the adiabatic approximation.

DOI
10.1103/physreva.48.2374

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A many-body theory of quantum fluctuations in the optical parametric oscillator is given. Dyson equations and diagrammatic representation of photon Green functions are derived from a combination of quantum Langevin and many-body techniques. Single-photon Green functions are used to compute dynamic spectra of the output using a self-consistent scheme which incorporates nonlinear, many-photon polarization effects. The latter scale with system size and their influence on the nonlinear quantum dynamics is addressed. We present results of dynamic output spectra and quantum-noise properties beyond the adiabatic approximation.
We analyze the influence of system-size effects on the quantum properties of a degenerate parametric oscillator below, at, and above the classical threshold using a Green's-function approach. The many-body technique permits a systematic analysis of finite-size corrections to standard linearization results. In particular we study a ``semiquantum'' limit, where even above threshold only few photons are in the subharmonic mode while the pump mode is highly populated and behaves quasiclassically. Substantial deviations from classical and linearization predictions are found. We show that the depletion of the pump mode is rather strong and the threshold of parametric oscillation is shifted to higher pump strength, when the subharmonic system is small.
Quantum properties of intracavity nonlinear optical processes such as parametric oscillation and second harmonic generation are limited to a linearized analytical treatment except for certain special cases. Perfect squeezing in one quadrature and the divergence in fluctuations of the conjugate quadrature predicted at threshold in parametric oscillation are a consequence of the approximate linearized analysis. We investigate nonlinear quantum fluctuation corrections by using nonequilibrium Green’s function methods1,2 in concert with quantum master equations, to assess when these system-size effects may be observable in the laboratory.
Nonlinear optical cavities are crucial both in classical and quantum optics; in particular, nowadays optical parametric oscillators are one of the most versatile and tunable sources of coherent light, as well as the sources of the highest quality quantum-correlated light in the continuous variable regime. Being nonlinear systems, they can be driven through critical points in which a solution ceases to exist in favour of a new one, and it is close to these points where quantum correlations are the strongest. The simplest description of such systems consists in writing the quantum fields as the classical part plus some quantum fluctuations, linearizing then the dynamical equations with respect to the latter; however, such an approach breaks down close to critical points, where it provides unphysical predictions such as infinite photon numbers. On the other hand, techniques going beyond the simple linear description become too complicated especially regarding the evaluation of two-time correlators, which are of major importance to compute observables outside the cavity. In this article we provide a regularized linear description of nonlinear cavities, that is, a linearization procedure yielding physical results, taking the degenerate optical parametric oscillator as the guiding example. The method, which we call self-consistent linearization, is shown to be equivalent to a general Gaussian ansatz for the state of the system, and we compare its predictions with those obtained with available exact (or quasi-exact) methods. Apart from its operational value, we believe that our work is valuable also from a fundamental point of view, especially in connection to the question of how far linearized or Gaussian theories can be pushed to describe nonlinear dissipative systems which have access to non-Gaussian states.
We present a method for the study of quantum fluctuations of dissipative structures forming in nonlinear optical cavities, which we illustrate in the case of a degenerate, type I optical parametric oscillator. The method consists in (i) taking into account explicitly, through a collective variable description, the drift of the dissipative structure caused by the quantum noise, and (ii) expanding the remaining---internal---fluctuations in the biorthonormal basis associated to the linear operator governing the evolution of fluctuations in the linearized Langevin equations. We obtain general expressions for the squeezing and intensity fluctuations spectra. Then we theoretically study the squeezing properties of a special dissipative structure, namely, the bright cavity soliton. After reviewing our previous result that in the linear approximation there is a perfectly squeezed mode irrespective of the values of the system parameters, we study which are the better squeezed modes at the different bifurcations that the cavity soliton can suffer, as well as the squeezing level attainable when the homodyne detection is made with a plane-wave local oscillator field. In this last case, the squeezing level is quite large in most of the existence domain of the cavity soliton. We then take into account the effect of the detector size and find that, in general, squeezing degrades as the detector is made smaller, except for detector sizes around the cavity soliton size, in which case there is an interesting deviation from the general trend.
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.