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Physical Review A

Effects of finite-system size in nonlinear optical systems:A quantum many-body approach to parametric oscillation

Oliver Veits, Michael Fleischhauer

Published Apr 1, 1997• 14 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.55.3059

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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.
The quantum fluctuations in a nondegenerate optical parametric oscillator are calculated without the standard linearization assumption. Analytical results are obtained with the help of a mean-field approximation, which is exact in the limit of a fast decaying subharmonic mode. This approach is equivalent to a linearization around a self-consistently determined pump field amplitude that takes into account fluctuations of the subharmonic mode. The range of validity of the mean-field approximation is analyzed, and the influence of the system size on field amplitudes and squeezing is discussed.
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.
A fully quantum treatment of the non-degenerate optical parametric oscillator in the extreme quantum limit of small photon numbers and very high nonlinear coupling strength is presented. When the nonlinear coupling constant becomes comparable with the cavity decay rate, the sharp threshold that is usually encountered disappears, a situation reminiscent of microlasers. Furthermore the output light exhibits unusual statistical properties such as strong super-Poissonian behaviour of the pump mode for (very) large coupling constants.
The degenerate parametric oscillator is an important source of squeezed light. If the pump mode is adiabatically eliminated, its operation is governed by two dimensionless parameters: the pump parameter λ and the coupling constant g. The pump parameter controls the threshold behavior predicted by semiclassical theory (threshold is reached for λ = 1). The coupling constant g determines the strength of the quantum fluctuations. The optical parametric oscillators used to generate squeezed light have g ~ 10–4 ≪ 1.1 Therefore, in these systems the quantum fluctuations only produce very small perturbations about semiclassical states. We have recently shown how the steady state of the degenerate parameter oscillator changes as a function of coupling strength.2 In the strong coupling limit, major departures from semiclassical states occur; for g2 ≫ 1 and 2λ2/g2 ≪ 1 the steady state approches the coherent state superposition | ψ + 〉 = | λ / g 〉 + | − λ / g .
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.