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Journal of Modern Optics

The non-degenerate optical parametric oscillator in the strong-coupling regime

Wieland Alge, Klaus M. Gheri, Monika A. M. Marte

Published Apr 1, 1997• 7 citations
Publisher

Abstract

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.

DOI
10.1080/09500349708230699

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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 .
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.
The quantum dynamics of the number of photons, of the quantum entropy, and the Wigner functions of field interacting modes are studied for the process of non-degenerate parametric decay in an optical cavity, where a quantum with energy ℏω3\hbar {{\omega }_{3}}ℏω3​ decays into two photons with energies ℏω2\hbar {{\omega }_{2}}ℏω2​ and ℏω1\hbar {{\omega }_{1}}ℏω1​ ( ℏω3=ℏω2+ℏω1\hbar {{\omega }_{3}} = \hbar {{\omega }_{2}} + \hbar {{\omega }_{1}}ℏω3​=ℏω2​+ℏω1​ ). It is shown that the mode with frequency ω3{{\omega }_{3}}ω3​ can localize in a squeezed state in the region of very strong coupling between the interacting modes, and the modes with frequencies ω1{{\omega }_{1}}ω1​ and ω2{{\omega }_{2}}ω2​ localize in states that are statistical mixtures of Fock states regardless of the value of the coupling coefficient
The optical parametric oscillator is one of the most well-characterized devices in nonlinear optics, with many useful applications, especially as a frequency converter and as a wide-band amplifier. Novel discoveries made with these devices in the quantum domain include demonstrations of large amounts of both single-mode and two-mode squeezing, and, in the two-mode case, significant quantum intensity correlations between the two modes and quadrature correlations, which provided the first experimental demonstration of the original Einstein–Podolsky–Rosen (EPR) paradox. In this chapter, a systematic theory is developed of quantum fluctuations in the degenerate parametric oscillator. The quantum statistical effects produced by the degenerate parametric amplifier and oscillator have been intensely studied. As we have seen, the amount of transient squeezing produced in an ideal lossless parametric amplifier was calculated using asymptotic methods by Crouch and Braunstein. It was found to scale inversely with the square root of the initial pump photon number. When the system is put into a cavity, and driving and dissipation are introduced, we find that the steady state of a driven degenerate parametric oscillator bifurcates. There is a single steady state at low driving strengths, but, as the driving field increases, a threshold value is reached and the single state splits and forms two stable states. Below the threshold, a linearized analysis of squeezing in the quantum quadratures is possible; while above this threshold, or critical point, the tunneling time between the two states due to quantum fluctuations can be calculated.
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.
Dynamical restrictions to the squeezing achievable by means of a degenerate optical parametric oscillator (OPO) are considered by quantum-statistical diagram techniques. We determine optimal pumping conditions which maximize the degree of squeezing in the output signal of the OPO. The maximal spectral squeezing in the OPO's output radiation is found to scale as ${\mathit{N}}_{2}^{\mathrm{\ensuremath{-}}2/3}$ where N2{\mathit{N}}_{2}N2​ is the number of photons in the pump mode.