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arXiv:quant-ph/0108030 · Physical Review A

Macroscopic quantum fluctuations in noise-sustained optical patterns

Roberta Zambrini, Stephen Mark Barnett, Pere Colet, Maxi San Miguel

Published Jan 15, 2002• 20 citations
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Abstract

We investigate quantum effects in pattern formation for a degenerate optical parametric oscillator with walk-off. This device has a convective regime in which macroscopic patterns are both initiated and sustained by quantum noise. Familiar methods based on linearization about a pseudoclassical field fail in this regime and new approaches are required. We employ a method in which the pump field is treated as a c-number variable but is driven by the c-number representation of the quantum subharmonic signal field. This allows us to include the effects of the fluctuations in the signal on the pump, which in turn act back on the signal. We find that the nonclassical effects, in the form of squeezing, survive just above the threshold of the convective regime. Further, above threshold, the macroscopic quantum noise suppresses these effects.

DOI
10.1103/physreva.65.023813

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We investigate quantum effects in pattern formation for a degenerate optical parametric oscillator with walk-off. This device has a convective regime in which macroscopic patterns are both initiated and sustained by quantum noise. Familiar methods based on linearization about a pseudoclassical field fail in this regime and new approaches are required. We employ a method in which the pump field is treated as a c-number variable but is driven by the c-number representation of the quantum subharmonic signal field. This allows us to include the effects of the fluctuations in the signal on the pump, which in turn act back on the signal. We find that the nonclassical effects, in the form of squeezing, survive just above the threshold of the convective regime. Further, above threshold, the macroscopic quantum noise suppresses these effects.
Summary form only given. Noise-sustained patterns have been recently predicted in several systems including a Kerr cavity and an optical parametric oscillator. In these systems a regime of convective instability can appear, in which a transverse pattern arises as a macroscopic manifestation of amplified and spatially structured quantum noise, with magnification factors of several order of magnitudes. We study quantum correlations in the convective regime of a degenerate type-I optical parametric oscillator (OPO) using a time dependent parametric approximation, in which the intracavity pump field is taken as a classical stochastic field.
The degenerate parametric oscillator above threshold is studied with phenomenological stochastic colored (nonwhite) pump noise for arbitrary pump to subharmonic relaxation rate. The current experimental limits of large intracavity threshold photon numbers (very small quantum noise) are considered, allowing for a semiclassical treatment of the system dynamics. A comparison between the effects of isotropic and squeezed pump noise on the internal transient and steady-state fluctuations is presented by simulation of the nonlinear semiclassical stochastic Langevin equations in the Wigner quadrature representation. It is found that the transient squeezing for the system starting in the unstable steady state (the vacuum) is not degraded by stochastic pump noise. A damped oscillatory behavior of the noise levels (periodic exchange of fluctuations between the squeezed quadrature of the signal and the pump) is observed for large damping of the signal in the turn on of pump depletion. Finally, it is shown that the limited squeezing above threshold in the steady state (50%) due to pump depletion can be enhanced if squeezed stochastic noise with sufficient significant spectral components (broadband squeezing) is fed to the pump. The above-threshold steady-state squeezing has been calculated analytically from the linearized stochastic equations and the effects of the time scale associated to the relaxation of the pump noise (the noise correlation time) compared to the dissipative time scale of the system and the pump to subharmonic loss ratio are presented.
Quantum fluctuations and squeezing of the signal field generated by a degenerate optical parametric oscillator are sensitive to the phase and amplitude fluctuations of the second-harmonic pump. The analysis reported here extends our previous calculations for a perfectly monochromatic noise-free pump to allow for finite pump amplitude and phase fluctuations. We treat the typically intense pump field with its fluctuations as a classical Gaussian stochastic process in two limits: first, when the pump has amplitude fluctuations but its phase is fixed and second, when its phase diffuses but its amplitude is fixed. Since pump amplitude noise affects the evolution of the signal field multiplicatively, it has a lesser impact on the quantum fluctuations of the signal than pump phase diffusion which compromises such fluctuations via the extreme phase sensitivity of the parametric amplification process. For the latter limit, our calculations extend the work of Gea-Banacloche and Zubairy [Phys. Rev. A 42, 1742 (1990)] to cavities with arbitrary transmission coefficients for the bounding mirrors.
We describe how a two-mode, above-threshold, optical parametric oscillator can generate nonclassical states of light with a large average number of photons: the fluctuation spectrum of the various fields is calculated, and several quantities are shown to have squeezed fluctuations. In particular, a perfect quantum noise suppression is predicted on the difference between the intensities of the two generated beams. We describe an experiment designed to demonstrate such an effect and show how it can be used to generate high-intensity amplitude-squeezed states.
A Langevin model is introduced to study quantum fluctuations below the threshold of pattern formation for optical parametric oscillators (OPO's). In particular we compare analytical and numerical results for the OPO with one and two transverse spatial dimensions and in the presence of either plane or spherical cavity mirrors. The far-field structure and the correlation functions of the fluctuating signal field anticipate the onset of a transverse spatial pattern which arises classically even in the presence of Gaussian input beams. Correlation functions also reveal the squeezed nature of the OPO field. Numerical simulations of the Langevin model describe the result of short time measurements and show that close to threshold the near-field signal is a noisy spot pattern evolving on a time scale longer than the inverse decay rate of the resonator. This and other far-field features should be experimentally accessible.