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Conference on Lasers and Electro-Optics

Quantum fluctuations in a continuous vectorial Kerr cavity model

Roberta Zambrini, M. Hoyuelos, A. Gatti, Pere Colet, Luigi A. Lugiato, Maxi San Miguel

Published Aug 29, 2005• 1 citation
Publisher

Abstract

Summary form only given. We consider a self-defocusing model for a Kerr medium in a planar resonator, which takes into account the vectorial character of the radiation field. For an intensity of the pump beam larger than a given threshold, a y polarized transverse stripe pattern appears. We analyze the spatial behavior of quantum fluctuations around this inhomogeneous steady state, using a Langevin treatment based on Wigner representation. The model is continuous, so that we avoid any restriction to a reduced number of spatial modes. The spatial distribution of the quantum fluctuations around the roll-pattern is dominated by the neutral (or Goldstone) mode, corresponding to rigid spatial displacements of the pattern. The spatial configuration of the field immediately outside the cavity input/output mirror depends on the time window over which fluctuations are averaged: only when the time window is on the order of the cavity lifetime the output field fluctuations are qualitatively similar to that of the intracavity field. We show the existence of a strong anticorrelation between the quantum fluctuations of the intensity of the x-polarized pump and the y-polarized field. We check the possibility to use a Kerr cavity like a quantum non-demolition device, which uses the tilted waves, corresponding in the far field to the y-polarized stripe pattern, as a meter to measure the intensity fluctuation of the pump beam.

DOI
10.1109/iqec.2000.907769

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Conference on Lasers and Electro-Optics•Aug 29, 2005•1 citation
Summary form only given. We consider a self-defocusing model for a Kerr medium in a planar resonator, which takes into account the vectorial character of the radiation field. For an intensity of the pump beam larger than a given threshold, a y polarized transverse stripe pattern appears. We analyze the spatial behavior of quantum fluctuations around this inhomogeneous steady state, using a Langevin treatment based on Wigner representation. The model is continuous, so that we avoid any restriction to a reduced number of spatial modes. The spatial distribution of the quantum fluctuations around the roll-pattern is dominated by the neutral (or Goldstone) mode, corresponding to rigid spatial displacements of the pattern. The spatial configuration of the field immediately outside the cavity input/output mirror depends on the time window over which fluctuations are averaged: only when the time window is on the order of the cavity lifetime the output field fluctuations are qualitatively similar to that of the intracavity field. We show the existence of a strong anticorrelation between the quantum fluctuations of the intensity of the x-polarized pump and the y-polarized field. We check the possibility to use a Kerr cavity like a quantum non-demolition device, which uses the tilted waves, corresponding in the far field to the y-polarized stripe pattern, as a meter to measure the intensity fluctuation of the pump beam.
We consider a model for a Kerr medium in a planar resonator, which takes into account the vectorial character of the radiation field. We analyze the spatial behavior of quantum fluctuations around a steady state, with a roll-pattern configuration in the beam cross section, using a Langevin treatment based on the Wigner representation. The spatial distribution of the quantum fluctuations around the roll pattern is dominated by the neutral (or Goldstone) mode, corresponding to rigid spatial displacements of the pattern. The spatial configuration of the field immediately outside the cavity input-output mirror depends on the time window over which fluctuations are averaged: only when the time window is on the order of the cavity lifetime the output field fluctuations are qualitatively similar to that of the intracavity field. The quantum correlations among the fields in play, as described by the full multimode model, turn out to be in good agreement with those predicted by a simple three-mode model.
We consider a vectorial Kerr-medium model including transverse spatial effects. We analyze cases in which, immediately above the threshold of the spatial instability, the homogeneous pump wave gives rise to two tilted waves corresponding to a stripe pattern in the near field. We analyze both the self-focusing and the self-defocusing case and we point out the existence of anti-correlations between the quantum fluctuations of the intensity of the pump and the sum of the intensities of the two tilted waves creating the transverse pattern in the near field. We also evaluate the efficiency of this scheme as a quantum nondemolition (QND) scheme which uses the tilted waves as a ``meter'' to measure the intensity fluctuations of the pump. Our results show the posibility of a QND measurement in the self-defocusing case. In this case, and for a linearly polarized pump, the output pump beam (uniform in the transverse plane) and the pattern have orthogonal polarization and could be easily separated experimentally.
Summary form only given. Transverse optical patterns are capable of allowing noteworthy aspects linked to quantum fluctuations. These issues have been mainly studied in stripe patterns, which present only two modes in the far-field. We address the study of the fluctuations and correlations in a hexagonal pattern in a ring cavity filled with a self-focusing Kerr medium. The dynamics of the electric field is described, in the mean-field approximation, by a Langevin equation for the scaled slowly varying amplitude of the field. We show that the pattern fluctuations can be understood in terms of the neutral modes of the pattern and modes with small damping. These modes are obtained from a stability analysis of the hexagonal pattern. We find strong correlations between the intensity fluctuations of the modes of the pattern, not only among the fundamental harmonics, but also with the higher order ones. We also find strong anticorrelations between the intensity fluctuations of the modes of the pattern and the homogeneous mode which are related to energy conservation.
Polarization squeezing [1] is attracting much attention in the last few years because of its potential applications in quantum information. We reconsider here the analysis of quantum fluctuations in dispersive optical bistability by generalizing the standard model in order to take into account fluctuations in the two orthogonal polarization components of the intracavity field, and not only in the pumped one as it is customary. In other words, we consider a relatively simple model for a vectorial Kerr cavity: The Kerr nonlinearity is the responsible for dispersive bistability, and two extra nonlinear terms, describing cross-phase modulation and four-wave mixing, couple the intense (pumped) intracavity field (with polarization parallel to that of the field injected in the cavity) with the orthogonally polarized mode. The weight of the new nonlinearities is controlled by the Maker-Terhune coefficients, which verily A + B!2 = 1 for an isotropic material, which we consider. This system exhibits a very rich variety of bifurcations [2], In this communication we describe the simplest results.
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.
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.