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Conference on Lasers and Electro-Optics

Transverse distribution of quantum fluctuations and correlations in a spatial soliton

Nicolas Treps, Claude Fabre

Published Aug 29, 2005• 0 citations
Publisher

Abstract

Summary form only given. The quantum properties of the interaction between an electromagnetic field and a nonlinear media have been widely studied, theoretically and experimentally, but so far these studies concerned mainly measurements performed on the total intensity of light beams, integrated over all the transverse plane. However, the details of the noise distribution inside the transverse plane, as well as the spatial quantum correlation between two different points, can be also very instructive, on a fundamental side in order to understand the propagation of the quantum noise, but also on a practical side to generate special configurations of the field which allow us to increase the spatial resolution of measurements in the transverse plane. We study here the spatial properties of the quantum fluctuations for a field freely propagating in a thick nonlinear medium, and in a case where the interaction distance is not limited by the diffraction : the spatial soliton regime, where diffraction effects are counterbalanced by nonlinearity and which exists in /spl chi//sup (2)/ and /spl chi//sup (3)/ media.

DOI
10.1109/iqec.2000.907773

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Summary form only given. The quantum properties of the interaction between an electromagnetic field and a nonlinear media have been widely studied, theoretically and experimentally, but so far these studies concerned mainly measurements performed on the total intensity of light beams, integrated over all the transverse plane. However, the details of the noise distribution inside the transverse plane, as well as the spatial quantum correlation between two different points, can be also very instructive, on a fundamental side in order to understand the propagation of the quantum noise, but also on a practical side to generate special configurations of the field which allow us to increase the spatial resolution of measurements in the transverse plane. We study here the spatial properties of the quantum fluctuations for a field freely propagating in a thick nonlinear medium, and in a case where the interaction distance is not limited by the diffraction : the spatial soliton regime, where diffraction effects are counterbalanced by nonlinearity and which exists in /spl chi//sup (2)/ and /spl chi//sup (3)/ media.
We give in this paper a general method by which to determine the transverse spatial distribution of quantum fluctuations in a light beam as it propagates through a nonlinear medium. We apply this method to the particular case of spatial solitons that propagate without distortion in ${\ensuremath{\chi}}^{(2)}$ or ${\ensuremath{\chi}}^{(3)}$ materials. We find, for example, that in the ${\ensuremath{\chi}}^{(2)}$ case the central part of the soliton has lower intensity quantum fluctuations than its outer part, and that in most cases strong correlations develop among the field quadrature components measured in different areas of the transverse plane.
We have studied quantum effects in a system governed by the spatial nonlinear Schrodinger equation derived for a two beams interaction in a quadratic nonlinear material. For a quadratic nonlinear material it has recently been demonstrated [2, 3] that, under suitable conditions, in the interaction between the fundamental and the second harmonic field, the fundamental field propagation can be described as a solution of a nonlinear Schroedinger equation, given in terms of spatial solitons [4,5]. In this work we are interested at the quantum properties of the fundamental beam in this kind of interaction. We use a quantum field description [1] where the spatial variable of propagation z plays the role of time in the standard quantum theory, and we apply it to the case of a strong plane wave that can be described as a classical field propagating through the nonlinear medium in the z-direction. We assume TE polarizations for all the monochromatics fields involved into the interaction in a type I material.
We investigate the spatial quantum noise properties of the one-dimensional transverse pattern formation instability in intracavity second-harmonic generation. The Q representation of a quasi-probability distribution is implemented in terms of nonlinear stochastic Langevin equations. We study these equations through extensive numerical simulations and analytically in the linearized limit. Our study, made below and above the threshold of pattern formation, is guided by a microscopic scheme of photon interaction underlying pattern formation in second-harmonic generation. Close to the threshold for pattern formation, beams with opposite direction of the off-axis critical wave numbers are shown to be highly correlated. This is observed for the fundamental field, for the second-harmonic field, and also for the cross-correlation between the two fields. Nonlinear correlations involving the homogeneous transverse wave number, which are not identified in a linearized analysis, are also described. The intensity differences between opposite points of the far fields are shown to exhibit sub-Poissonian statistics, revealing the quantum nature of the correlations. We observe twin beam correlations in both the fundamental and second-harmonic fields, and also nonclassical correlations between them.
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.
We develop a theory of the transverse spatial dynamics of two interacting photons in a Rydberg nonlinear medium. Extending the well-studied one-dimensional case, we explore both longitudinal and transverse correlations of photons propagating as Rydberg polaritons. We identify distinct behaviors and scaling laws for these correlations, arising from fundamentally different mechanisms in the two directions: diffraction for transverse correlations and diffusion for longitudinal correlations. We develop a model incorporating a Gaussian optical beam and an inhomogeneous atomic density distribution, from which we derive quantitative predictions for correlation functions in both directions. We further show that the propagation equations can be reduced to a single Schrödinger-like equation, allowing approximate solutions that are supported by full numerical results. Our findings indicate that the transverse correlation length is determined primarily by the blockade radius, whereas the longitudinal correlation length is limited by bandwidth. These results establish transverse Rydberg blockade as a distinct and measurable correlation mechanism and show that spatial photon correlations provide a direct means of measuring the blockade radius.
Nonclassical effects such as squeezing, antibunching, and sub-Poissonian statistics of photons have been attracting attention in quantum optics over the last decade. Up to now most theoretical and experimental investigations have been carried out exclusively in the time domain while neglecting the spatial aspects by considering only one spatial mode of the electromagnetic field. In many situations such an approximation is well justified. There are, however, problems that do not allow in principle a single-mode consideration. This is the case when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the direction of propagation of the light beam. Such an investigation requires a complete description of quantum fluctuations of light in both time and space and cannot be done within a single-mode theory. This space-time description brings about a natural generalization into the spatial domain of such notions as the standard quantum limit, squeezing, antibunching, etc. It predicts, for example, the possibility of generating a light beam with sub-Poissonian statistics of photons not only in time but also in the beam's transverse plane. Of particular relevance to the applications is a situation in which the cross section of the light beam contains several nonoverlapping areas with sub-Poissonian statistics of photons in each. Photodetection of such a beam produces several sub-shot-noise photocurrents depending on the number of independent areas with sub-Poissonian statistics. This is in marked contrast to the case of a single-mode sub-Poissonian light beam in which any attempt to collect light from only a part of the beam deteriorates the degree of shot-noise reduction. This property of multimode squeezed light opens a range of interesting new applications in optical imaging, optical parallel processing of information, parallel computing, and many other areas in which it is desirable to have a light beam with regular photon statistics across its transverse area. The aim of this review is to describe the recent development in this branch of quantum optics.