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Physical Review Letters

Spatial Quantum Signatures in Parametric Down-Conversion

Irene Marzoli, A. Gatti, Luigi A. Lugiato

Published Mar 17, 1997• 44 citations
Publisher

Abstract

We calculate the normal and time ordered spatial intensity correlation function of the signal field in a degenerate parametric oscillator below threshold, with spherical mirrors. In the far field, it exhibits a two peak structure, and the correlation is maximal between points opposite each other with respect to the axis of the system. This feature provides direct spatial evidence of the twin photon emission, and identifies states of the radiated field with local nonclassical squeezing properties.

DOI
10.1103/physrevlett.78.2092

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We calculate the normal and time ordered spatial intensity correlation function of the signal field in a degenerate parametric oscillator below threshold, with spherical mirrors. In the far field, it exhibits a two peak structure, and the correlation is maximal between points opposite each other with respect to the axis of the system. This feature provides direct spatial evidence of the twin photon emission, and identifies states of the radiated field with local nonclassical squeezing properties.
We analyze the quantum properties of the signal near field emitted by a degenerate parametric oscillator below threshold, including diffraction and its effects on the threshold for signal generation. The field is probed by a local oscillator field of arbitrary spatial configuration. The results are expressed in terms of an appropriate spectrum which describes how the level of squeezing varies with the angle from the direction of propagation. The results hold for cavities both with plane and with spherical mirrors.
We consider a quantum model for a degenerate optical parametric oscillator below threshold in a cavity with spherical mirrors. The spatial correlation function of a generic quadrature component of the signal field is calculated analytically in terms of an expansion over the Gauss-Laguerre modes. The critical behavior and the quantum image phenomenon, which emerge when threshold is approached, are described in detail. We calculate also the spatial correlation function of the intensity fluctuations, which can be more easily measured by direct detection and exhibits the same qualitative features.
We illustrate the spatial quantum features that emerge in a degenerate optical parametric oscillator below threshold, where the signal field is purely generated by quantum fluctuations. We analyze the behaviour of the spatial correlation function of a generic quadrature component of the signal field in both cases of a cavity with plane mirrors and a cavity with spherical mirrors. The correlation length diverges when the semiclassical threshold for signal field generation is approached. We focus on the novel concept of quantum image, which is analyzed both in the near field and in the far field.
In the first part of the paper we define the spatial spectrum of squeezing following an operational procedure, and we show in general how the spatial structure of squeezed states can be described in terms of correlation functions of quadrature components of the electric field. In the second part we formulate an appropriate quantum model for a degenerate optical parametric oscillator (OPO) with plane mirrors, and we analyze it extensively below the threshold for signal generation. The correlation length diverges when the OPO approaches threshold; this picture provides an ideal completion of the analogy with critical phenomena in second-order phase transition at equilibrium. The spatial configuration of the correlation function exhibits the phenomenon of ``quantum image:'' the signal field, which is purely generated by quantum noise, shows an ordered spatial structure which is observable via the spatial correlation function.
The signal and idler beams from a non-degenerate, above threshold, optical parametric oscillator have a strong quantum correlation, which are called twin beams. This correlation arises from the simultaneous generation of the signal and idler photons in the form of twin pairs of photons through the parametric down conversion, and the generated twin beams have exactly the same photon statistics. As a result, the fluctuations on the difference between the intensities of the twin beams are reduced with respect to the shot noise limit, that is, intensity-difference squeezing. We give a comprehensive overview of the generation of twin beams using the optical parametric oscillator, and the applications exploiting the quantum correlation of twin beams are presented.
Quantum images, that is inhomogeneous field distributions purely generated by quantum fluctuations, persist when passing from the degenerate to the non-degenerate case of optical parametric oscillators (OPO). Below the threshold for parametric oscillation where the near-field distributions are homogeneous both in intensity and phase, appropriate spatial correlation functions anticipate the transverse spatial pattern that appears above threshold. In particular, the angular dependence of the far field spatial correlation function is able to reveal the travelling-wave nature of the phase pattern above threshold typical of nondegenerate OPOs. Cross-correlation functions between signal and idler intensities show clear evidence of the non-classical nature of the output light.