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Journal of Modern Optics

Realization of the Einstein–Podolsky–Rosen paradox in the far-field of the optical parametric oscillator above threshold

F. Castelli, Luigi A. Lugiato

Published Apr 1, 1997• 14 citations
Publisher

Abstract

We consider an optical parametric oscillator above threshold; under appropriate conditions and in the far-field, the emitted signal is composed by two spatially separate beams, highly correlated on the quantum level. We show that this system can realize the Einstein–Podolsky–Rosen paradox for continuous and macroscopic observables. The pair of non-commuting obserables considered are the intensity and the phase of the two separate beams; the spectral variances of the fluctuation in the intensity difference and in the phase sum of the two beams are calculated. The conditions for which the product of the spectral variances is below the limit settled by the Heisenberg uncertainty principle are studied.

DOI
10.1080/09500349708230694

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We consider an optical parametric oscillator above threshold; under appropriate conditions and in the far-field, the emitted signal is composed by two spatially separate beams, highly correlated on the quantum level. We show that this system can realize the Einstein–Podolsky–Rosen paradox for continuous and macroscopic observables. The pair of non-commuting obserables considered are the intensity and the phase of the two separate beams; the spectral variances of the fluctuation in the intensity difference and in the phase sum of the two beams are calculated. The conditions for which the product of the spectral variances is below the limit settled by the Heisenberg uncertainty principle are studied.
SUMMARY Parametric processes in the interaction of electromagnetic field with a medium in an optical cavity are well known to produce states of light with nonclassical properties. The quantum correlations between these states can be employed for a realization of an EinsteinPodolsky-Rosen (EPR) paradox [l], in the original formulation using continuous variables, as proposed [2] and experimentally demonstrated [3] in the case of Nondegenerate Parametric Amplification below threshold. In this paper we study EPR-like behaviour between the signal and idler fields emitted by a degenerate optical parametric oscillator (OPO) above threshold; these fields, generated by conversion of a pump photon of frequency 2w0 in two photons of frequency w. , are spatially separated in the far field, showing non-local quantum correlations. The system is described by the full Hamiltonian for the operators associated with the pump and signal fields; the optical cavity introduces proper damping constants and corresponding vacuum field operators [4,5]. Above the threshold for the generation of the signal field, we consider the fluctuations in the intensity difference and in the phase sum of the signal beams, which are related to their degree of mutual correlation. By linearizing around the stationary state, we calculate the spectral variances of these quantities (i.e. the Fourier transform of the two time correlation functions). The Heisenberg uncertainty principle sets a lower limit to the product of the spectral variances for the fluctuations in the intensity and in the phase of each individual signal field; by using the idler beam to infer results of measurements on the correlated signal beam, we have an EPR paradox if the product of the spectral variances for the fluctuations in the intensity difference and phase sum of the fields, is below the uncertainty limit. This feature is apparent for a wide interval around zero frequency in the far field of the active OPO system, depending on the parameter space; in particular there is maximum correlation between signal beams, and therefore maximum breaking of the uncertainty limit, if the damping constant of the signal field is larger than that of the pump field.
The Einstein-Podolsky-Rosen paradox is demonstrated experimentally for dynamical variables having a continuous spectrum. As opposed to previous work with discrete spin or polarization variables, the continuous optical amplitudes of a signal beam are inferred in turn from those of a spatially separated but strongly correlated idler beam generated by nondegenerate parametric amplification. The uncertainty product for the variances of these inferences is observed to be 0.70\ifmmode\pm\else\textpm\fi{}0.01, which is below the limit of unity required for the demonstration of the paradox.
For measurements describable by a discrete set of possibilities, numerous demonstrations of the violation of Bell's inequalities have been reported in optical physics based on observations of polarization correlations for photon pairs. By contrast for observables with a continuous spectrum, there appears to be no general inequalities corresponding to those of Bell and no experimental realization of the original Einstein-Podolsky-Rosen paradox. Following Reid and Drummond,1 we are attempting to demonstrate the EPR paradox in the context of the measurement of continuous quantum variables with a subthreshold optical parametric oscillator operated in a frequency degenerate but polarization nondegenerate mode. In this case, the EPR paradox is formulated in terms of two sets of quadrature phase amplitudes for the spatially separated signal and idler fields. As the parametric gain is increased toward threshold, the quadrature amplitudes of the signal and idler beams become increasingly faithful quantum copies of one another. Our measurements of quadrature amplitudes are carried out using independent balanced homodyne detectors at two sites. When operated for the detection of squeezing at one site, noise reductions of 4 dB relative to the vacuum state limit are observed, which is most encouraging since noise reductions >3 dB in the difference photocurrent from the two sites are sufficient for inference at a distance of a signal observable from a measurement of the corresponding idler variable.
The paper reports on the investigation on to which extent the microscopic polarization properties survive at a macroscopic level, in presence of a continuous flux of photons, in a transverse spatially extended system. A prototype device to generate polarization entangled beams is a quadratic crystal in the type II phase matching configuration. The spectral brightness can be increased by means of a resonant cavity (optical parametric oscillator) . Quantum correlations below threshold is calculated analytically taking into account the walk-off effect between the (ordinary polarized) signal and the (extraordinary polarized) idler. In this regime the average of the signal field is zero, but the far-field (FF) intensity distribution is dominated by two undamped rings (quantum images, noisy precursors of the pattern), whose centers are displaced along the walk-off direction. In the first part, the effect of the selection of different polarization components on the Einstein-Podolsky-Rosen correlations is studied; this is a generalization of the first experimental realization of the EPR paradox for continuous variables in an OPO, taking now into account the transverse profile of the filed. In the second part, the correlations in other continuous variables, the Stokes parameters are then analyzed.