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Physical Review Letters

Quantum Correlations of Phase in Nondegenerate Parametric Oscillation

M. D. Reid, P. D. Drummond

Published Jun 27, 1988• 486 citations
Publisher

Abstract

The squeezing spectrum for nondegenerate parametric oscillation above threshold is calculated, including phase diffusion. A nonclassical correlation in phase and intensity occurs which is an example of the Einstein-Podolsky-Rosen paradox, even in fields of large photon number.
DOI
10.1103/physrevlett.60.2731

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We develop the quantum theory of a driven, damped intracavity parametric oscillator in the case of nondegenerate signal and idler modes. An analysis of the results of different types of local-oscillator measurements on the outgoing modes is presented. The frequency-domain squeezing is calculated below threshold including the effect of detunings and nonequal decay rates. We also calculate a nonclassical correlation in the quadrature amplitudes of the output beams. The existence of an Einstein-Podolsky-Rosen paradox is demonstrated through violations of the inferred Heisenberg uncertainty principle at causally unrelated locations in the output field. This provides a way of maintaining the commutation relations of the measured operators in the exact form envisaged originally by Einstein, Podolsky, and Rosen.
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.
We study the nondegenerate parametric oscillator in the limit of small fluctuations where a linearized theory is valid. Interest is focused on solutions above threshold. It has been shown that the field undergoes phase diffusion above threshold,1 and the theory incorporates this effect. We calculate the spectrum of fluctuations in the intensity difference between the two modes,2 allowing for differing cavity mode decay rates. Possibilities for quantum nondemolition experiments and other quantum properties are discussed. In particular, the effect of phase diffusion on the long-time heterodyne squeezing spectra is calculated. The linearized results are compared to the full nonlinear steady-state solutions in the limit of a very fast decaying pump.
We present a fully quantum mechanical treatment of the nondegenerate optical parametric oscillator both below and near threshold. This is a nonequilibrium quantum system with a critical point phase transition, that is also known to exhibit strong yet easily observed squeezing and quantum entanglement. Our treatment makes use of the positive PPP representation and goes beyond the usual linearized theory. We compare our analytical results with numerical simulations and find excellent agreement. We also carry out a detailed comparison of our results with those obtained from stochastic electrodynamics, a theory obtained by truncating the equation of motion for the Wigner function, with a view to locating regions of agreement and disagreement between the two. We calculate commonly used measures of quantum behavior including entanglement, squeezing, and Einstein-Podolsky-Rosen (EPR) correlations as well as higher order tripartite correlations, and show how these are modified as the critical point is approached. These results are compared with those obtained using two degenerate parametric oscillators, and we find that in the near-critical region the nondegenerate oscillator has stronger EPR correlations. In general, the critical fluctuations represent an ultimate limit to the possible entanglement that can be achieved in a nondegenerate parametric oscillator.
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.