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Physical Review A

Macroscopic quantum superposition states in nondegenerate parametric oscillation

M. D. Reid, Leo Krippner

Published Jan 1, 1993• 54 citations
Publisher

Abstract

Nondegenerate oscillation is studied in the limit of a fast-decaying pump mode. The study is facilitated by analytic solutions of the positive P representation. In the absence of all individual signal and idler losses, states of definite signal-idler photon-number difference are generated. Determination of this photon number enables preparation of a macroscopic superposition state.

DOI
10.1103/physreva.47.552

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Nondegenerate oscillation is studied in the limit of a fast-decaying pump mode. The study is facilitated by analytic solutions of the positive P representation. In the absence of all individual signal and idler losses, states of definite signal-idler photon-number difference are generated. Determination of this photon number enables preparation of a macroscopic superposition state.
In nondegenerate parametric amplification of light, one pump photon is destroyed and two photons are simultaneously created, one in each of two modes with different frequencies (the signal and idler). If m signal photons are counted in time t, what is the state of the idler conditioned on this result? The projection postulate cannot be used to answer this question as photon-counting measurements are not ``first-kind'' measurements. We use a theory of continual quantum-counting measurements to show that in general the idler field is left in a classical mixture of number states with at least m quanta. In an appropriate limit this mixture reduces to an exact number state with m quanta.
Nondegenerate parametric oscillation and four-wave mixing generate correlated twin-beams that exhibit squeezing in the intensity difference fluctuations. Using a simple theoretical model for parametric amplification we predict that the twin-beam devices generate macroscopic states that violate Bell’s inequality. In four-wave mixing, or parametric oscillation with an amplitude-squeezed pump, the twin-beams may also exhibit squeezing in the intensity of each beam. We predict that the use of these intensity-squeezed devices is advantageous in the preparation of the macroscopic quantum states. It is also possible to use nondegenerate systems to generate macroscopic superposition (Schrodinger cat) states.
We consider the model of a nondegenerate parametric oscillator, in the adiabatic limit of strongly damped pump mode, and find an exact analytic solution for the joint two-mode Wigner function in the steady-state regime. The one-mode Wigner functions for the individual signal and idler modes are derived as well. This gives an explicit and complete phase-space representation of this two-mode nonequilibrium quantum system. Simple examples are given illustrating the phase-space images of optical bistability and the phase diffusion effect.
A quantum analysis of the intensity fluctuations is given for the nondegenerate parametric oscillator both above and below threshold. Theoretical analysis and experiments by Heidmann, Horowicz, Reynaud, Giacobino, and Fabre [Phys. Rev. Lett. 59, 2555 (1987)] have shown that a noise suppression below the vacuum level is possible in the difference of the intensities of the signal and idler modes. The effect of nonequal cavity decay rates and intracavity losses is shown to modify significantly the fluctuation spectra for the difference in the intensity of the signal and idler beams. The results of exact nonlinear solutions for the internal cavity modes are discussed. We demonstrate that a very sensitive absorption detector may be made by operating the oscillator near threshold.
The quantum fluctuations in a nondegenerate optical parametric oscillator are calculated without the standard linearization assumption. Analytical results are obtained with the help of a mean-field approximation, which is exact in the limit of a fast decaying subharmonic mode. This approach is equivalent to a linearization around a self-consistently determined pump field amplitude that takes into account fluctuations of the subharmonic mode. The range of validity of the mean-field approximation is analyzed, and the influence of the system size on field amplitudes and squeezing is discussed.
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.