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Journal of the Optical Society of America B

Quantum phase fluctuations and squeezing in degenerate four-wave mixing

Gerard J Milburn, Daniel F. Walls, M. D. Levenson

Published Jun 1, 1984• 40 citations
Publisher

Abstract

We consider the generation of squeezed states in degenerate four-wave mixing, retaining all significant phase-matched terms in the third-order polarization. It is shown that, for output modes combined with appropriate phase shifts, ideal squeezed states are produced when the pump field is treated classically. An approximation scheme is used to investigate the effect of pump-field quantization. We show that for reasonable crystal lengths, the effect of pump-field quantization is insignificant in the region of significant squeezing.

DOI
10.1364/josab.1.000390

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We consider the generation of squeezed states in degenerate four-wave mixing, retaining all significant phase-matched terms in the third-order polarization. It is shown that, for output modes combined with appropriate phase shifts, ideal squeezed states are produced when the pump field is treated classically. An approximation scheme is used to investigate the effect of pump-field quantization. We show that for reasonable crystal lengths, the effect of pump-field quantization is insignificant in the region of significant squeezing.
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It is found that the field of the combined mode of the probe wave and the phase-conjugate wave in the process of degenerate four-wave mixing exhibits higher-order squeezing to all even orders. The degree of squeezing increases with the order N, and the higher-order squeeze parameter qN{q}_{N}qN​ may approach -1.
The suitability of degenerate four-wave mixing (DFWM) as a source of squeezed-state (two-photon coherent state) light is investigated, both theoretically and experimentally. In previous semiclassical theory, which neglected pump quantization and loss in the nonlinear medium, Yuen and Shapiro [Opt. Lett. 4, 334 (1979)] showed that such states would be generated by 50%-50% combination of the transmitted probe (TP) and the phase-conjugate (PC) output beams of DFWM. Here the effects of quantum fluctuations in the amplitudes and phases of the pump beams are calculated in a traveling-wave finite-interaction-length DFWM configuration. It is shown that in the limit of strong pump beams and weak nonlinear coupling, the classical-pumps assumption of Yuen and Shapiro is valid. With use of this assumption, loss in the nonlinear medium is incorporated into the model by coupling the interacting modes to a set of reservoir modes. It is shown that loss presents an absolute limit on the degree of squeezing that can be achieved with DFWM. Also, for comparison with experimental work, the photoelectron statistics of the TP, PC, and 50%-50% combination modes are calculated for lossy DFWM; significant deviations are found from the predictions for lossless operation. Quantum-noise measurements on the PC and TP beams generated in sodiumvapor DFWM are described. The mixer is pumped and probed by the nearly transform-limited pulses produced by a continuous-wave-oscillator-pulse-amplifier chain dye laser at 589 nm wave-length. The normalized second-factorial moment g2{g}_{2}g2​ of the PC and TP beams is determined from photoelectron-counting measurements. Inability to simultaneously achieve high DFWM reflectivity and acceptably low scattered-plus-fluorescent background, coupled with low photomultiplier quantum efficiency, restrict the experiment to a DFWM regime wherein theory predicts Poisson statistics. Experimental g2{g}_{2}g2​ values corroborate this prediction, indicating that systematic and excess noises have been eliminated. No attempt was made to measure the quantum statistics of the 50%-50% combination mode.
Abstract We analyze a simple model for degenerate four wave mixing, introduced by Savage and Walls. In the resonant case, the system exhibits a phase instability when the pump intensity is increased well beyond the threshold for generation of the signal mode. Still in the resonant configuration, we calculate the amplitude and phase squeezing spectra both for the signal and the pump mode. In the signal (pump) mode, the squeezing becomes especially large when the signal (pump) mode is much more damped than the pump (signal) mode. The signal field can exhibit both phase and amplitude squeezing, in distinct domains of the parameter space. On the other hand, the pump field can show squeezing only in the amplitude quadrature component. In general, the model exhibits sizable squeezing over very extended regions of the parameter space.