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arXiv:quant-ph/0105137

Limits to squeezing in the degenerate OPO

S. Chaturvedi, K. Dechoum, P. D. Drummond

Published May 28, 2001• 0 citations
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Abstract

We develop a systematic theory of quantum fluctuations in the driven parametric oscillator (OPO), including the region near threshold. This allows us to treat the limits imposed by nonlinearities to quantum squeezing and noise reduction, in this non-equilibrium quantum phase-transition. In particular, we compute the squeezing spectrum near threshold, and calculate the optimum value. We find that the optimal noise reduction occurs at different driving fields, depending on the ratio of damping rates. The largest spectral noise reductions are predicted to occur with a very high-Q second-harmonic cavity. Our analytic results agree well with stochastic numerical simulations. We also compare the results obtained in the positive-P representation, as a fully quantum mechanical calculation, with the truncated Wigner phase space equation, also known as semiclassical theory.

DOI
10.48550/arxiv.quant-ph/0105137

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We develop a systematic theory of critical quantum fluctuations in the driven parametric oscillator. Our analytic results agree well with stochastic numerical simulations. We also compare the results obtained in the positive-P representation, as a fully quantum-mechanical calculation, with the truncated Wigner phase-space equation, also known as the semiclassical theory. We show when these results agree and differ in calculations taken beyond the linearized approximation. We find that the optimal broadband noise reduction occurs just above threshold. In this region where there are large quantum fluctuations in the conjugate variance and macroscopic quantum superposition states might be expected, we find that the quantum predictions correspond very closely to the semiclassical theory.
We calculate the scale of the critical fluctuations in the quantum parametric oscillator at threshold. This allows us to calculate asymptotically the size of both the depletion of the pump mode and the amount of squeezing produced in the signal mode for large system size. In this driven, open quantum system, there is no known solution to the equilibrium density matrix. For this reason these results cannot be obtained using standard canonical ensemble methods. The calculations are compared to time dependent quantum stochastic simulations of the fully nonlinear coupled mode equations. We find that the critical squeezing variance in the output spectrum will scale as ${\mathit{N}}^{\mathrm{\ensuremath{-}}1/2}$, giving the feature that these fluctuations are reduced rather than enhanced near the critical point.
We calculate the scale of the critical fluctuations in the quantum parametric oscillator at threshold. This allows us to asymptotically calculate the size of both the depletion of the pump mode and the amount of squeezing produced in the signal mode for large system size. In this driven, open quantum system, there is no known solution to the equilibrium density matrix. For this reason these results cannot be obtained using standard canonical ensemble methods. The calculations are compared to time dependent quantum stochastic simulations of the fully nonlinear coupled mode equations. We find that the critical squeezing variance in the output spectrum will scale as N 1 2 , giving the novel feature that these fluctuations are reduced rather than enhanced near the critical point.
The degenerate parametric oscillator above threshold is studied with phenomenological stochastic colored (nonwhite) pump noise for arbitrary pump to subharmonic relaxation rate. The current experimental limits of large intracavity threshold photon numbers (very small quantum noise) are considered, allowing for a semiclassical treatment of the system dynamics. A comparison between the effects of isotropic and squeezed pump noise on the internal transient and steady-state fluctuations is presented by simulation of the nonlinear semiclassical stochastic Langevin equations in the Wigner quadrature representation. It is found that the transient squeezing for the system starting in the unstable steady state (the vacuum) is not degraded by stochastic pump noise. A damped oscillatory behavior of the noise levels (periodic exchange of fluctuations between the squeezed quadrature of the signal and the pump) is observed for large damping of the signal in the turn on of pump depletion. Finally, it is shown that the limited squeezing above threshold in the steady state (50%) due to pump depletion can be enhanced if squeezed stochastic noise with sufficient significant spectral components (broadband squeezing) is fed to the pump. The above-threshold steady-state squeezing has been calculated analytically from the linearized stochastic equations and the effects of the time scale associated to the relaxation of the pump noise (the noise correlation time) compared to the dissipative time scale of the system and the pump to subharmonic loss ratio are presented.
A full quantum treatment about the process of parametric down-conversion with frequency degenerate but polarization non-degenerate in an optical parametric oscillator (OPO) cavity is presented. Using the linearized Langevin equations and spectral matrix, we calculated the squeezing spectra of the coupled mode in the output field. The squeezing as a function of driving field and detection frequency is obtained. The results obtained, which are compared with those found semiclassically by Reynaud et al. , indicate that it is possible to generate a two-mode coherent squeezed state with large amplitude. The quantum correlation between the signal and the idler modes is also discussed. It is shown that there is an inseparable relationship between the two-mode squeezing and the intermode quantum correlation.