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Springer proceedings in physics

Langevin Equations for Squeezing by Means of Non-linear Optical Devices

Francisco Javier Bermejo, P. García Fernández, Pere Colet, S. Balle, Raúl Toral, Maxi San Miguel

Published Jan 1, 1991• 0 citations
Publisher
DOI
10.1007/978-3-642-76373-1_8

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Principal Squeezing in Optical Devices

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Springer eBooks•Jan 1, 1990•0 citations
We show that the Langevin equation for a nonlinear-optical system may be obtained directly from the Heisenberg equation of motion for the annihilation operators, provided a certain linearization procedure is valid. We apply the technique to the parametric oscillator used to generate squeezed light and compare our results to those obtained from Fokker-Planck-type equations. We argue that, only when the Wigner, as opposed to the P or Q, representation of quantum optics is used, do we get a correct description of the underlying stochastic process. We show how the linearization procedure may be carried out to describe the operation of the parametric oscillator both below threshold, where a squeezed vacuum state results, and above threshold, where we find a squeezed coherent state. In the region of the threshold a heuristic extension of the method leads to a possible description of the system by means of a nonlinear Langevin equation.
It is well known that the squeezing spectrum of the field exiting a nonlinear cavity can be directly obtained from the fluctuation spectrum of normally ordered products of creation and annihilation operators of the cavity mode. In this article we show that the output field squeezing spectrum can be derived also by combining the fluctuation spectra of any pair of s -ordered products of creation and annihilation operators. The interesting result is that the spectrum obtained in this way from the linearized Langevin equations is exact, and this occurs in spite of the fact that no s -ordered quasiprobability distribution verifies a true Fokker–Planck equation; that is, the Langevin equations used for deriving the squeezing spectrum are not exact. The (linearized) intracavity squeezing obtained from any s -ordered distribution is also exact. These results are exemplified in the problem of dispersive optical bistability.
The theory of a two-mode nondegenerate parametric amplifier in a cavity is reformulated in terms of quadrature-phase-amplitude variables. The corrrespondence with a genuine classical stochastic linear process is found (non-negative-definite diffusion matrices) for the case of a cavity device immersed in thermal or ordinary (nonsqueezed) vacuum sources. A special kind of squeezing, i.e., quadrature squeezing [B. L. Schumaker, Phys. Rep. 135, 318 (1986)], is found to be characteristic of the internal steady state for the case of a cavity model subjected to an additional phase-sensitive noise source coupled to one of the two internal modes. Finally, the usual squeezing spectrum of the output field is calculated in both cases by means of an input-output formalism based upon a symmetric ordering scheme of noncommuting operators, well adapted for the quadrature-phase description of the fields involved in the interaction.