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Physical Review A

Squeezing resulting from a fourth-order interaction in a degenerate parametric amplifier with absorption losses

P. Garca-Fernández, Pere Colet, Raúl Toral, Maxi San Miguel, Francisco Javier Bermejo

Published May 1, 1991• 9 citations
Publisher

Abstract

The squeezing properties of a model of a degenerate parametric amplifier with absorption losses and an added fourth-order nonlinearity have been analyzed. The approach used consists of obtaining the Langevin equation for the optical field from the Heisenberg equation provided that a linearization procedure is valid. The steady states of the deterministic equations have been obtained and their local stability has been analyzed. The stationary covariance matrix has been calculated below and above threshold. Below threshold, a squeezed vacuum state is obtained and the nonlinear effects in the fluctuations have been taken into account by a Gaussian decoupling. In the case above threshold, a phase-squeezed coherent state is obtained and numerical simulations allowed to compute the time interval, depending on the loss parameter, on which the system jumps from one stable state to the other. Finally, the variances numerically determined have been compared with those obtained from the linearized theory and the limits of validity of the linear theory have been analyzed. It has become clear that the nonlinear contribution may perhaps be profitably used for the construction of above-threshold squeezing devices.

DOI
10.1103/physreva.43.4923

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This thesis is primarily a theoretical study of degenerate parametric amplification as a means of generating squeezed-state light. i) A wideband traveling-wave formalism is developed for analyzing quantum mechanically a degenerate parametric amplifier. The formalism is based on spatial differential equations — spatial Langevin equations — that propagate temporal Fourier components of the field through the nonlinear medium. In addition to the parametric nonlinearity, the Langevin equations include absorption and associated fluctuations, dispersion, and pump quantum fluctuations. The dominant effects of dispersion and pump quantum fluctuations on the squeezing produced by a degenerate parametric amplifier are analyzed. ii) The wideband formalism of i) is used to carry out a more detailed analysis of the effects of phase mismatching. With the assumption of a lossless medium and a classical pump, we find that parametric amplification is capable of generating squeezed-state light over a wide band if materials with large χ(2) nonlinearities can be found, and that the squeezing bandwidth can be enhanced by phase mismatching away from degeneracy. iii) We consider again the effect of pump quantum fluctuations on the squeezing produced by parametric amplification. We perform discrete-mode calculations for a parametric amplifier with a quantum pump, and discuss some of the limitations of calculations of this sort in quantum optics. We derive stochastic differential equations (SDEs) for one- and two-mode parametric amplifiers, and from them obtain an iterative solution showing that pump quantum fluctuations impose a limitation on the degree of squeezing obtainable from a parametric amplifier. iv) A possible application of squeezing is considered; in particular, we study the effects of squeezing the intracavity noise in a laser oscillator. We solve the classical noise problem of a realistic laser model by making a bold — and possibly unrealizable — assumption, that the in-phase and quadrature Langevin sources which are responsible for the "noisiness" of the laser can be squeezed. We show that the effect of squeezing the in-phase quadrature is to reduce the phase noise, including the linewidth, of the laser but, due to amplitude-phase coupling, not to eliminate them altogether.

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Quantum fluctuation of nonlinear degenerate optical parametric amplification

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An analytical solution of the Fokker–Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).

Journal of Modern Optics•Jun 27, 2006•5 citations
The squeezing properties of a model of a degenerate parametric amplifier with absorption losses and an added fourth-order nonlinearity have been analyzed. The approach used consists of obtaining the Langevin equation for the optical field from the Heisenberg equation provided that a linearization procedure is valid. The steady states of the deterministic equations have been obtained and their local stability has been analyzed. The stationary covariance matrix has been calculated below and above threshold. Below threshold, a squeezed vacuum state is obtained and the nonlinear effects in the fluctuations have been taken into account by a Gaussian decoupling. In the case above threshold, a phase-squeezed coherent state is obtained and numerical simulations allowed to compute the time interval, depending on the loss parameter, on which the system jumps from one stable state to the other. Finally, the variances numerically determined have been compared with those obtained from the linearized theory and the limits of validity of the linear theory have been analyzed. It has become clear that the nonlinear contribution may perhaps be profitably used for the construction of above-threshold squeezing devices.
The squeezing properties of non-degenerate parametric amplification and oscillation with pump depletion and non-homogeneous absorption losses are studied. A method based in the Wigner representation is used which allows one, within a linearization scheme, to derive directly the Langevin equations for the involved optical fields from their quantum Heisenberg equations. Non-homogeneous absorption losses for the three beams are included in order to analyse the sensitivity of the squeezing spectra to the asymmetry of the losses. Intracavity squeezing spectra are calculated within this method both below- and above-threshold regimes. Special attention is given to the squeezing properties of the phase sum and amplitude difference of the correlated beams and amplitude of a single beam above threshold, where the role of loss asymmetry and pump losses is analysed in detail.
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.
This thesis is primarily a theoretical study of degenerate parametric amplification as a means of generating squeezed-state light. i) A wideband traveling-wave formalism is developed for analyzing quantum mechanically a degenerate parametric amplifier. The formalism is based on spatial differential equations — spatial Langevin equations — that propagate temporal Fourier components of the field through the nonlinear medium. In addition to the parametric nonlinearity, the Langevin equations include absorption and associated fluctuations, dispersion, and pump quantum fluctuations. The dominant effects of dispersion and pump quantum fluctuations on the squeezing produced by a degenerate parametric amplifier are analyzed. ii) The wideband formalism of i) is used to carry out a more detailed analysis of the effects of phase mismatching. With the assumption of a lossless medium and a classical pump, we find that parametric amplification is capable of generating squeezed-state light over a wide band if materials with large χ(2) nonlinearities can be found, and that the squeezing bandwidth can be enhanced by phase mismatching away from degeneracy. iii) We consider again the effect of pump quantum fluctuations on the squeezing produced by parametric amplification. We perform discrete-mode calculations for a parametric amplifier with a quantum pump, and discuss some of the limitations of calculations of this sort in quantum optics. We derive stochastic differential equations (SDEs) for one- and two-mode parametric amplifiers, and from them obtain an iterative solution showing that pump quantum fluctuations impose a limitation on the degree of squeezing obtainable from a parametric amplifier. iv) A possible application of squeezing is considered; in particular, we study the effects of squeezing the intracavity noise in a laser oscillator. We solve the classical noise problem of a realistic laser model by making a bold — and possibly unrealizable — assumption, that the in-phase and quadrature Langevin sources which are responsible for the "noisiness" of the laser can be squeezed. We show that the effect of squeezing the in-phase quadrature is to reduce the phase noise, including the linewidth, of the laser but, due to amplitude-phase coupling, not to eliminate them altogether.
In quantum information processing, control of quantum entanglement is of central importance, particularly for applications in quantum computing. One established method for generating entanglement is squeezing, which exploits the uncertainty principle: fluctuations in one of a pair of conjugate variables are suppressed at the expense of enhanced fluctuations in the other. A representative approach for squeezing is the parametric amplifier. Depending on the parametric excitation amplitude, the system exhibits regimes of stability and instability of the trivial solution. Conventional studies of squeezing have focused on the stable regime, where the squeezing is maximized near the instability threshold. When the parametric excitation amplitude is increased further and the system enters the unstable regime, the system can reach a steady state due to cubic nonlinearity. However, the squeezing properties of this state have not been fully clarified. In this study, as a first step toward a quantum-mechanical treatment, we investigate squeezing in a classical parametric amplifier in the unstable regime. To enable analytical treatment, we approximate the white noise by a narrowband resonant cosine component and neglect stochasticity. The analysis shows that the steady-state amplitude depends on the phase difference between the noise and the parametric excitation, as in the stable regime, indicating a squeezing effect. Moreover, the conditions under which the steady-state amplitude is maximized and minimized with respect to this phase difference are derived. In addition, we analytically determine how the parametric excitation and noise modify the cubic-nonlinearity-induced bifurcation structure. Based on these findings, we propose a squeezing mechanism that exploits the properties of the unstable regime, unlike conventional approaches. Numerical simulations incorporating stochastic noise reveal the emergence of squeezing, although the effect is weaker than in the stable regime. These results suggest that the unstable regime may offer a possible route for utilizing previously unexplored characteristics.
An analytical solution of the Fokker–Planck equation for the nonlinear degenerate optical parametric amplifier (DOPA) is presented, taking into account the influence of pump depletion on the generation of squeezed light. Results conform to those obtained using perturbation series expansion theory near threshold, and also apply to the whole region far away from threshold. When the nonlinear term(η → 0) is neglected, the solution transitions naturally to the linear approximation solution; when the nonlinear term is retained (∞ η), in the case μ → 0, the quantum fluctuations are close to vacuum fluctuations; in the case μ ≫ 1, squeezing increases, and tends to the result obtained using linear theory, 1/(1 + μ).