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Journal of Modern Optics

Non-classical Properties of Non-degenerate Parametric Amplifiers and Oscillators with Pump Depletion and Non-homogeneous Losses Using the Wigner Representation

Jesús Martínez-Linares, P. Garcia-Fernandez

Published Jan 1, 1994• 9 citations
Publisher

Abstract

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.

DOI
10.1080/09500349414550091

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In this paper, we first find out the analytic solution of the time-dependent Fokker-Planck equation of the non-degenerate optical parametric amplification (NOPA) system under the consideration of the dispersion, the loss and the pump depletion effects. Then, through the numerical calculation, we obtain the squeezing characteristic of the degenerate optical parametric amplification (DOPA) system with dispersion. the research indicates: the dispersion effect stems from the nonlinear susceptibility change from χ″ toχ″/{1+σ2/}/+2, with the increasing of the dispersion effect, the general feature of the squeezing curves beeps unchanged, and the curves contract toward left. The maximum squeezing approaches to the linear theory 1/(1+μ). Finally, we obtain the entanglement characteristic of the NOPA system with dispersion. We find out when σ is given, with the increasing of pump parameter μ, he corresponding phase makes a large change. The nonlinear susceptibility changes many times. When the polarity is positive, the system obtains the gain, when the polarity is negative, the system suffers the loss, but the gain is mainly dissipated by the loss, so the net gain is small, the squeezing is also small. The minimum variance V1 reduces gradually, and the whole curve moves to the right, approaches to the linear theory 1/(1+μ).

Acta Physica Sinica•Jan 1, 2010•0 citations
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.
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.
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.
Degenerate four-wave mixing (DFWM) in optical fibres can be used to obtain gain and to generate squeezed states, by utilizing a balanced Mach-Zehnder fibre interferometer (MZFI) with two non-linear arms, which behaves as a degenerate parametric amplifier. In a classical derivation for the optical gain; pump depletion and propagation loss are considered. Maximum achievable gain is calculated in the presence of loss. A numerical study compares the signal gain under different input pump-to-signal field ratios and its limit (undepleted pump), which is the assumption for the derivation of squeezing. Pump depletion and fibre loss hinder the validity of using large non-linear phase shifts to generate squeezed states. A semiclassical derivation provides the correct expression for the degree of squeezing in the absence of loss and pump depletion. Quantum-mechanical solutions provide the same result. The squeezing limitation due to loss is studied by a lumped-parameter model.
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.
In this paper, we first find out the analytic solution of the time-dependent Fokker-Planck equation of the non-degenerate optical parametric amplification (NOPA) system under the consideration of the dispersion, the loss and the pump depletion effects. Then, through the numerical calculation, we obtain the squeezing characteristic of the degenerate optical parametric amplification (DOPA) system with dispersion. the research indicates: the dispersion effect stems from the nonlinear susceptibility change from χ″ toχ″/{1+σ2/}/+2, with the increasing of the dispersion effect, the general feature of the squeezing curves beeps unchanged, and the curves contract toward left. The maximum squeezing approaches to the linear theory 1/(1+μ). Finally, we obtain the entanglement characteristic of the NOPA system with dispersion. We find out when σ is given, with the increasing of pump parameter μ, he corresponding phase makes a large change. The nonlinear susceptibility changes many times. When the polarity is positive, the system obtains the gain, when the polarity is negative, the system suffers the loss, but the gain is mainly dissipated by the loss, so the net gain is small, the squeezing is also small. The minimum variance V1 reduces gradually, and the whole curve moves to the right, approaches to the linear theory 1/(1+μ).