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Physical Review A

Effect of finite-bandwidth squeezing on inhibition of atomic-phase decays

A. S. Parkins, Crispin W. Gardiner

Published May 1, 1988• 100 citations
Publisher

Abstract

The inhibition of atomic-phase decays by squeezed light, and consequent line narrowing, is reexamined for the case of finite-bandwidth squeezing, as produced in the output of a degenerate parametric amplifier. The extension of previous work to incorporate the effect of nonwhite, or colored, noise is accomplished via an adjoint equation, which gives the time evolution of a matrix functional of the incoming field. Simple equations for atomic variables are derived in which noise operators are defined so as to commute with each other and with all other quantities. This facilitates the reduction of a purely quantum-mechanical problem to a classical problem, solvable by classical stochastic-approximation methods. In particular, small-noise-approximation methods and simulations are carried out for a wide range of parameters describing the squeezed-field input. The method of simulation is extended to the computation of correlation functions and spectra for a realistically achievable set of parameters, with consideration given to reflections of the input in the emitted light. The fundamental effect is confirmed, and modifications to previous results are given.

DOI
10.1103/physreva.37.3867

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The inhibition of atomic-phase decays by squeezed light, and consequent line narrowing, is reexamined for the case of finite-bandwidth squeezing, as produced in the output of a degenerate parametric amplifier. The extension of previous work to incorporate the effect of nonwhite, or colored, noise is accomplished via an adjoint equation, which gives the time evolution of a matrix functional of the incoming field. Simple equations for atomic variables are derived in which noise operators are defined so as to commute with each other and with all other quantities. This facilitates the reduction of a purely quantum-mechanical problem to a classical problem, solvable by classical stochastic-approximation methods. In particular, small-noise-approximation methods and simulations are carried out for a wide range of parameters describing the squeezed-field input. The method of simulation is extended to the computation of correlation functions and spectra for a realistically achievable set of parameters, with consideration given to reflections of the input in the emitted light. The fundamental effect is confirmed, and modifications to previous results are given.
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.
We derive stochastic density-matrix equations for an atom strongly driven by finite-bandwidth squeezed light. The quantum properties of the light are accounted for by a doubling of dimensions of the stochastic process for c-number electric field amplitudes. Saturation properties and the weak-field absorption spectrum of a two-level atom embedded in finite-bandwidth squeezed light and driven by a coherent field are calculated. We discuss the effect of finite bandwidth of the squeezed light in obtaining subnatural linewidth in the atomic absorption spectra, based on nonperturbative solutions of the stochastic optical Bloch equations.
The resonance fluorescence of a two-level atom driven by a coherent laser field and damped by a finite bandwidth squeezed vacuum is analysed. We extend the Yeoman and Barnett technique to a non-zero detuning of the driving field from the atomic resonance and discuss the role of squeezing bandwidth and the detuning in the level shifts, widths and intensities of the spectral lines. The approach is valid for arbitrary values of the Rabi frequency and detuning but for the squeezing bandwidths larger than the natural line-width in order to satisfy the Markoff approximation. The narrowing of the spectral lines is interpreted in terms of the quadrature-noise spectrum. We find that, depending on the Rabi frequency, detuning and the squeezing phase, different factors contribute to the line narrowing. For a strong resonant driving field there is no squeezing in the emitted field and the fluorescence spectrum exactly reveals the noise spectrum. In this case the narrowing of the spectral lines arises from the noise reduction in the input squeezed vacuum. For a weak or detuned driving field the fluorescence exhibits a large squeezing and, as a consequence, the spectral lines have narrowed linewidths. Moreover, the fluorescence spectrum can be asymmetric about the central frequency despite the symmetrical distribution of the noise. The asymmetry arises from the absorption of photons by the squeezed vacuum which reduces the spontaneous emission. For an appropriate choice of the detuning some of the spectral lines can vanish despite that there is no population trapping. Again this process can be interpreted as arising from the absorption of photons by the squeezed vacuum. When the absorption is large it may compensate the spontaneous emission resulting in the vanishing of the fluorescence lines.
A stochastic density-matrix equation for an atom strongly driven by finite-bandwidth squeezed light is derived. The quantum properties of the light are accounted for by a doubling of dimensions of the stochastic process for ccc-number electric field amplitudes. The possibility of obtaining reduced atomic linewidths in the absorption spectrum of the atom embedded in colored squeezed light is discussed, based on nonperturbative solutions of stochastic optical Bloch equations.
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.