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Optics and Spectroscopy

Field dispersion in parametrically excited oscillator upon relaxation

Vladimir P Bykov

Published Jul 1, 2010• 0 citations
Publisher

Abstract

A complete system of solutions of the equation describing the parametric excitation of an oscillator is presented. This system makes it possible to take into account losses in a parametrically excited oscillator in terms of the perturbation theory. It is shown that, although the oscillator losses are small (the relative spectral width of a typical laser cavity is 10 −8 ), they limit the achievable squeezing coefficients by values on the order of ten. At present, the progress in resonator technique allows one to achieve relative spectral widths on the order of 10 −11 –10 −12 . Therefore, it can be expected to achieve squeezing coefficients on the order of 10 3 .

DOI
10.1134/s0030400x10070040

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A complete system of solutions of the equation describing the parametric excitation of an oscillator is presented. This system makes it possible to take into account losses in a parametrically excited oscillator in terms of the perturbation theory. It is shown that, although the oscillator losses are small (the relative spectral width of a typical laser cavity is 10 −8 ), they limit the achievable squeezing coefficients by values on the order of ten. At present, the progress in resonator technique allows one to achieve relative spectral widths on the order of 10 −11 –10 −12 . Therefore, it can be expected to achieve squeezing coefficients on the order of 10 3 .
The main features of squeezed light are described. Specifically, we reveal that the evolution of the squeezed light excited in the degenerate regime of parametric oscillation considerably differs from that of the light excited in the nondegenerate regime. It is demonstrated that in the nondegenerate regime (which is the actual regime of parametric oscillation) the minimum variance of the quantum state decreases only at the initial stage of excitation; afterwards the minimum variance starts to rise (as the maximum variance), thus obstructing the increase in the squeezing coefficient. The difference between the excitation regime of a typical laser and that of a parametric oscillator is discussed. We show that in a parametric oscillator the competition between central and side frequencies of the cavity spectrum is practically absent due to the smallness of nonlinearity factor, and, as a consequence, the width of the generated spectrum line does not narrow. In order to increase the competition between the frequency components, a buffer regime of the parametric-oscillating excitation is suggested. With the help of numerical simulations we show that this competition is really increased in the buffer regime.
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The effect of squeezing on the degenerate parametric oscillator is studied. It is shown that when the ordinary-vacuum reservoir is replaced by a squeezed-vacuum reservoir, the maximum 50% limit of the second-order squeezing inside the cavity can be crossed. It is also shown that the mean number of photons increases considerably. An exact steady-state solution for the Fokker-Planck equation for the field mode is also presented. The photon distribution for the intracavity field and the higher-order variances have been obtained by using the solution of this equation.
The theory developed in the first two articles of this series, dealing with the interaction between the electromagnetic field in a cavity resonator and a number of two-level molecules, is generalized to include a Gaussian spread in the molecular frequency. The center of the molecular frequency distribution coincides with the cavity resonant frequency. There is a coherent driving field in the cavity at the same frequency, and cavity loss is taken into account.Using the formalism previously developed for a quantum-mechanical field in a lossy cavity, expressions are obtained by means of second-order perturbation theory for the expectation values of the field strength and field energy in the cavity, and of the power loss by the molecules. It is shown that the parts of the field energy resulting from induced and spontaneous emission, respectively, initially increase as the square of the time and approach steady-state values after (different, in general) transient periods, each of which is determined by two time constants: cavity relaxation time and inverse molecular frequency spread.It is also shown that both the induced and spontaneous emission power radiated by the molecules increase initially linearly with the time and approach steady-state values after transient periods. For the induced emission power, the transient period is determined by only one time constant, the inverse molecular frequency spread, while for the spontaneous emission power it is determined both by the inverse molecular frequency spread and the cavity relaxation time. The ratio of induced to spontaneous emission is initially nnn, and approaches a steady-state value $n{[\mathrm{exp}({r}^{2})(1\ensuremath{-}\mathrm{erf}r)]}^{\ensuremath{-}1},$ where nnn is the driving field energy in units of the photon energy, and rrr is the ratio of the cavity resonance width to molecular frequency spread. The seeming inconsistency of this value with the classical value of the ratio of the Einstein coefficients is discussed.
Abstract A new approach to the quantum analysis of the spectral linewidths of optical parametric oscillators (OPOs) is proposed. The approach is based on introducing a Hermitian frequency deviation operator. The spectral linewidths in the triply resonant OPO above-threshold regime have been calculated. It has been found that the width of the generated spectral line depends on the loss at all interacting frequencies, the photon number of frequency under study, and the threshold photon number of pump. The formula obtained for the spectral width is general; in particular cases, it yields the well-known results. The developed simple approach can be applied to the quantum analysis of the spectra of various oscillatory systems, for example, the multifrequency parametric and non-linear optomechanical interactions and lasers.
We study the evolution of the photon-added coherent state | α, m ⟩ (introduced by Agarwal and Tara [Phys. Rev. A43, 492 (1991)]) due to a time dependence of the frequency of the electromagnetic field oscillator in a cavity or a vibrational frequency of an ion inside an electromagnetic trap. We give explicit expressions for the photon distribution function, mean values and variances of the quadrature components and of the photon number, the Wigner and Q -functions, etc. We show that the parametric excitation leads to strong oscillations of the photon (phonon) distribution function and changes the subpoissonian photon statistics to the superpoissonian one. Besides, it enables to achieve a larger squeezing coefficient than in the usual squeezed states.