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Quantum Electronics

Variance of a two-mode field under parametric excitation conditions

Vladimir P Bykov

Published Nov 30, 1997• 1 citation
Publisher

Abstract

The quantum aspects of parametric excitation of two electromagnetic modes are considered. It is shown that a Gaussian wave packet, evolving in the two-dimensional space of these modes, is the exact solution of the Schroedinger equation. Equations are derived for describing evolution of the packet parameters with time. Perturbation theory is used to obtain explicit expressions for these parameters at low parametric excitation rates. It is found that the variance of the sum of the fields of these modes oscillates at the pump frequency. In contrast to parametric excitation of one mode, the minimum and maximum variance both increase with time. The increase in the minimum variance accounts for the small squeezing coefficients obtained experimentally for optical parametric oscillators. The effect is described by two squeezing coefficients.

DOI
10.1070/qe1997v027n11abeh001082

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The quantum aspects of parametric excitation of two electromagnetic modes are considered. It is shown that a Gaussian wave packet, evolving in the two-dimensional space of these modes, is the exact solution of the Schroedinger equation. Equations are derived for describing evolution of the packet parameters with time. Perturbation theory is used to obtain explicit expressions for these parameters at low parametric excitation rates. It is found that the variance of the sum of the fields of these modes oscillates at the pump frequency. In contrast to parametric excitation of one mode, the minimum and maximum variance both increase with time. The increase in the minimum variance accounts for the small squeezing coefficients obtained experimentally for optical parametric oscillators. The effect is described by two squeezing coefficients.
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.
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 .
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.
By using the Lewis–Riesenfeld quantum invariant theory and properly choosing Hermitian invariant operator, a closed solution of the Schrödinger equation is derived for two forced quantum oscillators with mixing of two modes, and the quantum fluctuations in the output fields are evaluated. For the initial two-mode squeezed number or squeezed coherent state, in some particular conditions, the time evolution of the oscillators can not only preserve the initial two-mode squeezing, but also produce squeezing in the individual modes; and exhibit a periodical squeezing behaviour. For the initial two-mode number state or coherent state, there is no squeezing in the individual and mutual quadrature phases of the two-mode fields. Furthermore, regardless of which state above being initially considered, the quantum fluctuations of all the quadrature phases in the output fields are all independent of the driving parameters. In particular, for the initial two-mode coherent state, the variances of the output fields are also independent of other parameters in the Hamiltonian, and always preserve their initial values 1/4.
We investigate continuous-variable entangling resources on the basis of two-mode squeezing for all operational regimes of a nondegenerate optical parametric oscillator with allowance for quantum noise of an arbitrary level. The results for the quadrature variances of a pair of generated modes are obtained by using the exact steady-state solution of the Fokker-Planck equation for the complex-PPP-representation quasiprobability distribution function. We find a simple expression for the squeezed variances in the near-threshold range and conclude that the maximal two-mode squeezing reaches 50% relative to the level of vacuum fluctuations and is achieved at the pump field intensity close to the generation threshold. The distinction between the degrees of two-mode squeezing for monostable and bistable operational regimes is cleared up.
A degenerate parametric oscillator operating above threshold is studied when the cavity is injected by a broadband squeezed light at the pump frequency. Using the solution of the quantum Langevin equations for the signal and pump modes, we have studied the quantum fluctuations in the intracavity and transmitted field quadratures for the signal and pump modes. For the intracavity modes we studied the variance in the field quadratures, and for the transmitted modes we studied the spectra for the variance in the squeezed quadratures. Our results have revealed that the same relative cavity damping rates between the signal and the pump modes that predict enhanced squeezing in the intracavity do not necessarily predict enhanced squeezing in the transmitted modes.