Exact quantum distribution for parametric oscillators.
Optical parametric oscillators (OPOs) are quantum mechanical devices with a definite threshold for selfsustained oscillations [1—4]. They have played a central role in squeezing and twin-beam noise reduction experiments [5,6]. Theoretical understanding of these properties of the OPOs has been based mostly on linearized treatments above and below the threshold of oscillation [7—9]. In the region of threshold where linearization fails, the complex-P distribution has been used [10]. The complexP distribution, unfortunately, does not have the character of a probability density and is of limited use for gaining insights into the coherence properties of the OPOs. Another distribution, closely related to the complex P, is the positive-P distribution which is a true probability density [11]. In this paper we present an analytic solution for the positive-P function for the optical parametric oscillators. With the help of analytic solutions tremendous insights into the coherence properties of other oscillators have been gained [12—14]. Our analytic treatment is based on the observation that the quantum dynamics of OPOs is naturally confined to a bounded region in an eight-dimensional phase space. The solution presented here provides us with an elegant picture of how the coherence properties of the OPOs are transformed in the threshold region. It also allows us to discuss quantum features that survive even as the field amplitudes grow up to macroscopic values above threshold. We model the parametric oscillator by two quantized field modes of frequencies co] and co2 interacting with a third mode of frequency co3 = ~& + co2 inside an optical cavity via a ~~ ) nonlinearity. Modes co] and ~2 experience linear losses characterized by the decay rates yl = y2 = y, and mode co3 suffers linear losses characterized by the decay y3. The cavity is excited by a classical pump at frequency ~3. In the interaction picture the microscopic Hamiltonian takes the form