VOLUME 74, NUMBER 12 PHYSICAL REVIEW LETTERS 20 MARcH 1995 Exact Quantum Distribution for Parametric Oscillators Recta Vyas and Surendra Singh Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701 (Received 15 August 1994) An exact quantum distribution for the nondegenerate parametric oscillators is presented and used to discuss their coherence properties. It is found that while each mode individually approaches a classical state, many quantum features exhibited by their combination survive even in the semiclassical limit. PACS numbers: 42. 50.Dv, 42. 50.Ar, 42.65.Ky Optical parametric oscillators (OPOs) are quantum mechanical devices with a definite threshold for self- sustained oscillations [1 — 4]. They have played a central role in squeezing and twin-beam noise reduction experi- ments [5, 6]. Theoretical understanding of these properties of the OPOs has been based mostly on linearized treat- ments 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 complex- P distribution, unfortunately, does not have the character of a probability density and is of limited use for gain- ing 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 probabil- ity density [11]. In this paper we present an analytic solution for the positive-P function for the optical para- metric oscillators. With the help of analytic solutions tremendous insights into the coherence properties of other oscillators have been gained [12 — 14]. Our analytic treat- ment is based on the observation that the quantum dy- namics of OPOs is naturally confined to a bounded region in an eight-dimensional phase space. The solution pre- sented 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 H = E ItK(at a2 a3 — a3 ala2) + i hy3e(a3 — a3) + Hl„, , (1) where a, and a, are the annihilation and creation opera- tors for the modes, ~ is the mode coupling constant, e is the pump field amplitude, and H~„, describes mode losses. This nonlinear quantum mechanical problem can be mapped into a classical stochastic process by using the positive-P representation [11]. Eliminating the pump mode adiabatically (y3 » y) we obtain the following set of Ito stochastic differential equations: 2 1 O. '2g CI l CI2 A'2g + np np ~np X Qa — 2nln2(gl + tie)2, 0 CVi = Ai + 0 2 l n2 = A2 + n]g n2nl Ct lg + np np Jnp X QO 2nl n2 (tll l 'g2), (2) (3) CJ' 2 l A j/ A lg + 0'2 n lg &2&2/ + np np np X Qlr 2nlsn2g ('l73 + l 'g4), 0 2 l A2g = A2g + A'] A'2gCIl CIlg + alp fLp P1p X Qcr 2nlgn2g (F13 l 7)4), (4) (5) where g; are white noise Gaussian random processes with zero mean and correlation functions given by (rl;(t)q, (t')) = 6;, 6(t — t'). Here time is measured in units of y ', np = 2yy3/K2 is parameter that sets the scale for the number of photons necessary to explore the nonlinearity of interaction, and o. = 2y3e/K is a dimen- sionless measure of the pump field amplitude scaled to give o. = np as the threshold condition. In the absence of mode losses adiabatic elimination of the pump mode is not justified [15]. Note that the adiabatic approximation does not amount to a neglect of the entanglement of pump and down-converted modes. Complex variables n; and n;. correspond to a; and a;, respectively. In the positive-P representation, a; and a;. are not complex conjugates of each other. Equations (2) — (5) describe trajectories in an eight- dimensional phase space. An examination of these equations reveals that the eight-dimensional phase space is naturally divided into two subspaces. If we consider the four-dimensional subspace n2 = (nl)*, n2. = (nl. )*, and (nl ( ( o. /2, (n2. (2 ( rr/2, we notice that the trajectories starting in this subspace initially remain confined to this subspace. In other words, the condition n2 = (nl)*, n2. = (nl. )* is preserved for all times if initially we start out in this subspace. The initial state, 2208 0031-9007/95/74(12)/2208(4)$06. 00 1995 The American Physical Society