PHYSICAL REVIEW A VOLUME 44, NUMBER 3 1 AUGUST 1991 Higher-order squeezing in nondegenerate four-wave mixing Nadeem A. Ansari and M. Suhail Zubairy Department of Electronics, Quaid i A--zam University, Islamabad, Pakistan (Received 27 December 1990; revised manuscript received 7 March 1991) Higher-order squeezing in nondegenerate four-wave mixing is evaluated by using the steady-state solu- tion of the Fokker-Planck equation in the Q representation. It is shown that under suitable conditions the amount of squeezing increases with the order of squeezing. I. INTRODUCTION Squeezed states have attracted a great deal of attention in recent years due to their potential application in com- munication, noise-free linear amplification, and weak- signal detection [1]. In particular, squeezing has been predicted in degenerate and nondegenerate four-wave mixing [2 — 4]. Experimentally second-order squeezing has been observed in a four-wave mixing by Slusher et al. [5] and Levenson et al. [6]. Hong and Mandel [7,8] in- troduced the concept of higher-order squeezing and dis- cussed it in many systems, such as degenerate parametric down conversion, second-harmonic generation, and reso- nance fluorescence. In this paper we have evaluated the fourth- and sixth- order squeezing in nondegenerate four-wave mixing. We take advantage of our recent calculations [9] in which we determined the exact steady-state solution of the Fokker-Planck equation in the Q representation. We use this solution to calculate the higher-order moments, which are essential in the calculation of fourth- and sixth-order squeezing. We then plot the higher-order moments versus mode spacing between the adjacent modes, for certain values of detuning of the pump fre- quency from the atomic transition frequency. This plot identifies the regions where higher-order squeezing can be achieved in nondegenerate four-wave mixing. II. FOKKER-PI. ANCE EQUATION AND ITS STEADY-STATE SOLUTION In four-wave mixing the pump mode is servo locked at frequency v2 and side mode frequencies are locked at their respective frequencies v& and v3 such that the mode-locking condition v2 — v, =v3 — v2 is satisfied. We consider the pump field to be arbitrarily intense and treat it classically. The side modes are considered weak and treated quantum mechanically. The atomic system con- sists of two-level atoms, and they are being pumped to the upper level only. The equation of motion for the reduced density matrix for the field modes of frequencies vi and v3 is [10] p= — [ A 1 ]sHB(pa i a 1 — a i pa 1 ) — ([&1]sHB+vi/2Q1)(a ia ip — a ipa 1 ) + [ Ci ]sHB(a 1a 3p a 3pa 1 ) + I D 1 ]sHB(pa ia 3 a 1 pa 3 ) + [ 1~3 ] + H' c' where v/Q is the cavity-loss rate for the given cavity configuration, a, and a; are the destruction and creation operators for the ith mode, and 1~3 represents the same term with subscript 1 and 3 interchanged. The expres- sions [ A, ]sHB, [8, ]sHB, [Ci ]sHB, and [D, ]sHB appearing in Eq. (1) are averaged over spatial hole burning and are given in Ref. [11]. By using Q representation for the field of modes 1 and 3 in terms of the field density matrix p, i.e. , eigenstate of a; (i = 1, 3) with eigenvalue a;, i.e. , a, ~ai, a3) =a, ~ai, a3) (i =1,3) . (3) In Ref. [9] we have derived an equivalent Fokker- Planck equation from the density matrix equation of motion for the field modes and calculated its exact steady-state solution. The solution is of the form =1 Q( 1»)= — e p[ — ( =1 Q(a„a, )= — (a„a3~p a„a, ), (2) +L, , 3a+a3L, *, 3aa)]3, (4) we can convert the density matrix equation of motion (1) into an equivalent Fokker-Planck equation. Here the disadvantage of using P representation for the nonclassi- cal states such as squeezed states is that P representation becomes nonpositive definite. In Eq. (2) ~ai, a3) is the where the normalization constant X is N= exp — L&j o& +133 (x3 +L13aia3+L»aia3 )]d a, d a3 2214 1991 The American Physical Society