Threshold and Linewidth of a Mirrorless Parametric Oscillator

Abstract

We analyze the above-threshold behavior of a mirrorless parametric oscillator based on resonantly enhanced four wave mixing in a coherently driven dense atomic vapor. It is shown that, in the ideal limit, an arbitrary small flux of pump photons is sufficient to reach the oscillator threshold. We demonstrate that due to the large group velocity delays associated with coherent media, an extremely narrow oscillator linewidth is possible, making a narrow-band source of non-classical radiation feasible.Stable and low-noise sources of coherent and nonclassical radiation are of interest in many areas of laser physics and quantum optics. Such sources have a wide range of applications such as frequency standards, optical magnetometry, gravitational wave detection, and highprecision spectroscopy.The present theoretical work is motivated by recent experiments demonstrating a phase transition to mirrorless oscillation of counter-propagating Stokes and antiStokes fields in resonant, double-Λ Raman media [1]. In contrast to earlier studies involving instabilities in alkali vapors [2][3][4][5], this oscillation could be achieved with pump fields of µW power (nano Joule pulse energy) and is accompanied by a dramatic narrowing of the beat signals between driving and generated fields. Oscillations of this kind are a clear manifestation of atomic coherence and interference effects, which have recently lead to many exciting developments in resonant nonlinear optics [6][7][8][9]. In particular, the unusual efficiency of the present processes is expected to lead to a new regime of quantum nonlinear optics in which interactions at a level of few light quanta are feasible. Furthermore, the photon pairs generated can possess nearly ideal quantum correlations, resulting in almost complete squeezing of quantum fluctuations [10].We here study theoretically the quantum dynamics of the mirrorless oscillator above threshold. We show that for an infinitely long lived atomic dark state an arbitrary small stationary flux of pump photons is sufficient to maintain the oscillation. We furthermore analyze frequency locking and linewidth narrowing of the beat note between oscillation-and pump frequencies. In particular, we show that the beat-note linewidth is given by an expression similar to the Schawlow-Townes formula for lasers where the cavity storage time is replaced by the group delay time τ gr in the medium. Due to the large linear dispersion associated with electromagnetically induced transparency (EIT) in optically thick media, the group delay can be extremely large [11][12][13] leading to a very small beat-note linewidth. This effect is analogous to the line-narrowing in intracavity EIT [14,15]. Since only very small pump powers are needed to reach threshold, ac-Stark shifts and the associated systematic effects on the beat-note frequency can be made very small. The combination of line-narrowing and small pump-power requirements makes the mirrorless parametric oscillator an interesting novel source of stable and narrow-linewidth non-classic...

Mirrorless Oscillation Based on Resonantly Enhanced 4-Wave Mixing: All-Order Analytic Solutions

“…Expanding these expressions into third order of the generated fields E 1 and E 2 reproduces the equations of [15]. Equations (23-26) together with the boundary-conditions…”

“…Close to the threshold of oscillation an almost perfect suppression of quantum fluctuations of one quadrature amplitude of a combination mode of the generated fields occurs [13,14]. Also sufficiently above threshold light fields with beat-frequencies tightly locked to the atomic Raman-transition and extremely low relative bandwidth are generated [15].…”

“…Not to far above threshold, the square root in (39) can be expanded and one recovers the third-order solution obtained in [15]:…”

“…The phase-locked emission of the second Stokes photon then closes the feedback loop. We have shown in [15] that phase-matching enforces a strong pulling of the beat-note of generated and pump fields to the atomic Raman transition. I will therefore assume here that both Λ systems are in perfect two-photon resonance.…”

Quantum theory of resonantly enhanced four-wave mixing: Mean-field and exact numerical solutions

“…A linear fluctuation analysis has shown that close to the threshold of oscillation an almost perfect suppression of quantum noise of one quadrature amplitude of a combination mode of the generated fields occurs [22,25]. In addition, sufficiently above threshold, light fields with beat-frequencies tightly locked to the atomic Raman-transition and extremely low relative bandwidth are generated [26].…”