Stochastic quantum dynamics of a continuously monitored laser

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Abstract
The stochastic dynamics of the quantum state of a laser conditioned on continuous intensity and phase-sensitive measurements on its output are examined. We first develop the essential model for a Poissonian laser, and then generalize it for regularly pumped lasers. We show that the rate of phase diffusion (which gives the laser linewidth) is not affected by the regularity of the pump. In both cases, heterodyne detection causes the phase variance in the conditioned laser state to become very small, yet remain significantly above (at least three times) that of a coherent state. The phase diffusion is manifest by a random walk undertaken by the mean phase, with the stochasticity arising from the local-oscillator shot noise. In contrast, intensity measurements have no effect on a Poissonian laser. For a perfectly regularly pumped laser, coarse-grained intensity measurements (for which an approximate theory is developed here) collapse the state to one with an arbitrarily small photon-number variance-to-mean ratio.