Moments of the intensity of a single-mode laser driven by coloured pump noises with a cross-correlation between the real and imaginary parts of the quantum noise

By means of the unified coloured noise approximation and phase lock, we study in this paper the stationary intensity distribution of the single-mode laser cubic model driven by coloured pump noise with cross-correlation between the real and imaginary parts of the quantum noise. We present a thorough discussion of how the cross-correlation \lambda_q between the real and imaginary parts of the quantum noise and the self-correlation time \tau of the pump noise determine the behaviours of the mean intensity 〈I〉 and variance \lambda_2 (0) for both below and above the threshold operation, and many new phenomena are discovered. When the laser is operated above the threshold, increasing the cross-correlation intensity \lambda_q makes the 〈I〉－\tau curves exhibit a "re-entrant phase transition". Whether the laser is above or below the threshold, the whole 〈I〉－\tau curve moves down as \lambda_q increases; however, when \lambda_q=1 (perfect cross-correlation), the curve abruptly runs up. A nonzero value of cross-correlation time \tau (\tau≠0) leads to the entire \lambda_2(0)－p′ curve being suppressed (here p′ is the pump noise intensity). This indicates the increasing precision of the laser beam due to the existence of the self-correlation time \tau. The behaviour of the \lambda_2 (0)－p′ curve as a function of \lambda_q is similar to that of the 〈I 〉－\tau curve against \lambda_q: that is, whether the laser is above or below the threshold, the \lambda_2 (0)－p′ curve moves up as \lambda_q increases; however, when \lambda_q=1, the curve suddenly moves down. Through the study in this paper, we can find a way to optimize for output laser intensity.