Correlations, transients, bistability, and phase-transition analogy in two-mode lasers

The general time-dependent problem associated with a two-mode-laser Fokker-Planck equation has been considered. Analytic expressions for the eigenvalues and the eigenfunctions have been derived for the below- and above-threshold operation of the system. It is shown that depending on the values of the pump parameters $a_1,a_2$ and the coupling constant $\xi$, a wide variety of dynamical behavior may be encountered. Thus for $\xi <1\mathrm{}$ interesting threshold effects may appear in the behavior of correlation times and linewidths as a function of excitation which may change their behavior whenever a mode passes through its threshold of oscillation. For $\xi =1\mathrm{}$ the correlation time of the stronger mode may pass through a maximum before it starts to decrease with increasing excitation. The weaker mode may show particularly interesting behavior for $\xi =1\mathrm{}$. Both the correlation time and the linewidth may approach a constant value independent of the excitation. For $\xi >1\mathrm{}$ the system may show bistability. Our analysis provides a deeper mathematical understanding of the onset of bistability and the analogy to the thermodynamic phase transition. Expressions for the switching time have been derived using several different procedures. We have also been able to derive closed-form expressions for the two-time joint probability densities. It should be possible to test some of these predictions in photon-counting experiments. Finally, we mention that our results and method should also be applicable in other similar problems, e.g., to the classical theory of coupled fields.