Quantum Dynamics of Interacting Modes in Intracavity Third Harmonic Generation Process

We study the quantum dynamics of the number of photons of the interacting modes, the dynamics of the quantum entropy, and the Wigner function of the states of the fundamental and the third harmonic modes for the process of intracavity third harmonic generation. It is shown that the quantum dynamics of the system strongly depends on the external resonant perturbation of the fundamental mode and on the coupling coefficient of the interacting modes. In the region of long interaction times, the modes of the field can be both in stable and in unstable states—depending on the above-mentioned quantities. In the paper, we also investigate the dynamics of transition of the system from stable to unstable states. 1. Introduction For certain optical processes, such as the intracavity generation of the second and third harmonics [1–3], stationary solutions for the dynamics of the number of photons are stable only for relatively small pump amplitudes. For these systems, a certain critical value of the pump field exists above which small fluctuations in the system do not decay, and the dynamics of the semiclassical value of the photon number changes to the regime of self-oscillations. Among the unstable optical systems mentioned above, the intracavity second harmonic generation (SHG) is rather well investigated. Studies [1, 2, 4–10] are devoted to the investigation of the behavior of the intracavity SHG above the bifurcation point of the optical system. As compared to the case of the SHG, the unstable behaviour of intracavity third harmonic generation (THG) is insufficiently studied. In [3], the Langevin equations for stochastic field amplitudes for the THG process were derived in the positive -representation. The bifurcation point of the system was found, and it was shown that, above this point, the dynamics of the number of photons of the interacting modes changes to the regime of self-oscillations. Then, in [11], the distribution functions for the phases of the fundamental mode and of the third harmonic mode above the bifurcation point of the system were studied in the positive -representation. The distribution functions were shown to have a two-component structure. In addition to this, the functions of joint distribution of the number of photons and phases of the interacting modes were studied. In [12], the distribution functions of the number of photons of the fundamental and the third harmonic modes above the bifurcation point of the system, as well as the joint distribution function of the number of photons of the interacting modes, were studied in the positive