The quantum-fluctuations of the optical parametric oscillator. I

Our treatment is based on a microscopically correct Hamiltonian which contains the Bose-operators of the light modes and the Fermi-operators of the optically active electrons in the medium. The coupling between modes and atoms is taken from quantum-electrodynamics. Besides that, the light modes may interact with external “heat baths” like the mirrors, scattering centers etc., while the atoms interact with lattice vibrations, incoherent light fields etc. Using recently developed methods the effect of these heatbaths is taken into account in a quantum mechanically consistent fashion. In the present paper we apply quantum mechanical Langevin equations for the field and electron operators which contain dissipation and fluctuation terms. The elimination of the electron operators by an iteration procedure finally leaves us with a set of coupled nonlinear field equations which are shown to be quantum mechanically consistent. They are solved in the Heisenberg picture below threshold by linearization and well above threshold by quantum mechanical quasi-linearization. The solutions show that the line width of the signal mode below threshold is due to the vacuum fluctuations in the idler and vice versa, whereas the thermal noise of the resonator and the spontaneous emission noise of the medium may be neglected. Above threshold the linewidth is caused by the undamped diffusion of the phase difference between signal and idler, to which the vacuum fluctuations of both modes contribute in equal parts. The phase sum of both modes adiabatically follows the slow phase diffusion of the external pump light, produced by a laser, and therefore contributes to the linewidth too. Well above threshold the amplitudes are stable. Correlation and cross-correlation functions of their small residual fluctuations are calculated.