Quantum phase fluctuations of coherent and thermal light coupled to a non-degenerate parametric oscillator beyond rotating wave approximation

The essence of the rotating wave approximation (RWA) is to eliminate the non-conserving energy terms from the interaction Hamiltonian. The cost of using RWA is heavy if the frequency of the input radiation field is low (e.g. below optical region). The well known Bloch-Siegert effect is the out come of the inclusion of the terms which are normally neglected under RWA. We investigate the fluctuations of the quantum phase of the coherent light and the thermal light coupled to a nondegenerate parametric oscillator (NDPO). The Hamiltonian and hence the equations of motion involving the signal and idler modes are framed by using the strong (classical) pump condition. These differential equations are nonlinear in nature and are found coupled to each other. Without using the RWA, we obtain the analytical solutions for the signal and idler fields. These solutions are obtained up to the second orders in dimensionless coupling constants. The analytical expressions for the quantum phase fluctuation parameters due to Carruther's and Nieto are obtained in terms of the coupling constants and the initial photon numbers of the input radiation field. Moreover, we keep ourselves confined to the Pegg-Barnett formalism for measured phase operators. With and without using the RWA, we compare the quantum phase fluctuations for coherent and thermal light coupled to the NDPO. In spite of the significant departures (quantitative), the qualitative features of the phase fluctuation parameters for the input thermal light are identical for NDPO with and without RWA. On the other hand, we report some interesting results of input coherent light coupled to the NDPO which are substantially different from their RWA counterpart. In spite of the various quantum optical phenomena in a NDPO, we claim that it is the first effort where the complete analytical approach towards the solutions and hence the quantum phase fluctuations of input radiation fields coupled to it are obtained beyond rotating wave approximation. To have the feelings of the analytical solutions, we give few numerical estimates of the quantum phase fluctuation parameters relevant to a real experimental situation.