Quantum parametric double Raman oscillators with co- and counterpropagating fields: Relative intensity squeezing and spatial photon correlations

We present a comprehensive study of co- and counterpropagating Stokes and anti-Stokes quantum fields in double Raman four-wave mixing parametric oscillators using full quantum Heisenberg-Langevin framework with noise operators. General analytical solutions of the fields operators at any point in the Raman medium are obtained for four cases: two possible copropagating (forward) and two counterpropagating (backward) Stokes and anti-Stokes. We analyze the symmetrical properties of the complex linear and nonlinear susceptibilities spectra of the quantum fields, nonclassicality of two-photon correlation functions, spatial variations of the quantum fields, and the two-mode relative intensity squeezing. We compare the results of forward and backward cases for several limiting double Raman schemes. We find interesting resonant effects of medium length, laser detunings and laser field strengths (Rabi frequencies) for backward-propagating geometries. Analysis of the solutions provide insights on the resonant conditions while computation over multiple variables enables us to identify the values of laser detuning, field strength, and propagation length that give enhanced nonclassical intensity squeezing and persistent correlations. The present work sets the crucial foundations for optimization of the nonclassicality of photons in double Raman systems and would be useful for quantum information storage, quantum nonlinear optics, and quantum spectroscopy.