Higher-order interactions in quantum optomechanics: Analytical solution of nonlinearity

A method is described to solve the nonlinear Langevin equations arising from quadratic interactions in quantum mechanics. While, the zeroth order linearization approximation to the operators is normally used, here first and second order truncation perturbation schemes are proposed. These schemes employ higher-order system operators, and then approximate number operators with their corresponding mean boson numbers, only where needed. Spectral densities of higher-order operators are derived, and an expression for the second-order correlation function at zero time-delay has been found, which reveals that the cavity photon occupation of an ideal laser at threshold reaches $\sqrt{6}-2$, in good agreement with extensive numerical calculations. As further applications, analysis of the quantum anharmonic oscillator, calculation of $Q-$functions, analysis of quantum limited amplifiers, and nondemoliton measurements.