Quantum description of nonlinearly interacting oscillators via classical trajectories

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Abstract
We investigate systems of few harmonic oscillators with mutual nonlinear coupling. Using classical trajectories—the solutions of Hamiltonian equations of motion for a given nonlinear system—we construct the approximate quasiprobability distribution function in phase space that enables a quantum description. The nonclassical effects (quantum noise reduction) and their scaling laws can be so studied for high excitation numbers. In particular, the harmonic oscillators represent modes of the electromagnetic field and the Hamiltonians under consideration describe representative nonlinear optical processes (multiwave mixings). The range of the validity of the approximation for Wigner and Husimi functions evolved within the classical Liouville equation is discussed for a diverse class of initial conditions, including those without classical counterparts, e.g., Fock states.