Stochastic Simulations and Discreteness of Light

Abstract

The stochastic simulations of nonlinear interactions are frequently employed method in quantum optics [1, 2]. The simulations use the continuous stochastic variables and one can put a question, whether it is possible to generate the effects of discreteness of light in the nonlinear systems. In addition, it is well known that for large variety of the nonlinear systems, the stochastic simulations exhibit an instability and generate invalid solutions [3] if the boundary terms in the phase-space become non-negligible. On the other hand, it will be presented here on an illustrative example that this instability is closely related to the omitting of a quantum discreteness in the continuous stochastic simulations. Because, if the discrete character is incorporated in a proper way, the stochastic simulations offer the correct results in agreement with the exact solutions. In summary, we suggest that the continuous stochastic simulations are not able to correctly describe a discreteness of light in the quantum nonlinear systems [4]. To illustrate this. we use the nonlinear photonic interaction (anharmonic oscillator) described by the interaction Hamiltonian, where κ is a real nonlinear constant and  is the annihilation operator of the light mode. An evolution of mean-value of the quadrature operator for initial coherent state |α〈 exhibits well-known periodical behavior with revival period Δt = κ/2π, as can be seen in Fig. 1(b), which is an exhibition of the interference between the discrete eigenstates of the photon-number operator [5].