Classical Model of Quantum Noise with the FDTD Method

Numerical models based on the finite-difference time-domain (FDTD) method have been developed to simulate thermal noise and spontaneous emission. Both types of noise may have effects on optical systems. Though their origin lies in quantum mechanics, macroscopic systems in which the discreteness of light can be ignored make it possible to simulate the noise using classical numbers. The absorbing boundary of a one-dimensional (1D) FDTD grid absorbs all incident fields and thus, can be considered a blackbody. For a blackbody to be in thermal equilibrium with its surroundings it must also radiate back into the system. Therefore, the extreme points of the 1D grid act as sources of thermal radiation penetrating into the grid. This method is applied to the study of a 1D leaky optical cavity in transition from the Markovian to a non-Markovian regime. The appropriate spectral properties are given to the noise and the standard result of the quantum Langevin equation is recovered. In a separate treatment, spontaneous emission, which is important to consider in laser dynamics, is simulated through a 1D FDTD based Maxwell-Bloch system. The coupling between the Maxwell and Bloch equations is achieved via a weakly coupled splitting scheme. This model is based on the c-number stochastic differential equations found through the use of the positive P representation. We validate our method by reproducing previous numerical results of superfluorescence. The gain atoms are initially inverted, so inversion-dependent contributions to the stochastic sources are dominant and thus the only sources of noise considered.