Numerical simulation of Kerr nonlinear systems; analyzing non-classical dynamics

Abstract

We simulate coherent driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and time evolving block decimation (TEBD) algorithm, and demonstrate how the numerical results are analogous to classical bistability. The comparison with analytics show that the TEBD numerics follow the quantum mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation . Comparing between two different numerical techniques, we see that the semi-classical Euler's method gives the dynamics of the system field of one among two coherent branches, whereas TEBD numerics generate the superposition of both of them. Therefore, the time dynamics determined by TEBD numerical method undergoes through a non-classical state which is also shown by determining second order correlation function.

Simulation of kerr nonlinearity: revealing initial state dependency

“…Euler's time propagation, and time-evolving block decimation (TEBD) methods, we simulate Kerr non-linear system, to investigate the impact of the initial state, under the influence of the different driving fields. Earlier, in [4], while simulating the dynamics of the Kerr nonlinear system using TEBD, we have shown the consistency between the analytical and the numerical results. Also, we have observed that the TEBD numerical result follows the quantum mechanical exact solution, whereas the time propagation of the system field obtained using Euler's method follows the semi-classical solution of the Heisenberg equation of motion.…”

“…non-linear quadratic electro-optic (QEO) response-had been in interest over centuries, for having some of the interesting quantum phenomenon e.g. photon switching in quantum interference [1], photon bunching and antibunching in bistable steady-state [2], dynamical optical bistability via bifurcation process [3] and the generation of non-classical states [4]. In general, the multistability of nonlinear systems offers a window for a long-range of applications.…”

“…This limitation of analytical treatment explicitly motivates us to simulate the time evolution numerically, which consists of transforming the environmental degrees of freedom to a one-dimensional (1D) many-body chain [27] and simulating the chain afterward. The computational method is composed of numerical diagonalization and renormalization process [4,28].…”

Simulation of Matrix Product States For Dissipation and Thermalization Dynamics of Open Quantum Systems

“…In conclusion, one can say that the numerically generated thermal bath shows promise, but, this requires a compromise between the quality of the result and the computation resources. The numerical scheme presented here was mainly motivated by an attempt to determine the exact solution in the case of nonlinear coupling between the system and the environment [26], non-classical dynamics of non-linear systems [27], and reach out single photon limit in optomechanical systems [24,25]. The combination of real and imaginary time evolution of open quantum system will allow us to investigate quantum Brownian motion of topological quantum matter [53,54].…”

“…The interesting effects, e.g. non-classical behavior of the nonlinear systems [27], are often overlooked when we cannot handle the interaction between two systems in a perturbative manner. Apart from nonlinear S/B coupling, the analytical model is also limited to providing the exact solution in case of non-Markovian dynamical phenomenon, for instance phase transition in a two level system (TLS) between dynamically localized and delocalized states, at zero temperature for Ohmic and sub-Ohmic reservoirs [3,[28][29][30].…”