Based on the path integral approach, we derive a new realization of the exact non-Markovian stochastic Schrödinger equation (SSE). The main difference from the previous non-Markovian quantum state diffusion (NMQSD) method is that the complex Gaussian stochastic process used for the forward propagation of the wave function is correlated, which may be used to reduce the amplitude of the non-Markovian memory term at high temperatures. The new SSE is then written into the recently developed hierarchy of pure states scheme, in a form that is more closely related to the hierarchical equation of motion approach. Numerical simulations are then performed to demonstrate the efficiency of the new method.Based on the path integral approach, we derive a new realization of the exact non-Markovian stochastic Schrödinger equation (SSE). The main difference from the previous non-Markovian quantum state diffusion (NMQSD) method is that the complex Gaussian stochastic process used for the forward propagation of the wave function is correlated, which may be used to reduce the amplitude of the non-Markovian memory term at high temperatures. The new SSE is then written into the recently developed hierarchy of pure states scheme, in a form that is more closely related to the hierarchical equation of motion approach. Numerical simulations are then performed to demonstrate the efficiency of the new method.

The Journal of Chemical PhysicsJun 10, 201626 citations

Stochastic Methods in Quantum Mechanics

Michel Caffarel

Springer eBooksJan 1, 198921 citations

Quantum-theoretical treatments of three-photon processes

M. K. Olsen, Lev I. Plimak, Michael Fleischhauer

We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency. As the generalized Fokker-Planck equation (GFPE) for the positive- $P$ function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency. As the generalized Fokker-Planck equation (GFPE) for the positive-$P$ function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.

Physical Review AApr 22, 200212 citations

A Monte Carlo method for propagating multi‐dimensional wave packets

Markus Schröder, Ulrich Kleinekathöfer

Abstract A method for propagating multi‐dimensional wave packets in time using a product ansatz for the wave function is presented. Other than in the time‐dependent Hartree approach which is based on a variational scheme, the current method is based on the stochastic Ito calculus. The correlations between the different one‐dimensional wave functions are taken into account via stochastic processes such that on average the original Schrödinger equation is reproduced. Numerical tests of the proposed scheme have been performed in two and four dimensions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)Abstract A method for propagating multi‐dimensional wave packets in time using a product ansatz for the wave function is presented. Other than in the time‐dependent Hartree approach which is based on a variational scheme, the current method is based on the stochastic Ito calculus. The correlations between the different one‐dimensional wave functions are taken into account via stochastic processes such that on average the original Schrödinger equation is reproduced. Numerical tests of the proposed scheme have been performed in two and four dimensions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

physica status solidi (b)Jun 22, 20042 citations

P

function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.