arXiv:quant-ph/9803079 · Physical Review Letters

Open System Dynamics with Non-Markovian Quantum Trajectories

Walter T. Strunz, Lajos Diósi, Nicolas Gisin

Published Mar 1, 1999• 363 citations

Abstract

A non-Markovian stochastic Schr\"odinger equation for a quantum system coupled to an environment of harmonic oscillators is presented. The ensemble average recovers the reduced density matrix without approximation and hence it allows one to determine open system dynamics with strong and non-Markovian environmental effects in a very efficient way. We demonstrate the power of our approach with several illustrative examples. First, we discuss a measurement-type situation, then a two-state system strongly coupled to a non-Markovian environment, exhibiting decays and revivals. Further examples showing the remarkable features of our new approach to non-Markovian open system dynamics are discussed, for instance, the possibility to shift the ``Heisenberg cut'' between system and environment.

DOI: 10.1103/physrevlett.82.1801

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As a first example, the dynamics of a dissipative molecular oscillator is investigated. Long time propagation - until thermalization is reached - is shown to be possible with the presented approach. The properties of the thermalized density are determined and they are ascertained to be independent of the system's initial state. Furthermore, the dependence on the bath's temperature and coupling strength is analyzed and it is demonstrated how a change of the bath parameters can be used to tune the system from the dissociative to the bound regime. A second investigation is conducted for a dissipative tunneling scenario in which a wave packet impinges on a barrier. The dependence of the transmission probability on the initial state's kinetic energy as well as the bath's temperature and coupling strength is computed. For both systems, a comparison with the high-temperature Markovian quantum Brownian limit is performed. The importance of a full non-Markovian treatment is demonstrated as deviations are shown to exist between the two descriptions both in the low temperature cases where they are expected and in some of the high temperature cases where their appearance might not be anticipated as easily.:1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Theory of Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Influence Functional Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Quantum Brownian Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Stochastic Unraveling of the Influence Functional . . . . . . . . . . . . . . . 20 2.4 Improved Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.1 Modified Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.2 Guide Trajectory Transformation . . . . . . . . . . . . . . . . . . . . 24 2.5 Obtaining Properly Correlated Stochastic Samples from Filtered White Noise 24 3 Unified Stochastic Trajectory Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Semiclassical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Guide Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1.2 Real Coherent State Center Coordinates . . . . . . . . . . . . . . . . 31 3.1.3 Propagation Scheme Including Stochastic Forces . . . . . . . . . . . 32 3.2 Stochastic Bohmian Mechanics with Complex Action . . . . . . . . . . . . . 33 3.2.1 Hydrodynamic Formulation of Bohmian Mechanics . . . . . . . . . . 33 3.2.2 Bohmian Mechanics with Complex Action . . . . . . . . . . . . . . . 34 3.2.3 Stochastic BOMCA Trajectories . . . . . . . . . . . . . . . . . . . . 38 3.3 Noise Distribution Preserving Removal of Adverse Samples . . . . . . . . . . 39 4 Dissipative Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Reservoir Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Harmonic Oscillator Analytic Expectation Values . . . . . . . . . . . . . . . 42 4.2.1 Ohmic Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.2 Drude Regularized Bath . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Sampling Strategies and Analytic Comparison . . . . . . . . . . . . . . . . . 44 4.4 Limits of the Markovian Approximation . . . . . . . . . . . . . . . . . . . . 45 5 Dissipative Vibrational Dynamics of Diatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 Molecular Morse Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Anharmonic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Transient Non-Markovian Effects . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 Trapping by Dissipation and Thermalization . . . . . . . . . . . . . . . . . . 53 6 Tunneling with Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1 Eckart Barrier Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2 Dissipative Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.3 Investigation of Markovianity . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Appendix A Conventions for Constants, Reservoir Kernels, and Influence Phases 69 Appendix B Stochastic Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 B.1 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 71 B.2 Position Verlet Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 B.3 Runge-Kutta Fourth Order Scheme . . . . . . . . . . . . . . . . . . . . . . . 73 Appendix CMorse Oscillator Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Appendix DPrerequisites of a Successful Stochastic Propagation . . . . . . . . . . . . . . 79 D.1 Hubbard-Stratonovich Transformation . . . . . . . . . . . . . . . . . . . . . 79 D.2 Kernels of the Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.2.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 D.2.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 89 D.3 Guide Trajectory Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.1 Quadratic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 D.3.2 Quartic Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 D.3.3 Strictly Ohmic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 92 D.4 Computation of Matrix Elements and Expectation Values . . . . . . . . . . 92 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

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