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Stochastic Description of Quantum Dissipative Dynamics

Physics and Chemistry in Quantum Dissipative Systems YITP, Kyoto University. Stochastic Description of Quantum Dissipative Dynamics. Jiushu Shao Beijing Normal University 11 August 2010. Outline. Motives Stochastic Formulation of Dissipative Systems Analytical and Numerical Results

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Stochastic Description of Quantum Dissipative Dynamics

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  1. Physics and Chemistry in Quantum Dissipative SystemsYITP, Kyoto University Stochastic Description of Quantum Dissipative Dynamics Jiushu Shao Beijing Normal University 11 August 2010

  2. Outline • Motives • Stochastic Formulation of Dissipative Systems • Analytical and Numerical Results • Summary

  3. Molecular Chirality Why are the chiral configurations stable?

  4. Quantum Control of ChiralityWang & JS, PRA49, R637 (1994); JS & Hanggi, PRA 56, R4397 (1997); JCP107, 9935 (1997)

  5. Multidimensional Dynamics • MD: large systems, no quantum effect • Difficulties of quantum dynamics • Schrödinger rep：memory bottleneck • Path integral: Sign problem Curse of Dimensionality

  6. Dynamics of Open Systems Projection Operator Nakajima (1958) Zwanzig (1960) Mori (1965) Influence Functional Feynman & Vernon (1963) Caldeira & Leggett (1983) Weiss’s Book (1993, 1999) Stochastic Description Kubo & Tanimura Stockburger & Grabert (2001) Shao (2004)

  7. Microscopic Description • Hamiltonian • Propagator of Whole System • Interaction Term

  8. Decoupling Interaction in Real Time EvolutionJS, JCP120, 5053 (2004); Castin, Dalibard, Chomaz Hubbard-Stratonovich Transformation

  9. PropagatorJS, JCP120, 5053 (2004); Chem Phys370, 29 (2010)

  10. Gaussian Fields • Statistical Properties for • Separated Hamiltonians White Noise

  11. Equation of Motion (EOM) • Initial Condition • Decoupled Equations of Motion • Change of Variables

  12. EOM • Reduced Density Matrix (RDM) • Trace of the Density Matrix for the Bath

  13. Girsanov Transformation • RDM • Change of Variables • EOM

  14. Primary Numerics

  15. Bath-induced Random Field • Caldeira-Leggett Model • Response Function

  16. Master Equation • Furutsu-Novikov Theorem • Exact “Master Equation”

  17. Formal Solution of Random Density MatrixJS, Chem. Phys.322, 187 (2006), 370, 29 (2010) correspond to

  18. Formal Solution of Auxiliary Operators • Time-Local Form • Time-Nonlocal Form

  19. Markovian Limit • Exact Relation • Approximation • Master Equation

  20. Spontaneous Decay of Two-State Atoms • Hamiltonian • Bath-Induced Field

  21. Number of Samplings：2^24

  22. Hierarchy SchemeYan, Yang, Liu, & JS, CPL395, 216 (2004), Tanimura, Cao, Yan • Memory Kernel • Auxiliary Quantities • EOM • Truncation

  23. Bath-Induced Field • Auxiliary Quantities

  24. Hierarchical Structure

  25. Truncation vs Dissipation StrengthZhou, Yan & JS, EPL72, 305 (2005), YiJing Yan

  26. Truncation vs Memory Length

  27. Rev. Mod. Phys. 59, 1 (1987)

  28. Mixed Random-Hierarchy ApproachZhou, Yan & JS, EPL72, 334 (2005)

  29. Special Case (α= 0.5)

  30. Decay Dynamics (α> 0.5)Zhou & JS, JCP128, 034106 (2008)

  31. Decay Rate

  32. Phase Diagram

  33. Summary • Establishing a stochastic formulation of quantum dissipative dynamics • Deriving master equations • Developing numerical techniques • Studying spin-boson model

  34. Acknowledgements • Dr. Yun-an Yan，Dr. Yun Zhou, Dr. Yu Liu，Fan Yang, and Dr. Wenkai Zhang • Profs. X.Q. Li, U. Weiss, Y.J. Yan • National Natural Science Foundation of China • Chinese Academy of Sciences

  35. Thank You

  36. Dissipative Systems

  37. Electron Transfer Yan, Yang, Liu, & JS, CPL395, 216 (2004) Model: Spectral Density Function A finite number Ne of exponentials will be used in numerical calculations.

  38. Transient Dynamics

  39. Rate Constants

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