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

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 Chirality Wang & JS, PRA 49, R637 (1994); JS & Hanggi, PRA 56, R4397 (1997); JCP 107, 9935 (1997)

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

7. 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)

8. Microscopic Description  Hamiltonian  Propagator of Whole System  Interaction Term

9. Decoupling Interaction in Real Time Evolution JS, JCP 120, 5053 (2004); Castin, Dalibard, Chomaz Hubbard-Stratonovich Transformation

10. Propagator JS, JCP 120, 5053 (2004); Chem Phys 370, 29 (2010)

11. Gaussian Fields  Statistical Properties for  Separated Hamiltonians White Noise

13. Equation of Motion (EOM)  Initial Condition  Decoupled Equations of Motion  Change of Variables

14. EOM  Reduced Density Matrix (RDM)  Trace of the Density Matrix for the Bath

15. Girsanov Transformation  RDM  Change of Variables  EOM

17. Primary Numerics

18. Bath-induced Random Field  Caldeira-Leggett Model  Response Function

19. Master Equation  Furutsu-Novikov Theorem  Exact “Master Equation”

20. Formal Solution of Random Density Matrix JS, Chem. Phys. 322, 187 (2006), 370, 29 (2010) correspond to correspond to

21. Formal Solution of Auxiliary Operators  Time-Local Form  Time-Nonlocal Form

22. Markovian Limit  Exact Relation  Approximation  Master Equation

23. Spontaneous Decay of Two-State Atoms  Hamiltonian  Bath-Induced Field

24. Number of Samplings ： 2^24

25. Hierarchy Scheme Yan, Yang, Liu, & JS, CPL 395, 216 (2004), Tanimura, Cao, Yan  Memory Kernel  Auxiliary Quantities  EOM  Truncation

26.  Bath-Induced Field  Auxiliary Quantities

27. Hierarchical Structure

28. Truncation vs Dissipation Strength Zhou, Yan & JS, EPL 72, 305 (2005), YiJing Yan

29. Truncation vs Memory Length

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

32. Mixed Random-Hierarchy Approach Zhou, Yan & JS, EPL 72, 334 (2005)

33. Special Case ( α = 0.5)

34. Decay Dynamics (α> 0.5) Zhou & JS, JCP 128, 034106 (2008)

35. Decay Rate

36. Phase Diagram

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

38. 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

39. Thank You

40. Dissipative Systems

41. Electron Transfer Yan, Yang, Liu, & JS, CPL 395, 216 (2004) Model: Spectral Density Function A finite number N e of exponentials will be used in numerical calculations.

42. Transient Dynamics

43. Rate Constants