Stochastic Description of Quantum Dissipative Dynamics

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

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 EvolutionJS, JCP120, 5053 (2004); Castin, Dalibard, Chomaz Hubbard-Stratonovich Transformation

10. PropagatorJS, JCP120, 5053 (2004); Chem Phys370, 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 MatrixJS, Chem. Phys.322, 187 (2006), 370, 29 (2010) 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 SchemeYan, Yang, Liu, & JS, CPL395, 216 (2004), Tanimura, Cao, Yan • Memory Kernel • Auxiliary Quantities • EOM • Truncation

26. Bath-Induced Field • Auxiliary Quantities

27. Hierarchical Structure

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

29. Truncation vs Memory Length

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

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

33. Special Case (α= 0.5)

34. Decay Dynamics (α> 0.5)Zhou & JS, JCP128, 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, CPL395, 216 (2004) Model: Spectral Density Function A finite number Ne of exponentials will be used in numerical calculations.

42. Transient Dynamics

43. Rate Constants