Stochastic Description of Quantum Dissipative Dynamics Jiushu Shao Beijing Normal University 11 August 2010 Physics and Chemistry in Quantum Dissipative Systems YITP, Kyoto University Outline Motives Stochastic Formulation of Dissipative Systems Analytical and Numerical Results Summary Molecular Chirality Why are the chiral configurations stable? Quantum Control of Chirality Wang & JS, PRA 49, R637 (1994); JS & Hanggi, PRA 56, R4397 (1997); JCP 107, 9935 (1997) Multidimensional Dynamics MD: large systems, no quantum effect MD: large systems, no quantum effect Difficulties of quantum dynamics Difficulties of quantum dynamics Schrdinger rep memory bottleneck Schrdinger rep memory bottleneck Path integral: Sign problem Path integral: Sign problem Curse of Dimensionality Dynamics of Open Systems Projection Operator Nakajima (1958) Zwanzig (1960) Mori (1965) Influence Functional Feynman & Vernon (1963) Caldeira & Leggett (1983) Weisss Book (1993, 1999) Stochastic Description Kubo & Tanimura Stockburger & Grabert (2001) Shao (2004) Microscopic Description Hamiltonian Propagator of Whole System Interaction Term Decoupling Interaction in Real Time Evolution JS, JCP 120, 5053 (2004); Castin, Dalibard, Chomaz Hubbard-Stratonovich Transformation Propagator JS, JCP 120, 5053 (2004); Chem Phys 370, 29 (2010) Gaussian Fields Statistical Properties for Separated Hamiltonians White Noise Equation of Motion (EOM) Initial Condition Decoupled Equations of Motion Change of Variables EOM Reduced Density Matrix (RDM) Trace of the Density Matrix for the Bath Girsanov Transformation RDM Change of Variables EOM Primary Numerics Bath-induced Random Field Caldeira-Leggett Model Response Function Master Equation Furutsu-Novikov Theorem Exact Master Equation Formal Solution of Random Density Matrix JS, Chem. Phys. 322, 187 (2006), 370, 29 (2010) correspond to correspond to Formal Solution of Auxiliary Operators Time-Local Form Time-Nonlocal Form Markovian Limit Exact Relation Approximation Master Equation Spontaneous Decay of Two-State Atoms Hamiltonian Bath-Induced Field Number of Samplings 2^24 Hierarchy Scheme Yan, Yang, Liu, & JS, CPL 395, 216 (2004), Tanimura, Cao, Yan Memory Kernel Auxiliary Quantities EOM Truncation Bath-Induced Field Auxiliary Quantities Hierarchical Structure Truncation vs Dissipation Strength Zhou, Yan & JS, EPL 72, 305 (2005), YiJing Yan Truncation vs Memory Length Rev. Mod. Phys. 59, 1 (1987) Mixed Random-Hierarchy Approach Zhou, Yan & JS, EPL 72, 334 (2005) Special Case ( = 0.5) Decay Dynamics (> 0.5) Zhou & JS, JCP 128, (2008) Decay Rate Phase Diagram Summary Establishing a stochastic formulation of quantum dissipative dynamics Deriving master equations Developing numerical techniques Studying spin-boson model 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 Thank You Dissipative Systems 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. Transient Dynamics Rate Constants