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Nonperturbative-NonMarkovian Quantum Dissipative Dynamics: Reduced Hierarchy Equations Approach

Nonperturbative-NonMarkovian Quantum Dissipative Dynamics: Reduced Hierarchy Equations Approach. Y. Tanimura, Kyoto University. Three important effects of the bath. Dissipation (relaxation) Fluctuation (heating) iii) Correlated effects (entanglement between the system & bath) Strong coupling

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Nonperturbative-NonMarkovian Quantum Dissipative Dynamics: Reduced Hierarchy Equations Approach

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  1. Nonperturbative-NonMarkovianQuantum Dissipative Dynamics:Reduced Hierarchy Equations Approach Y. Tanimura, Kyoto University

  2. Three important effects of the bath • Dissipation(relaxation) • Fluctuation(heating) iii) Correlated effects(entanglement between the system & bath) Strong coupling slow modulation Exist both in quantum & classical cases (correlation for colored noise) Balanced at equilibrium state fluctuation-dissipation theorem (has to be quantum version) important

  3. S-B coherence & external force S-B coherence is important to calculate response func.

  4. without secular approximation (fluctuation-dissipation) • colored noise bath (nonMarkovian) • strong interaction (nonperturbative) • correlated system-bath effect (unfactorized) Hierarchy Equation approach Tanimura & Kubo, J.Phys. Soc. Jpn 58, 101 (1989). R. X. Xu, and Y. J. Yan, J. Chem. Phys. 122, 041103 (2005). FMO: Ishizaki & Fleming, PNAS 106, 172(2009). LH2: Strümpfer &Schulten, JCP 131, 225101 (2009).

  5. Consider a molecular system coupled to an environment. The model Hamiltonian may be written as Quantum Fokker-Planck eq. potential coupling counter term

  6. where is the Feynman-Vernon influence functional. (All heat bath effects can be taken into account byinfluence functional.) If we combine the bath part

  7. The influence functional is calculated as where If the heat bath is an ensemble of harmonic oscillators, dissipation fluctuation

  8. Fluctuation and dissipation Approach to slow fast If we assume High T High T Low T Low T Dissipation YT JPSJ 75, 082001(2006). If temp. is high Fluctuation Cannot be delta-function

  9. Density matrix elements where Time derivative of each parts are

  10. Consider the time derivative of the density matrix: where Then

  11. We may evaluate by repeating the differentiation, then where is the density matrix for the element

  12. Wigner distribution function Density matrix elements complex variables (real for diagonal Spread out for weak damping Hard to set boundary condition Wigner dist. All real (no direct physical interpretation) Classical distribution in classical limit Wave packets are localized Periodical boundary condition, etc.

  13. G-M quantum Fokker-Planck eq Quantum Liouvillian terminator YT:JPSJ 75,082001(2006).

  14. Physical meaning of the hierachy elements (0th member: exact) (1st member: 1th lower) (2nd member: 2nd lower) (Nth member: Nth lower) member is grouped by characteristic time Dashed line represents the system-bath interactions Correlated initial condition can be set by Terminator

  15. Linear-Linear coupling case Linear+square-Linear coupling case Gaussian-Markovian QFP eq. Tanimura and P.G. Wolynes, PRA4131 (1991);JCP96, 8485 (1992).

  16. To obtain the above equation we assumed, For we can set In this limit, the above equation reduces to the QFP The temperature limitation of Gaussian-White F-P is much stronger than Gaussian-Markovian F-P equation. In classical limit

  17. A model Hamiltonian Hamiltonian for vibrational spectroscopy Stochastic theory (without dissipation, temperature can not be defined) LL + SL interactions Okumura & YT,PRE. 56, 2747(1997). Kato & YT JCP 117,6221(2002); 120,260 (2004).

  18. Oscillator system vs. Energy-level system LL T1 +T2 + SL + T2* (RWA) Non RWA form (positivity problem) Similar but different Steffen & YT, JPSJ 69, 3115(2000). YT & Steffen, JPSJ 69, 4095(2000).

  19. 3D IR spectroscopy Steffen & Tanimura, JPSJ(2000) ; JPSJ(2000). Tanimura, JPSJ 75, 082001 (2006)

  20. Model Hamiltonian(vibrational modes) LL interaction SL interaction T1 + T2 relaxation T2* relaxation Okumura & YT,PRE. 56, 2747(1997).

  21. Can we observe IR photon echo signal? • Homogeneous case (fast) inhomogeneous case (slow)

  22. MD VS. LL+LS Morse Osc. model HFliquid（MD） Fast Slow LL LL+SL system-bath int. LL + SL t2=0 + MD: Hasegawa &YT, JCP 128 (2008). SL Fokker-Planck TanimuraJPSJ 75, 082001 (2006) Kato & YT JCP120, 260 (2004).

  23. Multistates Q. F-P eq. Potential surfaces laser, nonadiabatic interactions The reduced density matrix is YT & Maruyama, JCP 107, 1779 (1997).

  24. The multistate quantum dynamics is described by replacements: We now consider the heat-bath. The Hamiltonian is The Wigner functions are defined by where Tanimura & Mukamel, JCP 101, 3049 (1994).

  25. Morse potentials system Linear absorption Tanimura & Maruyama, JCP 107, 1779 (1997).

  26. Wave Packet dynamics Tanimura & Maruyama, JCP 107, 1779 (1997).

  27. Pump-Probe spectra Tanimura & Maruyama, JCP 107, 1779 (1997). Maruyama & Tanimura, CPL 292, 28 (1998).

  28. Low temp. corrections of GM QFP eq. slow fast High T High T Dissipation Low T Low T Fluctuation YT JPSJ 75, 082001(2006). High (Matsubara) frequencies terms are approximated by delta func. Similar to GM case Matsubara freq. correct. terms

  29. Fluctuation Kernel at any temperature high temperature (in the high temperature limit) Influence functional is given by (at high temperatures) (at any temperature) where

  30. Density matrix element is at any temperature at high temperatures Time derivatives of system parts are Influence functional part is (at high temperatures) (at any temperature)

  31. Time derivative of the density matrices Tanimura,PRA 41, 6676 (1990). where

  32. For large Conditions: for ,or Example for K=2 Terminator 2 Nand K determine the hierarchy number slow modulationlarge N low temperaturelarge N and K Ishizaki and Tanimura, JPSJ 74, 3131 (2005); JCP 125, 084501(2006).

  33. Quantum Ratchet system P. Hanggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009)

  34. Quantum Ratchet system Under damping Wigner distribution (219 hierarchy ) Classical distribution (10 hierarchy)

  35. Quantum Ratchet system Current across the barrier Classical result Quantum result

  36. Brownian distribution (nonOhmic) Multi-level system with the BO distribution hierarchy equations for nonOhmic noise High temp. YT & Mukamel, JPSJ 63, 66 (1994). Low temp. Tanaka & YT, JPSJ 78, 073802 (2009). Tanaka & YT,JCP 132, 214502 (2010). ET reaction rates Stark effects

  37. HEOM for Brownian distribution Tanaka & YT, JPSJ 78, 073802 (2009).

  38. Terminator No limitation to temperature System-bath coupling oscillators configuration Non-adiabatic couplings Tanaka & YT, JPSJ 78, 073802 (2009).

  39. ET rate vs activation energy thermal quantum Sequential super-exchange calculated Tanaka & YT, JCP (2010).

  40. Summery • colored noise (Drude & Brownian distribution) • strong system-bath coupling • low temperature system • system-bath coherence • time-dependent external force • coordinate or/and energy state representation • system side of system-bath interaction can be any form Code for spin-boson system: NonMarkovian09 is available (feel free to request) Limitations • Coordinate description: 2D, spins: around 16-20 • At current stage, spectral distribution is Drude or Brownian

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