NonperturbativeNon Markovian Quantum Dissipative Dynamics Reduced Hierarchy Equations

Nonperturbative-Non. Markovian Quantum Dissipative Dynamics: Reduced Hierarchy Equations Approach Y. Tanimura, Kyoto University

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

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

Hierarchy Equation approach • • without secular approximation (fluctuation-dissipation) colored noise bath (non. Markovian) strong interaction (nonperturbative) correlated system-bath effect (unfactorized) 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). LH 2: Strümpfer &Schulten, JCP 131, 225101 (2009).

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

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

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

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

Density matrix elements where Time derivative of each parts are

Consider the time derivative of the density matrix: where Then

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

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.

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

Physical meaning of the hierachy elements (0 th member: exact) (1 st member: 1 th lower) (2 nd member: 2 nd 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

Linear+square-Linear coupling Linear-Linear coupling case Gaussian-Markovian QFP eq. Tanimura and P. G. Wolynes, PRA 4131 (1991); JCP 96, 8485 (1992).

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

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

Oscillator system vs. Energy-level system LL + SL T 1 +T 2 + T 2* (RWA) Non RWA form (positivity problem) Similar but different Steffen & YT, JPSJ 69, 3115(2000). YT & Steffen, JPSJ 69, 4095(2000).

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

Model Hamiltonian (vibrational modes) LL interaction T 1 + T 2 relaxation SL interaction T 2* relaxation Okumura & YT, PRE. 56, 2747(1997). 20

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

MD VS. LL+LS Morse Osc. model HF liquid（MD） Fast LL t 2=0 LL +LL+SL SL + MD: Hasegawa &YT, JCP 128 (2008). Fokker-Planck Tanimura JPSJ 75, 082001 (2006) Kato & YT JCP 120, 260 (2004). SL system-bath int. Slow

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

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).

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

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

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

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

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

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

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

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

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

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

Quantum Ratchet system Current across the barrier Classical result Quantum result

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

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

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

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

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: Non. Markovian 09 is available (feel free to request) Limitations • Coordinate description: 2 D, spins: around 16 -20 • At current stage, spectral distribution is Drude or Brownian