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

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

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 (nonMarkovian) strong interaction (nonperturbative) correlated system-bath effect (unfactorized) Tanimura & Kubo, J.Phys. Soc. Jpn 58, 101 (1989). FMO: Ishizaki & Fleming, PNAS 106, 172 (2009). LH2: Strümpfer &Schulten, JCP 131, (2009). R. X. Xu, and Y. J. Yan, J. Chem. Phys. 122, (2005).

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

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

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

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 Wigner dist. complex variables (real for diagonal Spread out for weak damping Hard to set boundary condition 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 G-M quantum Fokker-Planck eq Quantum Liouvillian terminator YT:JPSJ 75,082001(2006).JPSJ 75,082001(2006).

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

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

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

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

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

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

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

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

MD VS. LL+LS Morse Osc. model Fokker-Planck Tanimura JPSJ 75, (2006) JPSJ 75, (2006) Kato & YT JCP120, 260 (2004).JCP120, 260 (2004). FastSlow MD: Hasegawa &YT, JCP 128 (2008).JCP 128 (2008). t 2 =0 + LL LL + SL SL HF liquid （ MD ） LL+SL system-bath int.

Multistates Q. F-P eq. The reduced density matrix is Potential surfaces laser, nonadiabatic interactions YT & Maruyama, JCP 107, 1779 (1997).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).JCP 101, 3049 (1994).

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

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

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

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

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

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

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

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

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

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

Quantum Ratchet system Current across the barrier Classical result Quantum result

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

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

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

ET rate vs activation energy calculated Sequentialsuper-exchange thermal quantum Tanaka & YT, JCP (2010).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 Coordinate description: 2D, spins: around At current stage, spectral distribution is Drude or Brownian Code for spin-boson system: NonMarkovian09 is available (feel free to request) Limitations