Revisiting stochastic models: Anomalous Revisiting stochastic models: Anomalous relaxation effects in 2D spectroscopyrelaxation effects in 2D spectroscopy

FrantiFrantišek Šandašek Šanda11, Shaul Mukamel, Shaul Mukamel22

11 Charles University, Prague Charles University, Prague

22 UCI UCI

• Nonlinear response to three laser pulses

probes stochastic fluctuations of transition frequencies of a two level chromphore

Transition frequency undergoes spectral random walk

• Response in

phase matching direction

• Response in

phase matching direction

Solvable stochastic models

• (a) Gaussian process

• (b) Markovian process

• (c) Renewal dynamics (continuous time random walks)

We focus on (c) for Kubo-Anderson two state jumps between frequencies

Continuous time random walks

• Defined by waiting time distribution function the probability density for jump from frequency to frequency after t, and vice versa

• Consider algebraic long-time asymptotic

• Classification based on first two moments and of distribution

function

(a)Stationary ensembles, close to normal lineshapes

(b)stationary ensembles, but still anomalousfeatures in spectra(c)Nonstationary ensemble only, shows anomalous

effects including aging

• Special WTDF for the first jump is necessary to define stationary ensembles

Why solvable ?

• At the time of jump all memory is erased

• This renewal property makes CTRW solvable

• The memory effects enters through the time elapsed from the last jump

• Propagation between first and last jumps in each applicable interval can be summed up in frequency domain

Calculating nonlinear response function of CTRW spectral diffusion

• propagation over boundary (including coherence factor)

• Depending on number of jumps in each interval we have 8 type of paths

Stationary lineshapes

2 state jump of two level chromophore• Model has 3 timescales

(controls asymptotic)Observables• Frequency /frequency correlation plots

• Absorptive signal

Slow fluctuations

Plotted

SI,II diverges along lines

SA diverges at points (1,1),(-1,-1)

Asymptotic peak structure

• Along lines

• SA divergence at the peak

Fast fluctuations

Plotted

additional central peak (motional narrowing)

Time t2 evolution for slow limit

• Finite cross-peaks at (1,-1),(-1,1)

and algebraic relaxation

with t2 ;

showed at cross peak(-1,1)

(straight line in log-log plot)

• when properties of sample change with time

• RW is started t0 before the first pulse act on the sample;

• Response function depends on the initial delay t0

• Models:

(a)Nonstationary CTRW with diverging mean waiting time and

(b)Markovian process

with time-dependent rates

Aging

• We compare CTRW and aging Markovian models for symmetric two state jump

• Rates of Markovian master equation will be tailored to share particle density evolution with CTRW,

Response functions for Markovian spectral diffusion

• Calculated by solving stochastic Liouville equations

in the joint Liouville + bath space with use of Green’s function method

Aging in Markovian model

• Decreasing mobility of particles switch the lineshape from motional narrowing limit to static case

Aging (fast, nearly markovian)

• Diagonal (static peaks) occurs together with the motional narrowing central peak

• MME and CTRW shows different trajectory picture for the same master equation (for bath)

Conclusions

• Algorithm for an important class of nomarkovian processes

• Role of fluctuation timescale in 2D lineshapes

• Trajectory picture of stochastic fluctuations in 2D lineshapes

• References :

• F.Š., S.M, PRE 72,011103 (2006)

• F.Š., S.M, PRL 98,080603, (2007)

• F.Š., S.M, JCP 127, 154107 (2007),

•Acknowledgents

•GAČR, Ministry of Education