1. František Šanda1, Shaul Mukamel2 1 Charles University, Prague
Revisiting stochastic models: Anomalous relaxation effects in 2D spectroscopy
František Šanda1, Shaul Mukamel2
1 Charles University, Prague
2 UCI

2. Nonlinear response to three laser pulses
probes stochastic fluctuations of transition frequencies of a two level chromphore
Transition frequency undergoes spectral random walk

3. Response in
phase matching direction

4. Response in
phase matching direction

5. 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

6. 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

7. (a)
Stationary ensembles, close to normal lineshapes
(b)
stationary ensembles, but still anomalous
features in spectra
(c)
Nonstationary ensemble only, shows anomalous effects including aging
Special WTDF for the first jump is necessary to define stationary ensembles

8. 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

9. Calculating nonlinear response function of CTRW spectral diffusion
Propagation between first and last jumps in each applicable interval can be summed up in frequency domain

10. propagation over boundary (including coherence factor)
Depending on number of jumps in each interval we have 8 type of paths

11. Stationary lineshapes
2 state jump of two level chromophore
Model has 3 timescales
(controls asymptotic)
Observables
Frequency /frequency correlation plots
Absorptive signal

12. Slow fluctuations
Plotted
SI,II diverges along lines
SA diverges at points (1,1),(-1,-1)

13. Asymptotic peak structure
Along lines
SA divergence at the peak

14. Fast fluctuations
Plotted
additional central peak (motional narrowing)

15. 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)

16. Aging 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

17. 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,

18. 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

19. Aging in Markovian model
Decreasing mobility of particles switch the lineshape from motional narrowing limit to static case

20. Aging (fast, nearly markovian)

21. 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)

22. Conclusions Algorithm for an important class of nomarkovian processes
Role of fluctuation timescale in 2D lineshapes
Trajectory picture of stochastic fluctuations in 2D lineshapes

23. References :
F.Š., S.M, PRE 72, (2006)
F.Š., S.M, PRL 98,080603, (2007)
F.Š., S.M, JCP 127, (2007),
Acknowledgents
GAČR, Ministry of Education