Revisiting stochastic models: Anomalous relaxation effects in 2D spectroscopy

1. 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 72,011103 (2006) • F.Š., S.M, PRL 98,080603, (2007) • F.Š., S.M, JCP 127, 154107 (2007), • Acknowledgents • GAČR, Ministry of Education