Nonlinear Spectroscopy:Characterizing Fluctuations

Andrei TokmakoffMIT Department of Chemistry

2009

Fluctuations in lineshapes

System-Bath Interactions

• System: Spectroscopically Bright States• Bath: Dark States

• → Energy Gap Fluctuations• In spectroscopy … (Condon Approx.)

0 ,S i B j SB i jH H Q H q H Q q

2

0expeg

ti teg SBC t e i H t dt

S nH n E n

( )eg t

Energy Gap Fluctuations

• Energy Gap Correlation Function

• Dipole Correlation Function → Absorption lineshape• Second Cumulant approx.

2 eg

egi t g tC t e e

0 0

t t

egg t dt dt C t

2

10 0eg eg eg SB SBC H t H

A bath of arbitrary form

• Spectral Density

• Brownian Oscillator Model

Nonlinear response for fluctuating system

• Following earlier approach:

• Cumulant approximation:

1 3

34 * * *

1 3 1 2 2 3 1 2 1 2 3 expegg eg

iiR p g g g g g ge

1 3

34 * * * *

2 3 1 2 2 3 1 2 1 2 3 expegg eg

iiR p g g g g g ge

Two pulse echo

~1/D

Two pulse echo

Lens analogy

Three pulse echoes: Peak shift

2D Lineshapes

Spectral Diffusion and Waiting Time

Time-domain 2D Spectroscopy

Time-Domain: Echo shape

2D IR Relaxation Experiment Probes Spectral Diffusion

Ellipticity

2 2

2 2 2

a bEa b

2 2egE C

Correlation of Spectral Broadening

Correlation Effects in 2D IR Spectra

Correlated Energy Fluctuations

D D m n mn m n mn( ) (0) exp(- / )t t=Frequency Auto- and Cross-Correlation Functions: