Nonlinear Spectroscopy: Characterizing Fluctuations
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Nonlinear Spectroscopy:Characterizing Fluctuations
Andrei TokmakoffMIT Department of Chemistry
2009
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Fluctuations in lineshapes
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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
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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
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A bath of arbitrary form
• Spectral Density
• Brownian Oscillator Model
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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
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Two pulse echo
~1/D
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Two pulse echo
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Lens analogy
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Three pulse echoes: Peak shift
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2D Lineshapes
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Spectral Diffusion and Waiting Time
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Time-domain 2D Spectroscopy
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Time-Domain: Echo shape
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2D IR Relaxation Experiment Probes Spectral Diffusion
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Ellipticity
2 2
2 2 2
a bEa b
2 2egE C
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Correlation of Spectral Broadening
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Correlation Effects in 2D IR Spectra
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Correlated Energy Fluctuations
D D m n mn m n mn( ) (0) exp(- / )t t=Frequency Auto- and Cross-Correlation Functions: