Post on 09-May-2022
1
2D Femtosecond
Spectroscopy
Jesse Wilson Ph.D. Qualifying Exam
Advisor: Prof. Randy A. Bartels
SDG
2
Motivation
Who needs another dimension?
3
Broadening Mechanisms
Palese, et al. J. Phys. Chem. 1994.
Homogeneous Inhomogeneous
1D Raman
2D Raman
4
(ii)
(i)
Mode Coupling
*Okumura, et al. J. Chem. Phys. (1999)
Complex molecules
Raman-active dipoles
5
Mode Coupling (1D Raman)
*Okumura, et al. J. Chem. Phys. (1999)
6
2D Raman Spectrum
7
2D Raman Spectrum (Fundamentals)
8
2D Raman Spectrum (Coupling)
9
Another example
*Zhang, et al. J. Chem. Phys. (1999)
Linear Uncoupled
Coupled
10
2D IR Spectra Contain Structural
Information
Acetyleproline-NH2 in chloroform.
*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)
11
2D IR Spectra Contain Structural
Information
Magnitude
*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)
12
2D IR Spectra Contain Structural
Information
Real part
*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)
13
2D IR Spectrum Features
*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)
14
2D IR Spectrum: Broadening
*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)
Homogeneous/inhomogeneous width
15
2D IR Spectrum: Anharmonicity
*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)
Vibrational anharmonicity
16
2D IR Spectrum: Mode Coupling
*Hochstrasser, et al. Bull. Chem. Soc. Jpn. (2002)
Mode coupling
17
2D Spectroscopy Advantages
Discern homogeneous, inhomogeneous lines
Mode coupling
Vibrational anharmonicity
Structural information
18
Papers Reviewed
Steffen, Fourkas, and Duppen.
“Time resolved four-and six-wave mixing
in liquids. I. Theory.”
Journal of Chemical Physics (1996).
Blank, Kaufman, and Fleming.
“Fifth-order two-dimensional Raman
spectra of CS2 are dominated by third-
order cascades.”
Journal of Chemical Physics (1999).
19
Steffen, et al. (Part I. Theory)
20
Steffen, et al (Part II. Experiment)
21
Blank, et al.
22
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
23
1D Methods
…and their shortcomings
24
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
25
Time-resolved ISRS
t = 0
26
Time-resolved ISRS: Pumping
t = 0
pump
Stokes
27
Time-resolved ISRS: Ultrafast
t = 0
pump
wwpumpwstokes
E(w)
•Ultrafast pulse spectrum
•Short pulses: pulse << vib
•Impulsive excitation
Stokes
28
Time-resolved ISRS: Coherence
t
29
Time-resolved ISRS: Probing
t =
30
1D Experimental SetupLaser
Oscill
ato
r
Detector
pump
probe
signal
sample
31
32
Liquid Intermolecular
Modes
Homogeneous or Inhomogeneous?
33
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
34
CS2
Instantaneous response
Diffusive tail
Steffen and Duppen. J. Chem. Phys. (1997)
Librations
35
H2O
Palese, et al. J. Phys. Chem. 1994.
36
Broadening
Homogeneous: g(w) rapidly fluctuates
Inhomogeneous: g(w) changes slowly
w
ww )cos()()( tgtR)cos()( ttR w
37
Inhomogeneous Model
2
20
2
)(
)(
ww
ww
eg
= degree of inhomogeneity
)()3( tR
0
)3(
inh
)3(
inh ),()()( tRgdtR www
38
Model fit to data
Steffen and Duppen. J. Chem. Phys. (1997)
Experiment
Homogeneous limit
= 1.00 rad/ps
= 2.53 rad/ps
39
2D Provides Discrimination
*Palese et al. J. Phys. Chem. 1994.
Homogeneous
Inhomogeneous
Intermediate
40
2D Simulations
Homogeneous Inhomogeneous
41
42
Four Wave Mixing
Theory
Third-order, or one-dimensional theory…
43
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
44
3rd Order Response
)()()()](~),(~[
)()()(),,,(
32211221
32211321
ttttHtttti
ttttttttttR
)()()(),,,(ddd)( 321321
)3(
321
)3( tEtEtEttttRttttP
Raman
Hyper-polarizability (THG…)
45
Simplified 3rd Order Response
)()()()](~),(~[
)()()(),,,(
32211221
32211321
ttttHtttti
ttttttttttR
)]0(~),(~[2
)()( 111
iR
132
1 0
tt
t
46
Harmonic Oscillator
Unperturbed Hamiltonian:
Raising, lowering operators:
Displacement operator
)(21
0 aaH BO w
)(2
aam
qw
1~ a 1~ a
47
Polarizability Depends on q
2
21)( qqq
)(2
aam
qw
)2(2
)(2
22 aaaaaam
aam
qww
One-level transitions
Two-level transitions
Zero-level transition
48
3rd order HO Response
ww
ww
)2sin()()12(2
)sin(2
)()(
122
2
2
1
2
1
11
)3(
Pm
m
R
Hyperpolarizability
2
1
49
50
Double-Sided Feynman
Diagrams
Illustrating density matrix evolution
51
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
52
Optical Polarization
)(Tr)( tVtP
)( rrqV
53
1101
1000
Density Matrix
Population densities
Coherencesbra
ket
54
Liouville Equation
HHi
HHi
Hi
t
,
55
Perturbation Expansion
)()()()( )2()1()0( tttt
E2E 43 EE
56
Time evolution
*Mukamel, Principles of Nonlinear Optical Spectroscopy
]]]]),0([),([[),([),()( 0111
)(
VtVttVtrEi
t nn
n
n
57
Initial State
aaVV ),0(]),0([ 0
11
00
aa
58
Commutator
baV )0(
11
00
aa
,12,10,11
,02,01,00
)0( abaaV
59
Commutator
)0()0(),0(]),0([ 0 VaaaaVaaVV
baV )0(
baabV ]),0([ 0
60
Double-sided Feynman Diagrams
*Yee and Gustafson, Optics Communications (1977)
a
t
a a a
b a
0t
a b
baabVaaaaVaaVV )0()0(),0(]),0([ 0
61
Second order commutator
]]),0([),([ 01 VtV aaVtV ),0(),( 1
caabtV ),( 1
)()()()( 1111 tVcacatVtVababtV
bacbcbac
62
Liouville Space Paths
aa ba
ac bc
ca
ab
]]),0([),([ 01 VtV
)()()()( 1111 tVcacatVtVababtV
bacbcbac
63
Second Order Diagrams
a a
t
0t
c
1tt
b
a
a a
b
a
c
ca
a a
b
b
c
a
a a
a b
ca
64
Third order (1 of 8)
a at
0t
c
2tt
d
b
b
1tt
d
65
Polarization
a
at
0t
c
2tt
d
b
b
1tt
d
)(Tr)( tVtP
a
66
Polarization
a
at
0t
b
2tt
d
b
b
1tt
d
3tt c
a
b
a
)(Tr)( tVtP
67
Time-resolved ISRS
a
at
0t
b
2tt
d
b
b
1tt
d
3tt c
a
b
a
01 t 32 tt
68
Time-resolved ISRS
a
at
0t
b
t
a
a
a a
01 t 32 tt
69
Time-resolved ISRS
a
at
0t
b
t
a
a
a a
a
a b
a
b b
70
Coherence
a
at
0t
b
t
a
a
a a
a
a b
a
b b
Cohere
nce
w
ab
ba
i
EEi
e
e
/)(
71
Time-resolved ISRS
a
0t
0t
1
t
0
0
0 0
012
72
Time-resolved ISRS
a
0t
0t
1
t
0
0
0 1
012
01
w
10
01 /)(2
EE
73
Time-resolved ISRS
t
0t
t
10
w
10
01 /)(2
EE
0
0 1
0
1 1
74
Time-resolved ISRS
a
0t
0t
1
t
0
0
0 1
1001
www
10sin1010 ii
ee0
0 1
0
1 1
75
76
2D Methods
Tanimura and Mukamel (1993)
77
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
78
2D Femtosecond Spectroscopy
t = 0
1 2
79
2D Femtosecond Spectroscopy:
Impulsive pumping
t = 0
1 2
80
2D Femtosecond Spectroscopy:
Coherence propagation
t 1
1 2
81
2D Femtosecond Spectroscopy:
Rephasing pulse
t = 1
1 2
82
2D Femtosecond Spectroscopy:
Second coherence propagating
t 2
1 2
83
2D Femtosecond Spectroscopy:
Probing
t = 2
1 2
84
2D Experimental SetupLaser
Oscill
ato
r
Detector
1
signal
sample
2
2
1
85
Example: CS2
*Astinov, et al. Chemical Physics Letters (2000).
86
87
Six-wave Mixing Theory
Steffen, Fourkas, and Duppen
88
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
89
Fifth-order Response
0
21
2
1
2
21
)5(
2
0
1
)5( )()(),(dd)()( tEtERtEtP
)0(~,)(~),(~
4
)()0(~),(~
2
)()0(~),(~
4
)()(),(
1212
21
11
2121
)5(
i
i
i
R
6WM
Raman / 2nd hyper-Raman
90
Conditions
Nonlinear polarizability (NP)
Anharmonic coupling (AN)
ij
ji
qjii
i
qi
qqqq
00
2
02
1)()(
ijk
kjiijk
i
ii qqqqqV )3(2)2(
6
1
2
1)(
Coupling of ortho modesOrdinary Raman
Tokmakoff, et al. Chem Phys. (1998)
91
Harmonic Oscillator
92
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
93
Phenomenological Damping
Weak systembath coupling
ww i
*2/)(
State lifetime Decoherence
94
Full 3rd Order Damped Response
95
Full 5th Order Damped Response
96
State-independent Damping
State decay
One quantum coherences
Two quantum coherences
1
1
*
1,
2
2
*
2,
97
Damped 3rd Order Response
decoherence rates
98
Damped 5th Order Response
Undamped
Damped
99
Brownian Oscillator Bath
System / Bath linear coupling
Tanimura and Mukamel’s approach
Equivalent to phenomenological model when:
damping is state dependent!
21
221
1
100
Coupled Bath Dephasing
First excited state decayOne-quantum dephasing
101
Level-dependent dephasing leads to a
new term in R(5)
102
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
103
Inhomogeneous Damping
),,()(d),(
),()(d)(
21
)5(
0
21
(5)
inh
1
)3(
0
1
(3)
inh
www
www
RgR
RgR
104
One term is invariant to g(w
105
Rephasing Pathways
106
Initial State
1t 1
107
Pump Interaction
1t
0t
1 0
1
w10ie01
108
Rephasing Pulse
1t
0t
1 0
1
1 2
w10ie01
1t
21 w12ie
109
Probe Pulse
1t
0t
1
21 t
0
1
2 2
1 2
w10ie01
1t
21 w12ie
110
Phase Cancellation
1t
0t
1
21 t
0
1
2 2
1 2
w10ie01
1t
21 w12ie
)()( 21102110212110 wwwww
iiee
111
112
Third-order Cascades
Blank, Kaufman, and Fleming
113
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
114
CS2 Predicted 2D Response
115
CS2 Predicted 2D Response
116
What was measured:
117
Sequential Cascades
t = 0
chromophore a
chromophore b
2 4
118
Sequential Cascades
t = 0
chromophore a
chromophore b
2 4
119
Sequential Cascades
t 2
chromophore a
chromophore b
2 4
120
Sequential Cascades
t = 2
chromophore a
chromophore b
2 4
121
Sequential Cascades
t 2 + 4
chromophore a
chromophore b
2 4
122
Sequential Cascades
t = 2 + 4
chromophore a
chromophore b
2 4
123
Sequential response
124
Eliminating Sequential Cascades
Theory Measured
125
Parallel Cascades
t = 0
chromophore a
chromophore b
2 4
126
Parallel Cascades
t = 0
chromophore a
chromophore b
2 4
127
Parallel Cascades
t 2
chromophore a
chromophore b
2 4
128
Parallel Cascades
t = 2
chromophore a
chromophore b
2 4
129
Parallel Cascades
t 2+4
chromophore a
chromophore b
2 4
130
Parallel Cascades
t = 2+4
chromophore a
chromophore b
2 4
131
Parallel Response
132
Measured Parallel Response
Simulated Measured
133
134
Eliminating Cascades
135
Roadmap
1) One-dimensional Raman methods
a) Motion in Liquids
b) Third order response theory
c) Liouville space paths and Feynman diagrams
2) Two-dimensional Raman methods
a) Fifth order response theory
b) Homogeneous damping effects
c) Inhomogeneous damping and rephasing
d) Third order Cascades
e) Solutions to the problem of cascades
136
Diffractive Optics
Astinov, et al. Opt. Lett. (2000)
137
Diffractive Optic Results
Cascade
Astinov, et al. Opt. Lett. (2000)
138
More Accurate Phase Matching
sinc(Dkl/2) assumes
collinear propagation
Constant spatial overlap
Account for z-
dependent spatial
overlap:
Blank, et al. J. Chem. Phys. (2000)
139
Reduced Interaction Length
LEekL
REn
LiE kLi
s
s
D D
signal
2/
42
)5(5
signal2
sinc),( w
2
cascade
2/
2/
42
)3(
cas
5
int
int
cas
cas2
cascade
2sinc
2sinc
),(
LEeLk
eLk
REnn
iLE
Lkib
Lkia
b
a
D
D
D
D
ww
140
Heterodyne Detection
141
Heterodyne Detection Results
Astinov, et al. Chem. Phys Lett. (2000)
142
143
Conclusions
Steffen et al’s theory did not account for
parallel cascades
Blank, et al. found parallel cascades
dominate the signal
Eliminating parallel cascades reveals desired
signal
144
Thanks
145
Roadmap
1. One-dimensional Raman methods
1. Motion in Liquids
2. Third order response theory
3. Illustrating 3rd order response with Feynman
diagrams
2. Two-dimensional Raman methods
1. Fifth order response theory
2. Homogeneous damping effects
3. Inhomogeneous damping and rephasing
4. Third order Cascades
5. Solutions to the problem of cascades
146
147
2D Resonant IR (Part I)
148
2D Resonant IR (Part II)