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Photon Echo Technique
Quantum Mechanics of Ensembles
Described by the density matrix rather than a wavefunction.
1 2
1 2
1 2
/ /1 2
( ) /21 1 2
( ) / 21 2 2
=
= 1 + 2
1 0 1 = 2 =
0 1
=
iE t iE t
i E E t
i E E t
P
c e c e
c c c e
c c e c
Two level system
eg
11 12
21 22
=
= < > = x ( )
P N N Tr
The evolution of the electric field is governed by the polarization P.
couples the two states i.e.
aa bb ab ba = 0, = =
= ( )ba abP N
Calculating Nonlinear Signals
1
[H, ]t i
Time evolution of used to calculate the polarization P
Expand P as (1) (2) (3)
(n) (n)
2 (3)k
= < > + < + < .........
< = Tr ( )
( ) ( , , )
jj k
P P P P
P P
8 terms 48 terms
with
Long and tedious expressions.Help is at hand!
For a two level system only 4 terms and their complex conjugates survivethe definition of the density matrix
= Suggests we can represent these terms by diagrams in which we propagate the bra and ket separately.
Feynman diagrams & the density matrix
|g g|
|g e|
|e e|
|e g|
|g g|time
T
t
-k1
k2
k3
ks
ks = -k1+k2+k3
k3
k1
k2
1 0
0 0
|g
|e
e|g|-k1
k2
energy
|g
|e
k3
ks
density matrix
phase-matching direction
ks ks
R2
Geg(t3)
Gee(t2)
Gge(t1)
k3
k1
k2
R1
Geg(t3)
Gee(t2)
Geg(t1)
k3
k2
k1g g
ks
Geg(t3)
Ggg(t2)
Gge(t1)
k2k3
k1
ks
k3
k2
k1
Geg(t3)
Ggg(t2)
Geg(t1)
R3 R4
If k3 = k2 (same pulse) ks= k1 for R1 and R4
ks = 2k2-k1 for R2 and R3
Two level systems are described by four Feynman diagrams and their complex conjugates
Echo-Inhomogeneous broadening
from Erwin Hahn and Chris Noble
Lens Analogy for Photon Echoes
1egi te
0( )
t
egi de
1i te
1 1( )t t
After the first interaction we have a superpositionoscillating at the energy difference between and .g e
Optical frequency (1)
Homogeneous dephasing (2)
Inhomogeneous contributionleads to rephasing (3)
(3) Define electronic phase factor
Linear with slope determined by inhomogeneous parameter
For N molecules we get N lines with different slopes
Width amount of inhomogeneity
3i te
The second interaction produces a population—no ε-term in difference between |e> and |e>
Now the third pulse phasefactor is (sign change because now Ee-Eg not Eg-Ee), so now the slope of each ray will change sign but have the same magnitude.
e g
Rephasingresponsefunction
Non-rephasing response function
2 PE
t1 t2
DephasingSpectral diffusion
Dephasing
t1 t3t2
Refocusing gets poorerand poorer as t1, t2
increased.
Photon Echoes
Pulse 1 creates coherence(|g> AND |e>)
Pulse 2 creates population(|g> OR |e>)
Pulse 3 creates another coherence
1 2 3 echo
Oscillatory term during first (second) coherence: e -(+)iωegt
Slope of rays depends on ωeg in oscillator term
Top: CO asymmetric stretch of W(CO)6 in 2 methyl pentane.Bottom: CO asymmetric stretch of W(W)6 in dibutyl phthalate.The beats are at the anharmonic vibrational splitting, and arise because the pulsewidth (0.7ps) is less than in the top figure.
Figure 3. Temperaturedependence of the homogeneous line widths of the T|u CO stretching mode of W(CO)6 in 2-MTHF, 2-MP, and DBP determined from infraredphoton echo experiments using eq 9b.arrows mark the glass transition temperatures. Note the different temperature and line width scales.
Tokmakoff….Fayer J. Phys. Chem, 99 13310 (1995).
Absorption Linewidth
W(CO)6 in 2-MP
Two Pulse Electronic Echoes
2k1-k2 2k2-k1
20 fs transformlimited pulses
Deconvolution 20 fs decay
HITCI in
glycerol/water
(70/30)
900 950 1000 1050 11000.0
0.5
1.0
Nor
mal
ized
Abs
orba
nce
a.u.
Wavelength (nm)
(6, 5) E11
SWNT Peak
~0.75 nm
~800 nm
Exciton Dephasing in Semiconducting Carbon Nanotubes
• Only the (6,5) type SWNTs are resonantly excited, and the resulting 2-pulse photon echoes (2PEs) decays are measured
• 2PEs provides a direct method to determine dephasing times
• At RT, the FWHM of the inhomogeneous processes are ~6X the homogeneous width
Homogeneouscontribution
2D Spectroscopy of Aggregates
MOLECULAR AGGREGATES
WEAKLY COUPLED
STRONGLY COUPLED
LH2 Complex
Two-excitonBand 2e
One-excitonBand 1e
Ground state g
Linear chain of 2 level molecules with electrostatic dipole-dipole interaction
Absorption spectra of BIC monomer and J-aggregates
J-AGGREGATE HAMILTONIAN
Off-diagonalElectrostatic
DiagonalElectron-Phonon
N
nmnm
mn
N
nn nJmnn
1,1
)(qH
N
n
phelnn nqHn
1
)( )(qH ph
EXCITON BASIS: EXCITON WAVEFUNCTIONS
•Higher Exciton States are Strongly Delocalized•Exchange-Narrowing is Stronger for Higher (More Delocalized) Exciton States•Relaxation is Faster for Higher Exciton States
Diagonal Exciton-Phonon Off-Diagonal Exciton-Phonon
SITE BASIS:
Overlap Factors DefineRelaxation
Renormalization FactorsCause Exchange Narrowing
Photon Echo Technique
Integrated Three Pulse Photon Echo:Nile Blue in Acetonitrile
Origin of the Peak Shift
Non-rephasing side not influenced by spectral diffusion
Rephasing side as spectral diffusion occurs will become more and more like non-rephasing side
Eventually the echo signal will become symmetric around τ=0
Measuring inhomogenous broadening
Coherence Time, t (fs)-20 0 20 40 60
Po
pu
lati
on
Tim
e, T
(fs
)
0
50
100
150
200
Population Time, T (fs)
0 200 400 600 800
Peak
sh
ift,
(fs
)
0
5
10
15
20
25
30
Peakshift tracks the surface denoted by the blue line
IR 144 τ*(T) vs. T
32K
294K
Ethanol 294K
• Finite long time peak shift
• Inhomogeneous broadening
• Timescales of fluctuations in transition frequency.
What is the Peak Shift?
At high temperature it relates to the Stokes shift dynamicsand the ratio of dynamical and static contributions to the spectral broadening.
The long time value *( ( ))T allows the inhomogeneous width to
obtained: in
The time dependence gives ( )
2 2*
2 2
2 ( )( ( )( )
[ ( 2 ( )) 2 ( )]in in
in in
f t S TT
f T f T
Stokes Shiftinhomogeneouswidth
2 2 2*
2 2 2 2
( / )( )
[ ( 2 ( / ) ) 2 ( / )in in
in in
T
22 /
obtain inhomogeneouswidth, inM. Cho
( )S t
( ( ) ( ))S t M t
Solvation Dynamics IR144 in acetonitrile
Correlation function
Spectral Density
Peak Shift
Instantaneous Normal Mode Spectral Density
CH3CN
Solvation Spectral Density for Acetonitrile
Dielectric continuum models
from the experimental IR data of water
of hydrated lysozyme [S. C. Harvey and P. Hoekstra, JPC , 571 (1973)]76
from molecular dynamics (MD) simulations [W. F. Van Gunsteren JPC , 200 (1993)] et al. 97
from experimental dielectric data
[X. Song, R. A. Marcus, JCP , 7768 (1993)]99
c
m-1
)
c(cm-1)10-3 10-2 10-1 100 101 102 103
0
100
200
300
400
500
bulk water, bound water dynamic lysozyme
bulk water, static lysozyme
bulk water, static lysozyme , bound water
bulk water, dynamic lysozyme from MD
Model spectral densities
[X. J. Jordanides et al. J. Phys. Chem. B 103, 7995 (1999)]
Dielectric Response of Aqueous ProteinsLysozyme with eosin bound in the ‘hydrophobic box’
Eosin/lysozyme/water
Eosin/water
LH1 and Reaction Center of Purple Bacteria
Roszak, Howard, Southhall,Gardiner, Law, Isaacs & CogdellScience, 302, 1969 (2003).
Structure of the LH3 Complex
Rhodopseudomonas acidophila Strain 7050
K. McLuskey et al.: Biochemistry 40, 8713 (2001).
T(fs)
Pe
ak S
hif
t (f
s)
Photon Echo Peak Shift Measurements LH1 of Rb. sphaeroides vs. the B820 Subunit of LH1 of Rs. rubrum
Same parameters as LH1except no 90 fs EET component
B820 subunit of LH1
Inhomogeneous broadening90 fs energy transfertimescale
LH1
Light Harvesting Complex II
Wavelength/nm
Ab
sorb
ance
(n
orm
.)
Bacterial Light Harvesting
Bahatyrova, et al.Nature (2004) 430 1058
Hu, et al. J. Phys. Chem. B (1997) 101 3854
Population Period, fs
0 2000 4000 6000 8000 10000
Pea
k S
hif
t, f
s
0
5
10
15
20
Population Period, fs100 1000 10000
Pe
ak S
hif
t, f
s
0
5
10
15
20
Peak Shift on the B850 band of LH2 membranes (Rps. acidophila)
Intra-complex exciton relaxation or energy transfer
Energy Transfer between the complexes
Solubilized samples
Membrane samples
Membrane samples
Solubilized samples
Since the Peak Shift carries information abut the inter-complex energy transfer dynamics, we can say that the individual rings do not have the full disorder distribution that is observed in the absorption spectrum. Energy Transfer between the rings is estimated to be ~ 5 ps at room temperature.
In collaboration with C. N. Hunter,Sheffield
Pump Probe (Transient Absorption)
IR144 in MeOH
Pump-Probe (Transient Absorption)
k1and k2 come from same pulse
ks = -k1 + k1 + k3 = k3
signal along probe direction
P(3) heterodyned with probe field.
Measurement time window (t’) determined by the pulse duration of the probe.
• If the probe is short rephasing may not be detected.
• M(t) reflected in pump-probe signal (may be difficult to extract quantitatively).
• “coherence” spike not a coherent effect. Arises from dynamics.
gg
eg
ee
ge
gg
rephasing diagram
k3
ks
k2
k1
Detector
Probe
Pump
Contributions to Pump-Probe Signal
Pump Probe Signals (Calculation)
Transient Absorption
Coworkers
Taiha Joo
Minhaeng Cho
Yutaka Nagasawa
Sean Passino
Matt Lang
Xanthipe Jordanides
Xeuyu Song
Peak Shift IR144 in MeOH
1-Color Transient Grating Signals
0.0 0.2 0.4 0.6 0.8 1.0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1(a)
= 100, 200, 300, 400, 500 fs (from left to right)
1-C
TG
Sig
nal (
norm
aliz
ed)
Time (ps)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
= 100 fs
(b)
Transient birefringence (Re[P]2)
Transient dichroim (Im[P]2)
Total 1-C TG signal
1-C
TG
Sig
nal (
norm
aliz
ed)
Time (ps)
2[exp( / ) 0 5exp( 2 )]( )1 5
t tS t
Time unit: ps.
1600cm
Two Color Transient Grating Signals
0.0 0.2 0.4 0.6 0.8 1.0
2
4
6
8
10
12
14
16 (a)
W = 0, 200, 400, 600, 800, 1000, 1200, 1400 cm-1
(from top to bottom)
2-C
TG
Sig
nal (
arbi
trar
y un
it)
Time (ps)
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
(b)
= 100 fs
W = 800 cm-1
Transient birefringence (Re[P]2)
Transient dichroim (Im[P]2)
Total 2-C TG signal
2-C
TG
Sig
nal (
arbi
trar
y un
it)
Time (ps)
2[exp( / ) 0 5exp( 2 )]( )1 5
t tS t
1600cm
pump probeW
'downhill'W Positive
Negative 'uphill'W
At 11200W cmthe probe is at the bottom of the excited state well.
For large detuning the birefringentcontribution becomes similar to the dichroic contribution (at short times).
( 2 )
Two Color Transient Grating Signals. Homodyne Detection
0.0 0.2 0.4 0.6 0.8 1.04
5
6
7
8
9 = 100, 200, 300, 400, 500 fs (from left to right)
2-C
TG
Sig
nal (
arbi
trar
y un
it)
Time (ps)
Detuning = 800cm-1
2[exp( / ) 0 5(exp 2 )]( )1 5
t tS t
Maximum correlates well with Gaussian time constant, .
Experimental 1-Color and 2-Color TG Signals for DTTCI in MEOH
Downhill. Detuning =Probe close to minimum of excited state surface.
1833 ,cm 1430cm
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-50 150 350 550 750 950
Population Time (fs)
No
rma
lize
d I
nte
nsi
ty
800, 800, 800
750, 750, 750
800, 800, 750
750, 750, 800
Experimental One-Color and 2-Color TG Signals for IR144 in MEOH
Population Time, fs1 10 100 1000 10000
No
rmal
ized
Inte
nsi
ty
0.0
0.5
1.0
1
1
1500
833
cm
W cm
(downhill)
1C 750nm2C 750, 750 800nm
0
0.2
0.4
0.6
0.8
1
600 650 700 750 800 850 900
Wavelength (nm)
Inte
nsi
ty (
no
rmal
ized
)
0
0.2
0.4
0.6
0.8
1
600 650 700 750 800 850 900
Wavelength (nm)
Inte
nsi
ty (
no
rmal
ized
)
Two-Color three-pulse Photon Echoes
IR144 in Methanol
DTTCI in Methanol
-5
0
5
10
15
0 100 200 300 400 500
P o p u l a t i o n T i m e (fs)
P e
a k
s h
i f t
(f s
)
Type I Type II Difference
IR144 Methanol 750, 750, 800
gg
ge
ee
k1
k2
k3eg
gg
eg
ee
k1
k2
k3eg
ks
Type II scanFID (Non-Rephasing)(pulse sequence, 2-1-3)
Type I scanEcho (Rephasing)(pulse sequence, 1-2-3)
τI τII
TI TII
Population Period, fs
0 200 400 600 800 1000
Pe
ak
Sh
ift,
fs
-5
0
5
10
15
20
Type I
Type II
Population period, fs0 200 400 600 800 1000
Dif
fere
nc
e P
ea
k S
hif
t, f
s
0
1
2
3
4
5
6
7
Difference peak Shift = Type I - Type II
Δτ*(T) = τI*(TI) - τII*(TII)Two Colour
Difference Peak ShiftFor a fixed phase matching direction, i.e., k3 + k2 – k1
-5
0
5
10
15
0 100 200 300 400 500
P o p u l a t i o n T i m e (fs)
P e
a k
s h
i f t
(f s
)
Type I Type II Difference
IR144 Methanol 750, 750, 800
Population Period, fs
10 100 1000 10000
Dif
fere
nce
Pea
k S
hif
t, f
s)
0
1
2
3
4
5
IR144 in Methanol
DTTCI in Methanol
Pulse Sequence, 750-750-800 nm
Experimental Difference Peak Shift Data (downhill)
The Difference Peak Shift starts at a near zero value, then rises to a maximum value in ~ 200 fs and then decays to zero for both IR144 and DTTCI in methanol
Based on the turnover time, it is suggested that the ultrafast component in methanol is ~ 200 fs
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1000 -500 0 500 1000 1500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1000 -500 0 500 1000 1500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1000 -500 0 500 1000 1500
I
II
III
Population Period, fs
0 200 400 600 800 1000
Diffe
renc
e Pe
ak S
hift,
, f
s
0
2
4
6
8I
II
III
II
Spectral Models and the Two-Color Difference Peak Shift
Downhill Case
1 mode.Gaussian
M(t)
2 modes.
35 modes.
Two-Color 3PEPS as a Probe of Memory Transfer in Spectral Shift
= +probe
pump
Time-Dependent Spectral Shift
Tra
nsiti
on D
ensi
ty
Energy
Memory-Conserved Shift
Randomized Shift
Two-Color 3PEPS measures correlation dynamics (between transition energies in pumped and probed regions).
( )r pup td dee:
Homogeneous and Inhomogeneous Distributions of Transition Energies
• Homogeneous distribution
( )( ) 2hom( ; )
e g
e
g e gE Ej jj
s w j d w j j= - -å
A particular nuclear state in the ground electronic state
• Inhomogeneous distribution
Two Mechanisms for Existence of Non-Linear Signals of Two-Color Experiments
• Interactions of pump and probe lasers have to be made with the same molecule
• These two mechanisms are included in the response function formalism in a complicated way
A) Spectral Overlap due to Homogeneous Distribution
B) Spectral evolution due to Fluctuation of Inhomogeneous Distribution
( )hom( ) ( ; )g
g g gPj
s w s w j j= å
Statistical probability for a molecule to occupythe nuclear state gj
( ) ( ) ( )hom inhom
hom inhomhom inhom hom inhom
( ) ( )
( ) ( ) ( ) ( )pu pu pupr pr prt tP t P t
P t P t P t Pt
tde de dede de de= +
+ +
Total Signal = hom hom inhom inhom( ) ( ) ( ) ( )P t R t P t R t+
Total Correlation Function
At short times,
A Simple ad hoc Model for the Dynamics of Correlation Function
hom inhom( ) ( )P t P t>>
( ) ( )( ) ( ) ( ) ( )1 2 2 1 2 2 1 1 1inhom
1( ) ; ( )
( )pu pr pu absprupr p dt d t W E P t W E
N tw w w w w w w w wdde s we w= - -òò
Inhomogeneous distribution fluctuates with time due to random fluctuation of the statistical distribution of the nuclear states, which is described by a stochastic approach.
22 1
2 1 2 22 2
[( ) ( ) ( )]1( ; | ) exp
2 (1 ( ) )2 (1 ( ) )
M tP t
M tM t
Skinner et al, J. Chem. Phys. 106, 2129 (1997)
2 1 2 1( ;0 | ) ( )P
At longer times, hom inhom( ) ( )P t P t<<
0 100 200 300 400 500 600
No
rmal
ized
Dif
fere
nce
Pea
k S
hif
t
Rep
hasin
g C
apab
ility
Time (fs)
Dynamics of Conditional Probability for the Inhomogeneous Distribution
Full Response Function
( )inhomppr ut ed de
Homogeneous broadening domain
: No common transitions between the pump and the probe (no rephasing capability)
Rise in Two-Color Difference Peak Shift ~ Inertial Solvation Dynamics
Uphill and Downhill difference peak shifts should have distinct behavior for systems with a systematic red shift
Population Period, fs
0 200 400 600 800 1000
(T),
Dif
fere
nce P
eak S
hif
t, f
s
-2
0
2
4
6
8
10
500
300
200
100
50
Difference Peak Shift = TypeI - Type II
Model Calculations for Difference Peak Shift (downhill)
2( ) exp[( / )]gM T t
Empirical formula: 1 2 3~ { log( / ) },gturnoverT c c c
g Gaussian Time Constant, reorganization energy
Adding exponentials and vibrations does not alter the turnover time significantly. Therefore, we can extract information of the Gaussian parameters from the turnover time.
Frequency difference between the two pulses
Population Period, fs
0 100 200 300 400 500 600 700 800 900 1000
Dif
fere
nce
Pea
k sh
ift,
fs
0
1
2
3
4
5
0 1000 2000 3000 4000 5000
Dif
fere
nce
Pea
k S
hif
t0
1
2
3
4
5
Population Period, fs
Simulation model for the Difference Peak Shift
Simulation scheme: Type I and II peak shifts were calculated using a Gaussian (220 fs, = 150 cm-1) ,exponential 1 (2500 fs, = 75 cm-1), exponential 2 (9500 fs, 70 cm-1), 35 intramolecular modes ( tot ~ 400 cm-1)
IR144 in Methanol
Pulse Sequence: 750-750-800 nm
Two Color Peak Shift: Energy Transfer Systems
In an inhomogeneous energy transfer system, spectral overlap induces correlation between donors and acceptors.
Difference Peak Shift
Type I (rephasing)
Type II (nonrephasing)
1-and 2-Color (620, 620, 700nm) Photon Echo Peak Shift
• 1-color and 2-color peakshifts of LuPc2 are very similar
• Oscillation, of similar period in both measurements, but approximately π out of phase
Population Time (fs)
0 100 200 300 400 500
Pe
aksh
ift (
fs)
0
10
20
30
1-color
2-color
2LuPc
Theory for 2C3PEPS of Excitonically Coupled Molecules
• εA, εB = site energies • J = coupling
• θ = degree of mixing
• Cμν = theoretical renormalization coefficient for line broadening function
• C* = experimentally determined renormalization coefficient for line broadening function ratio.
eqB
eqA
J
2)2tan(
22 cossin2CC
)()(
)()(* TT
TTCC
onetwo
two
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