0

0.2

0.4

0.6

0.8

1

600 650 700 750 800 850 900

Wavelength (nm)

Inte

nsi

ty (

no

rmali

zed

)

0

0.2

0.4

0.6

0.8

1

600 650 700 750 800 850 900

Wavelength (nm)

Inte

nsi

ty (

no

rmali

zed

)

Two-Color three-pulse Photon Echoes

IR144 in Methanol

DTTCI in Methanol

Condensed Phase Dynamics

( ) ( )i

eg eg eg it tω ω δω ε= + +

In an ensemble of chromophores, the dynamics for an individual chromophore (i) can be expressed as

Full width half maximum of f( )ε = inhomogeneous width

Optical lineshapes and

echo measurements relate

to:

2

(0) ( )( )

tM t

δω δω

δω=

Stokes Shift and Solvation Dynamics

( ) ( )( )

(0) ( )t

S tυ υυ υ

− ∞=

− ∞

( )ieg eg i itω ω δω ε= + +

Optical Lineshapes

For the i th chromophore

egω

Optical lineshapes and echo

measurements relate to 2

(0) ( )( )

tM t

δω δω

δω=

Solvation Dynamics in C343 in Water

Jimenez, Fleming, Kumar and Maroncelli (1994) Nature 369, 471

0

0.2

0.4

0.6

0.8

1

600 650 700 750 800 850 900

Wavelength (nm)

Inte

nsi

ty (

no

rmali

zed

)

0

0.2

0.4

0.6

0.8

1

600 650 700 750 800 850 900

Wavelength (nm)

Inte

nsi

ty (

no

rmali

zed

)

Two-Color three-pulse Photon Echoes

IR144 in Methanol

DTTCI in Methanol

gg

ge

ee

k1

k2

k3eg

gg

eg

ee

k1

k2

k3eg

ks

Type II scan

FID (Non-Rephasing)

(pulse sequence, 2-1-3)

Type I scan

Echo (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

nce P

eak S

hif

t, f

s

0

1

2

3

4

5

6

7

Difference peak Shift = Type I - Type II

∆τ*(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

∆τ

∆τ

∆τ

∆τ∗∗ ∗∗

( ( ( (D

iffe

ren

ce P

ea

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 fsand 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

Two-Color 3PEPS as a Probe of Memory Transfer in Spectral Shift

= +ωωωωprobe

ωωωωpump

Time-Dependent Spectral Shift

Tra

nsitio

n D

ensity

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 j

j

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 occupy

the nuclear state gj

( ) ( ) ( )hom inhom

hom inhomhom inhom hom inhom

( ) ( )

( ) ( ) ( ) ( )pu pu pupr pr prt t

P 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= - -ò ò

Inhomogeneous distribution fluctuates with time due to random fluctuation of the statistical distribution of the

nuclear states, which is described by a stochastic approach.

2

2 1

2 1 2 22 2

[( ) ( ) ( )]1( ; | ) exp

2 (1 ( ) )2 (1 ( ) )

M tP t

M tM t

ω ω ω ωω ω

π

− − − = × −

∆ − ∆ − Skinner et al, J. Chem. Phys. 106, 2129 (1997)

2 1 2 1( ;0 | ) ( )P ω ω δ ω ω= −

At longer times,hom inhom( ) ( )P t P t< <

0 100 200 300 400 500 600

No

rmalize

d D

iffe

ren

ce P

eak S

hif

t

Rep

hasin

g C

ap

ab

ility

Time (fs)

Dynamics of Conditional Probability for theInhomogeneous 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

∆τ

∆τ

∆τ

∆τ∗∗ ∗∗

(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 tλ τ= −

Empirical formula: 1 2 3~ log( / ) ,gturnoverT c c cτ λ∆ +

gτ = Gaussian Time Constant, λ = 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

nc

e P

eak

sh

ift,

fs

0

1

2

3

4

5

0 1000 2000 3000 4000 5000

Dif

fere

nce

Peak

Sh

ift

0

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, λλλλ = 150 cm-1) ,exponential 1 (2500 fs, λλλλ = 75 cm-1),

exponential 2 (9500 fs, 70 cm-1), 35 intramolecular modes (λλλλ tot ~ 400 cm-1)

IR144 in Methanol

Pulse Sequence: 750-750-800 nm

Two Color Difference Peak Shift for Nile Blue

Population Time, fs

0 50 100 150 200

Detu

nin

g (

pum

p-p

robe),

cm

-1

-200

0

200

400

600

800

-200

0

200

400

600

800

0

3

6

9

0 50 100 150 200 250

-4

0

4

8

12

A B

C D

Experiment Simulation

Acetonitrile

Ethylene glycol

Ket State Transition

Bra State Transition

J

Eigen State RepresentationSite Representation

1. Electronic Mixing in Molecular Complexes

Two-Color Three-Pulse Photon Echoes

Two interactions with

0 µ→

0 υ→

one with

Detect at 0 υ→

frequency

A1

B1

A0 B0

µ µ

υ υ υ

0 0

2. Correlation between initial and final

states e.g. disordered energytransfer system

0 200 400 600 800 10000

5

10

15

20

25

30

35

40

Peak S

hift (

fs)

3PEPS

Population Time (fs)

0 200 400 600 800 1000-5

0

5

10

15

20

25

Peak

Shift (f

s)

3PEPS

Population Time (fs)

Energy Transfer system

( ) ( )S Sf G GG

= ( ) ( ( ))S S kG G

f G G» < > < >vs. <…..>Γ

Ensemble averages over distribution of static energies

Donors and acceptorsbecome correlated via energy transfer (even thoughinitially uncorrelated)

( )φ Γ phase factor( )S Γ population kinetics

2-colorAB

1-colorAB

Interaction with

1st pulse

2nd

(Population state)

3rd

One-color three-pulse photon echo peakshift (1C3PEPS)

Strong correlation between the two coherence states (long-lasting memory) large peakshift

Coherence state

Coherence state

|g⟩⟩⟩⟩

|e⟩⟩⟩⟩

Weak correlation between the two coherence states (quick memory loss) small peakshift

Interaction with

1st pulse 3rd

Strong correlation between the two coherence states (long-lasting memory) large peakshift

One-color three-pulse photon echo peakshift (1C3PEPS)

2

(0) ( )( )

tM t

ω ω

ω

∆ ∆=

∆

3PEPS profile closely follows the transition frequency correlation function

|g⟩⟩⟩⟩

|e⟩⟩⟩⟩

Coherence state

Coherence state

2nd

(Population state)

Pulses of two different colors generate photo echo signal associated with correlation between

∆ωH and ∆ωB

Energy fluctuation of two excitonically mixed states are correlated due to electronic coupling

Electronic coupling constant (J) can be determined by 2C3PEPS without knowing the site

energies (Eh and Eb)of the two chromophores

Two-color three-pulse photon echo peakshift (2C3PEPS)

|B⟩⟩⟩⟩

|g⟩⟩⟩⟩

|b⟩⟩⟩⟩

Ener

gy

|H⟩⟩⟩⟩

|h⟩⟩⟩⟩J

λ1=λ2≠λ3

Sensitive to coupling between two chromophores

λ1=λ2=λ3

Sensitive to environment around a chromophore

and energy transfer process

Two-color Three-pulse Photon Echo Peakshift

(2C3PEPS)

One-color three- pulse photon echo peakshift

(1C3PEPS)

Photon echo peakshift: coherence time (τ) where a photon echo signal is maximum

One- and two-color three-pulse photon echo peakshift experiment

Strongly Coupled:Exciton Dimer

Molecule 1

|1g>

|1e>

|2g>

|2e>

Molecule 2Dimer

|1e2e>

|1g2g>

c1 |1e2g> – c2 |1g2e>

c1 |1e2g> + c2 |1g2e>

0

0.2

0.4

0.6

0.8

1

1.2

1.4

550 600 650 700 750

0

0.2

0.4

0.6

0.8

1

1.2

1.4

550 600 650 700 750

Electronic Coupling, J, can be extracted from shifts of absorption peaks.

This requires knowledge of original site energies.

Pump Probe Spectroscopy of Bis-Phthalocyanine:Excited State and Wavepacket Dynamics

N. Ishikawa, Chuo University

LuPc2 Dimer

• Absorption spectra

shows monomer’s

single peak is split in

the dimer

• Inset: Sandwich

structure of Lutetium

bisphthalocyanineWavelength (nm)

400 500 600 700

E / 1

05 M

-1cm

-1

0.0

0.5

1.0

1.5

2.0

2.5

Dimer

Monomer

TwoTwo--Color Echoes on Excitonic Systems: Color Echoes on Excitonic Systems:

TheoryTheory

M. Yang, G. R. Fleming, J. Chem. Phys. 110, 2983-2990 (1999).

1 2(cos( ) sin( ) ) 0 0B B Bµµ θ θ+ + +≡ ⋅ + ⋅ ≡

1 2( sin( ) cos( ) ) 0 0B B Bνν θ θ+ + +≡ − ⋅ + ⋅ ≡1 2

2tan(2 )

eq eq

Jθ

ε ε=

−

Mixing between states with site energies ε1 and ε2 andcoupling J can be quantified by θθθθ.

Energy fluctuations at one monomer shift both exciton states in the same direction; the magnitude depends on the coupling.

Stronger mixing yields stronger correlation.

Coupling StrengthCoupling Strength

M. Yang, G. R. Fleming, J. Chem. Phys. 110, 2983-2990 (1999).

0 100 200 300 4000.0

0.1

0.2

0.3

Co

up

lin

g S

tren

gth

, C

µν

µν

µν

µν

Population Time, T (fs)

, ColorColor

Color

21

2

+=

*

( )( )

( ) ( )two

two one

TC C T

T Tµν τ

τ

τ τ

∗

∗ ∗≈ ≡

+

2 22 sin cosC Cµν νµ θ θ= = ⋅

Dependence of g(t) on Dependence of g(t) on

coupling causes the onecoupling causes the one--color color

and twoand two--color peak shift to color peak shift to

increase with increased increase with increased

coupling.coupling.

Coupling strength relates to the

renormalization coefficient, Cmn,

for the line broadening function, g(t).

Cµυµυµυµυ

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

akshift

(fs)

0

10

20

30

1-color

2-color

2LuPc

−

Cτ* (experiment)

Population Time (fs)

0 20 40 60 80 100

Cτ∗

(T)

0.0

0.2

0.4

0.6

10 100

0.1

0.3

0.5

• Experimental Cτ* is approximately 0.45

• Dip at longer time due to out-of-phase oscillation between 1- and 2-color

Electronic structure of 2

Lu(P )c−

Estimating Cµυ

EX-CRJEX + >

RC + >

g >

If all dipole strength comes from

the ratio of absorption intensities gives

EX ,+ >

2Tan ( ) 0.43Cµυθ ⇒

From wavefunction coefficients of

Ishikawa et. al 0.40Cµυ

The Mixing Coefficient

*

32

0

*

32

0

12 122

2

12

2

(1-C ) ( ) 2 ( )( ) =

(1 )

(C ( ) 2 ( )( ) =

(1 )

1 ( ) = ( ) (0)

= cos

one

tw o

T

vib

a T C N TT

C C

a T C N TT

C C

N T q T q

qe T

µν µν

µν

µν µν

µν

τπ

τπ

ω− Γ

+

−

−

−

< >h

h*

* *

( ) 2 ( ) = 1

( ) ( ) ( )two

one two

T N TC

T T a Tµν

τ

τ τ

−

+

Modulation of the Electronic Coupling

Minhaeng Cho

π phase

shift

( )( ) ( )a T real part of C t=

Time (fs)

20

15

10

5

0

-50 500 1000 1500

One-color PEPS

Two-color PEPS

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.10 0.5 1 1.5 2 2.5 3

Log(time) in fs

Rati

o

Time(fs)

Tim

e(f

s)

Wavelength (nm)

580 600 620 640 660 680 700 720 740

Po

pula

tio

n T

ime

(fs

)

0

200

400

600

800

1000

1200

1400

580 600 620 640 660 680 700 720 740

-0.010

-0.005

0.000

0.005

0.010

0.030

0.035

0.028

0.032

615 700

Wavelength resolved pump-probe signals for-2LuPc

The beats on the red side of the high energy band are exactly out of phase

with the beats on the red side of the low energy band, but are in phase withthe beats on the blue side of the low energy band

740 600 620 640 660 680 700 720 740

∆ECR+ = -60

∆EEX+ = 60

∆VEX-CR

= 30

F

Modulated Electronic Structure

Simple model with a single coordinate, x,coupledlinearly to the states , and the coupling betweenthese two states

o

EX EX EXE E E x+ + += + ∆o

CR CR CRE E E x+ + += + ∆

0

EX CR EX CR EX CRV V V x− − −= + ∆

EXE + CRE +

( ) cos( )exp( )ox t x t tω τ= −

EX CRV −

New,

time-dependent

potential

curves for

1 2E E+ +and

⇒ Non-Condon effects (modulation of transition

moments). If the signs of and

are opposite the motions of

are anti-correlated.

EXE +∆ CRE +∆

1 2E E+ +and

Physical Origin

Symmetric nitrogen-metal mode;

Fluctuating coupling

Ground State Wave-Packet

ω1

ω2

Bb

Ha

Ba

Hb

Fe2+

Qb Qa

P

~3ps

<1ps

~100fs

~150fs

XX

+

Uphill:

λ1=λ2=800nm,

λ3=750nm

Downhill:

λ1=λ2=750nm,

λ3=800nm

H band:

λ1=λ2=λ3=750nm

B band:

λ1=λ2=λ3=800nm

2C3PEPS1C3PEPS

|h⟩⟩⟩⟩

|gb⟩⟩⟩⟩

|b⟩⟩⟩⟩

Energy

|gh⟩⟩⟩⟩

|H⟩⟩⟩⟩

|B⟩⟩⟩⟩

Excitonic Coupling

1C and 2C3PEPS on P-oxidized reaction center of Rb. Sphaeroides

Simulation of 3PEPS based on density matrix

int( ) [ ( ), ( )] [ ( )]S

t i H H t t R tρ ρ ρ= − + −&

Directly simulate P(3)3PPE (t) from laser field-driven dynamics:

2(3)

3

0

( , ) ~ ( )PPES T dt P tτ∞

∫ Peakshift

Pros compared to conventional response function formalism:

• Pulse-overlap effects included by considering laser fields explicitly.

• Complete dynamics of the system treated by non-Markovian equations of motion.

Laser

Bath

6015050250H

60150

10

100

220

75

1500B

σΓΩλ λ: electron-phonon coupling constant

Ω: characteristic frequency

Γ: damping constant

σ: static disorder

Most of the dynamics happen in

less than 300fs due to the rapid

energy transfer (HBP+)

Proper inclusion of pulse-overlap

effects improved reproduction of

the peakshifts measured at small T

Determination of bath parameters

for H and B, respectively

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

Peaksh

fit(

fs)

T(fs)

H band (Experiment)

H band (Simulation)

B band (Experiment)

B band (Simulation)

Simulation for 1C3PEPS and obtained bath parameters

6015050250H

60150

10

100

220

75

1500B

σΓΩλ λ: electron-phonon coupling constant

Ω: characteristic frequency

Γ: damping constant

σ: static disorder

Most of the dynamics happen in

less than 300fs due to the rapid

energy transfer (HBP+)

Proper inclusion of pulse-overlap

effects improved reproduction of

the peakshifts measured at small T

Determination of bath parameters

for H and B, respectively

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

Peaksh

fit(

fs)

T(fs)

H band (Experiment)

H band (Simulation)

B band (Experiment)

B band (Simulation)

Simulation for 1C3PEPS and obtained bath parameters

Y.-C. Cheng et al., J. Phys. Chem. A 111, 9499 (2007)

J=250cm-1 gives best fit to both

measurements

Negative peakshift due to the rapid

dynamics BP+

0 50 100 150 200 250 300-5

0

5

10

15

P

eaks

hif

t(fs

)

T(fs)

0 50 100 150 200 250 300-10

-5

0

5

10

15

20

Peaks

hif

t(fs

)

T(fs)

Downhill

Uphill

Experiment

Simulation (J=200cm-1)

Simulation (J=250cm-1)

Simulation (J=300cm-1)

Determination of coupling constant J between H & B by 2C3PEPS

Y.-C. Cheng et al., J. Phys. Chem. A 111, 9499 (2007)

Nonlinear spectroscopy

incident fields signal fieldQM system

polarization P

different aspects probed by TA, TG, 3PEPS etc.

detection

~ i ω P(ω)

obtain complete information about system

by measuring signal intensity and phase

( ) ( )

0 1 1 1( ) ( , , ) | ( ) ( ) ( )n n

n n n sP d d E Eω ε ω ω χ ω ω ω ω δ ω ω∞ ∞

−∞ −∞

= −∫ ∫r r r

L K L

Third-order spectroscopy

1 2 3 sig

coh.time

pop.time

echotime

τ T t

homodyne detection: | Esig(ω) |2

or even frequency-integrated

heterodyne (time domain): | Esig(t) + ELO(t+tLO) |2

problem: tLO needs to be scanned

tLO

time-domainlocal oscillator

local oscillator = 4

| Esig(ω) + ELO(ω) exp-iωtLO |2heterodyne (frequency domain):

tLO

multichannel frequency detectionLO-position tLO fixed

freq.-dom.

1

2

3

4

delay 1delay 2

1 2

3 4

1&2

3&4

diffractive

optic (DO)

sample

2 f

sphericalmirror

spectro-meter

1 2 3 sig4=LO

coh.time

pop.time

echotime

τ T t

OD3

Two-Dimensional Heterodyne

Spectroscopy

Experimental setup

diffractive

optic (DO)sphericalmirrordelay 1

delay 2

sample

lens

Opt. Lett. 29 (8),

in press (2004)

Advantages of setup

first implementation of tunable2D spectroscopy in the visible region

small signals heterodyne advantage(record signals < 100 aJ)

phase stability diffractive-optics-based setup(2D background < 2%)

delay accuracy small-angled wedges(position repeatability: 5 as)

scattering subtraction by automated shutter

problem solution

raw data

cut & FFT-1

step 1a: slice

spectrum (for each τ)

FFT

step 1b: get

I(ω), Φ(ω)

2D data analysis

Re Imstep 2: FFT-1

(along τ)

Time-delay accuracy

0 min

5 min

10 min

20 min

15 min

lab time:

no shift of fringe pattern:

interferometric stability and reproducibility

Nile Blue heterodyned signal recorded within separate scans

Scattering subtraction

-3.27 -3.07

3.27

3.07

ωt/

fs-1

ωτ / fs-1

without subtraction

But heterodyned signal

involves scatter E1,…,E3:| E1+E2+E3+E4+Esig |2

-3.27 -3.07

ωτ / fs-1

with subtraction

Remove scattering by

| E1+E2+E3+E4+Esig |2

- | E3+E4 |2 - | E1+E2+E4 |2

Desired heterodyned signal comes from (E4* Esig)

Phasing (PP versus 2D)Nile Blue in acetonitrile (T = 0 fs)

positive signal:emission or

ground-state bleach

negative signal:excited-state

absorption

overall phase determines Re S2: absorption / emission

Im S2: refraction

2D spectrum

2: wait forpopulationtime T

“probability distribution”

S2(ωωωωττττ,T, ωωωωt)

1: excite at

frequency ωωωωττττ

3: emit or bleach

at frequency ωωωωt

here: inhomogeneoussystem “remembers”excitation after time T

“homogeneous”linewidth

“inhom.”linewidth

Nile Blue (experiment)

-3.27 -3.073.27

3.07

ωt/

fs-1

ωτ / fs-1

-3.17

3.17

3.27

3.07

ωt/

fs-1

3.17

-3.27 -3.07ωτ / fs-1

-3.17

Abs

Pha Im

Re

T = 0 fsT = 5 fsT = 10 fsT = 15 fsT = 20 fsT = 30 fsT = 40 fsT = 50 fsT = 60 fsT = 70 fsT = 80 fsT = 90 fsT = 100 fs

f

ωfe

dfe

gff(t)f-bath

,

( ) ( ) (0) ff fg fg ggv s

C t Tr U t U ρ=

Theory of 3-level systems

g

egee(t)

e-bath

laser: ω0, τp

ωeg

deg

g(t) has contributions from

vibrations: (ωv, λv , τv)

solvent: Gaussian (λsg , τsg)Exponential (λse, τse)

2

0 0

1( ) ( )

t

g t d d C

τ

τ τ τ′ ′= ∫ ∫h

,

( ) ( ) (0) ee eg eg ggv s

C t Tr U t U ρ=“regular”

energy-gap correlations

gfe(t)

,

( ) ( ) (0) fe fg eg ggv s

C t Tr U t U ρ=correlations betweenexcited states

Nile Blue (simulation)ImRe

0

fs

100fs

ImRe2-level simulation

3-level simulation

ωfe = ωeg+100 cm-1

gfe(t) = 0

3-level simulation

ωfe = ωeg+100 cm-1

gfe(t) = gee(t)

experiment

Dipole strength and gfe

ImRe

T = 0 fs T = 100 fs

ImRe

dfe = deg

gfe(t) = 0

ωfe = ωeg

dfe = 1.5 deg

gfe(t) = 0

ωfe = ωeg

dfe = 1.5 deg

gfe(t) = gee(t)

ωfe = ωeg

2D Spectroscopy of Aggregates

MOLECULAR AGGREGATES

WEAKLY COUPLEDSTRONGLY 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-diagonal

Electrostatic

Diagonal

Electron-Phonon

∑∑≠

==

+N

nmnm

mn

N

n

n nJmnn1,1

ε=)(qH ∑=

−N

n

phel

nn nqHn1

)(+ )(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

+

-

BIC J-Aggregates, Real 2D Spectra

Absolute Value 2D Spectra BIC J-Aggregates

70%

50%

Magnifiedcentral

part of 2Dspectrum

Diagonal Elongation: Static DisorderWidth Broadening at Higher Frequencies: Exciton RelaxationDetailed Width Profile: Frequency-Dependent Dynamics

Absolute Value 2D Plots

BIC J-aggregates

Absolute value of the 2D plots; population periods T=50, 500 fs.

Evolution of the 2D plots is mostly determined by the excitonrelaxation during the population period.

T = 50 fs T = 500 fs

Frequency-Dependent Exciton Relaxation

• Fast Mode, no Relaxation: Small Changes with T, Minimum Width at Higher Frequencies

• Simulation with relaxation & Experiment:- Minimum width to the lower Frequency of 2D Peak & Asymmetric Shape:

Relaxation induced “Life-Time Broadening” is stronger at Higher Frequency

- Increased Width: Exciton Relaxation in Population Period

- Reduced Width: No Exciton Relaxation

J-Aggregate HamiltonianSITE BASIS:

FREQUENCY- DEPENDENT DYNAMICS

Relaxation:Hopping Between States

Exchange Narrowing:

Reduction of theFluctuation Amplitude

Diagonal Exciton-Phonon Off-Diagonal Exciton-Phonon

EXCITON BASIS:

1

N

kn

n

k nϕ=

= ⋅∑

Off-diagonal

Electrostatic

Diagonal

Electron-Phonon

1 1,

N N

n mn

n m nm n

n n m J nε= =

≠

+∑ ∑=)(qH ∑=

−N

n

phel

nn nqHn1

)(+ )(qH ph+

|n> - monomer #n is excited

Experimental 2D Spectrum

2D Spectra From J-Aggregates for T=50fs and T=500fs

Broadening of Higher Frequency Part is Due to Exciton Relaxation

|ω1 |

Anti-Diagonal Width

Discrimination of Fluctuational and

Relaxational Dynamics

EXPERIMENT

SIMULATION

with RELAXATION

SIMULATION

without RELAXATION

a) fast mode

b) slower mode

Relaxation Broadens

High Frequency Part

Fluctuation Only

Dynamics

High Frequency Part

Shrinks

Frequency-Dependent

Delocalization,

Relaxation and Exchange-Narrowing

•Delocalization Varies from 5 to 30 monomers across the Absorption Band of

J-aggregates

•Relaxation is Increased at Higher

Frequencies

•Dynamic Fluctuations are Reduced at

Higher Frequencies by Exchange-

Narrowing

Best Fit: Experiment & Simulations

•Experiment is Reproduced

Theoretically over Complete R(3)

•Model Parameters(only 1 independent):

Disorder: σ = 250 cm-1

Gaussian Mode Correlation Function:

λ = 250 cm-1, τ = 50 fs (liquid water: 30…70fs)Coupling: J = 1080 cm-1 (from absorption)

(σ & λ determine absorption spectrum width)

•Frequency-Dependent Dynamics

is Correctly Reproduced

•Monomer Static Disorder and the

Approximate Correlation Function

are Retrieved

•Dynamics of One and Two Exciton

Contributions are Correctly

Reproduced

[Positive and Negative Spots in the

2D Real Spectrum ]

Solid lines are plotted at +10,20,…% and

dashed at -10,20,…% of the

absolute 2D spectrum maximum [at given time T ]