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