1. Photon Echo Technique

2. Quantum Mechanics of Ensembles
Described by the density matrix rather than a wavefunction.

3. Calculating Nonlinear Signals
Time evolution of used to calculate the polarization P
Expand P as
Long and tedious expressions.
Help is at hand!
For a two level system only 4 terms and their complex conjugates survive
the definition of the density matrix
Suggests we can represent these terms by diagrams in which we propagate
the bra and ket separately.

4. Feynman diagrams & the density matrix
phase-matching direction
energy
|e
ks = -k1+k2+k3 k3 k2
|g k1
time
density matrix
|g
g| ks
g|
e| t
|e
g|
-k1 k3 1 d
|g
|e
e| T
r = ks k2 k2 t
|g
e| d d
|e
-k1
|g
g| k3

5. Two level systems are described by four Feynman
diagrams and their complex conjugates ks ks R2
Geg(t3)
Gee(t2)
Gge(t1) k3 k1 k2 R1
Geg(t1)
Ggg(t2) R3 R4
If k3 = k2 (same pulse)
ks= k1 for R1 and R4
ks = 2k2-k1 for R2 and R3

6. Echo-Inhomogeneous broadening
from Erwin Hahn and Chris Noble

7. Lens Analogy for Photon Echoes
After the first interaction we have a superposition
oscillating at the energy difference between
and .
Optical
frequency
(1)
Homogeneous
dephasing
(2)
Inhomogeneous
contribution
leads 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

8. The second interaction produces a population—no ε-term in
difference between |e> and |e>
Rephasing
response
function
Now the third pulse phase
factor 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.

9. Non-rephasing response function
2 PE t1 t2
Dephasing
Spectral diffusion
Dephasing
Refocusing gets poorer
and poorer as t1, t2 increased. t1 t3 t2

10. Photon Echoes 1 2 3 Pulse 2 creates population (|g> OR |e>)
Pulse 1 creates coherence
(|g> AND |e>)
Pulse 3 creates another coherence
Oscillatory term during first (second) coherence: e -(+)iωegt
Slope of rays depends on ωeg in oscillator term

11. 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.

12. Figure 3. Temperature
dependence 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 infrared
photon echo experiments using eq 9b.arrows mark the glass transition temperatures. Note the different temperature and line width scales.

13. W(CO)6 in 2-MP Absorption Linewidth
Tokmakoff….Fayer J. Phys. Chem, (1995).
Absorption Linewidth

15. Two Pulse Electronic Echoes
HITCI in
glycerol/water
(70/30)
2k1-k2
2k2-k1
20 fs transform
limited pulses
Deconvolution
20 fs decay

16. Exciton Dephasing in Semiconducting Carbon Nanotubes
~800 nm
Homogeneous contribution
~0.75 nm
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

17. 2D Spectroscopy of Aggregates
MOLECULAR AGGREGATES
WEAKLY COUPLED
STRONGLY COUPLED
Absorption spectra of BIC monomer
and J-aggregates
LH2 Complex
Two-exciton
Band 2e
Linear chain of 2 level
molecules with electrostatic
dipole-dipole interaction
One-exciton
Band 1e
Ground state g

18. J-AGGREGATE HAMILTONIAN
SITE BASIS:
Off-diagonal
Electrostatic
Diagonal
Electron-Phonon
EXCITON BASIS:
EXCITON WAVEFUNCTIONS
Diagonal Exciton-Phonon
Off-Diagonal Exciton-Phonon
Renormalization Factors
Cause Exchange Narrowing
Overlap Factors Define
Relaxation
Higher Exciton States are Strongly Delocalized
Exchange-Narrowing is Stronger for Higher
(More Delocalized) Exciton States
Relaxation is Faster for Higher Exciton States

19. Photon Echo Technique

20. Integrated Three Pulse Photon Echo: Nile Blue in Acetonitrile

21. Origin of the Peak Shift
Rephasing side as spectral
diffusion occurs will become
more and more like non-
rephasing side
Non-rephasing side
not influenced by
spectral diffusion
Eventually the echo signal will become symmetric around τ=0

22. Measuring inhomogenous broadening
Peakshift tracks the surface denoted by the blue line

23. IR 144 τ*(T) vs. T Finite long time peak shift Inhomogeneous
broadening
Timescales of
fluctuations in
transition frequency.
32K
294K
Ethanol 294K

24. What is the Peak Shift?
At high temperature it relates to the Stokes shift dynamics
and the ratio of dynamical and static contributions to the spectral broadening.
The long time value
allows the inhomogeneous width to
obtained:
The time dependence gives
inhomogeneous
width
Stokes Shift
obtain
inhomogeneous
width,
M. Cho

25. Solvation Dynamics IR144 in acetonitrile
Correlation function
Peak Shift
Spectral Density

26. Instantaneous Normal Mode Spectral Density
CH3CN

27. Solvation Spectral Density for Acetonitrile

28. Dielectric Response of Aqueous Proteins
Lysozyme with eosin bound in the ‘hydrophobic box’
Eosin/lysozyme/water
Eosin/water

29. LH1 and Reaction Center of Purple Bacteria
Roszak, Howard, Southhall,
Gardiner, Law, Isaacs & Cogdell
Science, 302, 1969 (2003).

30. Structure of the LH3 Complex
K. McLuskey et al.: Biochemistry
40, 8713 (2001).
Rhodopseudomonas acidophila Strain 7050

31. Photon Echo Peak Shift Measurements
LH1 of Rb. sphaeroides vs. the B820 Subunit of LH1 of Rs. rubrum
Same parameters as LH1
except no 90 fs EET
component
B820 subunit of LH1
Peak Shift (fs)
Inhomogeneous broadening
90 fs energy transfer
timescale
LH1
T(fs)

32. Light Harvesting Complex II
Absorbance (norm.)
Wavelength/nm

33. Bacterial Light Harvesting
Bahatyrova, et al.
Nature (2004)
Hu, et al.
J. Phys. Chem. B (1997)

34. Peak Shift on the B850 band of LH2 membranes (Rps. acidophila)
Membrane samples
Intra-complex exciton
relaxation or energy transfer
Solubilized samples
In collaboration
with C. N. Hunter,
Sheffield
Energy Transfer between the complexes
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.

35. Pump Probe (Transient Absorption)
IR144 in MeOH

36. 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. ks gg k3 eg ee ge k2
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 k1
rephasing diagram
Probe
Detector
Pump

37. Contributions to Pump-Probe Signal

38. Pump Probe Signals (Calculation)

39. Transient Absorption

41. Coworkers Taiha Joo Minhaeng Cho Yutaka Nagasawa Sean Passino
Matt Lang
Xanthipe Jordanides
Xeuyu Song

43. Peak Shift IR144 in MeOH

44. 1-Color Transient Grating Signals
Time unit: ps.

45. Two Color Transient Grating Signals
Positive
Negative At
the probe is at the bottom
of the excited state well.
For large detuning the birefringent
contribution becomes similar to the
dichroic contribution (at short times).

46. Two Color Transient Grating Signals. Homodyne Detection
Detuning = 800cm-1
Maximum correlates well
with Gaussian time
constant,

47. Experimental 1-Color and 2-Color TG Signals for DTTCI in MEOH
Downhill. Detuning =
Probe close to minimum of excited state surface.

48. Experimental One-Color and 2-Color TG Signals for IR144 in MEOH
1C 750nm
2C 750, nm
(downhill)

49. Two-Color three-pulse Photon Echoes
IR144 in Methanol
DTTCI in Methanol

50. IR144 Methanol 750, 750, 800

51. Difference Peak Shift
For a fixed phase matching direction, i.e., k3 + k2 – k1 gg ge ee k1 k2 k3 eg ks k3 eg TI
TII ee τI
τII k1 eg gg k2
Type I scan
Echo (Rephasing)
(pulse sequence, 1-2-3)
Type II scan
FID (Non-Rephasing)
(pulse sequence, 2-1-3)
Difference peak Shift = Type I - Type II
Δτ*(T) = τI*(TI) - τII*(TII)
Two Colour

52. IR144 Methanol 750, 750, 800

53. Experimental Difference Peak Shift Data (downhill)
Pulse Sequence, nm
IR144 in Methanol
DTTCI in Methanol
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

54. Spectral Models and the Two-Color Difference Peak Shift
Downhill Case w I II
III
1 mode.
Gaussian
M(t) II
2 modes.
35 modes.

55. Two-Color 3PEPS as a Probe of Memory Transfer in Spectral Shift= +
Two-Color 3PEPS measures correlation dynamics
(between transition energies in pumped and probed regions).

57. Homogeneous and Inhomogeneous Distributions of Transition Energies
A particular nuclear state
in the ground electronic state
Statistical probability for a molecule to occupy
the nuclear state
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

58. A Simple ad hoc Model for the Dynamics of Correlation Function
Total Signal =
Total Correlation Function
At short times,
At longer times,
Inhomogeneous distribution fluctuates with time due to random fluctuation of the statistical distribution of the nuclear states, which is described by a stochastic approach.
Skinner et al, J. Chem. Phys. 106, 2129 (1997)

59. Dynamics of Conditional Probability for the Inhomogeneous Distribution
Full Response Function
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

60. Model Calculations for Difference Peak Shift (downhill)
Empirical formula:
Gaussian Time Constant,
reorganization energy
Frequency difference between the two pulses
Adding exponentials and vibrations does not alter the turnover time significantly.
Therefore, we can extract information of the Gaussian parameters from the
turnover time.

61. Simulation model for the Difference Peak Shift
Pulse Sequence: nm
IR144 in Methanol
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)

62. Two Color Peak Shift: Energy Transfer Systems
Difference Peak Shift
Type I (rephasing)
Type II (nonrephasing)
In an inhomogeneous energy transfer system, spectral
overlap induces correlation between donors and acceptors.

63. 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
1-color
2-color

64. 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.