Coherent Nonlinear Spectroscopy:

From Femtosecond Dynamics to Control Annu.rev.phys.chem., 52, 639

Marcos dantus

ⅠIntroduction

1. History of breaking time resolution limit

mid 1950 ; microsecond time resolution. First pump-probe work (porter)

1960s ; nano or picosecond time resolution

1985 ; zwail probe the ultrafast dynamics of isolated molecules with subpicosecond time resolution

1987 publishing the first time-resolved observation of transition states in a chemical reaction

2. relation between linear and nonlinear spectroscopy

① linear spectroscopy; emission and 1-photon absorption – output properties is linear input properties

② nonlinear spectroscopy; using the coherent interaction between sample and one or more of the laser pulse – output

properties is not linear input properties

3. “The goal of for active laser control is to devise electromagnetic fields”

① frequency resolved scheme (coherent control)

Utilizing the quantum interference between different rxn channel

Ex) chirped pulse enhancement of multiphoton ionization, optimal control

② time resolved scheme

‘the time dependent motion of wavepacket created by ultrafast laser pulses manipulates the outcome

of reaction’

Ex) Pump-probe, pump-dump, four wave mixing..

4. Schematic diagram of some techniques

Ⅱ Bound-free transitions ; concerted-elimination rxn

1-1. Concerted rxn ; multiple fundamental change (bond, formation, charge transfer)

Classification depends on the sensitivity of the chosen method to detect the ‘short-lived intermediates’

→rxn is rapid compared with the detection method

1-2. in the condensed phase

solvent provides ‘cage’

→ products are closed to each other

→ difficult to determine if a rxn proceeds by a concerted mechanism

1-3. Example case ; I2 elimination from CH2I2 (for more info ; j.chem.phys., 109, 4415)

① I/I* elimination channel is major one at 248.351 nm

② two photon absorption of 267 nm ; initiation of I2 elimination channel

③ at 310, 342, 369 nm (pump-probe spectroscopy)

→ I-C-I symmetric stretch, antisymmetric stretch, bending

④ PES(potential energy surface)

CH2I2

⑤ experimental result

I2(D’) ; more probable (rotation anisotropy is almost zero) → synchronous concerted

I2(f) ; less probable (anisotropy dependent) → asynchronous concerted

pump(312nm ; three photon excitation and molecular iodine dissociation)

+ probe (624nm ; I2 excitation from D’ to f )

⑥ kinetic model for the dissociation of CH2I2

Based on assumption of only two contributions of signal

⑦ What is the character of PES correlated with product??

If this process is direct, fast and pseudo-diatomic problem, then Eavail is almost same with Ekin. IVR (time is

longer than one vibration freq) isn’t observable.

L ; distance for bond breaking

τ; experimental dissociation time

then, if L1 = L2, Eavail = Ekin

comparison MeI2, BuI2 (equal rxn enthalpy)

Experimental τ2/τ1 = 1.85

τ2/τ1 estimated from eq = 1.85

→assumption is valid → ‘repulsive potential’

2. Chirped pulse enhanced multi-photon ionization

① chirped pulse

A pulse in which the wavelength changes during the duration of the pulse.

Intensity spectrum of a negatively chirped pulse

blue red

‘CPA(chirped pulse amplification)’

Positive group velocity dispersion

red blue

negative group velocity dispersion

②control of yield

This phenomena follows ‘Wave-packet following’ mechanism

As the wave-packet moves, the transition energy becomes time

dependent and a chirp that follows this dependence, while not necessarily

exactly on resonance, will be more effective in transferring population to

the higher state.

③ Another mechanism

- time delay resonance mechanism ; when considering negatively

chirped pulse, initially prepared wave-packet on B state, propagates until

it reaches a region of the PES, where it can be resonantly excited to the C

state by the trailing high frequency component.

- sequential resonance effect

This mechanism is suitable for spectroscopic property.

Suppose the first transition is resonant with the low frequency part of the pulse and the

second is resonant at high frequency. In this case, positively chirped pulse is more effective in

multiphoton transition. This effect only occurs for on sign of the chirp since the electronic

level spacing are uniquely determined

Ⅲ Bound-free transitions; photoassociation rxn

1.The difficulty of studying bimolecular rxn

① Two counters are required

→The encounters occur at random time, with random configurations, and random energy

② Experimental challenge

→to devise ways to determine or restrict the initial collision condition (impact parameter, orientation, collision

energy, time of collision)

→The traditional method is using the molecular beam where the energy of rxn can be regulated

2. Unimolecular dissociation rxn ; ‘half collision’

→“microscopic reversibility” ; Unimolecular photodissociation ≈ second half of a full collision

→ first half ; collision of the fragments

→ very specific initial condition (impact parameter & reagent energy) would reproduce the observed dissociation

dynamics

→ Unimolecular dissociation is small subset of the possible bimolecular pathway

3. The yield of bimolecular rxn

→ determined by the energy of collision, relative orientation, impact parameter

→ short pulse dissociation ; well determine life time, alignment of reagent

4. Photoassociation (excimer, free-bound transition)

① dispersed fluorescence

→ Only collision pairs that are in resonance with the binding laser at 312 nm (pump)

→ Only collision pairs oriented parallel to the polarization of the pulse are photoassociated to the D1u state.

→ The D→X fluorescence is used to monitored

→ 624 nm probe pulse ; depletion of D state

→ (b) Dispersed fluorescence spectrum resulting from excitation with a 60 fs laser pulse centered at 312 nm.

The peak at 407.8 nm is an atomic line resulting from two-photon excitation to the 71S0 state.

→ (c) Fluorescence spectrum resulting from the excitation of mercury vapor at 266 nm with a nanosecond laser

pulse. The D→X emission is blue-shifted compared to the emission produced by 312 nm excitation because

of the difference in excitation energy.

② Femtosecond transients

→ Femtosecond transients from the photoassociation of mercury at 312 nm. The heavy (thin) line corresponds to

parallel (perpendicular) polarization of the bind and probe pulses relative to each other.

→ the data is clearly anisotropic, indicating alignment of the photoassociated collision pairs.

→ Rotational anisotropy r(t) obtained from the experimental data. The heavy line is the best fit to the

Experimental

→ Rotational population of the photoassociated product, obtained from the fit to the rotational anisotropy.

③ rotational anisotropy

(exp) (theo)

→ ωi=4πBj (rotational freq) jmax ≈30 fwhm Δj ≈90

④ control of impact parameter

→ Frank-Condon factors dictate that transition probability is greatest when the laser wavelength is with energy

difference between ground and excited state

→ As a result, energy gap depends on the distance reactants

→ The wavelength of the binding pulse can thus be used to select a range of reactive impact parameter

→ relative collision E

V1(R′) ; potential energy of the ground state

R′ ; internuclear distance at which the laser is resonant

In order for photoassociation to occur, the relative collision energy of an atom pair with a given impact parameter b

should satisfy this condition

differential photoassociation cross section dóPA/db

P(b) is the opacity function.

-Figure 6 at 350 nm, only those collision pairs with very small impact parameters are photoassociated.

- As the binding laser is tuned to shorter wavelengths, the position of highest photoassociation probability shifts to

larger

-The opacity function reaches a limiting value P(b)=1 at high excitation energies, when V1(R′) approaches zero.

-Jh=μVb, where V is the relative velocity of the atoms when they are photoassociated.

Ⅳ Bound-bound molecular transition; vibrational dynamics and coherence

1. Transient grating

① three pulses has an identical pulse envelope ans frequency

components

② spatial modulation – constructive & destructive interference

③ the molecule in the interaction region experience varying

electric field intensities “according to their position”

④ The formation of the grating does not require that the two

crossing beams coincide in time as long as the coherence is

maintained in the sample

Grating ; any regularly spaced collection of essentially identical, parallel, elongated elements, but can consist of two

sets, in which case the second set is usually perpendicular to the first. When the two sets are perpendicular, this is

also known as grid

2. Phase matching

To ensure that a proper phase relationship between the interacting waves is maintained along the propagation

Direction

In phase-sensitive nonlinear process (freq doubling, sum & difference freq generation, four wave mixing) require

phase matching to be efficient

Ex) frequency doubling

“without chromatic dispersion k2=2k1”

- dispersion ; freq or mode dependence of the phase velocity in a medium

- phase velocity ; the velocity with which phase fronts propagate in a medium

- the chromatic dispersion of an optical medium is basically the freq dependence of the phase velocity

3. Density matrix

① diagonal block

ρee , ρgg ;population of each vibrational level

ρee’ , ρgg’ ; coherence of each vibrational level

② off diagonal block

(the vibronic coherence between the two electronic states)

cf) coherence ; a fixed phase relationship between the electric field values at different locations or at different

times

③ in the weak interaction limit (for vanishing multiphoton transition)

For four level system, we will include two electronic states with two vibrational levels each, |1> |2> |3> |4>

FWM signal

Population

Vibrational coherence

Each field, En, interacts linearly with the media, producing a change ρ(n) to the initial matrix

N is odd ; ρ(n) contains the changes in the probability amplitude of the electronic coupling

N is even ρ(n) represents the changes in the population and the coherence of the vibrational levels with each

electronic states

④ Density matrix calculation of nonlinear response functions

the density matrix is defined using the outer product of the state of the system |ket⟩ with its Hermitian conjugate ⟨bra|

a probability Pj to be in the state |Ψ j⟩ with ∑jPj = 1.

When Pj=1 for one state and is zero, otherwise, the system is in a pure state (a state with maximum information)

and can be described by a wave function. Otherwise, we have a mixed state that may not be described by a single wave

function. Adopting a basis set (æa), we have

the elements of the density matrix are given by

Solution of time dependent density matrix of n th interaction with electric field

Each term of the interaction operator has a well-defined direction (kn or -kn). Therefore each pulse interaction contributes in

a unique way to the phase matching direction of the nonlinear signal

Kn ; excitation ket

or deexcitation bra

-Kn ; deexcitation ket

or excitaqtion bra

⑤ vibrational wave packet description

(a) the initial wavepacket Ψ(0) in υ”=2 of the ground

electronic state is excited by field, Ea, to the excited

state B

(b) The resulting excited-state wavepacket Ψ(1) is

allowed to evolve until field Eb is applied at time τab at

τ=460 fs. The wavepacket located at the outer turning

point. Therefore the frank-Condon overleap with the

optically resonant level in the ground state is negligible

(c) The wavepacket is localized at inner turning point, providing a good overlap τab=610 fs ; resulting in a

significant ground state population

(d) Double-sided Feynman diagram

6. Example

①sient grating, Rverese transirnt grating

Gas phase I2

In PE is that field Eb acts first and is then

followed by fields Ea and Ec. PE processes

involve a rephasing of the coherence that is

lost owing to inhomogeneities in the sample

The wavepacket motion in the excited state

has a much wider range of internuclear

distance. This takes the wave packet in and

out of the franck-condon region. Therefore,

electronic polarization reflects

predominantly excited state dynamics

② Virtual echo, Stimulated photon echo

In the VE I measurement, when τab is 460 fs (3/2τe τe=2πωe) the dynamics show 307 fs oscillations, reflecting only

an excited-state contribution.

When τab is 614 fs (2τe), the dynamics show 160 fs oscillations, (ground)

In the PE I configuration, when τab is 460 fs, the dynamics reflect an excited-state contribution with 307 fs

oscillations ; no ground-state contribution is observed in this transient.

When τab is 614 fs, the 307 fs oscillations still dominate(excited); however, 2 ps later,

160 fs oscillations can be seen.

The selection between ground- or excited-state dynamics is much more effcient for the virtual echo set-up. The

observation of ground state has three laser interactions acting on the ket. This leads to high selectivity between the

two states.

For VE I, the appearance of ground-state dynamics arises from a wave packet being prepared in the excited

state, then pumped to the ground state and finally probed as a function of time, thereby giving a clear and intense

ground-state signal.

For PE I ground-state dynamics shows that the first two interactions are on the bra while the third interaction

is on the ket. This action on an unperturbed ground state by the third pulse leads to loss of the selectivity. The

reason for the small

Experimental data for VE (bold line) and PE (line)

measurements with τab=τba=460 fs.

out of phase with each other

In both cases, field Ec must interact with the wave packet

formed in the excited state by field Eb.

PE I case, Ψb(1) is at the outer turning point of the excited-

state potential when τ=0fs, minimizing the transition

probability when the third pulse is applied in the Franck-

Condon region.

③ variable time delay followed by followed time delay

For τba=460 fs, the transient shows 307 fs oscillations, corresponding to excited-state dynamics. The signal is

weaker and shows only a small background.

When τba=614 fs, the transient is dominated by 307 fs oscillations. The signal is stronger and shows a larger

background. Weak 160 fs vibrations are also observed.

Fourier transforms have confirmed that the τba=614fs transient shows a contribution of the ground state. For a

time delay of sba=614 fs, the observed background arises from the process depicted by the DSFD on the right.

The use of the fixed delay as a filter for the dynamics changes the ground-state contributions to the signal

slightly but does not give the same degree of control as is observed in the VE I case.

④ Mode suppression

Suppressing the contribution of excited-state

vibrational dynamics in order to improve relaxation

rate measurements in liquids.

They observed that when τ13 is in phase with the

excited-state dynamics, τ13=2πn/ωe(mode suppression

is on), the amplitude of the excited-state vibrations

was greatly reduced.

When τ13 was out of phase (mode suppression is off ),

the excited-state vibrations were very prominent.

When mode suppression is on, both R2 and R3

(photon echo) contribute to the signal.

Mode suppression is useful in liquid phase studies

because when mode suppression is on, R3 contributes a

large signal that overwhelms the excited-state

vibrational coherence.