Narrow transitions induced by broad band pulses |g> |f> Loss of spectral resolution.

Narrow transitions induced by broad band Narrow transitions induced by broad band pulsespulses

|g>

|f>

Loss of spectral resolution

53 1010

Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation

H

ti ˆ

tVHH ˆˆˆ0

tEtV ˆˆˆ

We solve the Schrödinger equation using perturbation theory.

The time dependent Schrödinger equation:

The interaction of the atom with the electric field

reˆˆ

ruEruH nnn 0ˆ


...,,,, )2(2)1()0( trtrtrtr

tVHH ˆˆˆ0

)0(0

)0(

ˆ H

dti

In the perturbative regime we can write H as:

Where is a varying parameter which characterizes the strength of the interaction.

We now seek a solution to Schrödinger equation in the form:

We require that all terms proportional to N satisfy the Schrödinger equation:

...3,2,1,ˆˆ )1()(0

)(

NVHdt

i NNN


)0(0

)0(

ˆ H

dti

...3,2,1,ˆˆ )1()(0

)(

NVHdt

i NNN

We assume that initially the atom is in the ground state so the solution for the zero's order is:

/exp,)0( tiErutr gg

We represent the Nth order contribution to the wavefunction as:

/exp,)( tiErutatr lll

Nl

N


...3,2,1,ˆˆ )1()(0

)(

NVHdt

i NNN

tiVtaia mlmll

Nl

Nm exp11

tiruVtatiruai lll

Nlll

l

Nl expêxp 1

/exp,)( tiErutatr lll

Nl

N

We get a set of equations:

We multiply by um(r) and integrate

lmml uVuV ˆ


tiVtaia mlmll

Nl

Nm exp11

tEtV

tEtV

mlml

ˆ

ˆˆ

This equation relates the amplitude of the Nth order to the amplitude of the N-1 order by a time integration.

'exp''' 11 titatVdtita mll

Nlml

tN

m

l

mlNlml

tN

m titatEdtita 'exp''ˆ' 11


tiE

d

tiEddti

titEdtita

mgmg

mg

mgmg

t

mg

t

mgm

expˆ

'expˆ'

'exp'ˆ'

1

1

11

1NlaFirst order: we include only one state - ga

l

mlNlml

tN



tiE

dta mgmg

mgm

expˆ11

m

The transient absorption is dictated by all frequency components

gmmgm Eta 1

g


tiE

dta mgmg

mgm

expˆ11

Second order:

lmg

mgml

lmlml

t

lmllml

t

m

tiEEdd

titiEEddti

titaEddtita

'exp'

1'ˆˆ'

'exp''exp1

'ˆˆ'

'exp'ˆ'

lglg

12

lglg

lg

12

112

l

mlNlml

tN



lmg

mgmlm tiEEddta 'exp

'

1'ˆˆ'

lglg

122

At t ' mg

g

l

m

If mg>>0 , mg=+’

lmgmlm EEdta

lglg

122 1ˆˆ


lmg

mgmlm tiEEddta 'exp

'

1'ˆˆ'

lglg

122

' mgAt t

g

l

m

E(t) is real, therefore E(-)= E()*

If mg<< 0 , mg=- ’

lmgmlm EEdta

lglg

*122 1ˆˆ


legele EEdta

lglg

122 1ˆˆ

lg

g

l

e

If Intermediate levels are far detuned

legmlm EEdta

lg0lg

122 ˆˆ1

egm EEdta

2

Two photon nonresonant transitionTwo photon nonresonant transition

eg

g

e

egm EEdta

2

egiegm eAAdta 2

All the paths are in phase

Transform limited pulses maximize the two photon absorption

Antisymmetric phase maintains the efficiency

2/ge

Two photon nonresonant transitionTwo photon nonresonant transition

eg

Transform limited pulses maximize the two photon absorption

2/geSpectral phase

Temporal envelope

Nonresonant TPA ControlNonresonant TPA Control

Experimental resultsExperimental results

Antisymmetric phase has no effect on transition probability Specific spectral phase mask can annihilate the absorption rate

Selective excitation

-2 -1 0 1 20.0

0.5

1.0

Step location/bandwidth

Meshulach & Silberberg, Nature, 396, 239 (1998),Phys. Rev. A 60, 1287 (1999)

Controlling The Spectrum of EControlling The Spectrum of E22

Transformed limited pulse

Shaped pulse

02022 Edttietta fg

Raman TransitionRaman Transition

gi

gv eAAdta 2

g

v

gv EEdta *2

2/ge

Transform limited pulses maximize the transition rate

Periodic phase functions maintain the efficiency

Raman TransitionRaman Transition

gi

gv eAAdta 2

g

v

2/ge

Spectral phase

Temporal envelope

CARS spectroscopyCARS spectroscopy

t

ccos25.1

Modulated spectral phase function

Fourier transform

Ba(NO3)2 (1048 cm-1)

Diamond (1333 cm-1)

Toluene (788, 1001, 1210 cm-1)

lexan

Spectrocopy in the fingerprint regionSpectrocopy in the fingerprint region

•Reduced nonresonant background

•Spectral resolution ~ 30 cm-1, 70 times the pulse band width

N. Dudovich, D. Oron and Y. Silberberg, J. Chem. Phys. 118, 9208 (2003).

Narrow transitions induced by broad Narrow transitions induced by broad band pulses: weak fieldsband pulses: weak fields

02022 Edttietta fg

00 NNN

fg Edttietta

00

dttietta fg

One photon transition:

Nth photon transition

Two photon transition

2

22 Edttietta fg

Raman transition

The transition is excited by a single frequency component of EN

Two Photon Resonant TransitionTwo Photon Resonant Transition

legele EEdta

lglg

122 1ˆˆ

g

0

e

If there is a single intermediate state:

gegeg

geggee

EEdEEi

EEdta

000

000

122

1

1ˆˆ

The transition is not maximized by a transform limited pulse

There is a destructive interference between frequencies below and above the resonance

On resonant Off resonant

Enhancement of resonant TPAEnhancement of resonant TPAamplitude shapingamplitude shaping

N. Dudovich, B. Dayan, S. M. Gallagher Faeder and Y. Silberberg, Phys. Rev. Lett., 86, 47 (2001).

1

0780 785 790 795

0

0.5

1

1.5

2

Higher Cutoff Wavelength [nm]

Flu

ores

cenc

e In

tens

ity [

a.u.

]

0.5

Pul

se p

ower

[a.

u.]

-400- -200- 0 200 400Time [fs]

Inte

nsit

y [a

.u.]

I(t)

g

f

Eliminate all frequency components that contribute destructively

blocker

Enhancement of resonant TPAEnhancement of resonant TPAphase shapingphase shaping

760 765 770 775 780 785 790 795 8000

1

2

3

4

5

6

7

Fluorescence Intensity [a.u.]

phase window center [nm]

g

f

Invert the sign around the resonance to induce constructive interference instead of destructive one

phase step

N. Dudovich, B. Dayan, S. M. Gallagher Faeder and Y. Silberberg, Phys. Rev. Lett., 86, 47 (2001).

Two photon absorptionTwo photon absorption

g

1f 2f

Two degenerate non-interfering paths

Angular momentum control

Px

Angular momentum controlAngular momentum control

1M

g

ExEx transitions

Two degenerate orthogonal states can be separately controlled

Ex

Ex

Ex

E-

Px P+

ExE+ transitions

P+Px

0M

212121,,

3 ˆˆ'exp'ˆ'expˆ

EEddtitEdttitP lglgmm

t

gmmglm

Four wave mixingFour wave mixing

'exp''ˆ'expˆ 213 titatEdttitP mm

t

gmmgm

303 ˆ p

3ma

tiEEddta glgll

2121212 expˆˆ

Assuming all intermediate levels are detuned,

tatitiutatiup mgmm

mgmmmggm

333 expêxpêxp


213321321,,

3 ˆˆêxpˆ

EEEtidddtP lglmmglm

3212133213 ˆˆˆˆ

EEEdddP lglmmg

212132133,,

3 ˆˆ'expˆ'expˆ

EEddtiEdtdtitP lglgmm

t

gmmglm

212121,,

3 ˆˆ'exp'ˆ'expˆ

EEddtitEdttitP lglgmm

t

gmmglm


1

2 3

1

2

3

1 2

3

The polarization is maximized by a transform limited pulse

The response is instantaneous

= 1+ 2- 3 = 1-2+ 3 = -1+2+ 3

3212133213 ˆˆˆˆ

EEEdddP lglmmg

Coherent Anti-Stokes Raman Scattering Coherent Anti-Stokes Raman Scattering (CARS)(CARS)

• In a CARS process a pump and a Stokes photon coherently excite a vibrational level. A probe photon interacts with the excited level to emit a signal photon.

• Large, directional and coherent signal (compare to Raman scattering).

• Attractive for microscopy applications

-provides a vibrational imaging with 3D

sectioning capability.

Four wave mixingFour wave mixing 303 ˆ p

3

tgiEEddtag

glll

2121

12*

212 exp

1ˆˆ

g

v

12*

212121,,

3 ˆˆ'exp1

'ˆ'expˆ

EEddtitEdttitP gllgmg

m

t

gmmglm

Assuming all intermediate levels are detuned, including one resonant level:

'exp''ˆ'expˆ 213 titatEdttitP mm

t

gmmgm


AEdPg

mmg

1

ˆˆ3

1*

11

21

EEdA

g

v

12*

212121

,,

3 ˆˆ1

ˆˆ

EEEddP gllg

mmglm

The response is not instantaneous – the nonlinear polarization can be enhanced

12*

2132121

33,,

3 ˆêxp1

ˆˆ

EEddtiEdtP gllg

mmglm

Four wave mixingFour wave mixingMultiplex CARSMultiplex CARS

g v

We can use the broad pulse to pump and a narrow probe to map the excitation

Can we probe with a broad band probe?


AEdPg

mmg

1

ˆˆ3

g

v

If E() is transform limited then:

gg AEP 3


AEdPig

immgi

1

ˆˆ3

igigi

AEP 3

g v

If there are several vibrational states:

Loss of spectral resolution


AEdPg

mmg

1

ˆˆ3

g

v

g

g

AEdAEPg

gg

13

g+ gate

Resonance enhancement around g+g

Extracted Raman spectraExtracted Raman spectra

Transform limited pulse

Phase-shaped pulse

•The resolution is dictated by the phase gate width (25 cm-1)

Narrow transitions induced by broad Narrow transitions induced by broad band pulses: weak fieldsband pulses: weak fields

02022 Edttietta fg

00 NNN

fg Edttietta

00

dttietta fg

One photon transition:

Nth photon transition


2

22 Edttietta fg

Raman transition

The transition is excited by a single frequency component of EN

Strong field coherent controlStrong field coherent control

...0440222 EcEcta fg


The transition depends on many orders of EN

g

e e e

Strong field coherent controlStrong field coherent control

The transition cannot be analyzed in a perturbative manner.

We cannot ignore coupling to all other levels in the system.

Adiabatic approach: If the transition rate is faster than the variation of the interaction we can find the new stationary states (dressed states) and then change them adiabatically with the laser field.

Adaptive search of the optimal solution

We have a high degree of control: we can shape the pulse using N free parameters.

Coherent control: Using shaped Coherent control: Using shaped ultrashort pulses to control the reactionultrashort pulses to control the reaction

Can an ultrashort pulse cause a molecule to vibrate in such a way as to break the bond of our choice?

Strong field coherent control – adaptive Strong field coherent control – adaptive algorithmalgorithm

Strong field coherent control – adaptive Strong field coherent control – adaptive algorithmalgorithm

CO

CH3

CO

CH3+

C

O

CH3

+

Manipulating the dissociation yields in acetophenone

Different pulse shapes can optimize different photo-fragments.

Levis and coworkers

1.6

1.4

1.2

1.0

0.8

0.6

20151050

Generation

Ratio:C7H5O/C6H5

No

rmal

ized

io

n i

nte

nsi

ty a

nd

rat

io

The absolute phase

Different absolute phases for a four-cycle pulse

Different absolute phases for a single-cycle pulse

With a pulse shaper we manipulate the envelope of the pulse, however for short pulses the absolute phase becomes

important

Ultrashort pulsesUltrashort pulses

L L/vg

When the group velocity is different than the phase velocity the absolute phase changes with time

gpc vvL

11

Ultrashort pulsesUltrashort pulses

nn

TtinTttAninTttAtE /expexp

n

repnn

ffATnATTnAE 0/2//2

If a pulse has more than one octave, it has both f and 2 f for some frequency, f. Interfering them in a SHG crystal yields two contributions at 2 f : that from the original beam and the SH of f.Simply measuring the spectrum is performing spectral interferometry yields a fringe phase:

0 0 02

Stabilizing the absolute phase

0 0 02

Stabilizing the absolute phase

Narrow transitions induced by broad band pulses |g> |f> Loss of spectral resolution.

Documents

perturbation solution

state slide

perturbation theory

electric field slide

zeros order

nth order contribution

ground state

time integration