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
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
...,,,, )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
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
)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
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
...3,2,1,ˆˆ )1()(0
)(
NVHdt
i NNN
tiVtaia mlmll
Nl
Nm exp11
tiruVtatiruai lll
Nlll
l
Nl expˆexp 1
/exp,)( tiErutatr lll
Nl
N
We get a set of equations:
We multiply by um(r) and integrate
lmml uVuV ˆ
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
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
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
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
m titatEdtita 'exp''ˆ' 11
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
tiE
dta mgmg
mgm
expˆ11
m
The transient absorption is dictated by all frequency components
gmmgm Eta 1
g
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
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
m titatEdtita 'exp''ˆ' 11
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
lmg
mgmlm tiEEddta 'exp
'
1'ˆˆ'
lglg
122
At t ' mg
g
l
m
If mg>>0 , mg=+’
lmgmlm EEdta
lglg
122 1ˆˆ
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
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
Perturbation solution to Schrödinger EquationPerturbation solution to Schrödinger Equation
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
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 expˆexpˆexp
Four wave mixingFour wave mixing
213321321,,
3 ˆˆˆexpˆ
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
Four wave mixingFour wave mixing
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
Four wave mixingFour wave mixing
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 ˆˆexp1
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?
Four wave mixingFour wave mixing
AEdPig
immgi
1
ˆˆ3
igigi
AEP 3
g v
If there are several vibrational states:
Loss of spectral resolution
Four wave mixingFour wave mixing
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
Two 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
Two photon transition
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
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
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