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DYNAMIQUE MOLECULAIRE QUANTIQUE: DE LA SIMULATION AU CONTRÔLE
Laboratoire de Photophysique Moléculaire du CNRS, Orsay, France
Permanents Doctorants
Osman ATABEK
Eric CHARRON
Arne KELLER
Roland LEFEBVRE
Annick WEINER
François DION
Catherine LEFEBVRE
Post-doctorante
Perola MILMAN
LPPM – CNRS, OrsayMPI, Germany
H. Figger
CEA, Saclay
C. Cornaggia
U. Laval, Canada
T.T.Nguyen Dang
U. Sherbrooke, Canada
A.D.Bandrauk
ACI, CERMICS
C. Le Bris
U. Umeå, Sweden
C. Dion
U. Bourgogne
D. Sugny
U. Besançon
G. Jolicard
LASER CONTROL STRATEGIES
Basic Mechanisms Optimal Control
Target State
Generic Strategies
BASIC MECHANISMS
Internal degrees of freedom: Nuclear vibrational dynamics
High frequency regime: field molω ω>>
Time independent picture
ATD*, BS*, VT*
Experimental verification
* Above Threshold Dissociation, Bond Softening, Vibrational Trapping
Field-free potentials
Field-dressedpotentials
Diagonalization of the radiative coupling:
0
0
1( )2det 0
1 ( )2
g
u
V V R E
E V V R
ω µ
µ
±
±
+ −=
−
h
Strong field mechanisms:
Bond Softening
Vibrational Trapping
COUPLED EQUATIONS IN FLOQUET FORMALISM
Floquet ansatz
Integration over r
Time periodicity
Integration over t
MULTI-PHOTON ABOVE THRESHOLD DISSOCIATION
A. Giusti-Suzor, X. He, O. Atabek and F.H. Mies, Phys. Rev. Lett., 64, 515 (1990);
P. Bucksbaum, A. Zavriyev, H.G. Muller and D.W. Schumacher, Phys. Rev. Lett., 64, 1883 (1990)
Field-freeSingle-photon dressedMulti-photon dressedMulti-photon adiabatic dressedMulti-photon ATD basic mechanisms
3D LIGHT-DRESSED POTENTIALS(Schematic view)
K. Yamanouchi, Science, 295, 1659 (2002)
BARRIER LOWERING – BOND SOFTENING
Aligned fragments
VIBRATIONAL TRAPPING
Misaligned fragments
ANGULAR RESOLVED KINETIC ENERGY DISTRIBUTIONS
2 22 2532( , ) ( , ) (1 )g unmH v H v H s Hλ+ + + + +
=Σ ⎯⎯⎯⎯→ Σ ⎯⎯→ +
R. Numico, A. Keller and O. Atabek, Phys. Rev. A , 60, 406 (1999)
,0g
,1u
,2g
,3u1hν1hν
2hν 3hν
2hν 3hν
EXPERIMENT vs THEORY
22 0785 16 /H nm I TW cmλ+ = =
• Electric discharge producing welldefined rovibrational populations of
• Spatial intensity distribution;
• Initial rovibrational distribution;
• Abel transformation
2H +
K. Sandig, H. Figger, T.W. Hänsch, Phys. Rev. Lett., 85, 4876 (2000)
V.N. Serov, A. Keller, O. Atabek, N. Billy, Phys. Rev. A68, 053401 (2003)
BASIC MECHANISMS
Internal degrees of freedom: Nuclear vibrational dynamics
Low frequency regime: field molω ω≈
Time dependent picture
DDQ*, BL*
Experimental verification
* Dynamical Dissociation Quenching, Barrier Lowering
DYNAMICAL DISSOCIATION QUENCHING
[ ]0 0 0( ) cos ( ) cos( )E t E t t E tω ω δ= − = +0δ =
tT
2πδ =
tT
1/4
0
0
( , ) cos( )det 0
cos( ) ( , )g
g
V W R t E tE t V W R t
µ ω δµ ω δ
±
±
− +⎡ ⎤=⎢ ⎥+ −⎣ ⎦
t=0
DYNAMICAL DISSOCIATION QUENCHING
t=T/4
tT
0δ =
tT
[ ]0 0 0( ) cos ( ) cos( )E t E t t E tω ω δ= − = +
2πδ =
1/4
0
0
( , ) cos( )det 0
cos( ) ( , )g
g
V W R t E tE t V W R t
µ ω δµ ω δ
±
±
− +⎡ ⎤=⎢ ⎥+ −⎣ ⎦
WAVE PACKET PROPAGATION
Time t=0
[ ]0 0 0( ) cos ( ) cos( )E t E t t E tω ω δ= − = +
0δ =2δ π=
WAVE PACKET PROPAGATION
Time t=4T
[ ]0 0 0( ) cos ( ) cos( )E t E t t E tω ω δ= − = +
0δ =2δ π= Quenching
Dynamical Dissociation Quenching
P=1
-Dis
soci
atio
n
Time (a.u.)
δ =
δ =
Quenching0δ =
Dissociation2δ π=
2H +
F. Châteauneuf, T.T. Nguyen Dang, N Ouellet and O. Atabek, J. Chem. Phys. 108, 3974 (1998)
EXPERIMENTAL VERIFICATION
Two-colour field
with c/ω1 =1280-1550 nmc/ω2 =2180-1700 nm
E(t)= E0 cos(ω1t) + E0 cos (ω2t)= E0 cos[(ω1+ω2)t/2] cos [(ω1-ω2)t/2]
OPA- Ti Sapphire
H. Niikura, P. B. Corkum, D. M. Villeneuve, Phys.Rev.Lett, 90, 203601 (2003)
Optimal gate timesVibrational periods
2 ( 2)16
H vT fs+ =
:
2 ( 2)22
D vT fs+ =
:
8gatet fs:
11gatet fs:
/ 2gatet T:
H. Niikura, P. B. Corkum, D. M. Villeneuve, Phys.Rev.Lett., 90, 203601 (2003)
[ ])22cos()cos()cos()( 0 φθγθδω ++Ω++Ω+= tttEtE
ωUV=400nm, ΩIR=635cm-1 (1.6x104 nm), I=5x1013 W/cm2
IR envelope Field
BASIC MECHANISMS
External degrees of freedom: Nuclear rotational dynamics
Alignment and orientation
Time dependent picture
Pendular states, ω+2ω, KickExperimental verification
CN
H
θ
No Orientation (θ > 0) Low reaction cross-section ε
Orientation (θ = 0) High reaction cross-section
CN H ε
MOTIVATION
Reaction cross sections Brooks (1976);
Laser induced isomerisation, isotope separationCharron, Giuisti-Suzor (1994);
Molecular trapping Seideman (1997);
High order harmonic generation Hay (2002);
Surface processing, catalysis Tenner (1991);
Nanotechnologies Sakai (1998).
EXPERIMENTAL MEASUREMENT
Breaking the molecule by MEDI (Multielectron Dissociative Ionization) measure the angular distributions of the ionizedfragments, Rosca-Pruna and Vrakking, Phys. Rev. Lett. 87, 153902 (2001);
RIPS (Raman Induced Polarization Spectroscopy): non-intrusiveobservation of a signal proportional to
Renard et.al. Phys. Rev.Lett. 90, 153601 (2003).
2cos 1 3θ< > −
TECHNIQUESBrute force, DC electric field (combined with lasers), Friedrich,
Herchbach (1991);
Microwave pulses in optimal control schemes, Judson, Lehmann, Rabitz, Warren (1990);
Picosecond two-color phase-locked lasers in coherent control, Vrakking, Stolte (1997);
Two IR lasers (ω+2ω) resonant with a vibrational transition of a polar molecule, Dion, Keller, Bandrauk, Atabek (1998);
Half-cycle pulses HCP, Dion, Keller, Atabek (2001);
Train of short pulses, Rabitz, Keller, Atabek (2004).
H. Stapelfeldt and T. Seideman, Rev. Mod. Phys., 75, 543 (2003)
. ~J θ∆ ∆ h
Efficiency
Duration
maxJ
Tim
e (a
.u.)
<cos
θ>
The Model
Rigid rotor Hamiltonian:
2 2 20
1cos ( cos )2
( ) ( )E EH t tBJ µ θ α θ α⊥= − − ∆ +
//α α α⊥∆ = −
Time dependent Schrödinger equation
( , , ) ( ) ( , , )i t H t tt
ψ θ φ ψ θ φ∂=
∂
Measure of alignment / orientation 222 2
0 0cos ( ) ( , , ) cos sint t d d
π πθ ψ θ φ θ θ θ φ= ∫ ∫
22
0 0cos ( ) ( , , ) cos sint t d d
π πθ ψ θ φ θ θ θ φ= ∫ ∫
Basis set methodBasis set of spherical harmonics
,0
( , , ) ( ) ( , )N
J J MJ
t c t Yψ θ φ θ φ=
= ∑
Coupled equations for ( )Jc t
'
'
12
'0 '
0
2 '12 '
2
0
2
( ) ( )
( ) , cos
( )
( ,
( ) , cos ,( )
1)
)
(J J
N
JJ
N
JJ
E tBJi c t c t
c t J M J M
c t
E t
E t J M J M
J α
µ θ
α θ
⊥
=
=
⎡ ⎤= −⎣ ⎦
−
+
− ∆
∑
∑
&h
with ( 0)Jc t = known
Split-operator methodWave packet propagation
' '( ) exp ( ) ( )t t
tit t H t dt t
δψ δ ψ
+⎡ ⎤+ = −⎢ ⎥⎣ ⎦∫h
For small time steps
( ) exp ( 2) ( )
exp exp exp2 2
it t tH t t t
i t i i ttV VT
ψ δ δ δ ψ
δ δδ
⎡ ⎤+ ≈ − +⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤≈ − − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
h
h h h
with H VT= +
Solved using Fast Fourier Transforms
PENDULAR STATES
Quantum Model Classical Model
/ θ∂ ∂
B. Friedrich and D. Herschbach, Phys. Rev. Lett., 74, 4623 (1995)
ADIABATIC vs SUDDEN PULSES
Time (ps)10 20
Pulse duration 20 ps
Adiabatic excitation of a single pendular state; isotropic distribution at theend of the pulse.
ADIABATIC vs SUDDEN PULSES
Pulse duration 1.7 psSudden excitation; post-pulse alignment
A. Keller, C. Dion and O. Atabek, Phys. Rev. A 61, 023409 (2000)
(ω+2ω) Mechanism
[ ]0( ) cos( ) cos(2 ) ; 0.5E t E t tω γ ω γ= + =
C.Dion,et.al, Chem. Phys.Lett.302,215 (1999)
The Kick Mechanism: Half Cycle Pulses
ε~150kV/cm (~ ), no risk of ionization damage;Non-negligible field components at rotational transition;Sudden excitation Marked asymmetry (12:1) between max and min.
8 210 /W cm
( ~ 1 ) ( ~ 20 )rop t st ps T p<<
D.You, R.Jones, P.Bucksbaum, D.Dykaar, Opt. Lett.18,290 (1993)
Impulsive limit: sudden approximation2( , , ) exp( ) ( , , )it BJ t tψ θ φ ψ θ φ= h%
2 20( , , ) exp( ) cos exp( ) ( ,( ) , )E tBJ BJi ii t t t t
tψ θ φ µ θ ψ θ φ∂ ⎡ ⎤= − −⎣ ⎦∂% %h h h
200
2
( , , )
exp( ) cos exp(( ) ( ,) , ) ( , , 0)p
p
t
t
E ti i it t tJ dB tBJ
ψ θ φ
µ θ ψ θ φ ψ θ φ
=
⎡ ⎤− +⎣ ⎦∫
%
% %h hh
22 exp( ) ~ 1p
i BJ tBJ
t << ⇒h
h
( , , ) exp( cos ) ( , , 0)pt iAψ θ φ θ ψ θ φ=% %
00
( )ptA E t dtµ
= ∫h
2( , , ) exp( )exp( cos ) ( , ,0)pit t t iABJψ θ φ θ ψ θ φ≥ = − h
The Kick Mechanism: Half Cycle Pulses
Sudden
Exact
( ~ 1 ) ( ~ 20 )rop t st ps T p<<2
~ 0.13 ( 1) 1p J JtBJ
+ <<h
Validity of the sudden limit
( 25 ) ( 25 ) 0.015J Jexact sudden
JMax P t ps P t ps= − = ≤
Final rotational distribution
Root mean-square η = 0.03
4 ps
C.Dion, A.Keller, O.Atabek, Eur. Phy. J. D14,249 (2001)
OPTIMAL CONTROL
• Molecular Orientation dynamics
• Attosecond pulse synthesis
Optimal Control
1
sin() )( ( )n n n
N
n
E t E t tω φ=
= +∑En(t)
Model
( , , ) ( ) ( , , )i t H t ttψ θ φ ψ θ φ∂
=∂
h
2 212
20 ( ) (( ) cos ( cos ) )H t J t E tB Eµ θ α θ α⊥= − − ∆ +
Evaluation function
cos ( ) costθ ψ θ ψ=
Optimization loop
Target(s)
1
2
2
cos ( )
1 cos ( )f rot
f
f
t T
trot
j t
j t dtT
θ
θ+
=
= ∫
Parameters
0 1 2 3
, , ,, , ,t t t t
E ω φ
Generating offspring
1tpsps
E0 9(10 / )V mω 1cm−
φ ( )radπ0t
2t3t ps
ps
0.0410310.76
0.00.310.61
1.19
0.041031
0.760.00.490.61
1.19
individual mutation
10310.04
0.76
0.00.310.61
1.19arithmetic crossover multi-point crossover
+ =
0.046321.30
0.00.44
1.00
1.25
+ = +
0.041031
0.040.04 0.04632 632 1031
0.76 1.300.0 0.0
1.30 0.760.0 0.0
0.31 0.44 0.31 O.440.61 1.00 1.00 0.61
1.19 1.25 1.19 1.25
0.04831
1.030.0
0.380.81
1.22
1j
E(t)
<cos
θ>(t)
NG number of generation
HCN
1 cos ( )fj tθ=
O.Atabek, C.Dion,A.B.H.Yedder, J.Phys.B 36,4667(2003)
1 cos ( )fj tθ=
HCN
NG=11
O.Atabek, C.Dion,A.B.H.Yedder, J.Phys.B 36,4667(2003)
1 cos ( )fj tθ=
O.Atabek, C.Dion,A.B.H.Yedder, J.Phys.B 36,4667(2003)
HCN
NG=57
1 cos ( )fj tθ=
O.Atabek, C.Dion,A.B.H.Yedder, J.Phys.B 36,4667(2003)
HCN
NG=245
1 cos ( )fj tθ=
O.Atabek, C.Dion,A.B.H.Yedder, J.Phys.B 36,4667(2003)
HCN
NG=383
1 cos ( )fj tθ=
O.Atabek, C.Dion,A.B.H.Yedder, J.Phys.B 36,4667(2003)
HCN
NG=500
n En (W/cm2) ω (cm-1) φ (π rad.)
1 1.01364E08 1389.541 1.12406
2 2.99976E12 500.051 0.31723
3 2.99989E12 500.000 1.31887
ATTOSECOND PULSE SYNTHESIS
A. Ben-Haj Yedder, O. Atabek, C. Le Bris, S. Chelkowski, A. Bandrauk, Phys. Rev. A69, 041802R (2004)
Chirped field
HHG: phases and amplitudes
Emitted field:
• without optimization;
•after optimization
HHG FOR 1D OPTICAL LATTICES
ε
izz
xk
1 11
( ,... , ; ,... ) ( , ; )N
N N i i ii
H x x t z z H x t z=
= ∑2
( , ; ) ( ) . ( , )2
ixi i i i i i
pH x t z V x ex E z t
m= + −
Schrödinger Equation
( , ; ) ( , ; ) ( , ; )i i i i i i i i iH x t z x t z i x t zt
ψ ψ∂=
∂h
Polarization
1
( , ) ( , ; ) ( , ; ) . ( )N
i i i i i i i ii
P z t x t z ex x t z z zψ ψ δ=
= −∑
Maxwell Equation2 2 2
2 2 2 2 2
( , ) ( , ) 4 ( , )E z t E z t P z tz c t c t
µ πµ∂ ∂ ∂− =
∂ ∂ ∂
TARGET STATE
•Projection of the observable into a finite dimensionalHilbert subspace.
•Eigenstate of the projected observable corresponding to its lowest eigenvalue (variational principle).
Train of kicks: the target state
<cos θ> replaced by ( ) ( ) ( ) ( )cos ; , ,m NN N N Nm m m m j m
C P P P j m j mθ +
== = ∑
Target state (eigenfunction of ( ) )NmC
2( ) 12 sin ,
2 2
m NN
mj m
j mj m
N Nχ π
+
=
+ −⎛ ⎞⎛ ⎞≈ ⎜ ⎟⎜ ⎟+ +⎝ ⎠ ⎝ ⎠∑
Finite subspace( ) : , ( , 1,... )NmH j m j m m m N= + +
Individual kick2cosA
Ai tBi J
JU e U eθ −= =
( ) ( )0 0cos cos
2N N
Nπχ θ χ ⎛ ⎞≈ ⎜ ⎟+⎝ ⎠
Efficiency
Duration
Choice of parameters to remain in ( )NmH ( ) 2cos ( ) cos ( ) ( )
0 0 0A Ai N i N Ne P e Pθ θ χ η− ≈
which, for small A, amounts to [ ]2 32( 2)A Nη π ⎡ ⎤≈ +⎣ ⎦ ( 1, 4) 0.02A N η= = ⇒ ≈
Train of kicks: the strategy
1st kick
2 kick
Maximum of<cosθ> after the
first kick
Maximum of <cosθ> after thesecond
kick
Train of kicks: the strategy
1st kick
2 kick
Train of kicks: the result
0.89
0.2
Efficiency = 0.89 Duration = 0.2 rotTfor LiCl 2 ps, for NaI 20 ps
D. Sugny, A.Keller, O.Atabek, D.Daems, C.Dion, S.Guérin, H.Jauslin, Phys. Rev. A 69, 033402 (2004)
Robustness
Robustness shifting timedelays by ±1% of the rotational period
Robustness varying thepulse energy by ±10%
Geometrical accuracy: 0.2 µ, corresponding to 0.6 fsec, as compared to Trot of the order of 10 psec.
BACK TO OPTIMIZATION
• Well defined target state
• Optimize the amplitudes and delays betweenthe HCP’s to reach the target state
Efficient, long-duration orientation with only 2 pulses
C. Dion, A. Keller and O. Atabek, Phys. Rev. Lett. (submitted)
Conclusions
No single solution arising from optimal control schemes (criteria, parameters,…)
Investigate basic mechanisms of laser-molecule dynamics(ω+2ω, kick,…)
Restricting the parameters sampling space, force optimal control to take advantage of these mechanisms
From the knowledge of the most relevant basic mechanisms, appropriately shape the laser pulse (train of kicks)
Perspectives: transposition to a generic system
Control objectives
Free dynamics under
Control of an observable
( upper or lower bounded, target: maximize or minimize )
0H( . . cos )e gO θ
2( . . )e g BJ
[ ]0, 0,H O O≠( )O t
Control strategy
Perturb the system with unitary
with:
the optimal target state is an eigenfunction of both
The optimum corresponds to a fixed point of the sequence
times where reaches its maximum under free evolution
cos( . . )iAAU e g U e θ=
[ ] 1, 0O U O U OU−= ⇒ =
O and U( )i iO O t=
it ( )O t
GENERIC STRATEGIES
0 1)) ((H H v Htt = +Pure state
0( 0) ;t Hαψ α α α α= = =
Example: 0j mα = = =
( )U Nunitaryα χ⎯⎯ ⎯→
Strategy : fixed point of convergent series
a) Robustness, experimental feasibility;
b) Optimal control with well-defined targets;
c) Other control objectives : logical gates;
d) Mathematical study of the fixed points
GENERIC STRATEGIES
0 1)) ((H H v Htt = +Mixed state
( ) ( ) ( )k k kk
t w t tρ ψ ψ= ∑ [ ]1( ) ,t i Ht
ρ ρ−∂= −
∂h
( 1) /1(0) , ,Bj j kT
j m
j m e j mZ
ρ − += ∑∑Example
Strategy :
( )( 0) ( )U Nunitaryt tρ ρ= ⎯⎯ ⎯→
[ ] [ ]( ) ( ) ; , 0O t Tr O t Oρ ρ= =specific ordering of the commoneigenvectors of andρ O
Use of different polarization schemes(elliptical); application to polyatomics
GENERIC STRATEGIES
0 1)) ((H H v Htt = +Molecular system interacting with an environment
Examples : collisions in a gas, frictions in a liquid
Strategy : Artificial reservoir against decoherence
( )( 0) ( )U Nnon unitaryt tρ ρ−= ⎯⎯ ⎯ ⎯→
E
As
What are the targets? Are they attainable?
