Reaching optimally oriented molecular states by laser kicks

0.15 MB·English

by  D. Sugny

Reaching optimally oriented molecular states by laser kicks
D. Sugny,∗ A. Keller,† and O. Atabek
Laboratoire de Photophysique Mol´eculaire du CNRS, Universit´e Paris-Sud,
Bˆat. 210 - Campus d’Orsay, 91405 Orsay Cedex, France
D. Daems‡
Center for Nonlinear Phenomena and Complex Systems,
Universit´e Libre de Bruxelles, CP 231, 1050 Brussels, Belgium
C. M. Dion
Department of Physics, Ume˚a University, SE-90187 Ume˚a, Sweeden
4 S. Gu´erin and H. R. Jauslin
0 Laboratoire de Physique de l’Universit´e de Bourgogne,
0 UMR CNRS 5027, BP 47870, 21078 Dijon, France
2
Wepresentastrategyforpost-pulsemolecularorientationaimingbothatefficiencyandmaximal
n
durationwithinarotationalperiod. Wefirstidentifytheoptimallyorientedstateswhichfulfillboth
a
J requirements. Weshowthatasequenceofhalf-cyclepulsesofmoderateintensitycanbedevisedfor
reaching these target states.
5
1
PACSnumbers: 33.80.-b,32.80.Lg,42.50.Hz

3
v
Molecular orientation plays a crucial role in a wide delivered by short HCPs, that allows to significantly en-
9
9 variety of applications extending from chemical reaction hancethedurationoftheorientation,maintainingahigh
0 dynamics, to surface processing, catalysis and nanoscale efficiency. Our constructionis first based on the identifi-
0 design[1,2,3,4]. Staticelectricfield[5]andstrongnon- cation of target states which fulfill the previous require-
1 resonantlonglaserpulses[6,7]havebeenshowntoyield ment. These states are characterized by the fact that
3
adiabatic molecular orientation which disappears when they only involve a limited number of the lowest lying
0
the pulseisoff. Noticeableorientationthatpersistsafter rotational levels and that they maximize the orientation
/
h theendofthepulse(andevenunderthermalconditions) efficiency within the corresponding restricted rotational
p
is of special importance for experiments requiring field- spaces. At a second stage, we show that these selected
-
t free transient orientation. It has recently been shown statescanbereachedbyatrainofkicks,actingatappro-
n
that very short pulses combining a frequency ω and its priately chosen times. The choice of the strength of the
a
second harmonic 2ω excite a mixture of even and odd pulses (taken equal for simplicity), together with the to-
u
q rotational levels and have the ability to produce such talnumberofkicksallowto approachthese targetstates
: post-pulse orientation [8]. But even more decisive has with good accuracy.
v
beenthesuggestiontousehalf-cyclepulses(HCPs),that The time evolution of the molecule (described in a
i
X through their highly asymmetrical shape induce a very 3D rigid rotor approximation) interacting with a lin-
r suddenmomentumtransfertothemoleculewhichorients early polarized field is governed by the time-dependent
a under such a kick after the field is off [9, 10]. Both the Schr¨odinger equation (in atomic units)
(ω+2ω) and the kick mechanisms have received a con-
∂
firmation from optimal control schemes [11]. The caveat i ψ(θ,φ,t)=[BL2−µ F(t)cosθ]ψ(θ,φ,t), (1)
0
∂t
is that the post-pulse orientation is maintained for only
short times. Recently, the use of a train of kicks to in- where L is the angular momentum operator, B the ro-
creasetheefficiencyoftheorientationhasbeensuggested tational constant, µ the permanent dipole moment and
0
in optimal control strategies [11] and applied to molec- F(t) the field amplitude. θ denotes the polar angle be-
ular alignment [12] and orientation of a 2D rotor [13]. tween the molecular axis and the polarization direction
However, due to the strength of the kicks used, only the oftheappliedfield. Themotionrelatedtothe azimuthal
efficiency of the process has beenoptimized, its duration angle φ can be separated due to cylindrical symmetry.
decreasing strongly. In the present letter, we propose a Fromnowon,weassumeasuddenapproximationdueto
control strategy using specially designed series of kicks the short durations τ of the HCPs, as compared to the
molecular rotational period T = π/B. For relatively
rot
lowl (wherellabelsthequantumeigenstatesofL2),this
amounts to the definition of a dimensionless, small per-
∗Electronicaddress: [email protected] turbative parameter ε = τB. This definition, together
†Electronicaddress: [email protected] with a rescaling of time s = t/τ (such that s ∈ [0,1]
‡Electronicaddress: [email protected]
during the pulse) leads to an equation suitable for the
2
application of time-dependent unitary perturbation the- 0.2
ory [14, 15]
0.9
∂
i ψ(θ,φ,s)=[εL2−E(s)cosθ]ψ(θ,φ,s), (2)
∂s N)> 0.15
widbnyihvgeaidrdneuyeaEnffla(emHsc)tCici=vPaelµiipm0niτscpFttaau(rnrτttessa)nia.seAoktuhitcsekloefwtvooolellsotutwhtoeiiorndnmgeoor[1plie5nec,ruεa1l,te6ot]hrde:eesiecAamrncieobrsiengθd--, (N)(χθχ<|cos|000.7 0.1∆ t/Trot
1
where A = E(s)ds is the total pulse area. Between
R0 0.5
two kicks, the molecule evolves under the effect of its
field-free rotation e−iεL2s.
The goalof the field drivenmolecularorientationis to 11 22 33 44 55 66 77 88 99 1100
N
maximize (or to minimize, depending on the choice of
theorientation)forthelongesttime duration,theexpec- FIG. 1: Maximal orientation efficiency (crosses) and associ-
tation value hcosθi(s)=hψ(θ,φ,s)|cosθ|ψ(θ,φ,s)i after atedduration(opencircles)asafunctionofN,whereN+1is
the pulse is over. An understanding of this process can thedimension of therotationally excited subspace H(N) (see
0
be obtainedby analysingthe moleculardynamics ina fi- text). Thesolidanddashedlinesarejusttoguidethelecture.
(N)
nite subspace H generatedby the first (N+1) eigen-
m
states of L2, i.e. |l,mi (l = |m|,|m|+1,...N +|m|) for
a molecule initially in the state |l ≥ m,mi. The justi- independent. From Fig. 1, we observe that, in order to
0
fication of such a reduction is in relation with the finite keep a duration of the order of 1/10 of the rotationalpe-
amountofenergythatafinitenumberofHCPsofagiven riod (which may amount to durations exceeding 10 ps,
area A can transfer to the molecule. The mathematical for heavy diatomics like e.g. NaI), N has to be limited
advantage it offers is the consideration of an operator to 5 or 6, which seems rather limiting. But this turns
out to be sufficient for very efficient orientation. N = 4
C(N) =P(N)cosθP(N), (3) already allows an orientation efficiency larger than 0.91.
m m m
Two basic questions are in order : which set ot pa-
(P(N) being the projector on the subspace H(N)) which, rameters and which number of kicks have to be chosen
m m
as opposed to cosθ, has a discrete spectrum. It turns to (approximately) remain in the subspace H(N), and
out that the state |χ(N)i, which maximises the orien- which strategies have to be followed to reach the maxi-
m
tation in the subspace H(N), is the eigenstate of C(N) mum possible efficiency within this subspace. The first
m m
question is in relation with the kick momentum transfer
with the highest eigenvalue. Using the approximation
operatoreiAcosθ,whichistheonlyevolutionoperator(as
hl,m|cosθ|l±1,mi ≃ 1/2 valid for l ≫ m, straightfor-
opposed to the free evolution) that rotationally excites
ward algebra leads to
the system and forces it to expand on a larger subspace.
We can estimate the loss outside H(N) assuming a pre-
l=|m|+N
χ(N) ≃ 2 1/2 sin πl+1−|m| |l,mi. liminar convergence to |χ(N)i (as shown by Fig. 2) by
(cid:12)(cid:12) m E (cid:16)N +2(cid:17) X (cid:16) N +2 (cid:17) looking for the smallness of the norm
(cid:12) l=|m|
(4)
2
Sincethequantumnumberm,relatedwiththeazimuthal eiAcosθ−P(N)eiAcosθP(N) χ(N) =η, (6)
(cid:13)(cid:16) (cid:17)(cid:12) E(cid:13)
angle is conserved,we will not write it explicitely, unless (cid:13) (cid:12) (cid:13)
(cid:13) (cid:12) (cid:13)
necessary. The maximal orientation in this subspace is which, for small A, amounts to
found to be
π (Aπ)2
χ(N)|cosθ|χ(N) ≃cos . (5) η ≃ . (7)
D m m E (cid:16)N +2(cid:17) 2(N +2)3
For a temperature T = 0 K (m = 0), Fig. 1 gath- This allows us to establish a relation between A and N
ers two informations relevant for the characterization of for a given loss. N ≃ 4 is found compatible with an
orientationas a function of N; namely hχ(N)|cosθ|χ(N)i η ≃0.02(notmore than2% ofthe rotationalpopulation
which is the maximum efficiency [approximatively given leavingthe subspaceH(4))asfarasAdoesnotexceed1.
by Eq. (5)] that can ideally be expected for a process The second question can be answeredby adapting the
that stays confined within the finite subspace H(N), and strategy suggested for the orientation of a 2D rotor in
∆t/T which measures the relative duration of the ori- Ref. [13], which consists in applying laser pulses each
rot
entationoverwhichhcosθiremainslargerthan0.5during time hcosθi reaches its maximum. The following argu-
the field-free evolution of |χ(N)i. The results, expressed ment shows that, if the dynamics stays within the sub-
as a fraction of the rotational period T , are molecule space H(N), such a strategy precisely converges to an
rot
3
optimal state |χ(N)i. This is done by approximating the 1
operators cosθ and eiAcosθ by C(N) and eiAC(N) respec-
tively. The interaction with a sudden HCP only alters 0.8
the slope of hC(N)i(s) and not its value as is clear from
(a)
the following relation
>0.6
θ
De−iAC(N)C(N)eiAC(N)E=DC(N)E. (8) cos
<0.4
Moreover,if a sudden pulse is applied at a time s when
i
hC(N)i(s) reaches its maximum C = hC(N)i(s ), the
i i 0.2
slope undergoes a change from zero to a finite value
d hC(N)i =i εL2,C(N) =0, 0
ds (cid:12)si−0 (cid:10)(cid:2) (cid:3)(cid:11)
d hC(N)i(cid:12) =i e−iAC(N) εL2,C(N) eiAC(N) 6=0.
ds (cid:12)(cid:12)si+0 D (cid:2) (cid:3) E 0.9
(9)
As hC(N)i is a periodic, continuously differentiable func- (b)
2
tion, it will reach within the rotational period, a max- >|0.6
imum value larger than the one obtained prior to the ψ(t)
aapnpilniccaretiaosninogfbtuhtebpouulnsed.edItaenradtitnhgertehfeorsetrcaotnevgeyr,gwenetgseet- (4)χ<|0
|
quenceofC ’s. ItslimitisafixedpointC =C ,corre- 0.3
i i i+1
spondingtothe eigenvectorsoftheimpulsivepropagator
eiAC(N) which are also the ones of C(N). Indeed, at a
fixed point, the slopes before and after the interaction
0
with the last pulse have to be zero. 0 0.1 0.2 0.3 0.4 0.5
t/T
Figure2givestwodifferentviewsoftheorientationdy- rot
FIG. 2: Orientation dynamics during the train of HCPs at
namics under the effect of a train of HCPs,separatedby T = 0 K: panel (a) for hψ(s)|cosθ|ψ(s)i and panel (b) for
time delays corresponding to the above discussed strat- |hχ(4)|ψ(s)i|2. Thesolid linecorrespondsto|ψ(s)icalculated
0
egy of maxima, with identical durations ε = 0.01 and exactly and thedashed line to the wave function propagated
pulse areas A = 1 (that is about 0.3 ps and a field am- inthesubspaceH(4). ThetrainofHCPsisdisplayedonpanel
0
plitude of 1.5·105 V cm−1 for LiCl [9]), which leads to (b) and theoptimal orientation is indicated by an horizontal
a dynamics that remains within the subspace H(4) for line on panel(a).
the considered numbers of kicks. From panel (a) it is
interesting to note that a single kick produces an orien-
tation of about 0.5, whereas the appropriate application Orientation is subject to a drastic decrease with tem-
of 15 kicks increases this efficiency up to 0.89, which is perature [10, 11]. This is basically due to the fact that,
almost the optimal limit as found from Fig. 1. On the withnon-zerotemperature,theinitialstateisasuperpo-
other hand, the comparison of the average hcosθi calcu- sition of a statistical ensemble of rotational states with
lated with the exact wave function |ψ(s)i and with the m 6= 0, which tends to misalign the molecule. The effi-
onepropagatedinthesubspaceH(4),arecloseenoughto ciencyoftheorientationischaracterizedbyanadditional
0 average of hcosθi over the density operators ρ (s)
support the claim that the rotational dynamics actually m
resides within H(4). Panel (b) shows the way the wave
0 hhcosθii(s)= Tr[ρ (s)cosθ]. (10)
function |ψ(s)i gets close to the optimally oriented state X m
m∈Z
|χ(4)i, showing thus the succesfull outcome of the pro-
0
cess. The difference between the two dynamics can be We recall that ρm(s) evolves according to the von Neu-
also estimated by Eq. (7). Here again, the close conver- mann equation
gence of the two calculations shows a coherent choice of
d
A, N and the number of kicks for appropriately describ- ρ (s)=i[ρ (s),εL2−E(s)cosθ], (11)
m m
ds
ingthedynamics. Thepost-pulsedynamics,whichisour
mainconcern,isdisplayedonFig. 3[panel(a)]. Aresult with as initial condition
expected from the previous analysis,but particularly re-
1
markable with respect to previous proposals,is obtained ρ (0)= |l,mie−Bl(l+1)/kThl,m|, (12)
m Z X
with an efficiency of about 0.89 and a duration of the l≥|m|
order of 2/ of the rotational period (that is about 2 ps
10
for a light molecule like LiCl and 20 ps for a heavy one, where Z = e−Bl(l+1)/kT is the partition
Pm∈ZPl≥|m|
like NaI). function and k the Boltzmann constant. Following our
4
1 |χ(N)iTr[ρ(N)]hχ(N)| with the constraint related to the
m m m
conservation of m, expressed as Tr[ρ(N)] to be kept con-
(a) m
stant
0.5 l=|m|+N
1
θs> Tr[ρ(mN)]= Z X e−Bl(l+1)/kT. (13)
o
c |m|
<
0 Theeffectofasuddenpulseonρise−iAcosθρeiAcosθ,and
because its optimal value corresponds to a fixed point of
the hhC(N)ii(s ) sequence, it is precisely the application
i
of a train of HCPs with individual pulses at times s
i
0.52 0.72 0.92 1.12 1.32 1.52 where hhcosθii(s ) reaches its maximum, that converges
i
1
to the best possible orientation within this model. The
(b) resulting dynamics is plotted in Fig. 3 [panel (b)], with
a maximum efficiency of about 0.75 and a duration of
about 1/ of the rotational period. To our knowledge,
0.5 20
this is the largest duration and efficiency achieved up to
>
> date for a thermal ensemble.
θ
s
o In conclusion, we have presented tools for controlling
c
< molecular orientation dynamics using a train of HCPs,
< 0
achieving both efficiency and duration of the post-pulse
orientation. Moreover, this scheme can be expected to
be transposable to a generic system, with free periodic
−0.5 dynamicsgovernedbyanHamiltonianH0,andforwhich
we are aiming to optimally control an observable O (i.e.
0.25 0.45 0.65 0.85 1.05 1.25
t/T maximize or minimize the average hOi(t) of an upper
rot
FIG.3: Post-pulseorientationdynamicsofthemoleculeLiCl or lower bounded operator O which does not commute
after interaction with a train of HCPs (the time t=0 corre- with H0). This could be done through a device that
sponds to the first kick) in the case A = 1, T = 0 K [panel perturbs the system according to a unitary operator U,
(a)] and A =2, T = 5 K [panel (b)]. Solid and dashed lines which commutes with O, such that its application does
correspond respectively to the averages calculated with the not alter hOi=hU−1OUi on one hand, and the optimal
exact wave function and the optimal state [|χ(04) > on panel target state is an eigenfunction of both O and U on the
(a) and ρ(7) on panel (b)]. other hand. This optimum corresponds to a fixed point
ofthesequenceO =hOi(t )wheret arethetimeswhen
i i i
hOi(t)reachesitsmaximum(orminimum)underthefree
previous analysis, we are looking for the optimal den- evolution. In particular, this scheme provides a compre-
sity operator which maximizes hhcosθii in the subspace hension of previous works on alignment and orientation
H(N), which is given by ρ(N) = ρ(N) where ρ(N) = of 2D and 3D rotors [12, 13].
Pm m m
[1] P.R. Brooks, Science 193, 11 (1976). 14, 249 (2001).
[2] F. J. Aoiz, Chem. Phys.Lett. 289, 132 (1998). [10] M.MachholmandN.E.Henriksen,Phys.Rev.Lett.87,
[3] T. Seideman, Phys. Rev.A 56, R17 (1997). 193001 (2001).
[4] H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. 75, [11] C.M.Dion,A.BenHajYedder,E.Canc`es,A.Keller,C.
543 (2003). L.Bris andO.Atabek,Phys.Rev.A65, 063408 (2002).
[5] L. Cai, J. Marango and B. Friedrich, Phys. Rev. Lett. [12] M.Leibscher,I.Sh.AverbukhandH.Rabitz,Phys.Rev.
86, 775 (2001). Lett. 90, 213001 (2003).
[6] M.J. J.Vrakkingand S.Stolte,Chem. Phys.Lett. 271, [13] I. Sh. Averbukh and R. Arvieu, Phys. Rev. Lett. 87,
209 (1997). 163601 (2001).
[7] S.Gu´erin,L.P.Yatsenko,H.R.Jauslin,O.Faucherand [14] D. Daems, A. Keller, S. Gu´erin, H. R. Jauslin and O.
B. Lavorel, Phys. Rev.Lett. 88, 233601 (2002). Atabek, Phys.Rev.A 67, 052505 (2003).
[8] C. M. Dion, A. D. Bandrauk, O. Atabek, A. Keller, H. [15] D. Sugny, A. Keller, O. Atabek, D. Daems, S. Gu´erin
Umeda and Y. Fujimura, Chem. Phys. Lett. 302, 215 and H.R. Jauslin, Phys.Rev.A submitted (2003).
(1999). [16] N. E. Henriksen, Chem. Phys.Lett. 312, 196 (1999).
[9] C. M. Dion, A. Keller and O. Atabek, Eur. Phys. J. D.