High Harmonic Generation in SF$_{6}$: Raman-excited Vibrational Quantum Beats

0.15 MB·English

by  Zachary B. Walters

High Harmonic Generation in SF : Raman-excited Vibrational Quantum Beats
6
Zachary B. Walters,1 Stefano Tonzani,2 and Chris H. Greene1
1Department of Physics and JILA, University of Colorado, Boulder, Colorado 80309-0440, USA
2Department of Chemistry, Northwestern University, Evanston, Illinois 60208-3113, USA
(Dated: February 2, 2008)
In a recent experiment (N. Wagner et al. [1]) on SF6, a high-harmonic generating laser pulse
7 is preceded by a pump pulse which stimulates Raman-active modes in the molecule. Varying the
0 timedelaybetweenthetwopulsesmodulateshighharmonicintensity,withfrequenciesequaltothe
0 vibrationfrequenciesoftheRaman-activemodes. Weproposeanexplanationofthismodulationas
2 a quantum interference between competing pathways that occur via adjacent vibrational states of
themolecule. TheRamanandhighharmonicprocesses actasbeamsplitters, producingvibrational
n
quantum beats among the Raman-active vibrational modes that are excited by the first pulse. We
a
introducearigorous treatment of theelectron-ion recombination process andtheeffect of theionic
J
Coulomb field in the electron propagation outside the molecule, improving over the widely-used
9
three-step model.

]
h PACSnumbers:
p
-
m High harmonicgeneration(HHG) is commonly under- rates into a product of uncoupled normal mode vectors
o stood as a 3 step process [2] in which an electron ion- ψ(t) (i) =a(i)(t) 0 (i)+a(i)(t) 1 (i) that can be treated
| i 0 | i 1 | i
t izes from a molecule, propagates in a strong laser field, individually.
a
. andthenrecombineswiththeparentionwhileemittinga After a Raman pulse with the intensity and duration
s
c photon. Accelerationbythelaserfieldallowstheelectron used in Ref.[1], only the 0 and 1 states of a given
| i | i
i to return with a large kinetic energy and emit photons normal mode have significant amplitude. The following
s
y with energy much higher than those of the driving laser. two-state, one-dimensional picture shows how individ-
h Although the HHG process is primarily electronic in ualRaman-activemodesaffectthehighharmonicsignal.
p
character, recent experiments have shown that vibra- Atomic units are used throughout this work.
[
tional degrees of freedom can play a role. In the ex- DuringtheRamanpulse,stimulatedRamanscattering

1 periment by Wagner et al., [1] a 25 fs, 2.4 1014 changes the vibrational state of the molecule from an
v W/cm2 HHGlaserpulsewasprecededbyaweake×r25fs, initial 0 into a coherent superposition a0 0 +a1 1 of
3 | i | i | i
5 1013W/cm2 pulse whichexcitedRaman-activevibra- the zeroth and first vibrational states. The vibrational
1 ×
1 tionsinthemolecule. Theintensityofthehighharmonic coefficients follow equations of motion given by
1 lightwasfoundtooscillatewiththeinterpulsedelaytime
0 at the excited molecular vibration frequencies. The sur- 1 1
ia˙ (t)=ω n + a (t) E (t)E (t)
07 prising result was that the breathing mode, overwhelm- ni i(cid:18) i 2(cid:19) ni − 2AX,B A B × (1)
inglydominantinconventionalRamanexperimentswith
/ α a +∂ α (√n +1a +√n a ) .
s thismolecule,isnolongerthestrongestmodeseeninthe AB ni i AB i ni+1 i ni−1
c
i HHG experiment. Here(cid:2)ω is the normal mode frequency, indices A (cid:3)and B
s i
y In this Letter, we interpret these oscillations as an in- run over x,y,z ,E (t) is the component of the electric
A
h terference between indistinguishable quantum pathways field in th{e (bod}y-frame) A direction at time t, Q is the
i
p
associated with different intermediate vibrational states normalized displacement associated with normal mode
:
v during the HHG process, illustrated in Fig. 1. We de- i and α (Q ,Q ,...) is the polarizability tensor of the
AB 1 2
Xi velop a quantum mechanical description of the recent molecule. These equations of motion have off-diagonal
r pump-probe SF6 experiments, using a framework that elements only if ∂iαAB (2mωi)−1/2∂αAB/∂Qi 0 = 0,
a substantially improves on the three-step model.[2] We which is the condition f≡or a mode to be Raman|ac6tive.
have included the effects of the ion’s Coulomb potential Thepolarizabilitytensoranditsderivativesarefoundby
on the propagating electron, and utilized a nonpertur- performing an unrestricted Hartree-Fock calculation [4]
bative electron-molecule scattering wavefunction [3] to using the aug-cc-pVTZ basis set. [5]
calculate the recombination amplitude. The resulting
Between laser pulses, 0 and 1 evolve as simple har-
calculationexhibits partialagreementwithexperimental | i | i
monic oscillator eigenstates, becoming ψ = a 0 +
vib 0
observations. a e−iωτ 1 at the beginning of the high harmonic |puilse
1
| i
We adopt a level of approximation in which all op- foraninterpulsedelayofτ,wherethenormalmodeindex
erators depending on the nuclear coordinates are ex- (i)isomittedforbrevity. Beforetheelectrontunnelsfree
panded to first order in the normal mode coordinates oftheion,thehighharmonicpulsestimulatesthenormal
Q ; then the full vibrational state vector Ψ(t) sepa- modefurtheraccordingtoEq. (1). Thisisapproximated
i
| i
2
by a unitary 2 2 transfer matrix M, where M is the The number of photons emitted in a given harmonic
nm
amplitude to en×din state i after startingin state j at is proportional to d~ d~∗ +d~ d~∗. The high harmonic
| i | i 0 · 0 1 · 1
the beginning of the pulse. intensity is a sum over all Raman active modes i:
Ionization and recombination, both electronic pro-
P(τ)=P +Σ P(i)cos(ω τ +δ ) (3)
cesses, are both modulated strongly by nuclear mo- 0 i 1 i i
tion. Taylor-expanding the tunnel-ionization operator Iˆ
. The static P primarily results from terms of the form
to first order in Qi, with the substitutions I0ˆI I eq a∗a , while P0results from terms of the form a a∗eiωτ
≡ | 0 0 1 0 1
a(2nmdωIi1()i)−1≡/2((2Amˆi+ωiA)ˆ−†i)1,/2yi∂eIlˆd/s∂tQhie|efiqrsatn-odrdthereeixdpenantistiyonQiint=o aan∗0Wda001aa∗0∗0ea−niωdτ.PD1ceofisn(iωngt+Wδ=) =M†12I(†aN∗1e†iR~ωτ†·WR~1N0aI0M+,Pc.0c.=).
raising and lowering operators, Iˆ = I ˆI + Σ I(i)(Aˆ + Since I and R are small relative to I and R , only
0 i 1 i 1 1 0 0
Aˆ†). Here ˆI is the identity operator, and Aˆ†,Aˆ are the their first-order terms are kept.
i i i
raisingandloweringoperatorsforthei thnormalmode. At this level of approximation, calculating Iˆand R~ˆ as
−
For each mode considered, the reduced mass m is equal functions ofthe nuclearcoordinatesandsubstituting the
tothemassofasinglefluorineatom. Therecombination expectation values for Q at ionization and recombina-
i
operator R~ˆ can be derived to first order using identical tion would give identical results. Nevertheless, tracking
logic. In both cases, dependence on nuclear positions the quantum mechanical pathways in this manner is in-
means that the vibrational state changes along with the formative, because it allows the prediction of other ob-
electronic state. servables less amenable to a “classical nuclear motion”
Theevolutionofthe ionicvibrational/electronicwave- analysis, like the relative populations in 0 or 1 after
| i | i
functionbetweenionizationandrecombinationisingen- recombination.
eralquitecomplicated,sincethethreedegenerateorbitals This framework is applied to real molecules, using an
of SF+, which have T symmetry, are coupled by vi- improvedversionof the 3 step model. For the ionization
6 1g
brational degrees of freedom [6, 7, 8]. The linear and step, a simple one-dimensional WKB tunneling picture
quadratic terms in the Jahn-Teller Hamiltonian [6, 7, 8], describesanelectrontunnelingonlyindirectionsparallel
which governs the coupled vibrational/electronic evolu- to the laser electric field. This is motivated by the “ini-
tion, are found for each Q by fitting the eigenvalues of tial value representation” [9, 10]. In the classically for-
i
the coupling matrix to the lowest 3 adiabatic energies bidden regionunder the barrierformedby the molecular
of SF+ for different displacements of the ion away from potential and the laser field, the tunneling wavefunction
6
the maximum symmetry configuration. The energies are equalsthevalueoftheunperturbedmolecularHOMOat
foundusingGaussian’sCASSCFmethodandacc-PVDZ theinnerturningpointtimesadecliningWKBexponen-
basisset[4]. Inthenotationof[6],V =.001209H/bohr, tial. With the direction of the electric field as zˆ, the
T2g −
V =.1420 H/bohr, N =-.0362 H/bohr2, K =.7288 wavefunction at the outer and inner turning points are
Eg 1 T2g
H/bohr2, K =1.8486 H/bohr2. For the A mode, related by
Eg 1g
which does not enter into the vibronic Hamiltonian, an
ψ (x,y,z ,t)=ψ (x,y,z ,t)
adiabatic potential E =V Q +1/2K Q2 , with t tp2 HOMO tp1 ×
A1g A1g A1g A1g ztp2 (4)
VA1g=.0645 H/bohr, KA1g=2.98 H/bohr2 gives the po- |C1/C2|1/6Bi(0)/Ai(0)exp[− dzk(z)],
tential energy surface for all three electronic states. An Zztp1
importantsimplificationisthattheoff-diagonalcoupling
where k(z)= 2(V +V E), Ai and Bi are Airy
betweendifferentelectronicstates,proportionaltoVT,is functions, z andmzol arelastehre−inner and outer turning
small and can be neglected for the short times between tp1p tp2
points, C and C are the z components of the slopes of
ionization and recombination. 1 2
2V(~r,t) at the two turning points, and the path integral
Neglecting off-diagonal coupling between electronic
is calculated alongthe z direction parallelto the applied
states, the adiabatic potential felt by the ion in a par-
electric field.
ticular electronic state is H = p2i + V Q + 1K Q2,
i 2m i i 2 i i After tunneling, the free electron wavefunction’s evo-
which can be rewritten to first order in the basis of
lutionisrelativelysimpleuntilitrescattersfromthepar-
oscillator states of the neutral molecule. Evolution of
ent ion. ψ (r,Ω,t), the continuum wavefunction at the
c
the vibrational state is given by a transfer matrix N =
instant just prior to the electron rescattering from the
exp( iH(t t )).
ret ion molecular ion, is found using Gutzwiller’s semiclassical
− −
In a two-state treatment, the i th vibrational wave-
propagator [11]:
−
function of the neutral molecule after recombinationhas
occurred is ψ =d 0 +d 1 , where K(~r,t;~r ,t )=(2πi)−3/2 C(~r,t;~r ,t )
vib 0 1 0 0 0 0
| i | i | i ×. (5)
d~0 d~1 = a0 a1e−iωτ MTITNTR~T (2) exp[iS(~r,t;~rp0,t0)−iφ]
HereS(~r,t;~r ,t )is the actionintegralS = L(q,q˙,t)dt
and AT i(cid:16)s the tr(cid:17)ansp(cid:0)ose of matrix(cid:1) A. calculatedfor0th0eclassicaltrajectorystartingat(~r,t)and
R
3
ending at (~r ,t ) and C(~r,t;~r ,t )= ∂2S is the pointsdescribesthephaseintheslowly-varyingregionof
0 0 0 0 |− ∂r0,A∂rB|
densityoftrajectoriesforgiveninitialandfinalpoints. φ interest.
is a phase factor equal to π times the number of conju- These stationary phase conditions are met by a tra-
2
gate points crossed by the trajectory. The semiclassical jectory that begins “downstream”of the molecule in the
continuum wavefunction is directionoftheelectricfieldwithzeromomentum,moves
only radially and parallel to the electric field until it
ψc(~r,t)= d3~r0 dt0K(~r,t;~r0,t0)ψt(~r0,t0) (6) reencounters the parent molecule at the scattering state
Z Z energy. Only the incoming-wave part of the scattering
When the electron recollides with the parent ion, its statesgivesanonvanishingcontributiontotheexpansion
wavefunction is distorted strongly by the molecular po- coefficients. Performing the resulting gaussian integrals
tential and by exchange effects with the other electrons, about the initial and final points, and about the initial
which can dramatically change amplitudes for recombi- time, yields expansion coefficients
nation with respect to the plane wave approximation.
TechniquesdescribedinRefs[3,12,13]determineacom- 4π2 ∂r ∂H −21
pinlegtestsaetteso.fsBteaytioonndartyhefierladn-fgreeeofeltehcetrmono-lmecoulelacrulpeostceantttiearl-, Al,m(E)= i√2s(cid:12)∂p0A,B(cid:12)(cid:18)∂t0(cid:19) fE−l∗(r)Yl∗,m(zˆ) (9)
(cid:12) (cid:12)
the scattering states are given in terms of incoming and e(cid:12)(cid:12)i[S(~r,t,~r(cid:12)(cid:12)0,t0)−φ]ψt(~r0)e[−iEHOMOt0]
outgoing Coulomb radial functions f±(r) and the scat-
El
tering S-matrix as for a stationary phase trajectory that starts at (z0,t0)
and ends at (z,t), where zˆis the direction of the electric
1
ψ (~r)= f−(r)Y (θ,φ) field at the time of ionization.
E,lm i√2 El l,m − The recombination amplitude is now D~(E) =
(7)
i√12 fE+l′(r)Yl′,m′(θ,φ)Sl′,m′;l,m(E). thel,mreAcolm,mb(iEn)ad~til,omn(aEm),plwithuedreecda~Elc,ulmlat=edhfoψrge|~axc|hψEin,ldmiviidis-
lX′,m′ P
ualscatteringstate. Thisconnectstothequantumpaths
During the electron-molecule scattering, when re- framework when I =ψ (~r ,t ), and R~=D~(E) calculated
t 0 0
combination occurs, the electron wavepacket is ex- for I = 1. Both quantities are calculated at the equilib-
panded in terms of these scattering states as ψS = riumgeometry,thenforageometrydistortedby0.1bohr
dE l,mAl,m(E)ψE,lme−iEt, where in the normal mode coordinate to find Iˆ,R~ˆ and their de-
R P rivitives. Thisinvolvesrecalculatingthescatteringstates
A (E)=eiEt d3~rψ∗ (~r)ψ (~r,t). (8)
l,m E,lm c and recombination dipoles for the distorted geometry.
Z Themodulationofthe39thharmonic,whichtheJILA
For a chosen time of projection onto the scatter-
experiment considered in detail, is calculated for com-
ing states, Eqs. (6) and (8) together define a seven-
parison with experiment. This harmonic is close to the
dimensional integral over initial and final positions and
measured cutoff, so it can only be produced by a laser
initial times. However, the integrand oscillates rapidly
half-cycle where the maximum of the electric field falls
almost everywhere, causing cancellations. Stationary
closetothemaximumoftheGaussianenvelope. Accord-
phase techniques identify the regionwhere the integrand
ingly, only the modulation for a single half-cycle where
oscillates slowly, which permits evaluation.
these two maxima coincide was used. In the expression
The semiclassical action S is expanded to second or-
for the harmonic intensity at some harmonic order, Eq.
der around the starting and ending points of some clas-
(3),allthequantitiesdependonmolecularorientation. A
sical trajectory, and about the starting time. Near
rotationalaverageofP andP werecalculated,sincethe
0 1
the starting position and time, ψ (~r +δ~r ,t +δt ) =
t 0 0 0 0 JILAexperimentwasperformedonagasjetofmolecules
ψt(~r0,t0)exp[i(~k0δ~r0−iEHOMOδt0)],whilefE±,l(r+δr)= which had no preferred orientation. Only polarizations
f (r)exp[ k (r)δr], where k (r) = 2(E V (r)), perpendicular to the propagatingbeam were included in
E,l E,l E,l l
∓ −
near the final position. Angular derivatives of the spher- these averages. Although each of the T and E modes
p 2g g
ical harmonics are neglected. modulates the harmonic intensity strongly at particular
The integrandoscillatesmostslowlywhen its complex orientations,the phase offset δ in P cos(ωτ+δ) changes
1
phase is nearly constant. This happens when the linear with orientation, cancelling some of the observed oscil-
termsintheS expansionarecanceledbythelinearterms lation. The more symmetric A mode experiences less
g
in the complex phases of the initial and final states. For cancellation because the initial Raman pulse stimulates
suchatrajectory,thecontributionsofnearbytrajectories it equally for all molecular orientations. The trends in
with nearly the same ~x ,t ,~x,t to the integral will tend the peak intensities of different normal modes are, how-
0 0
to add constructively. These “stationary phase” trajec- ever, similar with and without the phase. Fig. 2 com-
toriesarenottheonlytrajectoriesofinterestintheprob- pares the spherically averaged peak-to-peak modulation
lem, but expanding S about their beginning and ending withphaseinformationincluded,withphaseinformation
4
e-
excluded by setting δ = 0 for all orientations, and the
experimentally measured modulations for each mode for
the tworunsfor whichallthreemode modulations could
be distinguished from the background [14]. It should be
a e-iwt M I N R d
1 11 0 11 0 1
noted that although the relative peak intensities are not a V=1 M01V=1 V=1 N01V=1 V=1
inagreementwithexperiment,the intensities forthe dif- 1 I1 R1
ferentpeaksareallthesameorderofmagnitude,whereas a0 M10 N10 d0
in the Raman spectrum the Ag mode is 20 times more V=0 a0 V=0 M00 V=0 I0 V=0 N00 V=0 R0 V=0
intense than the others. [1]
Since the scatteringwavefunctionis expandedinfield-
FIG. 1: (Color online) The vibrational interference model in
free scattering states, the calculated modulation varies
one dimension. The molecule ends the Raman pulse in a su-
slightly depending on the time at which the semiclassi-
perposition of the v=0 and v=1 vibrational states. After a
cal wavefunction is projected onto the scattering states. timedelay,thetwovibrationalstatesaremixedbystimulated
Additional uncertainty may arise because of the various Ramanscattering, “hopping”duringionization andrecombi-
approximationsmadeinthetreatmentofmolecularscat- nation,andevolutionoftheionicwavefunctionwhiletheelec-
tering, detailed in [3, 12, 13]. Fig. 2 shows modulations tronisaway. Interferencebetweenadjacentvibrationalstates
modulates the high harmonic signal.
calculated when this projection is made at ωt = 3.9 ra-
dians, when the short electron trajectories return to the
vicinityoftheparentmoleculewiththecorrectenergyto
yield 39th harmonic photons upon recombination. Experiment 1
Experiment 2
In the present work, we have combined for the first
T mode, 525 cm-1 Phase information included
timearigoroustreatmentofelectron-moleculescattering 2g Phase information excluded
withasemiclassicaldescriptionofelectronicpropagation, A1g mode, 775 cm-1
rheassulbtienegn iunseadflteoxtibreleatanadcormobpulestx immoplleecmuleentwaittihonmtahnayt Modulation 0.1
itlnahrtagetrenissaylustndepemgrer.ceWeedseeonhftaefvdreeeisndhootmwhinsthoaoreawalteohvfeerliensotefearrsncoahplhfdoisertgisrcueaectshioonaf Peak-to-Peak 0.05 Eg mode, 642 cm-1
freedom allowfor phenomena which have no analoguein
atomic systems,demonstrating that the HHG signalcan
be modulated interferometrically by a molecule’s vibra-
tionalstate. ThusHHGcouldservebothasaninterfero- 0
500 550 600 650 700 750 800
metricprobeofchemicaldynamics(similartothewayin Wavenumbers (cm-1)
which rearrangement dynamics are investigated in Ref.
[15]) and as a complementary tool to traditional spec-
troscopic techniques, offering great potential for future FIG. 2: (Color online) Peak-to-peak modulation of the high
harmonic signal vs. wavenumber, comparing theory to the
investigations.
two experimental runs for which data is available. Modula-
Acknowledgments: TheauthorswouldliketothankN.
tions corresponding to the same frequency have been placed
Wagner, A. Wu¨est, M. Murnane, and H. Kapteyn for
side-to-side for purpose of comparison.
many stimulating discussions. This work was supported
in part by the Department of Energy, Office of Science,
andinpartbythe NSFEUVEngineeringResearchCen- (1985).
ter. [7] W. Moffitt and W. Thorson, Phys. Rev. 108, 1251
(1957).
[8] I. Bersuker, The Jahn-Teller effect and vibronic interac-
tions in modern chemistry (Plenum Press, 1984).
[9] W. H. Miller, J. Phys.Chem. A 105, 2942 (2001).
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X. Zhou, M. M. Murnane, and H. C. Kapteyn, Proc. [11] M. C. Gutzwiller, Chaos in Classical and Quantum Me-
Natl. Acad.Sci. U.S.A.103, 13279 (2006). chanics (Springer, New York,1990).
[2] M. Lewenstein, P. Balcou, M.Y. Ivanov, A. L’Huillier, [12] S.TonzaniandC.H.Greene,J.Chem.Phys.124,054312
and P.B. Corkum, Phys.Rev.A 49, 2117 (1994). (2006).
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(2005). [14] N. Wagner, privatecommunication.
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[5] T. Dunning,J. Chem. Physp.1007 (1989). Smith, C. C. Chirila, M. Lein, J. W. G. Tisch, and J. P.
[6] S. Estreicher and T. L. Estle, Phys. Rev. B 31, 5616 Marangos, Science 312, 424 (2006).