Stimulating the production of deeply bound RbCs molecules with laser pulses: the role of spin-orbit coupling in forming ultracold molecules

0.71 MB·English

by  Subhas Ghosal

Stimulating the production of deeply bound RbCs molecules with laser pulses:
the role of spin-orbit coupling in forming ultracold molecules
Subhas Ghosal,1,∗ Richard J. Doyle,1 Christiane P. Koch,2,† and Jeremy M. Hutson1,‡
1Department of Chemistry, Durham University, South Road, DH1 3LE, United Kingdom
2Institut fu¨r Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany
(Dated: January 30, 2009)
We investigate the possibility of forming deeply bound ultracold RbCs molecules by a two-color
photoassociationexperiment. WecomparetheresultswiththoseforRb2 inordertounderstandthe
9 characteristic differences between heteronuclear andhomonuclear molecules. Themajor differences
0 arise from the different long-range potential for excited states. Ultracold 85Rb and 133Cs atoms
0 collidingontheX1Σ+ potentialcurveareinitiallyphotoassociated toform excitedRbCsmolecules
2 in the region below the Rb(5S) + Cs(6P ) asymptote. We explore the nature of the Ω = 0+
1/2
n levels in this region, which have mixed A1Σ+ and b3Π character. We then study the quantum
a dynamicsofRbCsbyatime-dependentwavepacket(TDWP)approach. Awavepacketisformedby
J excitingafewvibroniclevelsandisallowed topropagateonthecoupledelectronicpotentialenergy

0 curves. Wecalculate thetime-dependenceoftheoverlap between thewavepacket and ground-state
3 vibrational levels. For a detuning of 7.5 cm−1 from the atomic line, the wavepacket for RbCs
reachestheshort-rangeregioninabout13ps,whichissignificantlyfasterthanforthehomonuclear

h] Rb2 system; this is mostly because of the absence of an R−3 long-range tail in the excited-state
potential curves for heteronuclear systems. We give a simple semiclassical formula that relates the
p
- time taken to the long-range potential parameters. For RbCs, in contrast to Rb2, the excited-
m state wavepacket shows a substantial peak in singlet density near theinner turning point, and this
produces a significant probability of deexcitation to form ground-state molecules bound by up to
o
1500 cm−1. The short-range peak depends strongly on nonadiabatic coupling and is reduced if
t
a the strength of the spin-orbit coupling is increased. Our analysis of the role of spin-orbit coupling
. concerns the character of the mixed states in general and is important for both photoassociation
s
c and stimulated Raman deexcitation.
i
s
y
I. INTRODUCTION both magnetoassociation and photoassociation [4, 5, 6],
h
p and there have been considerableadvances in using such
[ The study of cold molecules, below 1 K, and ultracold methods to produce heteronuclear alkali metal dimers

molecules, below 1 mK, offers many new opportunities [7, 8, 9, 10, 11, 12, 13, 14]. However, both photo- and
2
magnetoassociation initially produce molecules in very
v forchemicalandmolecularphysics. Coldmoleculesopen
high vibrationallevels,which are only very weakly dipo-
3 up a new regime for molecular collisions in which clas-
0 sical physics breaks down completely and a quantal de- lar even for heteronuclear species. There is therefore
7 scription is needed for all degrees of freedom. Collisions great interest in producing ultracold molecules in low-
5 lying vibrational states. For example, Sage et al. [8, 9]
in this regime are dominated by long-range forces and
0. exhibit resonance phenomena that can be controlled by have succeeded in producing small numbers of ultracold
1 applied electric and magnetic fields. Cold molecules also RbCs molecules in the ground vibronic state by a four-
8 photon photoassociation scheme using continuous-wave
presentnewopportunitiesforprecisionmeasurementand
0 lasers, while Deiglmayr et al. [14] have produced LiCs
openuppossibilitiesformeasuringquantitiessuchasthe
v: dipole moment of the electron and the time-dependence molecules with a two-photon scheme. Both these ap-
i of fundamental “constants”. proaches include a spontaneous emission step, but Ni et
X
al.[15]haveveryrecentlyproducedKRbmoleculesinthe
There is particular interest in ultracold polar
r lowest vibrational levels of the lowest singlet and triplet
a molecules,becausedipolarspeciesinteractmorestrongly
electronicstatesbymagnetoassociationfollowedbyaco-
and at much longer range than non-polar species. Dipo-
herent two-photon process (stimulated Raman adiabatic
lar quantum gasesare predicted to exhibit new phenom-
passage, STIRAP). Danzl et al. [16] have carried out
ena such as anisotropic Bose-Einstein condensation [1]
analogousexperimentsonthehomonuclearmoleculeCs ,
and may have applications in quantum information pro- 2
while Lang et al. [17] have produced Rb molecules in
cessing[2]. Theyalsoprovideopportunitiesforengineer- 2
the lowest level of the lowest triplet state.
ing highly correlated quantum phases [3].
Molecules can be formed in ultracold atomic gases by
Heteronuclear molecules differ from homonuclear
molecules in several ways. The most important differ-
enceisthattheexcited-statepotentialcurvescorrelating
with 2S + 2P atoms have anR−6 behaviorat long range
∗Electronicaddress: E-mail: [email protected]
†Electronicaddress: E-mail: [email protected] in the heteronuclear case but an R−3 behavior in the
‡Electronicaddress: E-mail: [email protected] homonuclearcase,duetotheresonantdipoleinteraction.
2
12500 A1Σ+ RbCs (5s + 6p) Rb(5S) + Cs(6P1/2) asymptote forms an excited-state
molecule around internuclear distance R , as shown in
L
10000 v’ Detuning (δat) Fig. 1. When spin-orbit coupling is included there are 8
L
electronic states that correlate with the excited 5S +
-1m) 7500 6P and5S +6P asymptotes: two0+ states1,/2two
c ω ω 1/2 1/2 3/2
nergy ( 25500000 b3Π+ dump puLmp at T0−hessteactoesr,retlharteeeatΩsh=or1trsatantgeeswaintdhtohneeAΩ1Σ=+2,bst3aΠte,B[215Π].
E and c3Σ+ electronic states in Hund’s case (a) labelling.
v=54
0 Absorption of a photon from the ground X1Σ+ (0+)
R Rb(5s) + Cs (6s)
L state can produce excited-state molecules in 0+ or 1
-2500 X1Σ+ states, while absorption from the a3Σ+ state (with 0−
5 10 15 20 25 30 35 40 and 1 components) can produce 0+, 0−, 1 and 2 states.
Internuclear distance (a)
0 We focus here on the 0+ excited states, which may be
formed by combining the A1Σ+ and b3Π states. The
FIG. 1: The potential curves and laser pulse scheme consid-
c3Σ+ state does not contribute because it has only 0−
ered here. The potential curves correspond to the ground
X1Σ+ and the excited A1Σ+ and b3Π states. A picosecond and1componentsandtheB1Πstatedoesnotcontribute
because it has only an Ω = 1 component. Rotational
laser pulse is used to excite few vibroniclevels in theexcited
state to form a wavepacket. A second laser pulse is used couplingsthatwouldconnectstatesofdifferentΩarene-
to dump the wavepacket to the ground X1Σ+ state after an glected. This 2-state approximation is analogous to the
appropriate time delay. The vibrational wavefunction of the approachused for the 0+u states in Cs2 and Rb2 [26], for
target level of thedump pulse, v=54, is also displayed. which the strongly mixed singlet-triplet character of the
vibronic wavefunctions was found to produce enhanced
formation of ground-state molecules with binding ener-
Because of this, the Franck-Condon factors for photoas- gies on the order of 10 cm−1 [21, 27].
sociation are quite different. In addition, the density of Thepumppulseproducesanon-stationarystatemade
vibrationallevelsintheelectronicallyexcitedstateisdif- upoflong-rangeexcited-statelevelsofmixedsingletand
ferent for 1/R6 and 1/R3 potentials. triplet character. The corresponding wavepacket prop-
Magnetoassociationmust be carried out in tight traps agates towards short range under the influence of the
at very low temperatures. However, there would be ad- excited-state potentials. After a suitable time delay,
vantages in producing molecules in low-lying states at when a sufficient amount of the wavepacket has reached
the somewhat higher temperatures (in the microkelvin the short-range region, a dump pulse is activated to
regime) that are available in magnetooptical traps transferthemoleculesintovibrationallevelsoftheX1Σ+
(MOTs), where the number of atoms available is of- electronic ground state.
ten much larger. A very promising approach for this is We simulate the entire pump-dump process using a
photoassociation using shaped laser pulses. Short-lived time-dependent wavepacket (TDWP) approach. Our
molecules are formed in the excited electronic state by goalistooptimizetheparametersofthepumpanddump
photoassociation using a short laser pulse (pump pulse) pulsestomaximizetheproductionofmoleculesindeeply-
during the collision of two ultracold atoms. The excited bound levels of the ground state. We find that a crucial
molecules are then stabilized by stimulated emission us- factor is the magnitude of the spin-orbit coupling near
ing a second laser pulse (dump pulse) into the bound the avoided crossing. If the spin-orbit coupling is suffi-
vibrational levels of the ground electronic state. Luc- ciently strong, then the levels that lie on the lower adi-
Koenig et al. [18, 19] and Koch et al. [20, 21] have sim- abatic curve, which correlates with 5S + 6P at long
1/2
ulated this process for homonuclear diatomic molecules range,havemainlytripletcharacteratshortrange. Such
such as Cs and Rb , and initial experiments on these levels have low intensities for deexcitation. We therefore
2 2
systems have been reported by Salzmann et al. [22, 23] form the excited-state wavepacket by selective photoas-
and Brown et al. [24]. Similar experiments can be envi- sociationintovibroniclevelsoftheA1Σ+-b3Πelectron-
sioned for heteronuclear molecules, with the use of evo- ically excited states that are strongly mixed by nonadi-
lutionary algorithms or other strategies to maximize the abatic coupling and have enhanced singlet character at
rateofproductionofground-statemolecules. However,a shortrange. Weestimatetheparametersrequiredforthe
theoretical study of short-pulse photoassociation of het- dump pulse and the time delay by analyzing the time-
eronuclear molecules has not yet been carried out. In dependence of the overlap between the singlet part of
this paper, we consider RbCs as a prototype heteronu- the wavepacket and the vibrational levels of the ground
clear molecule and we study the possibility of forming electronic state.
ground-state RbCs molecules through pump-dump pho- We find an important difference between photoassoci-
toassociation. ationfor heteronuclearandhomonuclearmolecules. The
We consider the photoassociationof a colliding pair of excited states of heteronuclear molecules have a lower
85Rb and 133Cs atoms, initially in their ground state. density of vibrational levels near dissociation than those
Absorption of a photon red-detuned from the atomic of homonuclear molecules, because of the long-range
3
R−3 term in the excited-state potentials for homonu- 1000
clearspecies. Becauseofthis,theatomsofheteronuclear
mctinhaoganlnteelcynhueolfreamgssyto.penrrTuocchdlleeuaasctsreiidmcmaealotldaeeccllocuaenlyelgesrrreacaqtrnieuoaginrteeetddeoxwbapeatetrrwtdiheesneeocnsenatemhaaeesnipbogutinnhmidfiepr-- -1y (cm) 680000 -1Energy (cm)----6543000000000000 dadiaibabataitcic
anddumppulsesisthussmallerintheheteronuclearcase nerg 400 WSΣOΠ 6 Inte8rnuclea1r0 distac1e 2(a) 14
(about 13ps for RbCs with a detuning of7.5 cm−1 from E WΠΠ 0
SO
the atomic line)thaninthe homonuclearcase. However,
200
even for heteronuclear molecules, there is little probabil-
ity ofproducingground-statemoleculeswithinthe dura-
0
tion of a single pulse of less than a few ps. 5 10 15 20 25
Internuclear distance (a)
Thepaperisorganizedasfollows. InsectionII,wedis- 0
cuss the potential energy curves and spin-orbit coupling FIG. 2: The diagonal and off-diagonal components WΠΠ(R)
SO
functions used in this study. In sections III and IV we and WΣΠ(R) of the spin-orbit coupling as a function of R.
SO
explorethedependenceofthezero-energys-wavescatter- The avoided crossing between the two adiabatic potentials is
inglengthsontheinteractionpotentialsanddiscusstheir shown in the inset.
influenceontheFranck-Condonfactorsforphotoassocia-
tion. In section V we describe wavepacketstudies of the
photoassociation process and the subsequent formation for the A and b states respectively.
of ground-state molecules. At each stage, the results for Spin-orbitcouplingbetweentheA1Σ+ andb3Πstates
RbCsarecomparedwiththoseforRb inordertoestab- plays an important role. Following Marinescu and Dal-
2
lishthesimilaritiesanddifferencesbetweenheteronuclear garno[34]andAubert-Fr´econet al. [35],theeffectofthe
and homonuclear molecules. spin-orbitcoupling for the 0+ states is included in terms
of a 2 2 electronic Hamiltonian matrix,
×
II. POTENTIAL CURVES Hel = VA1Σ+(R) √2WSΣOΠ(R) ,
(cid:18)√2WSΣOΠ(R) Vb3Π(R)−WSΠOΠ(R)(cid:19)
SeveralstudiesofthepotentialenergycurvesforRbCs (1)
have been published. Obtaining accurate potentials is
not a straightforward task, because electronic structure where V (R) and V (R) are the Born-
A1Σ+ b3Π
calculations are very difficult for such heavy atoms. Ex- Oppenheimer potentials neglecting spin-orbit coupling.
perimental results are available for some states [28, 29], The diagonal and off-diagonal spin-orbit coupling ele-
but are sparse or nonexistent for others. Only some of ments are denoted WΠΠ(R) and WΣΠ(R) respectively
SO SO
the theoretical potentials include spin-orbit coupling ef- and are functions of the internuclear separation R as
fects; for example the relativistic curves of Kotochigova shown in Fig. 2. At long range,
andTiesinga[25]includeavoidedcrossingswhichareab-
sent in the older pseudopotential results of Allouche et ∆ECs
WΠΠ(R )=WΣΠ(R ) = SO,
al. [30]. SO →∞ SO →∞ 3
The potential curves for the electronic states of
RbCsconsideredhere,neglectingspin-orbitcoupling,are where ∆ECs is the spin-orbit splitting of the P state of
SO
showninFig.1. Fromtheviewpointofphotoassociation, atomic Cs. Bergeman et al. [36] give a non-zero matrix
themostimportantpartsofthecurvesarethelong-range element connecting the Ω = 0 components of the b3Π
tails. In this study we have chosen for simplicity to use and c3Σ+ states but this is appropriate only for the 0−
the ab initio results ofAllouche et al. [30] at shortrange componentanddoes notinfluence the 0+ states. We use
(bondlengthsR<15a ). Forthe long-rangepotentials, thespin-orbitcouplingfunctionsobtainedbyFellowsand
0
in the ground state, we use the experimentally derived Bergeman [37], which were obtained by fitting to results
parametersofFellowset al.[28],withC replacedbythe from Fourier-transform spectroscopy [29]. The diagonal
6
more recent value of Derevianko et al. [31]. These pa- and off-diagonal spin-orbit couplings show a significant
rameters are equivalent to column V of Table 2 in Ref. dip from their asymptotic values near the crossing be-
32. The Le Roy radius for the ground X 1Σ+ state is tween the A1Σ+ and b3Π potential energy curves. The
24.4a ,butasmoothmatchbetweenthelong-rangeand two adiabatic potentials obtained by diagonalizing the
0
short-range potentials was best achieved at 16.5 a . For electronic Hamiltonian of Eq. (1) are shown as an insert
0
theexcitedsingletandtripletstates,weusethetheoreti- in Fig. 2.
cal long-rangeparameters of Marinescu and Sadeghpour For Rb the potentials are more accurately known. In
2
[33], corresponding to the Rb(5S) + Cs(6P) asymptote. the present work, the ground-state X1Σ+ potential is
g
For the excited states the matching between long-range obtained by combining the short-range results of Seto et
andshort-rangecurveswasachievedat16.5a and24a al. [38] with the long-range coefficients of Marte et al.
0 0
4

400 70
(a)0 a)0 60
gth 200 R> ( 50
n
e < 40
g l 0
n 30
ri
e
att -200 20
c
s
0.05
-400
0.04
0.99 1.00 1.01 1.02
F
scaling factor C0.03
F

FIG.3: RbCss-wavescatteringlengthfortheX1Σ+ state,as 0.02
afunction of ascaling factor used tomultiplythepotential.
0.01
0.00
[39]. The excited-state potentials are taken from Berge-
man et al. [40], who obtained potential curves and spin- 0 20 40 60 80 100
orbit coupling functions for the A1Σ+ and b3Π states -1
u u Binding Energy (cm )
by combining photoassociation spectroscopy with short-
rangeab initioresults[41]. Thecorrespondingcurvesfor
Rb (not shown here) are qualitatively similar to those FIG.4: Upperpanel: AveragebondlengthhRiforvibrational
2 levelsofRbCsinitsA1Σ+ stateasafunctionofthebinding
for RbCs, except for the important difference in long-
energy. Lowerpanel: Franck-Condonfactors forphotoassoci-
range behavior.
ation (in arbitrary units), calculated for a ground-state scat-
tering length a = 66.3 a0. Note that the largest factors are
off thetop of thescale in thisfigure.
III. SCATTERING LENGTH
Rb shows similar behavior.
A crucial parameter affecting low-energy scattering 2
propertiesandthepositionsofhigh-lyingexcitedstatesis
thes-wavescatteringlengtha. CalculationsbyJamieson
et al. [32]producedvaluesfora rangingfrom380to 1a IV. FRANCK-CONDON FACTORS AND THEIR
0
fortheX1Σ+ stateofRbCs,dependingonthesubtleties DEPENDENCE ON SCATTERING LENGTHS
ofthepotentialparameters. Therearefurtheruncertain-
ties due to the choice of hyperfine states involved. In WefirstcalculatetheFranck-Condonfactors(FCFs)of
addition, evenif the potential is knownaccurately,there relevanceto photoassociationexperiments for85Rb133Cs
isthe possibility ofcontrolling the scatteringlengthwith and compare the results with those for the homonuclear
applied fields. We therefore investigate the dependence 85Rb system. The purpose of this Section is to un-
2
ofphotoassociationonthescatteringlengthsforboththe derstand the qualitative differences between RbCs and
ground and excited states. Rb , which stem from the different long-range tails of
2
Scattering lengths were calculated by solving the theexcited-statepotential(proportionaltoR−6forRbCs
Schr¨odinger equation for zero-energy s-wave scattering andR−3 forRb ). Thesequalitativedifferencesaremost
2
numerically using Numerov integration. The long-range simplyillustratedusingcalculationsthatneglectthecou-
wavefunctionwasmatchedtoBesselandNeumannfunc- plingbetweenthe A1Σ+ andb3Πstates,andthatisthe
tions of fractionalorder [42] to take account of the long- approach we use in this Section. However, coupled cal-
range R−n potential and avoid the need to propagate to culations of FCFs have been carried out by Luc-Koenig
excessively large distances. We adjusted the scattering [43],andindeedthecalculationsdescribedinSectionVC
length by scaling the whole potential curve by a con- to VE below use fully coupled wavefunctions.
stant factor. As is well known, the scattering length is The FCFs for photoassociation are calculated as the
extremely sensitive to such scaling and passes through a squares of overlap integrals between bound vibrational
pole whenever there is a bound state at exactly zero en- states in the excited A1Σ+ electronic potential and a
ergy. For 85Rb133Cs a complete cycle is achieved within low-energy scattering wavefunction on the ground-state
a scaling of just over 1%, as can been seen from Fig. 3. potential. Both the bound and scattering states were
5

0.005 250
a = 135.8 a0
0.004 a = 66.3 a0 a)0200
0FCF .003 aa == 3-638.5.8 a a00 <R> (150
100
0.002
50
0.001
0
0.000
0 100 200 300 400 500
-1 0.4
Binding Energy (cm )
F
C
F

0.035 0.2
0.030 a = 135.8 a0
a = 66.3 a0
0.025
a = 33.5 a0 0.0
F
0FC .020 a = -68.8 a0 0 20 40 60 80 100
0.015 -1
Binding Energy (cm )
0.010
0.005
FIG.6: Upperpanel: AveragebondlengthhRiforvibrational
0.000 levels of Rb2 in its A 1Σ+u state as a function of the binding
energy. Lowerpanel: Franck-Condonfactors forphotoassoci-
0 10 20 -1 30 ation. The scale is arbitrary but is consistent with that for
Binding Energy (cm ) RbCs in Figs. 4 and 5. Note that the largest factors are off
thetop of thescale in this figure.

14
12 a = 135.8 a0 that the relative values of the resulting Franck-Condon
a = 66.3 a0 factorsshouldbevalidforexcitationtostatesdominated
10
a = 33.5 a0 by distances R<100 a0.
F
FC 8 a = -68.8 a0 Figure 4 shows the calculated FCFs for RbCs with
a = +66.3 a as a function of binding energy (detuning
6 0
from the Cs atomic line), and the corresponding values
4 of R forthe excitedvibrationalstates. TheFCFsshow
h i
2 atypicaloscillatingpatternastheboundandcontinuum
states come into and out of phase with one another. For
0
RbCs each peak of the envelope encompasses around 4
0.00 0.05 0.10 0.15 0.20 0.25 vibrational levels of the excited state. The most intense
-1
Binding Energy (cm ) peak is just below the dissociation threshold, and the
state at the center of this peak has R 71 a . The
0
h i ≈
FIG. 5: Franck-Condon factors for RbCs photoassociation, nextpeaksinorderofdecreasingintensity correspondto
calculated for a variety of ground-state scattering lengths. R 47, 37, 32, 29 and 26 a0. The largest FCF occurs
h i≈
The scale is arbitrary but is consistent between graphs. for the least-bound state, with R = 150 a0, which is
h i
not partof an envelope. However,this state is bound by
only 0.0007 cm−1, and it is likely that frequencies this
obtained by Cooley-Numerov propagation [44] for rota- close to the atomic line would be blocked in an actual
tional angular momentum N = 0. A LeRoy-Bernstein experiment.
analysis [45] confirmed that every bound state had been To investigate the effect of adjusting the ground-state
found. The s-wave scattering function was calculated at scattering length, we have repeated the Franck-Condon
a collision energy of 1 mK and normalized by setting its calculations for a = 135.8, 33.5, and 68.8 a , corre-
0
−
maximum amplitude to 1 as R . The ground-state sponding to potential scaling factors of 0.9965, 1.0015
→ ∞
potential does not die off to 1 mK until R 110 a , so and 1.0039 respectively. The results are shown in Fig.
0
∼
6
5. It may be seen that changing the scattering length
shifts the positions of the peaks in the envelope of the
intensity distribution. Nevertheless, the peak intensi- 1
ties themselves follow a curve that is only very weakly 0.1
potential-dependent. The peak intensities are thus al-
0.01
most a single-valued function of binding energy.
1E-3
Scaling the potential curve for the excited state, while
F
keeping the ground-state scattering length constant, C1E-4
F
shifts the positions of the individual lines but does not 1E-5
alter the positions of the peaks in the intensity distri-
1E-6
bution. This effect is shown for a = +66.3 a in Fig. 8:
0
1E-7
thefrequenciesthatcorrespondtovibrationallevelsshift
as the potential is scaled, but the envelope of the FCFs 1E-8
does not change. Since a laser pulse will always cover a
10 100 1000
rangeofbindingenergies,itisthisenvelopethatmatters -1
Binding Energy (cm )
morethanthespecificpositionofthelevelsandtheover-
alltransitionprobabilitieswillnot be greatlyaffected by
changes in the excited-state scattering length.
The Franck-Condon factors for photoassociation to 1
form Rb2 are shown in Fig. 6. The potential curve used 0.1
here has a = 157.9 a . Once again the FCFs show an
0 0.01
oscillatory structure. The major difference from RbCs
1E-3
is that the oscillations in the envelope of line intensities F
are slower as a function of binding energy and the lines C1E-4
F
themselvesaremuchmoredenselypackedbecauseofthe 1E-5
greater density of states for an R−3 potential. The over-
1E-6
all values of the FCFs are also somewhat larger in the
homonuclear case. 1E-7
Another significant difference between the heteronu- 1E-8
clear and homonuclear cases is shown in Fig. 7. On a
10 100 1000
log-log plot, the intensities of the peak envelopes vary -1
Binding Energy (cm )
almost linearly with binding energy E E near disso-
D v
−
ciation. This corresponds to a power-law dependence on
FIG. 7: Franck-Condon factors for different values of the
bindingenergy. However,thepowersinvolvedareclearly
different for the two cases: about (E E )−1 for the ground-state scattering length, shown on a log-log plot for
homonuclearcaseand(E E )−1.6forDt−hehveteronuclear RbCs (upper panel) and Rb2 (lower panel). Note that the
D− v peak values lie on a straight line in each case, but with a
case. This will be significant in designing pulsed-laser
differentslope,correspondingtoadifferentpower-lawdepen-
experiments, because the intensity will fall off consider-
denceon bindingenergy.
ably faster with detuning (from the atomic line) in the
homonuclearcasethanintheheteronuclearcase. Thusa
broadpulsewillexciteawavepacketinwhichtherelative whereν isthe classicalvibrationalfrequency. Thevibra-
population of deeper levels is lower in the heteronuclear tional spacing dE /dv may be calculated exactly for a
v
case. particularpotentialcurvebutitisinstructivetoconsider
itsvalueinlong-rangetheory. Ifonlytheleadingtermin
the long-range potential is included, V(R) = C R−n,
n
−
thevibrationalperiodmaybewrittensemiclassically[46]
V. TIME-DEPENDENT CALCULATIONS
dE
v =K (E E )(n+2)/2n, (3)
A. Vibrational periods n D v
dv −
where
In section VB we study the photoassociation process
in a time-dependent wavepacket approach. However, we K (2π)1/2h nΓ(1+1/n)
n
K = = , (4)
can make a rough estimate of the time needed for the n µ1/2C1/n µ1/2C1/n (cid:18)Γ(1/2+1/n)(cid:19)
wavepackettoevolvefromlongtoshortrangebyconsid- n n
ering the vibrational half-period, µ is the reduced mass and Γ is the gamma function. Le
RoyhastabulatednumericalvaluesofthecoefficientsK
n
1 h [46]. The times obtained for wavepackets corresponding
t = = , (2)
1/2 2ν 2 dEv to each of the main peaks in Fig. 4 are given in Table I
dv
(cid:0) (cid:1)
7
at small detunings to under 1 ps for a detuning of 315
cm−1. Although the positions of the individual ma−xima
0.040 are sensitive to details of the potential, the time taken
is analmostsingle-valuedfunction of the binding energy
0.035
(laserdetuning)foraparticularlong-rangepotentialand
0.030
reduced mass.
0.025 ThetimestakenforRbCsaresignificantlyshorterthan
F
C those for Rb , shown in Table II. For example, for a
F 0.020 d etuning of 225 cm−1 from the atomic line, the classical
0.015
time is about 13 ps for Rb but 5 ps for RbCs.
2
0.010
TABLE II: Vibrational half-periods for an Rb2 molecule at
0.005 energies corresponding to maxima in the intensity distribu-
0.000 tion for photoassociation. The index m labels the successive
maxima, starting at thedissociation limit.
0 30 60 90
-1 m binding energy hRi relative t
Binding Energy (cm ) 1/2
(cm−1) (a0) intensity (ps)
1 −0.017 500 12 6300
FIG.8: Franck-Condonfactorsfordifferentscalingsoftheup-
2 −0.038 380 8.5 3200
perpotentialwiththeunscaledground-statepotential. Black,
blue and red symbols are for scalings of 1.0, 1.001 and 1.005 3 −0.10 270 5.2 1400
respectively. 4 −0.49 160 2.4 380
5 −5.1 75 0.36 54
6 −27 43 0.043 13
TABLE I: Vibrational half-periods for an RbCs molecule at
7 −56 34 0.014 7.3
energies corresponding to maxima in the intensity distribu-
tion for photoassociation. The index m labels the successive 8 −91 29 0.007 4.9
maxima, starting at thedissociation limit. 9 −135 26 0.004 3.5
m binding energy hRi relative t
1/2
(cm−1) (a0) intensity (ps) Theclassicalmodelusedhereisofcourseconsiderably
1 −0.0063 71 ∼1.5 290
oversimplifiedandneglectsfeaturessuchasthespreading
2 −0.76 47 0.13 54 of a wavepacket as it propagates inwards. A full treat-
3 −2.9 37 0.038 22 mentofthe time evolutionrequiresquantum-mechanical
4 −7.3 32 0.016 12 calculations as described below.
5 −15 29 0.0080 7.5
6 −26 26 0.0045 5.2
B. Wavepacket model and methods
7 −41 24 0.0027 3.8
8 −57 23 0.0018 3.0
In the time-dependent wavepacket approach, the pho-
9 −82 20 0.0013 2.4
toassociation reaction starts with two cold atoms in
10 −113 20 0.00096 1.93
the ground electronic state with relative kinetic energy
11 −150 19 0.00073 1.56
E . Absorption of a photon red-detuned by an energy
kin
12 −194 18 0.00056 1.26 ~ω from the atomic resonance line at ~ω produces a
L at
13 −245 17 0.00041 1.13 molecule in the excited electronic state as shown in Fig.
14 −315 17 0.00028 0.97 1. The Hamiltonian describing optical transitions be-
tweentheX1Σ+ electronicgroundstateandthecoupled
A1Σ+ and b3Π excited states can be represented in the
for 85Rb133Cs. The half-periods vary from over 100 ps diabatic basis as
Tˆ +V (R) µ(R) E(t) 0
Hˆ = µ(R)X1ΣE+(t) Tˆ +V ·(R) ~ω √2WΣΠ(R)  , (5)
 0· √A21WΣ+SΣOΠ(R−) L Tˆ +Vb3Π(R)−SWOSΠOΠ(R)−~ωL 
whereTˆ isthe kinetic energyoperatorandV (R)arethe androtating-waveapproximations,thecouplingbetween
i
respective potential energy curves. Assuming the dipole
8
the X1Σ+ and A1Σ+ electronic states is represented by eigenfunctionsoftheadiabaticstates,definedbythenon-
thescalarproductbetweenthetransitiondipolemoment, crossing upper and lower adiabatic curves. In this case
µ(R),andthe polarizationvectorofthelaserfield,E(t). the adiabatic states themselves change character from
In the present work the R-dependence of µ(R) is ne- onesideofthecrossingtotheother,sothatinthepresent
glected and it is represented by its asymptotic value. casethezeroth-orderstatesarepredominantlysingleton
The A1Σ+ and b3Π excited states are coupled by the onesideofthecrossingandpredominantlytripletonthe
spin-orbit interaction. The hyperfine interaction can be other. The couplings between the adiabatic states are
neglectedbecausetheassociatedtimescaleismuchlonger providedbynonadiabaticcouplings(whicharerelatedto
than the femtosecond or picosecond laser pulses, so that off-diagonalmatrix elements of d/dR and d2/dR2).
thehyperfineinteractionisnotresolvedfortheprocesses The wavefunction for a mixed vibronic state may be
considered in the present study. written
The Hamiltonian, Eq. (5), is represented on a Fourier
ϕv(R)ψ +ϕv(R)ψ (7)
gridwithavariablegridstep[26]. Detailsofthemapped 1 1 3 3
Fourier Grid Hamiltonian (FGH) method are described
where the functions ϕv(R) and ϕv(R) are vibrational
in refs. 47 and 48. Using the FGH method, we are able 1 3
wavefunctionsandψ andψ areelectronicfunctions for
to extend the spatial grid to R 103 a using only 1023 1 3
∼ 0 the singlet and triplet excited states respectively. It is
gridpoints; this is sufficientboth to representthe bound
useful to compare the eigenstates that are obtained for
vibroniclevelsandto approximatethe scatteringcontin-
twodifferentcases,asshowninFig.9. Thetwopanelson
uum in the ultracold temperature regime by box states
the left show RbCs eigenstates calculated with the spin-
[49].
orbit coupling functions fixed at their asymptotic value,
To describe the short-pulse photoassociation pro-
∆ECs = 554.04 cm−1 (corresponding to WΣΠ = 184.68
cess, we solve the time-dependent Schr¨odinger equation SO SO
cm−1, which is considerably larger than the experimen-
(TDSE),
tally derived value [37] around the crossing point of the
diabatic curves (WΣΠ 100 cm−1, see Fig. 2). These
i~∂ Ψ(t)=Hˆ(t)Ψ(t), (6) willbe referredto aSsOAS≈O (asymptoticspin-orbit)calcu-
∂t
lations. The two levels shown are for vibrational indices
by expanding the evolution operator exp[ iHˆt/~] in v′ =359,with binding energy 22.19cm−1, andv′ =365,
− with binding energy 10.01 cm−1. It may be seen that
terms of Chebychev polynomials [50]. Because of the
for both ASO wavefunctions there is a switchover from
largeextentofthe gridandtheverysmallkinetic energy
singlet to triplet character around the avoided crossing
ofthe system,itis notnecessaryto enforceanabsorbing
(R 10 a ), but in opposite directions. This occurs be-
boundary condition at the edge of the grid. ∼ 0
cause the states are nearly adiabatic: the function for
v′ =365liesonthe upper adiabaticcurve,sohasmainly
singlet character inside the crossing and mainly triplet
C. Analysis of the coupled vibronic wavefunctions character outside. The function for v′ = 359 lies on
the lower adiabatic curve, so that it has mostly triplet
Diagonalization of the Hamiltonian of Eq. (5) with character at short range. Outside the crossing, it has
E(t) set to zero yields the binding energies and vibronic mainly singlet character in the region where the singlet
wavefunctions and allows the calculation of FCFs, rota- and triplet curves are separated by more than the spin-
tional constants, etc. For RbCs it produces 136 bound orbit coupling. At long range (R > 20 a ), however,
0
vibronic levels for the X1Σ1 ground electronic state and it reacquires the triplet admixture c∼haracteristic of an
398 levels for the coupled A1Σ+ and b3Π electronic Ω = 0 state at the 5S + 6P threshold, yielding a 2:1
1/2
states. Theeigenfunctionsoftheexcitedelectronicstates ratio of triplet to singlet probability. It stretches to con-
areperturbedbythe spin-orbitcouplingandhavemixed siderablygreaterinternucleardistancebecauseofthedif-
singlet and triplet character. The highest excited-state ferent turning points for the two adiabatic states.
level has a binding energy of 0.002cm−1, which differs The near-adiabatic vibronic levels obtained for the
slightly from the result in Section IV because the latter ASOcaseprovideacontexttounderstandtheeigenstates
neglected the coupling between the singlet and triplet obtained with the R-dependent spin-orbit (RSO) cou-
states. plingfunction,showninthepanelsontherightofFig.9.
When two electronic states interact and their curves In the RSO case the levels with index numbers v′ =359
cross,therearetwoquitedifferentlimitingcasesthatmay and365havebindingenergies17.79cm−1and7.53cm−1,
beconsideredtobe“uncoupled”. Ifthecouplingnearthe respectively. Thev′ =359levelexistsmostlyonthelower
crossingpointis veryweak(comparedtothe localvibra- adiabatic curve, and has triplet character with only a
tional spacings), the crossing is only weakly avoidedand smallsingletadmixture atshortrangeand switches over
theeigenstatesofthecoupledsystemareclosetothoseof to singlet character (with the same characteristic triplet
the individual diabatic (crossing)electronic states. Con- admixture as before) outside R = 10 a . Nevertheless,
0
versely, if the coupling is strong, the crossing is strongly thereissignificantnonadiabaticmixingwhichintroduces
avoided and the best zeroth-order picture is to consider upper-state character, shown by the peaks in probabil-
9
0.7 0.2
b3Π
0.6 ASO RSO A1Σ+
b3Π
0.5 A1Σ+ 0.15
2 2
)| )|
R 0.4 R
( (
5 5 0.1
6 6
=30.3 =3
v v
ϕ ϕ
| |
0.2
0.05
0.1
0 0
0.25 b3Π
ASO b3Π 0.20 RSO A1Σ+
A1Σ+
0.20
2 2
)| )| 0.15
R R
( (
90.15 9
5 5
3 3
v= v=0.10
ϕ ϕ
0.10
| |
0.05
0.05
0.00 0.00
10 20 30 40 10 20 30 40
R (a ) R (a )
0 0
FIG. 9: Eigenfunctions of the coupled excited-state potentials corresponding to v = 359 and 365. Left-hand panels: results
obtainedwithconstantasymptoticspin-orbitcoupling(ASO).Right-handpanels: resultsobtainedwithR-dependentspin-orbit
coupling (RSO).
ity densities aroundthe outer turning pointof the upper In an alternative approach, the nonadiabatic mixings
curvenearR 17a . Thev′ =365levelismorestrongly could be shown in an adiabatic representation, display-
0
∼
mixed, with a larger contribution from the upper adia- ing the components of the vibrational wavefunctions on
batic state that makes it predominantly singlet at short thetwo0+ states. Thiswouldemphasizethenearlyadia-
u
range. However, this state too has substantial lower- baticcharacteroftheASOwavefunctionsbutwouldhide
state character so that it does have density out to the theirswitchoverfromsinglettotripletcharacternearthe
outermost turning point (R 33 a in this case). It is avoided crossing.
0
∼
notable that both RSO eigenstates have significantly en-
The character of the mixed levels, as a function of vi-
hancedsingletdensityneartheouterturningpointofthe
brational quantum number, may be seen very clearly in
upper curve(R 17a ). This is the featurethat inRb
0 2 the calculated rotational constants for the coupled lev-
∼
wasresponsibleforenhanceddeexcitationtolevelsofthe els (B = h/(8π2cµR2) ), which are shown in Fig. 10
groundstate bound by upto 10 cm−1 [21, 27]. However, v h i
for both the ASO and RSO cases. In the ASO case, the
thewavefunctionforv′ =365alsohassubstantialsinglet
rotationalconstantsformtwoalmostindependentseries,
density near the inner turning point of the singlet state;
with the higher values corresponding to levels that are
this is a feature that was not observed for Rb (see Fig.
2 predominantly on the upper adiabatic curve. The up-
2 of ref. 21), and as will be seen below it allows deexci-
per state has a larger vibrational spacing than the lower
tation to even more deeply bound levels. The fact that
state. IntheRSOcase,thepeaksinrotationalconstants
thewavefunctioncanhavesignificantsingletdensityboth correspond to levels with a significant contribution from
at this turning point and at the outermostturning point
the upper adiabatic state. However, it may be seen that
arisesfromnonadiabaticcouplingbetweentheupperand
inthiscasethenonadiabaticcouplingisstrongenoughto
lower adiabatic states.
spread the character of each upper-curve vibronic state
10
0.03
RbCs R-dependent SO TABLE III: The parameters for the pump and dump pulse:
Asymptotic SO detuning δat from the atomic line, intensity I , temporal
L L
widthτ ,spectralbandwidthδ ,andintegratedpulseenergy
L ω
perarea.
0.02 v=365
-1m) pulse δLat IL τL δω/(2πc) Energy/Area
c (cm−1) (W cm−2) (ps) (cm−1) (J m−2)
B (v pump (ASO) −10.01 16.86 5.00 2.94 0.095
0.01
pump (RSO) −7.53 16.86 5.00 2.94 0.095
v=365
dump (RSO) 1427.61 98.32 1.00 14.72 0.474
v=359
1.5×10-5
0
0 10 20 30 40 50 60 70
-1
Binding Energy (cm ) Dump
s)
FspIoGn.d1in0g: tRootthaetiovnibarlocnoicnslteavnetlss oBfv′th=e chhou/p(8leπd2cAµR1Σ2)+i+cobr3reΠ- c unit 1.0×10-5
excited electronic states for RbCs. mi
o
at
E(t) ( 5.0×10-6
0.01 Pump
Rb Delay
2
0.008
0.0
0 5 10 15 20 25 30 35
Time (ps)
-1m) 0.006
c FIG. 12: Schematic diagram of the laser pulse sequence con-
B (v0.004 sideredinthepresentstudy. Thepumpanddumppulsesare
centered at 12.0 ps and 25.8 ps, respectively, with a time de-
layof13.8ps(notethat1×10−5 atomicunitsofelectricfield
0.002 correspond to 5.14 MV/m).
0
0 10 20 30 40 50
D. Photoassociation with a short laser pulse
Binding Energy (cm-1)
FIG. 11: Rotational constants B′ = hh/(8π2cµR2)i corre- The pump-dump scheme considered here is shown
v
sponding to the vibronic levels of the coupled A1Σ+ +b3Π schematically in Fig. 12. Each laser pulse is assumed
excited electronic states of Rb2. to be transform-limited,
E(t)=E f(t)cos(ω t), (8)
0 L
withaGaussianprofilef(t)=exp α(t tc)2. Thepulse
across several eigenfunctions of the coupled problem, so − −
has central frequency ω /(2π) and maximum field am-
L
that the peaks are not very large. There are no eigen-
plitude E . The full width at half maximum (FWHM)
0
states of the coupled problem that are predominantly on of the intensity profile E2f(t)2 is τ = (2ln2/α)−1/2.
the upper adiabatic curve. 0 L
ThecorrespondingFWHMfrequencywidthisδ /(2π)=
ω
2ln2/(πτ ). The Gaussian envelopes are centered at
The rotational constants for RbCs may be compared L
times t = t and t for the pump and dump pulses re-
with those for Rb [26, 40], shown in Fig. 11. Rb ex- p d
2 2
spectively.
hibits nonadiabatic mixing that is intermediate between
The parameters used for the laser pulses in this study
theASOandRSOcasesforRbCs. ForRb theextentof
2
are given in Table III. We choose the parameters of the
the mixing hasbeenshownto be isotope-dependent[51].
pump pulse to excite a few excited-state vibronic levels
The existence of bound states with singlet character v′ close to the maxima in Fig. 10 where nonadiabatic
both at long range and near the inner turning point will coupling is strongest. The central frequency is chosento
alsobeimportantindesigningfixed-frequencyschemesto beresonantwiththelevelv′ =365,withalaserdetuning
populate deeply-bound ground-state levels by STIRAP from the atomic line δat = 7.53 cm−1 in the RSO case
L
andrelatedmethods. Suchschemeswillbemostefficient and 10.01 cm−1 in the ASO case. The temporal width
when they proceed via mixed levels with population on of the pulse is chosen to give an energy spread that will
both the upper and lower adiabatic curves. excite5to10levelsnearv′ =365. Theintensityischosen