Inelastic collisions of ultra-cold heteronuclear molecules in an optical trap

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by  Eric R. Hudson

Inelastic collisions of ultra-cold heteronuclear molecules in an optical trap
Eric R. Hudson,∗ Nathan B. Gilfoy, S. Kotochigova,† Jeremy M. Sage,‡ and D. DeMille
Department of Physics, Yale University, 217 Prospect Street, New Haven, CT 06511, USA
Ultra-coldRbCsmoleculesinhigh-lyingvibrationallevelsofthea3Σ+ groundelectronicstateare
confined in an optical trap. Inelastic collision rates of these molecules with both Rb and Cs atoms
are determined for individual vibrational levels, across an order of magnitude of binding energies.
A simple model for the collision process is shown to accurately reproduce the observed scattering
rates.
8
0
The electric dipole-dipole interaction provides a long-
0
2 range, tunable anisotropic interaction between polar
molecules. This is fundamentally different from most in-
n
teractions studied between ultra-cold atoms, which are
a
J typically isotropic and comparatively short-ranged. Fea-
1 tures of the dipole-dipole interaction can lead to many
3 novelandexcitingphenomena,suchasfield-linkedstates
[1], long-range topological order [2], quantum chemistry

] [3, 4], and the possibility for quantum computation
h
[5, 6]. Furthermore, the presence of closely spaced in-
p
- ternal levels of the molecules, e.g. Ω-doublet, rotational,
m and vibrational levels, presents a host of new possibil-
o ities for precision measurement of fundamental physics
t [7, 8, 9, 10, 11]. Producing ultra-cold samples of polar
a
. molecules will facilitate trapping and, thus, the required
s
high densities and long observation times for observing
c
i these phenomena.
s FIG.1: (Color Online)Schematic of theexperiment showing
y Techniques such as Stark deceleration [12] and buffer the overlap of the DSM with the relevant beams and rela-
h gas cooling [13] are capable of producing cold samples tiveposition of theion detectorutilized in thestate-selective
p from a wide range of molecular species; however, the REMPI detection.
[
temperaturesanddensitiescurrentlyattainableviathese

1 “direct cooling” methods are not sufficient for observ-
v ingmanyoftheinterestingphenomenamentionedabove.
estimatemolecule-moleculeinelasticscatteringratesand
4 Conversely,theassociationofultra-coldatoms,eithervia
discuss implications for producing trapped samples of
0 aFeshbach[14]oropticalresonance[15],restrictsexperi- X1Σ(v =0) RbCs molecules.
8
mentstoalimitedclassofmolecules–namely,thosecom-
4 The apparatus used in this work is shown in Fig. 1.
posed oflaser cooledatoms. Nonetheless,these methods
. Briefly, 85Rb and 133Cs atoms are cooled and collected
1 are approaching temperatures and densities appropriate
0 in a dual-species, forced dark-spot magneto-optical trap
for observing the aforementioned phenomena.
8 (DSM) [15, 16]. Using absorption imaging along two or-
0 In this Letter, we report the optical confinement thogonaldirectionsweco-locatethespeciesandmeasure
: of ultra-cold, vibrationally excited RbCs molecules in the atomic density, n, and atom number, N, as n =
v Rb
i the a3Σ+ ground electronic state, produced via photo- 4(2)×1011cm−3,NRb =9(1)×107,andnCs =5(1)×1011
X association (PA) of laser-cooled 85Rb and 133Cs atoms. cm−3, N = 2(1)×108. The temperature, T, of each
Cs
r We utilize the long observation times afforded by the species in the DSM was measured by time-of-flight ex-
a
optical trap to determine the inelastic scattering rate pansion to be T = (80 ± 25) µK and T = (105
Rb Cs
for specific vibrational levels of these molecules, with ± 40) µK. Our optical trap is a quasi-electrostatic trap
both 85Rb and 133Cs atoms, across an order of magni- (QUEST) realized in a 1-D lattice configuration. The
tude of binding energies. We show that a simple model QUEST, represented by the green beam in Fig. 1, is
for the collision process accurately reproduces the ob- formed by focusing and retro-reflectingthe beam from a
served scattering rates. We also extend this model to vertically-aligned100WCO laser,operatingwitha10.6
2
µm wavelength. An acousto-opticalmodulator placed in
thebeampathallowsfortherapidturn-offoftheQUEST
(τ < 1 µs) and serves as an optical isolator for light re-
∗Electronicaddress: [email protected] flectedbackintothe laser. The e−2 intensitybeamwaist
†Department of Physics, Temple University, Philadelphia, Penn-
at the focus is ∼75 µm, yielding a peak intensity of ∼3
sylvania19122-6082, USA
‡Lincoln Laboratory, Massachusetts Institute of Technology, Lex- MW/cm2 and trap depths of ∼4 mK, ∼6 mK, and ≥9
ington,MA,02420,USA mK for Rb, Cs, and a3Σ+ RbCs, respectively. Because
2
RbCs Only; = 404 ms – 47 ms
1.00
] RbCs with Cs; = 68 ms – 12 ms
.
u RbCs with Rb; = 130 ms – 3 ms

.
b Cs Atoms only: = 384 ms – 35 ms
ar 0.75
[

l
a
n
g
i 0.50
S

n
o
I
+ 0.25
s
C
b
R
0.00
0 200 400 600 800 1000 1200 1400 1600
Time [ms]
FIG.3: (ColorOnline)Typicalmolecularlifetimedata. Here
thenumberofmoleculesinthea3Σ+(v=v0)statewithbind-
ing energy EB= - 5.0 ± 0.6 cm−1 is observed in theQUEST
as a function of time. The presence of inelastic collisions be-
tween theatoms and molecules is evidenced by thedramatic
reduction of the molecular lifetime when atoms are present.
With noatoms we observemolecule lifetimes consistent with
the background gas limited lifetime seen for isolated atomic
clouds in the trap.
ber v . In ourpreviouswork[18]we tentativelyassigned
0
v = 37, but this value has an uncertainty of several
0
units since the depth of the a3Σ+ state potential well is
FIG. 2: (Color Online) Formation and detection processes not accurately known.
for ultracold RbCs. (a) The PA process excites colliding
ExperimentaldataistakenbyloadingtheDSMfor5s
atom pairs into bound RbCs∗ molecules, which (b) decay
from background alkali vapor, provided by heated get-
into a range of vibrational states of the a3Σ potential. (c)
Metastable a3Σ(v) molecules are excited to level i, then (d) ters, in the presence of both the PA laser and QUEST.
Atoms are not efficiently loaded into the lattice directly
ionized and subsequently detected via time-of-flight mass
spectrometry. The application of Stokes light (e) instead of fromthe DSM due to the lowtrapdepth(∼100-200µK)
theionizingpulse(d)canbeusedtoproducemoleculesinthe away from the focus. Because the QUEST is substan-
absolute ground state X1Σ(v=0). tiallydeeperforthemoleculesthantheatoms(≥300µK),
the PA process efficiently loads RbCs into the lattice.
Thus, molecules are loaded into the QUEST simply by
we utilize a lattice configurationfor our QUEST, we are applying the PA beam during the DSM loading process.
not restricted to trapping at the focus. We find it ad- FromthemeasuredatomicdensitiesandknownPArates
vantageous to trap atoms and molecules away from the [18], we estimate thatwe trapN ≈ 104 moleculesat
RbCs
focus, where the trap volume is much larger. By mov- adensityofn ≈109cm−3 andtemperatureofT
RbCs RbCs
ing the point of overlap between the QUEST and DSM, ≈ 100 µK with roughly 7% in the a3Σ(v =v ) state. To
0
we trap ∼9 mm away from the focus, where the waist is study atom-molecule collisions we load atoms into the
∼400µmandthetrapdepthsarereducedby∼30. Inad- lattice more efficiently via an optical molasses cooling
ditiontoprovidingalargertrappingvolume,thismethod stage for the desired atomic species, after the DSM is
mitigates the effects of QUEST-induced light-shifts [17]. loaded. Theopticalmolassesstageisperformedbyshift-
The energy level scheme relevant to the RbCs forma- ing the detunings, ∆, of the DSM trap lasers to ∆ =
Rb
tion is shown in Fig. 2. The PA laser has an intensity of -6Γ, ∆ = -16Γ for 10 ms, where Γ is the transition
Cs
∼2kW/cm2,anditsfrequencyislockedtoanΩ=0−,J naturallinewidth. While thehyperfinedepumping beam
P
= 1+ level, located 38.02 cm−1 below the Rb 5S (F= ofthe DSM remains onduring the entire molassesstage,
1/2
2) + Cs 6P (F = 3) atomic asymptote [15]. Sponta- the DSM hyperfine re-pumping beam is extinguished for
1/2
neousdecayofthisstateprimarilyproducesmoleculesin thelast100µstoensurethatalltrappedatomsareinthe
the a3Σ+ state vibrationallevel with binding energyE lowest(dark) hyperfine state. Loading the lattice in this
B
= - 5.0± 0.6cm−1, to whichwe assignvibrationalnum- way leads to typical densities of n = 2(1)×1011 cm−3
Rb
3
and n = 6(1)×1011 cm−3, and temperatures of T =
Cs Rb
(20 ± 11) µK and T = (20 ± 15) µK in a volume of
Cs
≈ π(89 µm)2× 960 µm, which is roughly a factor of 40
increase in density and factor of 5 reduction in temper-
ature compared to loading directly from the DSM. Af-
ter the molasses stage, we apply resonant ‘push’-beams
for 10 ms to remove any undesired atoms from the lat-
tice. After the ‘push’-beam sequence, all beams except
the QUEST are shuttered, and the molecules and any
deliberately trapped atoms are held in the lattice.
After a variable delay time, the QUEST is switched
off and the trapped molecules are state-selectively ion-
izedusingResonance-EnhancedMulti-PhotonIonization
(REMPI), as shownin Fig. 1. The resulting ions are de-
tected using time-of-flight mass spectrometry [19]. In
this manner, we use the observed trap-lifetime of the
molecules in the QUEST as a direct measurement of the
molecular collision rates.
FIG. 4: (Color Online) . Molecular trap-loss scattering rate
TypicallifetimedataisshowninFig. 3formoleculesin
the a3Σ+(v =v0) state. As can be seen, the presence of Kvibcroantisotnaanltlvevs.elsbionfdtinhgeean3eΣr+gystEaBte,.foVribmroalteicounlaelssitnatsepelacbifiecl
atoms in the lattice significantly shortens the lifetime of
appearsbeloweachdatapoint. Despitemorethananorderof
the trappedmolecules,whichisotherwiselimitedbycol- magnitude difference in binding energy, the scattering rates
lisions with background gas. We attribute this behavior appear to be identical within experimental precision. The
to inelastic collisions between the atoms and molecules. error bars on each point are the results of the uncertainties
Theselossesarelikelyduetoro-vibrationalquenchingor in the density and lifetime measurements. Note that the Rb
hyperfine changing collisions. Each of these degrees of data’serrorbarshavebeenangledforclarity. Theblackcross-
freedom carries sufficient energy that its relaxation cre- hatched box and red hatched box are the prediction of the
simple model described in thetext for collisions with Cs and
ates enough kinetic energy to remove both the molecule
Rb,respectively. Thewidthoftheboxesisduetouncertainty
and the atom from the trap, e.g. one vibrational quan-
in the collision temperature.
tum is ∼2 K. The number of trapped molecules, N ,
RbCs
evolves in time according to:
sured rate constants are identical within experimental
dN
RbCs = −Γ N precision, despite over an order of magnitude of varia-
BG RbCs
dt
tion in E . Since the molecule size and vibrational en-
B
β
−Γ N − N2 . (1) ergy spacing both change substantially over this range
atom RbCs V RbCs of E , the lack of dependence of the scattering rate on
B
molecular vibration hints at a unitarity limited process,
Here Γ is the loss rate due to collisions with back-
BG
wherethe detailsofthe short-rangeinteractionpotential
ground gas, Γ is the loss rate due to inelastic col-
atom
are unimportant. This view is supported by the agree-
lisions with atoms, β is the molecular 2-body loss rate,
ment of the data with the results of a simple model of
and V is the trap volume occupied by the molecules.
the collision process, shown in Fig. 4 as hatched boxes.
Since two-body processes are negligible compared with
Thismodel[20],whichis detailedbelow,simplyassumes
background gas collisions (βn /Γ ≪ 1), we use a
RbCs BG
fit of the data to the form of N (t) = N e−t/τ with thatanycollisionwhichpenetratestoshort-rangeresults
RbCs o
τ−1 =Γ +Γ ,toextractthevalueofΓ . Γ inaninelastictraplossevent. Thus,itrepresentsamore
atom BG atom atom
accurateestimateoftheupper-boundofaninelasticpro-
is related to the energy-dependent cross-section, σ(E),
cess than the usual Langevin (unitarity-limited) cross-
and the relative velocity, v, as Γ = n hσ(E)vi,
atom atom
section [21], which yields a scattering rate constant that
where h i denotes thermal averaging. Hence, knowledge
is roughly two to three times larger [27].
of the densities and temperatures allows the determina-
Fortwocolliding particles,the energy-dependentscat-
tionofthescatteringrateconstant,K(T)=hσ(E)vi,for
tering cross-section for the ℓth partial wave, with pro-
these collisions.
jection m, from state i to state f in all outgoing partial
Because the initial PA process populates several vi-
waves ℓ′,m′ is generally written as
brational states in the a3Σ+ state, we utilize the state-
sdealteactsivimitiylaprrotovitdheadtbiyntFhieg.RE3MfoPrIadreatnecgteioonf tvoibmraetaiosunrael σℓ,m(E,i→f)= kπ2 X |Tℓ,m,ℓ′,m′(E,i→f)|2. (2)
ℓ′,m′
states and therefore, a range of binding energies, E ≈
B
-0.5 cm−1 to -7 cm−1. The results of the collision mea- Here Tℓ,m,ℓ′,m′(E,i → f) is the ‘T-matrix’, whose el-
surements are summarized in Fig. 4 as a function of E ements represent the probability amplitude for a tran-
B
(measured relative to the a3Σ+ asymptote). The mea- sition from the incoming spherical wave, Ψ , to the
i,ℓ,m
4
TABLE I: Calculated C6 coefficients for a3Σ+ RbCs(v) col-
liding with various partners given in atomic units.
Collision Type (v0−1) (v0) (v0+2) (v0+6)
RbCs(v) + RbCs(v) 65745 65086 64310 61291
Rb + RbCs 16991 16920 16869 15960
Cs + RbCs 19688 19604 19541 18482
outgoingwave,Ψf,ℓ′,m′ andk =p2µE/~2 is the magni-
tude of the wave-vector at collision energy E. Since our
experiments are sensitive to the total cross-section for
collisions that remove molecules from the trap, we must
sum overallfinalstates f that leadto traploss,in addi-
tion to the normal sum overℓ,m. If we assume that any
collision that penetrates to short-range is inelastic with
unit probability, we can re-write Eq. (2) as
π FIG.5: (ColorOnline)Numericallycalculatedscatteringrate
σ(E,i)= σ (E,i→f)= (2ℓ+1)P (E,ℓ),
X ℓ,m Xk2 T constant vs. energy for the three types of collisions. The p-
f,ℓ,m ℓ wave barrier height for each collision is represented by the
(3) vertical lines with the same plot-style (color).
where P (E,ℓ) is simply the probability of transmis-
T
sion to short range. We calculate P (E,ℓ) by numeri-
T
cally solving the Schr¨odinger equation for the potential
V(r,ℓ)= ~2ℓ(ℓ+1) − C6, and assuming that any flux that ratescalesasv−1. Fromthepredictedmolecule-molecule
2µr2 r6 scatteringrate,weexpectaninitialtwo-bodylossrateof
does not reflect off the potential is transmitted to short- βN = Kn ≈ 0.1 Hz. Given that Γ ≈ 2 Hz,
V RbCs RbCs BG
range and then completely lost to inelastic processes. ourmeasurementofbackgroundgaslimiteddecayforthe
Thissimpletechniqueisapplicabletoanyhighly-inelastic pure RbCs sample is consistent with the predicted scat-
process; it requires only the knowledge of the long-range tering rate.
partofthescatteringpotential,whichinourcaseisgiven
In conclusion, we have demonstrated the trapping of
entirelybythevalueofthevanderWaalscoefficient,C ,
6 heteronuclear RbCs molecules in vibrationally excited
and reduced collision mass, µ.
levels of the a3Σ+ electronic ground state. Observations
In general, the C constant for two colliding particles
6 of the molecules in the trap have revealed strong inelas-
is given by the integral over imaginary frequency of the
tic collisions, presumably due to a combination of vibra-
product of the particles’ dynamic polarizabilities [22].
tional, rotational and hyperfine quenching. A simple,
Since the dynamic polarizability of RbCs was calculated
extendable model, which relies only on the knowledge of
inRef. [23]andtheatomicvaluesarewell-known,theC
6 the van der Waals coefficient, accurately reproduces the
constant as a function of vibrational level is straightfor-
observedratesbyassumingthatallshort-rangecollisions
wardto calculate. These are shown for reference in Tab.
are inelastic for all observed vibrational levels and both
I.
colliding atomic species.
Using these values, P , σ, and K are calculated.
T
We are currently working towards transferring these
The results are shown in Fig. 5, where the scatter-
trapped molecules into their absolute ground state, via
ing rate, K = hσ(E,i)vi, is plotted versus collision en-
the transfer scheme previously demonstrated in our lab
ergy for the three classes of collisions. Note that the
[19]. It appears that with minimal improvements, e.g.
average center-of-mass frame collision velocity is given
implementing an adiabatic transfer [26] instead of the
as hvi = q8πkmbT11 + 8πkmbT22 = q8kπbµTµ, where Ti (Tµ) is stimulated emission pumping used in [19], a sample of
thelaboratoryframe(center-of-massframe)temperature > 104 absolute ground state molecules X1Σ(v = 0) at a
and m is the mass of the colliding particles. Since the temperature of 20 µK and density of ≥109 cm−3 can be
i
ℓ = 1 (p-wave) barrier heights lie at E/k ≈ 25 µK, 15 created. Fromcalculationsofthe adiabatictransferwith
B
µK,and5µKforRbCscollidingwithRb,Cs,andRbCs, available laser powers, we estimate the transfer process
respectively, the collision rates fall to the values given can take place in ∼100 µs; thus, we anticipate negligible
solely by their s-wavecontributionatthe lowestenergies loss of population due to the inelastic collisions studied
on the graph. Interestingly, quantum reflection from the in the present work. In fact, it appears these inelastic
ℓ=0potentialisfoundtoscalelinearlywithkasE →0, collisions could serve a useful purpose. If Cs atoms are
reproducing the Wigner threshold law [24] for low tem- deliberately loaded into the lattice, any molecule not in
perature inelastic scattering of σ ∝ 1/k [25]. Thus the the absolute ground state will quickly (∼100 ms) be re-
calculatedscatteringrateremainsfiniteatzerotempera- moved from the trap by these inelastic collisions. By
ture,despitethefactthattheunitaritylimitedscattering contrast, X1Σ(v = 0) molecules, which cannot undergo
5
inelastic collisions with Cs atoms, are unaffected by the + Cs. After the excited state molecules have been re-
presence of the atoms – even groundstate molecules can moved, resonant ‘push’-beams can eject the remaining
have inelastic collisions with Rb atoms via the energet- atoms from the trap, leaving behind a pure sample of
ically allowed substitution reaction RbCs + Rb → Rb ground state molecules.
2
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