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Inelastic Collisions of Ultracold Polar Molecules
John L. Bohn∗
JILA, National Institute of Standards and Technology and University of Colorado, Boulder, CO 80309-0440
(February 2, 2008)
The collisional stability of ultracold polar molecules in
electrostatic traps is considered. Rate constants for collisions
that drive molecules from weak-field-seeking to strong-field-
seeking states are estimated using a simple model. The rates
are found to be quite large, of order 10−12-10−10 cm3/sec,
and moreover to grow rapidly in an externally applied elec-
tric field. It is argued that these results are generic for po-
lar molecules, and that therefore polar molecules should be
trapped by other than electrostatic means.
Recently Bethlem et al. broadened the scope of ultra-
cold AMO physics by cooling and electrostatically trap-
ping ND3 molecules [1,2]. This achievement is notable
both for the complexity of the species trapped and for
the generality of the Stark slowing technique, which could
in principle cool any polar molecule. This technique is
therefore now on a par with other experimental methods
for producing cold molecular gases, such as photoassoci-
ation [3] and buffer-gas cooling [4], as well as photopro-
duction of molecular ions in a Paul trap [5].
The electrostatic trap demonstrated in Ref. [2] raises
anew questions of collisional stability that are familiar in
the context of magnetic trapping of atoms [6]. An elec-
trostatic trap can only confine dipoles that are in their
weak-field seeking states, since Maxwell’s equations per-
mit a local field minimum but not a field maximum. Thus
the dipoles are succeptible to orientation-changing colli-
sions that populate the strong-field seeking, untrapped
states. In the case of magnetic trapping of alkali atoms
a standard remedy against collisional losses is to pre-
pare the atoms in their stretched spin states, whereby
the dominant spin-exchange collisional processes are ab-
sent. Atomic spins can then only change their orientation
via spin-spin dipolar processes, which are weak because
of the inherent weakness of magnetic dipolar interactions.
The purpose of this Rapid Communication is to point
out that polar molecules are not as immune to dipolar re-
laxation as are magnetic atoms, simply because electric
dipoles have a much stronger interaction. Indeed, the
force between a pair of d = 1 Debye (0.39 atomic units)
electric dipoles is ∼ 3×103 times larger than that between
a pair of µ = 1µB magnetic dipoles. The basic physics
of electric dipolar relaxation lies in the competition be-
tween the dipoles’ interaction with the electric field when
they are far apart, −~d · ~E , and with each other when they
are closer together, ∼ ~d1 · ~d2/R3. At small values of in-
termolecular separation R the dipoles will tend to lock
on their intermolecular axis rather than on the lab-fixed
axis set by ~E ; the competition between these tendencies
scrambles the orientation of the molecular dipoles. The
resulting state-changing collisions can in principle be sup-
pressed by a field strong enough to maintain the dipolar
orientation. This will happen, roughly, if dE > d2/R3
for small values of R. Still, for d = 1 Debye dipoles at
a typical collision distance R ≈ 10 atomic units (a.u.),
an electric field of 106 V/cm would be required to main-
tain dipolar orientation. Thus very large laboratory fields
may exert some mitigating influence, but are unlikely to
arrest relaxation altogether.
To quantify this general argument this paper presents
detailed calculations using a simplified model of colli-
sions. In general the physics of cold molecular colli-
sions will be quite complex, intertwining rotational, elec-
tronic, nuclear spin, and perhaps even vibrational de-
grees of freedom. However, to establish orders of magni-
tude for dipolar relaxation rate constants it suffices to
focus on orientational degrees of freedom, and to ac-
count only for the dominant dipole-dipole interaction be-
tween molecules. Accordingly, a simplified “toy” model
is used here, which has zero spin and nuclear spin. The
molecules are assumed to be diatomic rigid rotors with
electric dipole moments d = 1 Debye along their molec-
ular axes. The electronic ground state of the molecules
is assumed to be 1Π, so that it possesses a Λ-doublet of
parity eigenstates. The splitting of this doublet is as-
sumed to have a “typical” value of ∆ = 10−3 cm−1, and
the lower-energy state is assumed to have even parity.
At ultracold temperatures and in zero electric field
the molecules occupy parity eigenstates, hence exhibit
no permanent dipole moment. The dipole moments only
become apparent when the field is large enough to sig-
nificantly mix states of different parity, thus “activat-
ing” the dipoles. This occurs at a field value where the
Stark effect transforms from quadratic to linear, at ≈ 100
V/cm in the present model (Figure 1). For fields below
this value the molecules are fairly weakly interacting,
whereas above this value the molecules have extremely
strong dipole couplings. Thus in the Λ-doubled state col-
lisions can be manipulated using modest electric fields,
in contrast to the ∼ 105 V/cm fields required to influence
cold atomic collisions [7]. These arguments also apply to
molecules with 2S+1Σ electronic symmetry when S > 0
and the molecule exhibits an Ω-doubling [8], as well as
to ND3.
Figure 1 shows the electric field dependence of the
lowest-lying energy levels in the model molecules. Al-
though both N = 1 and N = 2 rotational levels are
1

shown, the calculations below focus exclusively on the
N = 1 levels. Their low-E behavior is shown in the inset,
labeled by the pair of quantum numbers |MN |, p. Here p
stands for the parity in zero field, while |MN | denotes the
magnitude of the molecular rotation’s magnetic quantum
number referred to the laboratory axis; the MN = 1 and
MN = −1 levels are degenerate even in an electric field.
In an electrostatic trap of the kind used by Bethlem et al.
the trapped states are the weak-field seekers, i.e., those
whose Stark energy rises with rising field. These are the
||MN |, p〉 = |1,−〉 states in the model.
The Hamiltonian for collisions between the molecules
consists of four terms in this model,
Hˆ = Tˆ + Hˆfs + Hˆfield + Hˆdip−dip. (1)
Here Tˆ represents the kinetic energy, Hˆfs the molecular
fine structure including the Λ-doubling, and the last two
terms are the electric field interaction and the dipole-
dipole interaction between molecules:
Hˆfield = −(~d1 + ~d2) · ~E , (2)
Hˆdip−dip =
~d1 · ~d2 − 3(Rˆ · ~d1)(Rˆ · ~d2)
R3 , (3)
where Rˆ denotes the orientation of the vector joining
the centers-of-mass of the molecules. Dispersion and ex-
change potentials are neglected here since they are of sec-
ondary importance to dipolar interactions at large R. To
avoid problems with the singularity of 1/R3 at R = 0,
vanishing boundary conditions are imposed at a cutoff
radius R0 = 10 a.u., where the potentials are deep com-
pared to ∆.
In the scattering calculation the molecules are assumed
to be identical bosons, so that only even partial waves are
relevant. Only the partial waves L = 0 and L = 2 are
included explicitly, even though in principle all partial
waves are coupled together by strong anisotropic interac-
tions. However, the neglected higher-L partial waves can
be shown, in the Born approximation, to fall off rapidly
with L [9,10]. Cross sections for processes that change
molecular channel |i〉 into channel |i′〉 are given as in Ref.
[11]:
σi→i′ =
π
k2i
∑
LMLL′M ′L
|〈i, LML|T |i′L′M ′L〉|2, (4)
where T is the T-matrix for scattering and ki is the inci-
dent wave number. Because of the incoherent sum in (4),
contributions arising from L = 0 and L = 2 incident par-
tial waves can be presented individually. All results, for
both elastic and state-changing collisions will be reported
as event rate coefficients,
Ki→i′ = viσi→i′ , (5)
where vi is the incident relative velocity of the collision
partners.
Although state-changing collisions are primarily of in-
terest here, it is useful first to consider elastic scattering
of polar molecules, to illustrate the enormous influence
of the electric field. Figure 2 shows the elastic scat-
tering rate constant Kel for molecules in their |MN , p〉
= |1,−〉 state. Figure 2(a) depicts the low-field limit
(E = 0 V/cm), while Fig. 2(b) shows a higher-field limit
(E = 200 V/cm), where the Stark effect is linear. These
rate constants are separated into their s-wave (solid line)
and d-wave (dashed line) contributions.
In both the low- and high-field regimes the s-wave rate
has the familiar Kel ∝ E1/2 threshold behavior as a func-
tion of collision energyE, arising from the Wigner thresh-
old law. For the d-waves, however, the threshold depen-
dence changes dramatically, falling off as Kel ∝ E5/2 at
low field but as Kel ∝ E1/2 at higher fields. This be-
havior has to do with qualitative differences in the long-
range intermolecular potentials. In the low-field limit,
the asymptotic molecular states are parity eigenstates,
hence have no dipole moment. At large values of R,
where the dipole-dipole interaction energy becomes less
than the Λ-doubling energy, the dipole-dipole 1/R3 po-
tential is thus effectively absent. By contrast, in the high-
field regime parity eigenstates are mixed at large R and
the 1/R3 potential is activated. It is well known [12,13]
that a 1/R3 potential contributes a long-range scattering
phase shift that is proportional to the wave number k
for all partial waves L > 0, thus yielding a Kel ∝ E1/2
threshold law.
This “switching on” of the dipolar interaction in the
presence of a field also manifests itself in the inelastic rate
constants, which are shown in Figure 3. This figure shows
the sum Krelax =
∑
i′ Ki→i′ of all the rate constants for
collisions of |MN , p〉= |+1,−〉molecules in which at least
one molecule relaxes into one of the strong-field-seeking
states ||MN |, p〉 = |0,−〉, |0,+〉, or |1,+〉 (see Fig. 1).
Again both the E = 0 V/cm and E = 200 V/cm cases
are shown, and the rates are separated into L = 0 and
L = 2 initial states. These relaxation rates are domi-
nated by exothermic processes which exhibit their own
characteristic threshold behavior. The rates for L = 0
partial waves become independent of energy at thresh-
old, while the behavior of L = 2 partial waves transforms
from the usual Wigner result Krelax ∝ E2 at low field, to
a Krelax ∝ E behavior at high field when the dipoles are
activated [14].
More significantly, the values of the loss rate constant
are substantially boosted by the presence of an electric
field, even for s-wave collisions. This is simply because
the strong dipolar interactions that drive inelastic tran-
sitions are also made stronger in an electric field. Figure
4 shows the threshold relaxation rate as a function of
electric field. Even at low field the rates are fairly large,
since dipole interactions are still present at small R. As
the field grows to E ∼ 100 V/cm, where the dipoles turn
on, the rates rise sharply. In the particular model con-
sidered here the rates are boosted nearly two orders of
magnitude by the field.
2

Weak-electric-field seeking states should quite gener-
ally suffer these large loss rates. Quantum mechanically
this follows from the fact that the large-R Hamiltonian
Hˆfs+ Hˆfield is diagonal in the laboratory frame, while the
interaction Hamiltonian Hˆdip−dip is diagonal in the body
frame that joins the centers-of-mass of the two molecules.
The former is stronger at large separations R, while the
latter dominates at small R. Molecules that start out in
eigenstates of Hˆfs+Hˆfield are thus distributed over all the
different states of Hˆdip−dip during the collision, and re-
assembled into an assortment of eigenstates of Hˆfs+Hˆfield
as the molecules separate. Since the asymptotic states
are completely deconstructed and reconstructed during
this collision, in general it is expected that the probabil-
ity for inelastic scattering is roughly the same as that for
elastic scattering, and that therefore the rates are com-
parable.
This is of course the same kind of physics that gov-
erns spin-exchange collisions in the alkali atoms, which
are driven by the competition between hyperfine-plus-
magnetic field interactions at large R, and exchange po-
tentials at small R [15]. In the case of alkali atoms it is
possible that the short-range phase shifts, from singlet
and triplet total electronic spin states, can interfere in
such a way as to eliminate probabilities for inelastic pro-
cesses [16]. For molecules this coincidence seems unlikely,
however, since there are are many degrees of freedom at
short range, all of which would have to contribute nearly
identical scattering phase shifts in order cancel loss rates.
In conclusion, dipolar molecules electrostatically
trapped in weak-field-seeking states are succeptible to
state-changing collisions that can rapidly deplete the
trapped gas. Moreover, these rates can grow in the pres-
ence of the trapping electric field, which effectively turns
on the full dipolar coupling at large intermolecular sepa-
ration. Although this conclusion has been demonstrated
using a particular toy model, the physics is quite gen-
eral and should apply to any polar species. It is there-
fore recommended that dipolar molecules be trapped in
strong-field-seeking states, where inelastic channels are
absent at low temperatures. This kind of trapping cannot
be achieved in a static trap, but would require a time-
varying electric field. Magnetic dipoles in strong-field
seeking states have indeed been confined in such traps,
using either time-varying fields [17] or microwave cavi-
ties [18]. A more conventional magnetic trap may also
be useful, although the influence of the electric dipoles
on losses in magnetic traps would have to be explored.
More broadly, an externally applied electric field is seen
to have a profound influence on the collision dynamics of
ultracold polar molecules, even to the extent of altering
the threshold behavior. Preliminary results on the prop-
erties of quantum degenerate gases with dipolar inter-
actions have been reported in the literature [9,19]. More
detailed scattering calculations are required to help shape
the study of these unusual substances [10].
This work was supported by the National Science
Foundation. I acknowledge useful discussions with E.
Cornell and C. Greene.
∗ Electronic address: bohn@murphy.colorado.edu
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[13] H. R. Sadeghpour et al., J. Phys. B 33, R93 (2000).
[14] N. F. Mott and H. S. W. Massey, The Theory of Atomic
Collisions (Oxford, The Clarendon Press, third ed.,
1965), p. 403.
[15] H. T. C. Stoof, J. M. V. A. Koelman, and B. J. Verhaar,
Phys. Rev. B 38, 4688 (1988).
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Lett. 67, 2439 (1991).
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051601(R), (2000).
3

0 1x105 2x105 3x105 4x105 5x105
-16
-12
-8
-4
0
4
8
12
|MN|,p
0,-
0,+
1,-
1,+
N=1
N=2
En
er
gy
/k
B
(K
)
Electric Field (V/cm)
0 100 200 300
2.876
2.880
2.884
FIG. 1. Stark energy levels for the model molecules, which
have 1Π electronic symmetry. This paper concentrates on
collisions between molecules in their |MN , p〉 = |+1,−〉 states,
which are weak-field seekers (see inset).
10-6 10-5 10-4 10-3
10-15
10-12
10-9
= 200 V/cm( (b)
L=0
L=2
K
el

(cm
3 /s
ec
)
Collsion Energy E/kB (K)
10-6 10-5 10-4 10-3
10-15
10-12
10-9
= 0 V/cm( (a)
L=0
L=2
FIG. 2. Elastic scattering rate constants for |MN , p〉
= | + 1,−〉 molecules. Shown are results for low-field
[E = 0 V/cm in (a)] and high-field [E = 200 V/cm in (b)]
regimes. Notice that in the presence of an electric field the
d-wave (L = 2) rate is significantly boosted, and acquires a
Kel ∝ E1/2 threshold behavior.
10-6 10-5 10-4 10-3
10-15
10-12
10-9
= 200 V/cm( (b) L=0
L=2
K
re
la
x
(cm
3 /s
ec
)
Collision Energy E/kB (K)
10-6 10-5 10-4 10-3
10-15
10-12
10-9
= 0 V/cm( (a)
L=0
L=2
FIG. 3. Dipolar relaxation rate constants for |MN , p〉
= | + 1,−〉 molecules, at the same field values as in Fig.
2. The rates are substantially higher in the presence of an
electric field, as explained in the text.
4

0 100 200 30010
-12
10-11
10-10
10-9
K
re
la
x
(cm
3 /s
ec
)
Electric Field (V/cm)
FIG. 4. Electric field dependence of the total relaxation
rates for molecules initially in their |MN , p〉 = |+1,−〉 state,
in the zero-collision-energy limit. These rates rise sharply by
E = 100 V/cm, where the molecular dipoles are “activated”
by a suitable admixture of parity eigenstates.
5