# Controlling collisional loss and scattering lengthsof ultracold dipolar
molecules with static electric fields

Bijit Mukherjee  Joint Quantum Centre (JQC) Durham-Newcastle, Department of
Chemistry, Durham University, South Road, Durham, DH1 3LE, United Kingdom.
Jeremy M. Hutson<EMAIL_ADDRESS>Joint Quantum Centre (JQC) Durham-
Newcastle, Department of Chemistry, Durham University, South Road, Durham, DH1
3LE, United Kingdom.

(August 31, 2024)

###### Abstract

Trapped samples of ultracold molecules are often short-lived, because close
collisions between them result in trap loss. We investigate the use of
shielding with static electric fields to create repulsive barriers between
polar molecules to prevent such loss. Shielding is very effective even for
RbCs, with a relatively low dipole moment, and even more effective for
molecules such as NaK, NaRb and NaCs, with progressively larger dipoles.
Varying the electric field allows substantial control over the scattering
length, which will be crucial for the stability or collapse of molecular Bose-
Einstein condensates. This arises because the dipole-dipole interaction
creates a long-range attraction that is tunable with electric field. For RbCs,
the scattering length is positive across the range where shielding is
effective, because the repulsion responsible for shielding dominates. For NaK,
the scattering length can be tuned across zero to negative values. For NaRb
and NaCs, the attraction is strong enough to support tetraatomic bound states,
and the scattering length passes through resonant poles where these states
cross threshold. For KAg and CsAg, there are multiple bound states and
multiple poles. For each molecule, we calculate the variation of scattering
length with field and comment on the possibilities for exploring new physics.

##  I Introduction

Trapped samples of ultracold polar molecules offer opportunities to study
important physical phenomena that range from quantum simulation [1, 2] to
quantum magnetism [3, 4]. However, close collisions between dipolar molecules
often result in trap loss due to short-range processes such as chemical
reaction [5], inelastic collisions, 3-body recombination [6] or laser
excitation of collision complexes [7]. Because of this, there is much interest
in shielding molecules from close collisions by engineering long-range
barriers between them. This may be done with static electric fields [8, 9, 10,
11, 12], near-resonant microwaves [13, 14, 15], or lasers [16]. Both microwave
shielding and static-field shielding have been demonstrated experimentally
[17, 18, 19], and microwave shielding has recently been used to achieve Fermi
degeneracy for fermionic NaK [20]. Bose-Einstein condensation has not yet been
achieved for dipolar molecules, but is the subject of active experimental
efforts [21, 22].

Shielding relies on the dipole-dipole interaction between two polar molecules.
This is proportional to $1/R^{3}$ at long range, where $R$ is the
intermolecular distance. Two different states of a molecular pair are tuned
into near-degeneracy with an electric, microwave or laser field. If the two
pair states are connected by the dipole-dipole interaction, they repel one
another. It is thus possible to engineer an effective interaction potential
for the upper pair state that has a repulsive component at moderately long
range.

We have recently developed an efficient computational approach to simulating
shielding with static fields [12]. This is based on coupled-channel
calculations of inelastic processes, with an absorbing boundary condition [23,
24] to take account of short-range loss without simulating it in detail. Our
approach is conceptually similar to earlier methods for static-field [9] and
microwave [13] shielding, but is made far more efficient by using a basis set
of field-dressed molecular functions and by taking account of energetically
well-separated functions by a Van Vleck transformation [25, 26]. We have
applied this approach to shielding of CaF, which we found to be very effective
at electric fields of approximately 23 kV/cm.

Ultracold collisions are often characterized by a scattering length $a$ [27,
28], which describes the phase shift for completed low-energy collisions. The
scattering length governs both 2-body and many-body properties, and in many-
body physics is often represented by a zero-range contact interaction [29].
For dipolar systems, many-body properties arise from an interplay between the
scattering length and long-range anisotropic interactions; the latter depend
on the space-fixed dipole moment $d$ and may be affected by the geometry of
the system. The mean-field ansatz usually adopted [30, 31] involves
supplementing a contact potential based on an effective scattering length
$a_{\textrm{eff}}$ with a long-range dipole-dipole interaction based on $d$.
This approach has been validated by Ronen _et al._ [32], who compared the
results of the mean-field prescription with those of fully correlated
calculations. An important point is that $a_{\textrm{eff}}$ is affected by
long-range forces and depends on $d$, so Ronen _et al._ referred to it as the
_dipole-dependent_ scattering length.

In the absence of a long-range well, the repulsion due to shielding would be
expected to produce a positive scattering length. However, we have previously
shown [12] that, for CaF, the scattering length is large and negative in the
region where shielding is effective. This arises because there is in fact a
long-range well outside the shielding barrier; this adds a negative
contribution that overcomes the positive one from shielding.

In this paper, we explore the tunability of scattering lengths for a variety
of polar molecules shielded with static electric fields and outline the
scientific possibilities that this offers. We show that the scattering length
depends on the applied field in a non-trivial way, and not just on the dipole
moment $d$ induced by the field. Indeed, the same scattering length can often
be achieved for different fields that induce different dipole moments. This
opens the way to investigating many-body properties as independent functions
of scattering length and dipole moment.

A wide range of behavior is possible for different molecules. For some
molecules, such as RbCs, the long-range attractive well is shallow and the
scattering length has moderate positive values. For others, such as NaK, the
attractive well is deeper and the scattering length can be tuned to zero; this
may allow the production of a quantum gas with purely dipolar interaction.
There are also some systems, such as NaRb and NaCs, where the attractive well
is deep enough to support one or more bound states; in this case, adjusting
the electric field across the threshold for creation of a bound state allows
“dialling up” any desired scattering length. It may also allow
“electroassociation” to form long-range tetraatomic molecules.

The structure of the paper is as follows. Section II describes our theoretical
approach, including the Hamiltonians and basis-set expansions used, and
describes how the properties of individual molecules affect shielding and the
ability to control scattering lengths. Section III presents our results for
each of the representative molecules mentioned above. Finally, Section IV
presents conclusions and perspectives.

##  II Theory

###  II.1 Coupled-channel approach

The theoretical approach used here is similar to that in ref. [12]. If
vibrational motion and spins are neglected, the Hamiltonian for a single
linear polar molecule $k$ in a $\Sigma$ state, in an electric field $\bm{F}$,
is

$\hat{h}_{k}=b_{k}\hat{\bm{n}}_{k}^{2}-\bm{\mu}_{k}\cdot\bm{F},$ (1)

where $\hat{\bm{n}}_{k}$ is the operator for molecular rotation, $b_{k}$ is
the rotational constant, and $\bm{\mu}_{k}$ is the dipole moment along the
molecular axis. It is convenient to represent the resulting energy levels in
reduced units, $E/b$ for energy and $F\mu/b$ for field, so that they can be
applied to any molecule in a $\Sigma$ state. Figure 1(a) shows the single-
molecule energy levels as a function of electric field; we label the levels
$(\tilde{n},m_{n})$; here $\tilde{n}$ is a quantum number that correlates with
the free-rotor quantum number $n$ at zero field and $m_{n}$ represents the
conserved projection of $n$ onto the space-fixed $z$ axis. Figure 1(b) shows
the corresponding energy of a _pair_ of noninteracting molecules. Shielding
can occur when the dipole-dipole interaction connects two pair states that are
close together in energy. In Figure 1(b), it can occur when two molecules in
the state (1,0) collide at fields just above $F\mu/b=3.244$, where (1,0)+(1,0)
lies just above (0,0)+(2,0). Shielding is most effective at reduced field
$F\mu/b\approx 3.4$, when the space-fixed dipole moment $d$ of the state (1,0)
is about $-0.13\mu$ [11].

Figure 1: Energy of (a) a single molecule in a $\Sigma$ state and (b) a pair
of molecules as a function of electric field. The initial pair state that can
be shielded from close collisions, ($\tilde{n},m_{n})$ = (1,0)+(1,0), is shown
in red. The state (0,0)+(2,0) that is responsible for shielding is shown in
blue, and crosses the initial state near $F\mu/b=3.244$.

The Hamiltonian for a colliding pair of molecules is

$\hat{H}=\frac{\hbar^{2}}{2\mu_{\textrm{red}}}\left(-R^{-1}\frac{d^{2}}{dR^{2}}R\right)+\hat{H}_{\textrm{int}},$
(2)

where $R$ is the intermolecular distance and $\mu_{\textrm{red}}$ is the
reduced mass. The internal Hamiltonian for the pair is

$\hat{H}_{\textrm{int}}=\hat{h}_{1}+\hat{h}_{2}+\hat{H}_{\textrm{cent}}+V_{\textrm{int}},$
(3)

where

$\hat{H}_{\textrm{cent}}=\frac{\hbar^{2}\hat{\bm{L}}^{2}}{2\mu_{\textrm{red}}R^{2}}$
(4)

is the centrifugal Hamiltonian and $\hat{\bm{L}}^{2}$ is the angular momentum
operator for rotation of the two molecules about one another. The interactions
involved in shielding occur at long range, so here we approximate the
interaction potential $V_{\textrm{int}}$ by the dipole-dipole operator,

$\hat{H}_{\textrm{dd}}=[3(\bm{\mu}_{1}\cdot\hat{\bm{R}})(\bm{\mu}_{2}\cdot\hat{\bm{R}})-\bm{\mu}_{1}\cdot\bm{\mu}_{2}]/(4\pi\epsilon_{0}R^{3}),$
(5)

where $\hat{\bm{R}}$ is a unit vector along the intermolecular axis.

The total wavefunction is expanded

$\Psi(R,\hat{\bm{R}},\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})=R^{-1}\sum_{j}\Phi_{j}(\hat{\bm{R}},\hat{\bm{r}}_{1},\hat{\bm{r}}_{2})\psi_{j}(R),$
(6)

where $\hat{\bm{r}}_{k}$ is a unit vector along the axis of molecule $k$. We
use a basis set of functions $\\{\Phi_{j}\\}$,

$\Phi_{j}=\phi^{\tilde{n}_{1}}_{m_{n1}}(\hat{\bm{r}}_{1})\phi^{\tilde{n}_{2}}_{m_{n2}}(\hat{\bm{r}}_{2})Y_{LM_{L}}(\hat{\bm{R}}),$
(7)

symmetrized for exchange of identical bosons. Here
$\phi^{\tilde{n}_{1}}_{m_{n1}}(\hat{\bm{r}}_{1})$ and
$\phi^{\tilde{n}_{2}}_{m_{n2}}(\hat{\bm{r}}_{2})$ are field-dressed rotor
functions that diagonalize $\hat{h}_{1}$ and $\hat{h}_{2}$, respectively, and
$Y_{LM_{L}}(\hat{\bm{R}})$ are spherical harmonics that are the eigenfunctions
of $\hat{\bm{L}}^{2}$. The field-dressed functions
$\phi^{\tilde{n}}_{m_{n}}(\hat{\bm{r}})$ are themselves expanded in free-rotor
functions $Y_{nm_{n}}(\hat{\bm{r}})$.

Substituting the expansion (6) into the total Schrödinger equation produces a
set of coupled equations in the intermolecular distance $R$. These equations
are solved separately for each value of the projection of the total angular
momentum, $M_{\textrm{tot}}=m_{n1}+m_{n2}+M_{L}$, which is a conserved
quantity. The eigenvalues of $\hat{H}_{\textrm{int}}$ as a function of $R$
form a set of adiabats $U_{i}(R)$ that represent effective potentials for
relative motion. The computational method of ref. [12] does not rely on these
adiabats and takes full account of transitions between them, but the adiabats
are nevertheless conceptually valuable tools for understanding the scattering
properties.

The dipole-dipole operator is off-diagonal in the partial-wave quantum number
$L$, and converged calculations typically require substantial values of
$L_{\textrm{max}}$. The basis sets can thus become very large if substantial
numbers of pair functions $(\tilde{n}_{1},m_{n1},\tilde{n}_{2},m_{n2})$ are
included explicitly. The problem become extreme when electron and nuclear
spins are included. We therefore include only a relatively small number of
“class 1” pair functions (with labels a, b, $\ldots$) explicitly in the basis
set, with the remaining “class 2” functions (with labels $\alpha$, $\beta$,
$\ldots$) taken into account through a Van Vleck transformation as described
in ref. [12]. This contributes matrix elements _between_ the channels in class
1 of the form

$\displaystyle\langle a$ $\displaystyle|\hat{H}_{\textrm{dd,VV}}|b\rangle$
$\displaystyle=\sum_{\alpha}\frac{1}{2}\left[\frac{\langle
a|\hat{H}_{\textrm{dd}}|\alpha\rangle\langle\alpha|\hat{H}_{\textrm{dd}}|b\rangle}{(E_{a}-E_{\alpha})}+\frac{\langle
a|\hat{H}_{\textrm{dd}}|\alpha\rangle\langle\alpha|\hat{H}_{\textrm{dd}}|b\rangle}{(E_{b}-E_{\alpha})}\right].$
(8)

We make the further approximation of replacing the energies in the
denominators with their asymptotic values, so that they are independent of $R$
and $\hat{H}_{\textrm{dd,VV}}$ is proportional to $R^{-6}$.

We use a basis set with $\tilde{n}$ up to 5 to calculate the monomer
properties. For the pair functions, the Van Vleck treatment allows us to
restrict the basis set of functions in class 1 to those with $\tilde{n}\leq
2$, and to include the remaining functions up to $\tilde{n}=5$ in class 2. In
this way, we achieve an enormous reduction in computational cost without
compromising the convergence [12]. We include functions with even $L$ up to
$L_{\textrm{max}}=20$.

###  II.2 Cross sections and scattering lengths

Colliding molecules may be lost from a trap in two ways. First, colliding
pairs may undergo a transition to a lower-lying pair state. This releases
kinetic energy that is usually sufficient to eject both molecules from the
trap, and we refer to it as inelastic loss. Secondly, any pairs that penetrate
through the engineered repulsive barrier and reach small intermolecular
distance are also likely to be lost, through short-range inelasticity, laser
absorption, or 3-body collisions. We refer to this as short-range loss and to
the sum of inelastic and short-range loss as total loss.

To model these processes, we solve the coupled equations with a fully
absorbing boundary condition at short range [23, 24]. The numerical methods
used are as described in ref. [12]. This is done separately for each electric
field $F$ and collision energy $E_{\textrm{coll}}$, producing a non-unitary S
matrix $\bm{S}$ and cross sections $\sigma_{\textrm{el}}$,
$\sigma_{\textrm{inel}}$ and $\sigma_{\textrm{short}}$ for elastic scattering,
inelastic scattering and short-range loss, respectively. The corresponding
rate coefficients $k$ at collision energy $E_{\textrm{coll}}$ are related to
the cross sections $\sigma$ through $k=v\sigma$, where
$v=(2E_{\textrm{coll}}/\mu_{\textrm{red}})^{1/2}$. We also obtain a complex
energy-dependent scattering length [33],

$a(k_{0})=\alpha(k_{0})-i\beta(k_{0})=\frac{1}{ik_{0}}\left(\frac{1-S_{00}(k_{0})}{1+S_{00}(k_{0})}\right),$
(9)

where $k_{0}=(2\mu_{\textrm{red}}E_{\textrm{coll}}/\hbar^{2})^{1/2}$.

The interaction potential between two polar molecules is typically deep and
strongly anisotropic at short range. However, shielding occurs due to dipole-
dipole interactions that occur at intermolecular distances $R\gg 100\ a_{0}$.
At these distances, the chemical interactions that dominate at short range
make very little contribution, and they are neglected in the present work.
However, there are also effects due to dispersion interactions, which are
proportional to $R^{-6}$ at long range. These are of two types. Rotational
dispersion interactions arise from matrix elements of $\hat{H}_{\textrm{dd}}$
off-diagonal in monomer rotational quantum numbers, which are included
directly in the coupled equations. In addition, there are electronic
dispersion interactions, arising from dipole-dipole matrix elements off-
diagonal in electronic state. We take these into account through an additional
interaction $V_{\textrm{disp}}^{\textrm{elec}}=-C_{6}^{\textrm{elec}}/R^{6}$.
This makes only a small contribution to loss rate, at the level of 10% for
RbCs but much less for the other molecules considered here.

###  II.3 Reduced units

Dipole-dipole scattering may be cast in a dimensionless form, with lengths
expressed in terms of a dipole length
$R_{3}=(2\mu_{\textrm{red}}/\hbar^{2})(\mu^{2}/4\pi\epsilon_{0})$ and energies
expressed in terms of a corresponding energy
$E_{3}=\hbar^{2}/(2\mu_{\textrm{red}}R_{3}^{2})$ [11]. A crucial quantity for
shielding with a static field is the reduced rotational constant of the
molecule, $\tilde{b}=b/E_{3}$. For ultracold molecules of current interest,
this ranges from ${\sim}10^{6}$ for NaLi to ${\sim}10^{11}$ for YO [11], or
even larger for some molecules containing Ag. Static-field shielding is
generally most effective for molecules with larger values of $\tilde{b}$.
Table 1 summarizes the key quantities for the molecules considered in the
present work.

Table 1: Key quantities for the molecules considered in this study. Molecular dipole moments and rotational constants are taken from refs. [34, 35, 36, 37, 38, 39, 40, 41, 42].  |  $b/\mu$ (kV/cm) |  $R_{3}$ ($a_{0}$) |  $E_{3}$ (K) | $\tilde{b}$ |  $C_{6}^{\textrm{elec}}/(E_{3}R_{3}^{6})$111Values of $C_{6}^{\textrm{elec}}$ taken from Ref. [43].
---|---|---|---|---|---
87Rb133Cs | 0.795 | $9.3\times 10^{4}$ | $9.1\times 10^{-11}$ | $2.6\times 10^{8}$ | $9.5\times 10^{-11}$
23Na39K | 2.085 | $1.3\times 10^{5}$ | $1.7\times 10^{-10}$ | $8.2\times 10^{8}$ | $3.0\times 10^{-12}$
23Na87Rb | 1.297 | $3.2\times 10^{5}$ | $1.6\times 10^{-11}$ | $6.4\times 10^{9}$ | $1.8\times 10^{-13}$
23Na133Cs | 0.726 | $9.9\times 10^{5}$ | $1.1\times 10^{-12}$ | $7.4\times 10^{10}$ | $3.5\times 10^{-15}$
KAg 222Weighted-average atomic masses are used. | 0.467 | $3.0\times 10^{6}$ | $1.3\times 10^{-13}$ | $7.3\times 10^{11}$ |  $3.3\times 10^{-17}$ 333Calculated $C_{6}^{\textrm{elec}}$ not available, so we use a value of 10000 $E_{\textrm{h}}a_{0}^{6}$.
CsAg 22footnotemark: 2 | 0.164 | $6.5\times 10^{6}$ | $1.7\times 10^{-14}$ | $2.2\times 10^{12}$ |  $2.5\times 10^{-18}$ 33footnotemark: 3

###  II.4 Adiabatic curves

()  ()  ()  ()

Figure 2: Adiabats correlating asymptotically to partial waves $L=0,2,4,6$ at
the incoming threshold (1,0)+(1,0). Thick lines in each panel is the adiabat
for the s-wave channel. Panels (a) and (b) show adiabats for a value of
$\tilde{b}$ appropriate for RbCs at two different values of the reduced
electric field $F\mu/b$. Panels (c) and (d) show adiabats for a larger
$\tilde{b}$ (appropriate for NaRb) with the same two values of $F\mu/b$. The
insets show expanded views of the attractive region of the incoming s-wave
channel.

Figure 2 shows the adiabats correlating with the incoming pair state
(1,0)+(1,0), in reduced units, for values of $\tilde{b}$ appropriate for NaRb
and RbCs. The adiabats are shown for reduced electric fields of 3.4 (where
shielding is typically most effective) and 3.7 (where shielding is less
effective for RbCs). When (0,0)+(2,0) lies slightly below (1,0)+(1,0), as
here, the adiabats for (1,0)+(1,0) are repulsive at distances of a few hundred
bohr due to mixing with the lower threshold. However, the adiabat for s-wave
scattering ($L=0$) is attractive at larger distances. This arises because
$\hat{H}_{\textrm{dd}}$ mixes channels with $\Delta L=\pm 2$ arising from the
same threshold; since $\hat{H}_{\textrm{dd}}$ is proportional to $R^{-3}$ and
the separation between such channels is proportional to $R^{-2}$, this
produces an attraction of the form $-C_{4}/R^{4}$ at long range, with
coefficient $C_{4}=(-2/15)(d/\mu)^{4}E_{3}R_{3}^{4}$, where $d$ is the space-
fixed dipole induced by the electric field.

In reduced units, the coefficient of the long-range attraction is the same for
any polar molecule at the same reduced field. However, the repulsion depends
on the separation of rotational levels, and in reduced units the leading term
is proportional to $\tilde{b}^{-1}$. Because of this, the attraction persists
to shorter reduced distance for larger $\tilde{b}$, and the resulting well is
deeper in reduced energy units. Figure 2 shows that, in reduced units, it is 2
orders of magnitude deeper for NaRb than for RbCs. This is the origin of the
molecule-dependent effects on scattering length described below. However, it
should be noted that the effects on range and depth are often reversed in
absolute units.

The height of the repulsive barrier also depends on $\tilde{b}$ and reduced
field. The finite height arises because there are additional adiabats that
come down from higher thresholds and undergo avoided crossings with those from
(1,0)+(1,0). The top of the barrier is the lower limb of such an avoided
crossing. The relative energies of the higher thresholds, in reduced units,
are proportional to $\tilde{b}$ and decrease as a function of reduced field.
The barrier maximum is thus at larger reduced distance for smaller $\tilde{b}$
or larger reduced field, and the barrier is correspondingly lower. Figure 2
shows that, in reduced units, the barrier is about a factor of 40 higher for
NaRb than for RbCs at $F\mu/b=3.7$. This is responsible for the more effective
shielding for larger $\tilde{b}$ described below.

##  III Results

This section describes the results of coupled-channel calculations of shielded
collisions for a variety of systems. We consider a sequence of bosonic alkali-
metal diatomic molecules, in order of increasing $\tilde{b}$. This corresponds
to increasing depth of the long-range well when expressed in reduced energy
units. We consider both the effectiveness of shielding and the elastic
scattering properties, including the real part $\alpha$ of the scattering
length $a$. For many-body properties, this will play the same role as the
dipole-dependent scattering length of Ronen _et al._ [32]. In each case we
identify interesting new physics that is made accessible by the special
properties of the system concerned.

We use a fixed collision energy $E_{\textrm{coll}}=10$ nK$\times
k_{\textrm{B}}$ in all calculations in the present work. This is a reasonable
lower limit for the temperatures that are likely to be experimentally
accessible, and where Bose-Einstein condensation may occur. It is nevertheless
quite large compared to the characteristic energies of dipolar collisions of
molecules with high $\tilde{b}$, such as NaCs and the Ag systems; at the
fields where shielding is most efficient, the space-fixed dipole $d$ is about
$0.13\mu$, so the characteristic energy is about $(\mu/d)^{4}E_{3}\approx
3500E_{3}$ [11].

The rate coefficients for inelastic and short-range loss are generally
dominated by s-wave scattering; they vary weakly or decrease with collision
energy between 10 nK and 1 $\mu$K. Those for elastic scattering have an s-wave
contribution that may depend strongly on the field, together with substantial
contributions from higher partial waves; both contributions are proportional
to $E_{\textrm{coll}}^{1/2}$ at low energy, though the s-wave contribution may
drop below this at $E_{\textrm{coll}}>3500E_{3}$ [12]. The ratio $\gamma$ of
elastic scattering to loss thus increases with collision energy in most cases.

###  III.1 RbCs

Figure 3: (a) Rate coefficients for two-body elastic collisions and loss
processes for RbCs as a function of electric field, and (b) real part
$\alpha(k_{0})$ of scattering length calculated at collision energy
$E_{\textrm{coll}}/k_{\textrm{B}}=10$ nK.

Figure 3a shows the calculated rate coefficients for inelastic, short-range
and total loss for RbCs as a function of electric field, in the shielding
region, together with the rate coefficient for elastic collisions. The results
shown are for 87Rb133Cs, but other isotopologues would give results almost
identical on the scale of the Figure. At a field of 2.7 kV/cm, the rate
coefficient for total loss is more than three orders of magnitude lower than
that for elastic collisions. At a density of $10^{13}$ cm-3, plausible for a
BEC, the calculated loss rate corresponds to a lifetime around 30 s.

For shielded collisions, even elastic collisions are governed by shielding
barriers that are mostly due to dipole-dipole forces and occur at long range.
The scattering lengths are therefore predicted reliably by the current
calculations. Figure 3b shows the real part $\alpha$ of the scattering length
as a function of field.

The attractive well for RbCs is not deep enough to support a bound state.
Despite this, the scattering length is positive in the shielding region, with
a value near 200 $a_{0}$ near 2.7 kV/cm. This arises because, in this system,
the inner turning point of the adiabat for $L=0$ is near 750 $a_{0}$; the
resulting excluded volume makes a positive contribution to the scattering
length, which is larger than the negative contribution due to the long-range
attraction. The overall value is favorable for sympathetic cooling, without
being so large that the sample will enter the hydrodynamic regime at low
densities. It is also favorable for producing a condensate that is stable (and
does not collapse) over a wide range of geometries. Furthermore, the lack of a
bound state outside the shielding barrier implies that 3-body recombination
cannot occur. However, 3-body inelasticity (to produce separated molecules
with one of them in a lower rotational state) is still possible.

###  III.2 NaK

Figure 4: (a) Rate coefficients for two-body elastic collisions and loss
processes for NaK as a function of electric field, and (b) real part
$\alpha(k_{0})$ of scattering length calculated at collision energy
$E_{\textrm{coll}}/k_{\textrm{B}}=10$ nK.

Figure 4a shows the calculated rate coefficients for elastic scattering and
loss for bosonic NaK. The results shown are for 23Na39K, but 23Na41K would
give results almost identical on the scale of the Figure. At a field of 7.1
kV/cm, the rate coefficient for total loss is about four orders of magnitude
lower than that for elastic collisions.

The real part $\alpha$ of the low-energy scattering length for NaK is shown in
Figure 4b. In this case $\alpha$ is negative at the center of the shielding
region, indicating that the negative contribution from the long-range
attraction is sufficient to overcome the positive contribution from the
shielding core. However, this is not simply because the attraction is stronger
for NaK than for RbCs. Indeed, in reduced units the attractive coefficient
$C_{4}/(E_{3}R_{3}^{4})$ is the same for all systems at the same reduced field
$F\mu/b$. However, as described in section II.4, the shielding repulsion is
proportional to $\tilde{b}^{-1}$, so that the attraction dominates to lower
reduced distance for larger $\tilde{b}^{-1}$. For NaK at 7.1 kV/cm, the inner
turning point of the lowest adiabat is at $R_{\textrm{in}}=0.005R_{3}=650\
a_{0}$, compared to $0.008R_{3}$ for RbCs at the same value of $F\mu/b$.

The negative value of $\alpha$ implies that condensates of NaK shielded by a
static field of 7.1 kV/cm will probably be unstable with respect to collapse,
except perhaps in strongly anisotropic traps dominated by side-to-side dipolar
repulsions [32]. However, the scattering length is controllable, with zeroes
near 6.90 kV/cm and 7.75 kV/cm. Shielding is still effective at 6.90 kV/cm,
with a ratio $\gamma\sim 2500$ between the rate coefficients for elastic
scattering and loss, but less effective at 7.75 kV/cm, with $\gamma\sim 25$.
It may be noted that $\gamma$ usually increases approximately as
$E_{\textrm{coll}}^{1/2}$ at low energy [10]. The space-fixed dipole of NaK is
$-0.38$ D at 6.90 kV/cm and $-0.29$ D at 7.75 kV/cm. This should permit the
formation of dipolar condensates with positive or slightly negative scattering
lengths, and allow exploration of their stability as a function of geometry.
The regions close to zeroes are particularly interesting, because they
correspond to purely dipolar condensates with only weak non-dipolar
interactions, so that the dipolar interactions will be particularly important.

A further possibility arises from tuning the molecular dipole moment itself.
Vibrationally excited states of NaK with vibrational quantum number $v$ have
modified dipole moments $\mu_{v}$, usually reduced from those of the ground
state. These may be estimated using the dipole moment functions of ref. [44].
The rotational constant $b_{v}$ varies with $v$ as well [37]. The key quantity
is $\tilde{b}_{v}$, which is proportional to $b_{v}\mu_{v}^{4}$. Figure 5
shows a calculation for $v=25$, with $b_{v}\approx 0.85b_{0}$ and $\mu_{v}$
estimated as $0.95\mu_{0}$, corresponding to $\tilde{b}_{v}=5.7\times 10^{8}$.
The ratio $b_{v}/\mu_{v}$ is smaller for $v=25$ than for $v=0$, so effective
shielding occurs at somewhat lower fields. In this case the zeroes in $\alpha$
occur at 6.31 and 6.70 kV/cm, where shielding is very effective ($\gamma\sim
3200$ and $650$, respectively).

Figure 5: (a) Rate coefficients for two-body elastic collisions and loss
processes for NaK ($v=25$) as a function of electric field, and (b) real part
$\alpha(k_{0})$ of scattering length calculated at collision energy
$E_{\textrm{coll}}/k_{\textrm{B}}=10$ nK.

Fermionic molecules are also of great interest. Schindewolf _et al._ [20] have
recently succeeded in cooling fermionic 23Na40K to quantum degeneracy by
evaporative cooling assisted by microwave shielding. Figure 6 shows the
calculated rate coefficients for elastic scattering and loss for this system
in the presence of a static electric field. The calculation is the same as for
bosons, except that symmetrization with respect to exchange is handled
differently and the sum over $L$ is taken over odd values. As expected,
shielding is even more effective than for bosonic NaK, because s-wave
scattering is forbidden and all channels have a centrifugal barrier that
itself suppresses close collisions. In this case $\gamma$ reaches $10^{7}$ at
electric fields near 7.1 kV/cm.

Figure 6: Rate coefficients for two-body elastic collisions and loss processes
for ground state fermionic NaK as a function of electric field calculated at
collision energy $E_{\textrm{coll}}/k_{\textrm{B}}=10$ nK.

###  III.3 NaRb and NaCs

()  ()  ()  ()

Figure 7: Rate coefficients for two-body elastic collisions and loss processes
as a function of electric field for (a) NaRb and (b) NaCs calculated at
$E_{\textrm{coll}}/k_{\textrm{B}}=10$ nK. For NaCs at 2.43 kV/cm $<F<2.53$
kV/cm the results are limited by numerical precisions and the dashed lines
show upper bound to the loss rates. Real part $\alpha(k_{0})$ of scattering
length for (c) NaRb and (d) NaCs. The binding energies $E_{\textrm{b}}$ of the
tetramer bound states are shown as grey lines.

Very recently, evaporative cooling has been achieved for bosonic NaRb [22] and
NaCs [21] by means of microwave shielding. Here we consider the effect of
static-field shielding for these two molecules. NaRb and NaCs are more polar
than RbCs and NaK and thus have higher values of $\tilde{b}$, which in turn
produces more effective shielding. Figs. 7a and 7c show the calculated rate
coefficients for NaRb and NaCs as a function of electric field. The results
shown are for 23Na87Rb and 23Na133Cs, but other isotopologues would give
results almost identical on the scale of the Figure. For NaRb at an electric
field of 4.5 kV/cm, the rate coefficient for total loss is about 7 orders of
magnitude lower than that for elastic collisions. For NaCs at an electric
field of 2.5 kV/cm, the corresponding ratio $\gamma$ is at least 10 orders of
magnitude (and is probably substantially more, because our calculated loss
rates are limited by numerical precision between 2.43 kV/cm and 2.53 kV/cm).

Figures 7b and 7d show the real parts $\alpha$ of the low-energy scattering
lengths for NaRb and NaCs as a function of electric field. For both these
molecules, the long-range attraction is strong enough that the potential well
supports a bound state at the center of the shielding region. The well depth
decreases towards the edge of the shielding regions, and there are poles in
the scattering lengths at the fields where the state crosses threshold. For
NaRb, these appear at 4.38 and 4.57 kV/cm, with $\gamma\sim 10^{7}$ in both
cases. For NaCs they appear at 2.37 and 2.97 kV/cm, with $\gamma\sim 10^{8}$
and 4, respectively. In addition, for NaCs there are zeroes in the scattering
length at 2.395 and 2.725 kV/cm, where $\gamma\sim 10^{9}$ and $10^{4}$,
respectively. These will offer similar opportunities to those afforded by the
zeroes in NaK, but in regions of higher shielding and in the presence of a
single weakly bound state. There is also good shielding ($\gamma\sim 5000$) at
the zero at 2.356 kV/cm, where there is no weakly bound state and 3-body
recombination cannot occur. However, this is very close to the field where the
pair levels (1,0)+(1,0) and (0,0)+(2,0) cross; as a result the derivative
$d\alpha/dF$ is very large, about 4000 $a_{0}$/(V/cm), so precise field
control would be needed to study this region experimentally.

Tuning the electric field across one of the poles will convert pairs of
diatomic molecules into weakly bound tetraatomic molecules, termed field-
linked states by Avdeenkov and Bohn [45]. This process has been discussed
theoretically by Quéméner _et al._ [46] for the case of microwave shielding,
and has been achieved experimentally for fermionic NaK by Chen _et al._ [47]
by varying the ellipticity of microwave radiation across an analogous
resonance. Quéméner _et al._ coined the term electroassociation, which seems
even more appropriate for association with a static electric field than with a
microwave field.

Figures 7c and 7d show the binding energies of the bound states for NaRb and
NaCs as a function of electric field. These are obtained by coupled-channel
bound-state calculations [48], with dissociation to the open channels
prevented by hard-wall boundary conditions at 50 and 20000 $a_{0}$ [49]. These
states are interesting because they offer a possible route into ultracold
cluster chemistry with 4-atom (and perhaps larger) metal clusters. However,
the existence of such weakly bound states may enhance 3-body recombination in
regions of field where they exist, and thus inhibit evaporative cooling at
those fields.

The states of the tetraatomic collision complexes that causes the resonances
are formally quasibound rather than bound. They have finite lifetimes, even
below threshold, because they can decay through inelastic and short-range loss
processes. Under these circumstances there is no actual pole in $\alpha$, but
instead it undergoes an oscillation of amplitude $\pm a_{\textrm{res}}/2$
[33]. Here $a_{\textrm{res}}$ is very large, about $4\times 10^{10}$ and
$3\times 10^{10}$ $a_{0}$ for the low-field and high-field features in NaRb
and $4\times 10^{7}$ and 50000 $a_{0}$ for the corresponding features in NaCs.
This still provides a very wide range of tuning of $\alpha$, and for
simplicity we refer to the features as poles in the current work. The lifetime
$\tau$ of the state below resonance is

$\tau=\frac{\hbar}{\Gamma_{\textrm{loss}}},$ (10)

where $\Gamma_{\textrm{loss}}$ is the energy width of the quasibound state. We
have obtained $\Gamma_{\textrm{loss}}$ by locating the resonant signature in
the eigenphase sum from a coupled-channel scattering calculation below the
incoming threshold, as described in ref. [50]. At the field where the binding
energy is greatest, we obtain lifetimes greater than 100 ms for NaRb and 1000
s for NaCs.

###  III.4 KAg and CsAg

()  ()  ()  ()

Figure 8: Rate coefficients for two-body elastic collisions and loss processes
as a function of electric field for (a) NaRb and (b) NaCs calculated at
$E_{\textrm{coll}}/k_{\textrm{B}}=10$ nK. Results at electric fields within
the range 1.52 kV/cm to 1.68 kV/cm for KAg and 0.547 kV/cm to 0.559 kV/cm for
CsAg are limited by numerical precision and the dashed lines show upper bound
to the loss rates. Real part $\alpha(k_{0})$ of scattering length for (c) KAg
and (d) CsAg. These systems support up to 3 and 4 tetramer bound states,
respectively; the binding energies of the deeper states are shown as grey
lines in panels (c) and (d).

Diatomic molecules formed from an alkali-metal atom and Cu or Ag are
isoelectronic with alkali-metal dimers, but are expected to have larger dipole
moments [42] and correspondingly larger values of $\tilde{b}$. In this
subsection we investigate electric-field shielding of KAg and CsAg. We
consider bosonic molecules and use weighted-average atomic masses for K and
Ag. Figures 8a and 8b show the resulting rate coefficients as a function of
electric field. The rate coefficient for total loss is at least 12 orders of
magnitude lower than that for elastic scattering.

Figures 8c and 8d show the real parts $\alpha$ of the scattering lengths for
KAg and CsAg. There are 3 bound states for KAg in the middle of the shielding
region and 4 for CsAg. These bound states enter and leave the potential well
as a function of electric field. However, the outermost poles in each case are
strongly suppressed by loss processes, and the corresponding states will be
short-lived in this region. Nevertheless, there are multiple poles and zeroes
for each system in the range of fields where shielding is effective, so the
scattering length is highly tunable. For KAg, for example, the scattering
length passes through zero at $F=1.54$ kV/cm, where $\gamma>10^{12}$.

##  IV Conclusions

We have investigated the use of a static electric field to shield polar
molecules from short-range collisions that cause trap loss. The shielding is
due to dipole-dipole interactions between the molecules, which can be made
repulsive by engineering a near-degeneracy between two pair states with
different rotational quantum numbers [8, 9]. We have explored the behavior of
a representative range of ultracold polar molecules of current experimental
interest: RbCs, NaK, NaCs, KAg and CsAg. Static-field shielding has distinct
differences from microwave shielding [13, 14, 15], and may be advantageous in
some cases.

Important differences between molecules arise because the dipole-dipole
interaction produces a long-range attraction in the effective potential
(adiabat) for s-wave scattering. This attraction exists at intermolecular
distance _outside_ the repulsion that produces shielding. Without the
attraction, the repulsive core of the shielding interaction would produce a
large positive scattering length. However, the attraction adds a negative
contribution and allows substantial tuning of the scattering length with
electric field. If the attraction is strong enough, the resulting long-range
well can support weakly bound states.

The attractive and repulsive parts of the interaction are affected differently
by applied field. In many cases, the same scattering length can be achieved
with several different applied fields, corresponding to different space-fixed
dipole moments. This will make it possible to investigate many-body properties
as independent functions of scattering length and dipole moment.

For RbCs, microwave shielding is impractical, at least on the lowest
rotational transition, because the required wavelength is very long and the
antennas needed to generate the radiation are very large. In addition, very
strong microwave fields are needed. However, static-field shielding occurs at
easily accessible electric fields (near 2.7 kV/cm) and is quite effective,
with a ratio of 2-body elastic to loss rates $\gamma$ about $10^{3}$ at 10 nK
and $10^{4}$ at 1 $\mu$K. For RbCs, the long-range attraction is sufficient to
reduce the scattering length to moderate positive values (${\sim}200\ a_{0}$)
that are very favorable for evaporative cooling towards quantum degeneracy.
Such values are also favorable for forming a dipolar Bose-Einstein condensate
(BEC) that will not collapse [32]. Moreover, the absence of a long-range bound
state means that 3-body recombination cannot occur.

For NaK, static-field shielding requires larger fields than for RbCs (about 7
kV/cm) and is even more effective than for RbCs ($\gamma>10^{4}$ at 10 nK).
The long-range attractive well is deeper than for RbCs, but not deep enough to
support a bound state. The scattering length is negative in the region where
shielding is strongest, but there are zeroes in the scattering length as a
function of field, including one at 6.90 kV/cm where $\gamma$ is ${\sim}2500$.
This may provide opportunities to study the stability of a dipolar BEC as the
scattering length is tuned from positive to negative values. In excited
vibrational states of NaK, with smaller rotational constants, the zeroes in
the scattering length move towards the fields where shielding is most
effective. Once again, the absence of a long-range bound state precludes
3-body recombination.

For NaRb and NaCs, the attractive well is deep enough to support a single
bound state at the center of the shielding region. This state is very shallow
for NaRb (binding energy $E_{\textrm{b}}$ up to 60 Hz$\times h$), but rather
deeper for NaCs ($E_{\textrm{b}}$ up to 40 kHz$\times h$). For each molecule
there are two poles in the scattering length as the bound state crosses
threshold, and for NaCs there are additional zeroes in the scattering length
between the poles, in the region where a long-range bound state exists. The
poles may allow electroassociation to form tetraatomic molecules and the study
of Efimov physics in dipolar systems, while the additional zeroes provide
opportunities to study the stability of dipolar BECs in the presence of 3-body
recombination.

For KAg and CsAg, and other molecules with very large dipoles, the attractive
well is deep enough to support multiple bound states. There are multiple poles
in the scattering length where the states cross threshold and multiple zeroes
between the poles. Such systems will provide opportunities to study the
behavior of dipolar BECs under a wide range of conditions, and to study the
effects of scattering length and space-fixed dipole independently.

In conclusion, static-field shielding provides an important tool for shielding
dipolar molecules from destructive collisions and for controlling their
scattering length. It is complementary to microwave shielding, with some
similarities but also important differences. Different molecules show quite
different behavior, and static-field shielding offers access to much new
physics that is so far unexplored.

## Rights retention statement

For the purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising
from this submission.

## Data availability statement

Data supporting this study are openly available from Durham University [51].

## Acknowledgement

We are grateful to Matthew Frye and Ruth Le Sueur for valuable discussions.
This work was supported by the U.K. Engineering and Physical Sciences Research
Council (EPSRC) Grant Nos. EP/W00299X/1, and EP/V011677/1.

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