# Rotational state dependence of interactions between polar molecules

Etienne F. Walraven  Institute for Molecules and Materials, Radboud
University, Nijmegen, The Netherlands  Tijs Karman<EMAIL_ADDRESS>Institute for Molecules and Materials, Radboud University, Nijmegen, The
Netherlands

###### Abstract

The long-range electrostatic interactions between molecules depend strongly on
their relative orientation, which manifests as a rotational state dependence.
Interactions between molecules in the same rotational quantum state are well-
known attractive rotational van der Waals interactions. Interactions in
rotational states that differ by one quantum show resonant dipole-dipole
interactions. We show that where molecules are in rotational states that
differ by more than one quantum, they exhibit repulsive van der Waals
interactions. At temperatures below a millikelvin, this effect can reduce
collisional loss by multiple orders of magnitude. These repulsive interactions
lead to applications in quantum simulation and impurity physics with ultracold
polar molecules.

##  I Introduction

Well-separated molecules interact with one another through the Coulomb
potential set by their electric fields. This electric field can be described
as an expansion in terms of multipoles, such as the molecular charge, dipole
moment, quadrupole moment, and so on [1, 2]. The electrostatic interaction
between two molecules can then be expanded similarly, giving rise to dipole-
dipole, dipole-quadrupole, quadrupole-quadrupole interactions, and so forth.
Interactions involving higher multipole moments decay with higher inverse
powers of the intermolecular distance $R$. For instance, the interaction
between two charges scales with $1/R$ and dipole-dipole interactions with
$1/R^{3}$. More generally, when we label the two multipoles with ranks $l_{A}$
and $l_{B}$, then the interaction between two multipoles scales as
$1/R^{l_{A}+l_{B}+1}$. From here on, we only consider neutral molecules such
that dipole-dipole is the dominant contribution.

The electrostatic interaction depends on the relative orientation of the
molecules [3]. For example, the dipole-dipole interaction is attractive for
head-to-tail orientations, whereas it is repulsive for side-to-side approach.
Orienting or aligning molecules in space requires external fields [4, 5].
Without external fields, molecules are allowed to rotate freely such that in
the frame of the lab their average dipole moment is zero. This means that for
molecules in pure rotational states, the first-order dipole-dipole interaction
vanishes. One exception to this is when two molecules are in rotational states
that differ by one quantum. In this case the rotational excitation can reside
on either molecule, and the dipole-dipole interaction couples these two states
leading to a _resonant_ dipole-dipole interaction [6].

The dipole-dipole interaction can couple to excited states of the molecule-
molecule dimer, which in second-order perturbation theory gives rise to a van
der Waals interaction given by $-C_{6}/R^{6}$ [7] with van der Waals
coefficient $C_{6}$. We can distinguish between the contributions of virtual
excitations to electronically excited states, $C_{6}^{\mathrm{elec}}$, and
contributions of virtual excitations to different rotational states of both
molecules within the electronic ground state, $C_{6}^{\mathrm{rot}}$ [8]. The
electronic contribution is what is typically thought of as the van der Waals
interaction [1]. For many bialkali molecules the rotational contribution is
two orders of magnitude larger [9], due to the small rotational excitation
energy (0.1 cm-1) compared to electronic ones (10 000 cm-1). The electronic
contribution to the van der Waals interaction is almost always attractive,
which is a result of the spectrum of electronic states which generally becomes
denser for more highly excited states. We show that while the rotational van
der Waals attraction is also attractive for two molecules in the same
rotational state, the interaction between molecules in different rotational
states is always repulsive. This results directly from the structure of the
rotational energy levels which –unlike the electronic spectrum– becomes
sparser for higher excitations.

In a thermal gas of molecules, many different rotational states may be
populated and collisions between molecules happen at energies higher than
rotational energies. In such a situation, the collision dynamics is determined
by interactions at relatively short distances, where they become comparable to
the collision energy, and hence are strong compared to the energy difference
between rotational states. Therefore one typically does not think of the
interaction as first and second-order perturbative couplings between pairs of
rotational states [1]. However, it has become possible to prepare molecules in
a single rotational state and at ultracold temperatures of around a
microkelvin while having precise control over the molecules using external
fields [10, 11, 12, 13]. At these incredibly low temperatures and collision
energies, the collision dynamics is determined by long-range interactions. Of
particular interest are molecules that posses a sizeable permanent dipole
moment (e.g., NaK, KCs). For these molecules the rotational contribution to
the van der Waals interaction is by far dominant over the electronic one [9].

In order to create stable quantum gases of molecules that can be used as a
platform for quantum technologies, one needs to suppress the loss of molecules
due to collisions [14, 15, 16, 17, 18, 19]. Shielding molecules from
collisions by creating repulsive interaction potentials has been realized
using static electric fields [20, 21, 22, 23] and microwaves [24, 25, 26, 27,
28]. We show in this paper that loss can also be suppressed in collisions
between two molecules in different rotational states without the need of
external field control. We consider exploiting these interactions in several
applications. First, an exciting idea is to leverage these interaction towards
efficient forced evaporation of a fermi mixture of molecules in different
rotational states. This intuitively appears possible since collisions between
molecules in the same rotational state are suppressed by the $p$-wave
centrifugal barrier, and loss in $s$-wave collisions between distinguishable
molecules in different rotational states is strongly suppressed by the
repulsive interactions, while these still lead to elastic collisions and
thermalization. Unfortunately we find that realistically the ratio of good-to-
bad collisions remains limited. Promising applications that we consider are in
suppressing loss in on-site encounters of fermionic molecules in an optical
lattice for quantum simulation, as well as studying impurity physics with
controlled dipolar interactions while suppressing collisional loss.

The structure of this paper is as follows: Section II discusses qualitatively
the dipole-dipole interaction between molecules in different rotational states
in more detail, quantitatively calculates strength of the rotational van der
Waals interaction, describes higher multipoles as well as the electronic
contribution to the van der Waals interaction, and characterizes pairs of
rotational states that are degenerate for which the collision dynamics is
vastly different; Section III provides computational details, illustrates
interaction potentials for NaK and KCs for various rotational pairs, computes
collisional loss rates by means of quantum scattering calculations, shows that
a dipole can still be induced using static electric fields and microwaves, and
explains the loss rates observed for degenerate rotational pairs; Section IV
describes potential applications utilizing repulsive interactions for the
evaporation of fermi mixtures, fermions in optical lattices and impurity
physics; and Section V concludes this paper.

##  II Molecule-molecule interactions

###  II.1 Dipole-dipole interaction

The leading term in the multipole expansion of the electrostatic interaction
between two neutral molecules is the dipole-dipole interaction. Dipolar
interactions only couple rotational states $j$ and $j\pm 1$. For a pair of
molecules, this selection rule holds for each molecule separately. In second-
order perturbation theory, the interaction strength is determined by virtual
dipole-dipole (de-)excitation to other states. This strength is proportional
to the square of the coupling and inversely proportional to the energy
difference between the states. This means that the interaction is dominated by
coupling to states closest in energy and that the sign of this energy
difference determines whether the interaction is attractive or repulsive.

The rotational energy ladder is given by $E_{j}=Bj(j+1)$, with rotational
constant $B$. In contrast to electronic states, where the excited levels
become more closely spaced, excited rotational levels become sparser. This
unique structure results in attractive interactions when $\Delta j=0$ and
repulsive ones when $\Delta j\geq 2$, where $\Delta j=|j_{A}-j_{B}|$ denotes
the difference in rotational states between the two molecules. We discuss the
separate cases in more detail.

The van der Waals interaction is always attractive for two molecules in the
same rotational state $j$. This is easiest to see when both molecules are in
their rotational ground state. When $j_{A}=j_{B}=0$, the only possible second-
order coupling is to a state higher in energy, where both molecules are
virtually excited to $j=1$. This results in an attractive interaction where
$C_{6}^{\mathrm{rot}}=d^{4}/6B$ with dipole moment $d$. If both are in the
same $j>0$ state, there are also contributions from de-excitation of one or
both molecules. Excitation of a molecule always yields a larger energy
difference $E_{j+1}-E_{j}=2B(j+1)$ than de-excitation $E_{j}-E_{j-1}=2Bj$ due
to the structure of the rotational energy ladder. The pair state that lies
closest to the $j_{A}=j_{B}=j$ state is the one where $j_{A}$ is excited and
$j_{B}$ is de-excited, or vice versa, which is higher in energy by $2B$, e.g.,
$E_{1}+E_{1}<E_{0}+E_{2}$. In second order, this leads to an attractive van
der Waals interaction when both molecules are in the same rotational state.

When $\Delta j=1$, e.g., the pair state $j_{A}+j_{B}=0+1$, they experience
dipole-dipole interactions in first order due to resonant interactions with
$1+0$. This interaction scales with $R^{-3}$ and results in fast collisions.

Figure 1: Schematic representation of rotational energy levels for (a) two
molecules in different rotational states (here, $j=1$ and $j=3$) and (b) the
combined pair (here, $j+j^{\prime}=1+3$). Different colors represent the
various ways of virtually (de-)exciting the molecules via dipole-dipole
interactions, creating strong second-order coupling to a state barely below
the original state.

For $\Delta j\geq 2$, the interaction is again second order. Since the
molecules are in different rotational states, it now matters whether one
excites the one in the lower rotational state $j_{<}$ and de-excite the higher
rotational state $j_{>}$, or vice versa. The smallest energy difference is
found to be excitation of the $j_{<}$ molecule and de-excitation of $j_{>}$.
This state lies below the state of interest by $\Delta E=-2B(j_{>}-j_{<}-1)$.
Figure 1 shows this schematically. In second order, this therefore results in
a repulsive interaction.

In addition to this qualitative understanding, we quantitatively determine the
rotational van der Waals coefficients, $C_{6}^{\mathrm{rot}}$, for various
pairs of rotational states. The dipole-dipole interaction between two
molecules is given by

$\hat{V}_{\mathrm{dd}}(R)=\frac{1}{4\pi\epsilon_{0}R^{3}}\left(\hat{\bm{d}}^{(A)}\cdot\hat{\bm{d}}^{(B)}-3\hat{d}_{z}^{(A)}\hat{d}_{z}^{(B)}\right)\,,$
(1)

in the body-fixed frame where the intermolecular axis lies along the $z$ axis.
The $C_{6}^{\mathrm{rot}}$ coefficient between rotational states $j_{A}$ and
$j_{B}$ can then be calculated using second-order perturbation theory as

$\sum_{n^{\prime}\neq n}\frac{\left|\langle
n^{\prime}|\hat{V}_{\mathrm{dd}}(R)|n\rangle\right|^{2}}{E_{n}-E_{n^{\prime}}}=-\frac{C_{6}^{\mathrm{rot}}}{R^{6}}.$
(2)

In practice we used a space-fixed basis set with functions
$|n\rangle=|(j_{A},j_{B}),j_{AB},L;JMp\rangle$ and energies
$E_{n}=B[j_{A}(j_{A}+1)+j_{B}(j_{B}+1)]$, see Sec. III.1 for a detailed
description. Here, the quantum number $j_{AB}$ represents the coupled angular
momentum of $j_{A}$ and $j_{B}$, $L$ the angular momentum number associated
with end-over-end rotation, $J$ the quantum number of the total angular
momentum, $M$ the total angular momentum along a space-fixed quantization
axis, and $p=\pm 1$ corresponds with the permutation symmetry of bosons ($+1$)
and fermions ($-1$) such that if $j_{A}=j_{B}$ only basis functions where
$p=(-1)^{j_{A}+j_{B}-j_{AB}+L}$ contribute. Since $j_{A}=j_{B}$ channels
couple by dipole-dipole only to channels with $|j_{A}-j_{B}|=0$ or $2$, the
differences between bosons and fermions disappear for $\Delta j>2$. The
electrostatic interactions are diagonal in $J$, $M$, $p$, and are independent
of $M$. For $L=0$ channels, $j_{AB}=J$ and hence $j_{AB}$ is effectively a
good quantum number within the $s$-wave channels.

Table 1: Rotational van der Waals $C_{6}^{\mathrm{rot}}$ coefficients for pairs of bosonic and fermionic molecules in various rotational states $j_{A}$ and $j_{B}$, where $j_{A}\geq j_{B}$, in units of $d^{4}/B$. Different values per pair correspond to various $j_{AB}$ in increasing order. |  | $j_{A}$
---|---|---
|  | 0 | 1 | 2
|  | bosons | fermions | bosons | fermions | bosons | fermions
$j_{B}$ | 0 | $+0.167$ | - |  |  |  |
1 | - | - |  | $+0.290$
---
$+0.104$
$+0.230$ |  |
|  |  |  |  | $+0.298$ |  $+0.274$ $+0.163$
2 | $-0.097$ | $-0.070$ | - | - | $+0.228$
|  |  |  |  | $+0.085$
|  |  | $-0.095$ | $-0.093$ |  |
3 | $-0.033$ | $-0.033$ | $-0.062$ | $-0.076$ | - | -
|  |  | $-0.062$ | $-0.027$ |  |
|  |  |  |  | $-0.097$ | $-0.097$
|  |  | $-0.036$ | $-0.036$ | $-0.083$ | $-0.086$
4 | $-0.019$ | $-0.019$ | $-0.027$ | $-0.027$ | $-0.074$ | $-0.065$
|  |  | $-0.019$ | $-0.019$ | $-0.042$ | $-0.061$
|  |  |  |  | $-0.053$ | $-0.014$
Figure 2: Rotational van der Waals $C_{6}^{\mathrm{rot}}$ coefficients on a
logarithmic scale for pairs of bosonic molecules in various rotational states
$j_{A}$ and $j_{B}$. For each pair, an average is taken over the various ways
of coupling both together.

We calculate $C_{6}^{\mathrm{rot}}$ for the $L=0$ states, although in the
basis we also include $L=2$ to account for dipole-dipole coupling. The values
for different pairs of $j_{A}$ and $j_{B}$ in units of $d^{4}/B$ are given in
Table 1 and a visual representation is shown in Fig. 2. Due to the anisotropy
of the dipole-dipole interaction, the second-order interaction depends on
$j_{AB}$ for $j_{A},j_{B}>0$. In the figure we have averaged over the various
possible ways of coupling to $j_{AB}$ for each pair and only show results for
bosonic molecules, although the case of fermionic molecules is qualitatively
very similar. Moreover, since the dominant interaction for $\Delta j=1$ is
first order, not second order, these values are not given.

From Table 1 and Fig. 2 we can clearly see the attractive van der Waals
interaction when $j_{A}=j_{B}$. For all other pairs where $\Delta j\geq 2$ the
interaction is repulsive. This decreases in strength as $\Delta j$ increases,
since the energy difference between the prepared and virtually (de-)excited
states increases.

###  II.2 Higher multipoles and electronic contribution

Higher-order multipole-multipole interactions also play a role, although these
are typically weaker than the dipole-dipole term. For a multipole of rank $l$,
the selection rules that couple a rotational state $j$ to some state $k$ are
given by the the triangular condition $|j-k|\leq l\leq j+k$ and the parity
rule that $j+k+l$ must be even. Therefore, if both $j,k>0$, there are various
multipoles that contribute to the full electrostatic interaction.

At first glance, it would seem that for states where both $j_{A},j_{B}>0$
quadrupole-quadrupole coupling is possible in first order, which would scale
as $R^{-5}$. However, in the $s$-wave channel this anisotropic interaction has
zero expectation value and the interaction couples the $s$-wave to the
$g$-wave channel such that in second order the interaction only scales as
$R^{-8}$.

Besides the electrostatic contribution to the van der Waals interaction, there
is still the electronic contribution. For bialkalis with a sizeable dipole
moment, typical electronic and rotational contributions to the van der Waals
coefficients are $C_{6}^{\mathrm{elec}}\sim 10^{4}$ a.u. and
$C_{6}^{\mathrm{rot}}\sim d^{4}/B\sim 10^{6}$ a.u., such that the electronic
contribution is generally two orders of magnitude smaller, although still
present [9]. This holds for molecules such as NaK and KCs, but not for KRb due
to its small dipole moment. In this work we include this as an isotropic
interaction that is the same for all rotational states. When $\Delta j\geq 2$,
the strength of the repulsive interaction decreases for larger $\Delta j$,
while the electronic contribution remains constant. Therefore, for larger
$\Delta j$, the electronic contribution is important and can completely
suppress the repulsive part of the rotational contribution.

###  II.3 Pair degeneracies

For a pair of linear molecules of the same species, the combined states
$j+j^{\prime}$ can become degenerate with another pair of $k+k^{\prime}$
rotational states. When this is the case, the coupling between the two states
does not occur in second order, but in first order, which drastically changes
the interaction and resulting collision rates. Here, we derive which pairs of
rotational pair states are degenerate, and we discuss the physically most
relevant degeneracies.

If two molecules are the same species, i.e., both have the same rotational
constant $B$, then degeneracy, which we denote by $j+j^{\prime}\equiv
k+k^{\prime}$, holds if $E_{j}+E_{j^{\prime}}=E_{k}+E_{k^{\prime}}$, which
gives

$j(j+1)+j^{\prime}(j^{\prime}+1)=k(k+1)+k^{\prime}(k^{\prime}+1)$ (3)

for which we can find integer solutions. By completing squares and using
$a^{2}-b^{2}=(a-b)(a+b)$ one arrives at

$(j-k)(j+k+1)=(k^{\prime}-j^{\prime})(k^{\prime}+j^{\prime}+1)\,.$ (4)

We then define $l=j-k$ and $l^{\prime}=k^{\prime}-j^{\prime}$, which
correspond to the lowest rank multipoles that couple the rotational states. We
assume without loss of generality that $j>k$ and $k^{\prime}>j^{\prime}$. Both
sides of Eq. (4) equal the same integer, where the left-hand side is divisible
by $l$ and the right-hand side by $l^{\prime}$. This means that any factors
that divide $l$ but not $l^{\prime}$, should divide $k^{\prime}+j^{\prime}+1$.
Equation (4) must therefore equal an integer of the form
$nll^{\prime}/\mathrm{gcd}(l,l^{\prime})$, where $n$ is any positive integer
and $\mathrm{gcd}(l,l^{\prime})$ the greatest common divisor of $l$ and
$l^{\prime}$. This leads to $j+k+1=nl^{\prime}/\mathrm{gcd}(l,l^{\prime})$ and
$k^{\prime}+j^{\prime}+1=nl/\mathrm{gcd}(l,l^{\prime})$, and together with the
definitions of $l$ and $l^{\prime}$ this yields

$\displaystyle j$ $\displaystyle=k+l\,,$ (5) $\displaystyle j^{\prime}$
$\displaystyle=\frac{1}{2}\left(\frac{nl}{\mathrm{gcd}(l,l^{\prime})}-l^{\prime}-1\right)\,,$
$\displaystyle k$
$\displaystyle=\frac{1}{2}\left(\frac{nl^{\prime}}{\mathrm{gcd}(l,l^{\prime})}-l-1\right)\,,$
$\displaystyle k^{\prime}$ $\displaystyle=j^{\prime}+l^{\prime}\,.$

These only give integer solutions if

  1. 1.

both $l,l^{\prime}$ are odd and $n$ is even, or

  2. 2.

$l$ is odd, $l^{\prime}$ is even (or vice versa) and $n$ is odd, or

  3. 3.

both $l,l^{\prime}$ are even, containing the same power of 2 in their prime
factorization, and $n$ is odd.

The last requirement shows, for example, that for $l=2$ and $l^{\prime}=4$
there are no degeneracies, while for $l=2$ and $l^{\prime}=6$ there are.
Positivity of all quantum numbers is ensured in both cases if
$n\geq\mathrm{gcd}(l,l^{\prime})\max(\frac{l+1}{l^{\prime}},\frac{l^{\prime}+1}{l})$
and $l,l^{\prime}\geq 1$. Given $l$ and $l^{\prime}$ as restricted above, one
can then generate a family of degenerate states parametrized by $n$.

Equation (3) can be extended to higher degrees of degeneracies, where finding
different degenerate pairs becomes mathematically equivalent to finding the
different ways of writing an integer as a sum of two squares of odd numbers.
For example, for a three-fold degeneracy, the lowest integer for which this
holds is 650, resulting in $8+9\equiv 5+11\equiv 2+12$.

We find all degenerate pairs that are coupled by dipole-dipole interactions,
where $l=l^{\prime}=1$. From the recipe above these are given by

$\displaystyle 1+0$ $\displaystyle\equiv 0+1\,,$ $\displaystyle 2+1$
$\displaystyle\equiv 1+2\,,$ $\displaystyle 3+2$ $\displaystyle\equiv
2+3\,,\text{ etc.}$

In general, for $l=l^{\prime}$, one finds the trivial solutions
$j+j^{\prime}\equiv j^{\prime}+j$, which correspond to the known resonant
multipole-multipole interactions.

For dipole-quadrupole coupling, where $l=1$ and $l^{\prime}=2$, we find non-
trivial degenerate pairs. Following the recipe above, we must take
$n=3,5,7,...$ . The degenerate pairs are then of the form
$n+\frac{n-3}{2}\equiv\frac{n+1}{2}+(n-1)$. So, for increasing $n$ we find the
degenerate pairs

$\displaystyle 3+0$ $\displaystyle\equiv 2+2\,,$ $\displaystyle 5+1$
$\displaystyle\equiv 3+4\,,$ $\displaystyle 7+2$ $\displaystyle\equiv
4+6\,,\text{ etc.}$

Exact degeneracies as described above only hold for rigid rotors, where no
centrifugal distortion is taken into account. In reality, centrifugal forces
pull the atoms apart and the usual equation for the rigid rotor energy levels
is extended to

$E_{j}=Bj(j+1)-Dj^{2}(j+1)^{2}\,,$ (6)

with $D\ll B$ the centrifugal distortion constant. This splits the
‘degenerate’ states by a small amount compared to the rotational level
spacing. The interaction potentials then contain a crossing between the two
coupled states, which can significantly change the dynamics of the collision
compared to non-degenerate states.

Of special interest is the $3+0\equiv 2+2$ degeneracy. The centrifugal
distortion lifts this degeneracy such that the $3+0$ state is slightly lower
in energy. The dipole-dipole interaction leads to an attractive van der Waals
interaction for the $2+2$ state and a repulsive van der Waals interaction for
the $3+0$ state. The dipole-quadrupole coupling between these nearly
degenerate states creates a repulsion between them, possibly stabilizing the
upper $2+2$ state and destabilizing the $3+0$ state. We explore this idea
quantitatively in section III.5.

##  III Interaction potentials and collisions

###  III.1 Computational details

The colliding molecules $X=A,B$ are modeled as rigid rotors with monomer
Hamiltonian

$\hat{H}^{(X)}=B\left[\hat{j}^{(X)}\right]^{2}-D\left[\hat{j}^{(X)}\right]^{4}-\hat{\bm{d}}^{(X)}\cdot\bm{E}_{\text{static}}+\hat{H}_{\text{ac}}^{(X)}\,,$
(7)

where the first term describes the rotational kinetic energy, the second term
the centrifugal distortion, the third the Stark interaction between the
molecular dipoles and a static electric field, and the fourth the interaction
with a microwave field. These last two terms are only included in those
calculations where a static electric field or a microwave field is used. For a
more detailed description of this Hamiltonian and subsequent scattering
calculations we refer to the Supplemental Material of Ref. [24]. Note that we
have not included hyperfine structure in these calculations as the hyperfine
energy scale is negligible compared to rotational energy-level structure.

The total dimer Hamiltonian is given by

$\displaystyle\hat{H}=$
$\displaystyle-\frac{\hbar^{2}}{2\mu}\frac{1}{R}\frac{\mathrm{d}^{2}}{\mathrm{d}R^{2}}R+\frac{\hat{L}^{2}}{2\mu
R^{2}}+\hat{H}^{(A)}+\hat{H}^{(B)}$
$\displaystyle+\hat{V}_{\mathrm{dd}}(R)+\hat{V}_{\mathrm{dq}}(R)+\hat{V}_{\mathrm{qq}}(R)-\frac{C_{6}^{\mathrm{elec}}}{R^{6}}\,,$
(8)

where $\mu$ is the reduced mass, $R$ the intermolecular distance, $\hat{L}$
the angular momentum operator associated with the end-over-end rotation, and
the dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole interactions
given by

$\displaystyle\hat{V}_{\mathrm{dd}}$
$\displaystyle=-\frac{\sqrt{30}d^{2}}{R^{3}}\left[\left[C_{1}\left(\hat{r}^{(A)}\right)\otimes
C_{1}\left(\hat{r}^{(B)}\right)\right]^{(2)}\otimes
C_{2}\left(\hat{R}\right)\right]^{(0)}_{0},$
$\displaystyle\hat{V}_{\mathrm{dq}}$
$\displaystyle=\frac{\sqrt{105}dQ}{R^{4}}\Bigg{\\{}\left[\left[C_{1}\left(\hat{r}^{(A)}\right)\otimes
C_{2}\left(\hat{r}^{(B)}\right)\right]^{(3)}\otimes
C_{3}\left(\hat{R}\right)\right]^{(0)}_{0}$ $\displaystyle-$
$\displaystyle\left[\left[C_{2}\left(\hat{r}^{(A)}\right)\otimes
C_{1}\left(\hat{r}^{(B)}\right)\right]^{(3)}\otimes
C_{3}\left(\hat{R}\right)\right]^{(0)}_{0}\Bigg{\\}},$
$\displaystyle\hat{V}_{\mathrm{qq}}$
$\displaystyle=\frac{\sqrt{630}Q^{2}}{R^{5}}\left[\left[C_{2}\left(\hat{r}^{(A)}\right)\otimes
C_{2}\left(\hat{r}^{(B)}\right)\right]^{(4)}\otimes
C_{4}\left(\hat{R}\right)\right]^{(0)}_{0},$ (9)

where $C_{l}(\hat{r})$ is the rank $l$ tensor whose spherical components are
the Racah normalized spherical harmonics depending on the polar angles of the
vector $\hat{r}$, and

$\displaystyle\left[\hat{A}_{k_{A}}\otimes\hat{B}_{k_{B}}\right]^{(k)}_{q}=\sum_{q_{A},q_{B}}\hat{A}_{k_{A},q_{A}}\hat{B}_{k_{B},q_{B}}\langle
k_{A}q_{A}k_{B}q_{B}|kq\rangle$ (10)

is the $q$ spherical component of the rank $k$ irreducible tensor product of
the rank $k_{A}$ spherical tensor $\hat{A}$ and the rank $k_{B}$ spherical
tensor $\hat{B}$.

We make use of an uncoupled basis set, where we express the basis functions of
the monomers using angular momentum states $|jm_{j}\rangle$. The end-over-end
rotation of the molecules about one another is described by the state
$|LM_{L}\rangle$. Lastly, in the case where we include microwaves, the basis
also contains a photon state $|N\rangle$ representing the number of photons
taken up by the complex. The full dimer basis functions are then of the form

$|j^{(A)}m^{(A)}\rangle|j^{(B)}m^{(B)}\rangle|LM_{L}\rangle|N\rangle.$ (11)

The resulting basis can be reduced by adapting it to permutation symmetry and
restricting good quantum numbers [24]. We perform separate calculations for
the two parities of $(-1)^{j_{A}+j_{B}+L}$ as well as for different values of
$M_{\mathrm{tot}}=m^{(A)}+m^{(B)}+M_{L}$. For convergence, we have used a
basis up to $L_{\mathrm{max}}=7$, $|M_{L,\mathrm{in}}|_{\mathrm{max}}=1$ and
$j_{\mathrm{max}}=\max(j^{(A)}_{\mathrm{in}},j^{(B)}_{\mathrm{in}})+2$ for all
calculations, with ‘in’ representing the initially prepared rotational state.
For states where $\Delta j=1$, we have converged the results up to
$|M_{L,\mathrm{in}}|_{\mathrm{max}}=5$ due to the strong resonant coupling.
When microwaves are included the basis also contains $N=0$ and $-1$. For the
results including electric fields or microwaves results are converged by
including up to $j_{\mathrm{max}}=3$ and
$|M_{L,\mathrm{in}}|_{\mathrm{max}}=3$.

Using the Hamiltonian as defined in Eqs. (7) and (8) and the dimer basis set
(11), we calculate loss rates by coupled-channels scattering calculations
using the renormalized Numerov method [29]. We propagate two sets of linearly
independent solutions on a radial grid. At short range we impose absorbing
boundary conditions to model the short-range losses of ultracold molecules and
at long range $S$-matrix boundary conditions [30]. Calculations were converged
with respect to the radial grid, which ranges from 50 to 10 000 $a_{0}$ with
at least 10 points per local de Broglie wavelength. After propagation, this
results in a scattering $S$-matrix from which we get all state-to-state
scattering probabilities. Collisional loss depends on the probabilities of
short range and inelastic collisions. The cross sections for these loss
mechanisms follow from the $S$-matrix as

$\displaystyle\sigma_{\mathrm{short}}$
$\displaystyle=\frac{\pi}{k^{2}}\sum_{r,L,M_{L}}\left|T_{r;i,L,M_{L}}^{\mathrm{(short)}}\right|^{2}\,,$
(12) $\displaystyle\sigma_{\mathrm{inel}}$
$\displaystyle=\frac{\pi}{k^{2}}\sum_{f\neq
i,L^{\prime},M_{L}^{\prime},L,M_{L}}\left|T_{f,L^{\prime},M_{L}^{\prime};i,L,M_{L}}^{\mathrm{(long)}}\right|^{2}\,,$
(13)

where $\bm{T}=\bm{1}-\bm{S}$ and $k=\sqrt{2\mu E}$. The label $r$ stands for
all adiabatic channels that are classically allowed at the capture distance
and the labels $i$ and $f$ for the initial and final rotational states of the
molecules. When both molecules are in the same $|jm_{j}\rangle$ state, they
are indistinguishable and the cross sections should be multiplied by 2. The
energy-dependent cross-section,
$\sigma(E)=\sigma_{\mathrm{short}}+\sigma_{\mathrm{inel}}$, can then be
averaged over a Maxwell-Boltzmann distribution for a given temperature $T$ to
yield thermal rate coefficients

$k_{2}=\sqrt{\frac{8k_{B}T}{\pi\mu}}\frac{1}{(k_{B}T)^{2}}\int_{0}^{\infty}\sigma(E)\exp\left(-\frac{E}{k_{B}T}\right)E\;\mathrm{d}E\,.$
(14)

We integrate using 7 logarithmically spaced collision energies between $0.1T$
and $10T$. All results in this paper are given for $T=1$ $\mu$K.

The molecules that we study here are NaK and KCs [15, 31, 32]. Both have a
large dipole moment [33, 34], a small electronic contribution to the long-
range interaction [9], and both are non-reactive [35]. Moreover, due to the
available isotopes of K, both bosonic and fermionic NaK and KCs can be
produced. Note that the theory presented above applies generally for other
dipolar, linear molecules for which the rotational contribution to the van der
Waals interaction is dominant. The molecular parameters used for NaK are
$B=9.41\times 10^{-2}$ cm-1, $D=8\times 10^{-7}$ cm-1 [36], $d=2.72$ Debye
[33], $Q=10.6$ a.u. [37] and $C_{6}^{\mathrm{elec}}=8.28\times 10^{3}$ a.u.
[9], and for KCs we used $B=3.00\times 10^{-2}$ cm-1, $D=2.4\times 10^{-8}$
cm-1 [38], $d=1.92$ Debye [34], $Q=6.1$ a.u. and
$C_{6}^{\mathrm{elec}}=1.84\times 10^{4}$ a.u. [9], where for KCs we have
approximated $Q\approx dR_{e}$ with $R_{e}$ the equilibrium bond distance.

###  III.2 Interaction potentials

Figure 3: Interaction potential curves for bosonic (a) NaK $0+0$, (b) NaK
$0+1$, (c) NaK $0+2$, (d) NaK $1+3$, (e) NaK $1+4$ and (f) KCs $0+2$, with
dimer rotational states labeled as $j+j^{\prime}$ where
$m_{j}=m_{j^{\prime}}=0$. All potentials are referenced to the energy of the
initial state (solid blue lines) at infinite separation. Due to strong first-
order interactions for $0+1$, the energy is scaled by 100 in panel (b). The
pure second-order rotational interaction curve is given by the dashed light
blue curve. Addition of an isotropic electronic $C_{6}^{\mathrm{elec}}$
interaction is given for (c) and (f) by the dash-dotted green curve. Addition
of a quadrupole moment for dipole-quadrupole and quadrupole-quadrupole
interaction is represent in (c)-(f) by the dotted red curve.

We find adiabatic interaction potentials by diagonalizing the dimer
Hamiltonian (8), excluding the radial kinetic energy term, at fixed
intermolecular distances $R$. For a number of rotational state pairs for
bosonic NaK and KCs these potential energy curves are shown in Fig. 3. In all
panels, the solid blue curve corresponds to the incoming $j+j^{\prime}$ state
with $L=0$, whereas the thin gray curves correspond to higher $L$, or
different couplings between $j$ and $j^{\prime}$ that have higher barriers.
These solid lines are calculated including only the dipole-dipole interaction
but excluding the dipole-quadrupole, quadrupole-quadrupole and electronic
terms.

With both molecules in the ground rotational state ($j+j^{\prime}=0+0$) in
Fig. 3(a) we find the well-known attractive van der Waals interaction with
$C_{6}^{\mathrm{rot}}=d^{4}/6B$. For the $0+1$ pair, panel (b) demonstrates
the strong resonant $R^{-3}$ dipole-dipole interaction. Note that the energy
scale for this panel has been extended by two orders of magnitude. In panels
(c)-(e) we show the pairs $0+2$, $1+3$ and $1+4$, respectively, where $\Delta
j\geq 2$. As discussed in section II, this results in a repulsive rotational
van der Waals interaction at long range as shown by the dashed second-order
$R^{-6}$ curve. However, there is a distance below which the perturbative
approach breaks down. At very short distances, the molecules polarize one
another, and ultimately the lowest interaction potentials become attractive.
At intermediate distances, this results in a potential barrier that can shield
collisions at low enough collision energy. The main loss mechanism at low
collision energy then stems from tunneling through this barrier and reaching
the short range, which depends on the height and width of the barrier as well
as the mass of the colliding molecules. Panel (f) finally shows the $0+2$ pair
for bosonic KCs, where a similar, though lower barrier is compared to NaK.
Potential curves including the isotropic electronic van der Waals interaction
are shown in panels (c)-(f) as the dash-dotted lines. This decreases the
barrier heights and widths, though not by so much as to remove the barrier in
these cases. For state pairs with larger $\Delta j$, where the repulsive
$C_{6}^{\mathrm{rot}}$ becomes small compared to the attractive
$C_{6}^{\mathrm{elec}}$, the barrier disappears and higher losses are again
expected.

In Figs. 3(c) and (f) we also include dipole-quadrupole and quadrupole-
quadrupole interactions, which decrease the barrier width and height slightly.
This occurs even in cases where we expect no contribution to the $s$-wave
potential from perturbation theory. The electrostatic interaction cannot be
described perturbatively at the position of the barrier. The various states
then mix, where especially the rotational quantum numbers $j$ and $L$ cease to
be good quantum numbers.Level repulsion between the various states lowers the
lowest potential curve, and lowers the height of the potential barrier of
interest.

###  III.3 Collisional loss rates

Figure 4: Loss rates for various rotational state pairs $j+j^{\prime}$ with
$m_{j}=m_{j^{\prime}}=0$ for (a) bosonic NaK, (b) bosonic KCs, (c) fermionic
NaK and (d) fermionic KCs. Panel (a) contains loss rates where only the
dipole-dipole interaction is taken into account (dashed lines). The full
calculations further include contributions from dipole-quadrupole, quadrupole-
quadrupole, isotropic electronic $C_{6}^{\mathrm{elec}}$ and the centrifugal
distortion constant.

We calculated loss rates for a range of rotational state pairs $j+j^{\prime}$
with $m_{j}=m_{j^{\prime}}=0$ for bosonic and fermionic species of NaK and KCs
at 1 $\mu$K, which are shown in Fig. 4. The dashed line in panel (a) only
contains dipole-dipole interactions, whereas the full calculations in all
panels further include dipole-quadrupole, quadrupole-quadrupole, isotropic
electronic $C_{6}^{\mathrm{elec}}$ and centrifugal distortion terms. The last
term is important for the degenerate $0+3(\equiv 2+2)$ and $1+5(\equiv 3+4)$
states due to strong dipole-quadrupole interactions. We explore this further
in section III.5.

For $0+0$ collisions we find loss rates close to universal loss [39], as
expected. For bosonic and fermionic NaK, universal loss rates at 1 $\mu$K are
$6\times 10^{-10}$ and $6\times 10^{-11}$ cm3/s, respectively, and for KCs
$3\times 10^{-10}$ and $1\times 10^{-10}$ cm3/s. For fermions, the universal
loss rate scales linearly with temperature due to the presence of the $p$-wave
centrifugal barrier, and hence the rate is lower than for bosons. Similar loss
rates are found for $1+1$ collisions. The highest loss rates we find when
$\Delta j=1$ due to strong resonant dipole-dipole interactions, which holds
here for $0+1\equiv 1+0$ and $1+2\equiv 2+1$ pairs. For the states where
$\Delta j\geq 2$ one can see from the dipole-dipole calculations that loss
rates decrease rapidly by orders of magnitude as a function of $\Delta j$ as
the attractive interactions occur at shorter distances, creating a higher and
wider barrier.

Dipole-quadrupole, quadrupole-quadrupole and electronic terms lower the
barriers or even cause them to disappear, especially at larger $\Delta j$,
where the rotational contribution decreases. We find that the loss rates can
be orders of magnitude smaller than universal loss. Since the barrier for KCs
is generally lower than for NaK, loss rates for KCs at larger $\Delta j$ are
higher than for NaK.

Figure 5: Spread of loss rates for rotational state pairs $1+j^{\prime}$ for
different $m_{j}$ and $m_{j^{\prime}}$ for bosonic NaK. The green curve
corresponds to the dashed green curve in Fig. 4(a) and accounts only for the
dipole-dipole interaction.

The interaction between molecules is anisotropic for nonzero $j$ and
$j^{\prime}$, which leads to a dependence on $m_{j}$ and $m_{j}^{\prime}$. We
explore this dependence by calculating loss rates for bosonic NaK
$1+j^{\prime}$ states with different $m_{j}$ and $m_{j^{\prime}}$, as shown in
Fig. 5. The solid line in Fig. 5 corresponds to the dashed line in Fig. 4(a).
The various markers then show the spread in loss rates, which are at most an
order of magnitude, which still shows that irrespective of the choice of
$m_{j}$ and $m_{j}^{\prime}$, loss rates are suppressed for $\Delta j\geq 2$.

Figure 6: Scaling of loss rates for $j+j^{\prime}=0+2$ of (a) bosonic NaK and
(b) bosonic KCs with respect to mass, rotational constant, dipole moment and
quadrupole moment. These results also include an isotropic
$C_{6}^{\mathrm{elec}}$ contribution.

We investigate the dependence on various molecular parameters by scaling the
mass, the rotational constant, the dipole moment and the quadrupole moment.
Results are given in Fig. 6, where the two panels show results for NaK and
KCs, respectively. We note that both panels exhibit qualitatively similar
behavior. As the rotational constant is increased, the barrier shifts to
shorter distances with decreasing width and increasing height, changing the
loss rates only by a small factor. The effect of scaling of the dipole moment
is most pronounced and shows that for both NaK and KCs the physical dipole
moment results in smallest losses. Decreasing the dipole moment has the effect
that the quadrupole couplings and the contribution from
$C_{6}^{\mathrm{elec}}$ become relatively more important, such that losses
increase. Increasing the dipole moment increases the repulsive van der Waals
interaction, but also strengthens the attractive dipole-dipole interactions at
short distances. As a result, the barrier shifts to larger distances,
resulting also in faster losses. Reducing the quadrupole moment has no effect
as it does not determine the barrier. When we increase the quadrupole moment,
the quadrupole couplings between the potential curves become strong enough
such that level repulsion lowers the barrier of the lowest adiabatic potential
and increases loss rates.

Surprisingly, we find that the dependence on mass is relatively weak. There
are actually two competing effects. First, the loss is determined by the rate
of tunneling through the repulsive potential, and for a mass-independent
potential one expects a steep suppression of the loss rate as the mass is
increased. The other effect is that, perhaps surprisingly, even the $L=0$
potential is mass dependent at short enough distances. As mentioned in the
previous section, the barrier arises due to coupling by anisotropic
interactions to other nearby states, including higher $L$ states. The
centrifugal barrier for higher $L$ states depends on mass and when the
perturbative treatment breaks down, the coupling to such states results in the
barrier also becoming mass-dependent. This is in stark contrast with the
$s$-wave interaction potential for isotropic interactions, which is mass
independent. As mass increases, the barrier height decreases, resulting in
increasing loss rates. The two effects are opposite and their near-
cancellation results in the weak mass dependence observed in Fig. 6.

###  III.4 Adding electric fields and microwaves

The applications of ultracold molecules rely on their permanent dipole moment.
These dipole moments, however, are not oriented in the lab frame. To access
the molecules’ dipole moments, one can apply a static electric field to orient
the molecules and thereby align the dipole moments along the field axis [40,
41, 16]. Alternatively, microwaves can be used to induce an oscillating dipole
moment along the polarization direction, leading to an equivalent time-
averaged interaction [40, 42]. Ideally, these molecules that exhibit long-
range dipole-dipole interactions would also be stable with respect to
collisions. Above, we have demonstrated that, in the absence of such fields,
collisional loss of ultracold molecules can be suppressed by preparing
molecules in $j+j^{\prime}=0+2$ rotational states. Here, we investigate
whether the suppression of loss rates of NaK in $j+j^{\prime}=0+2$ persists in
the presence of external fields.

Including a static electric field, we show in Fig. 7(a) loss rates for bosonic
and fermionic NaK $0+2$ states as a function of electric field strength. We
find that for field strengths up to around 2 kV/cm both bosons and fermions
retain low collision rates until losses rise up to around universal loss
again. In Fig. 7(b) we show the dipole moment that is induced for different
rotational states as the electric field strength is increased. Within the
range of low collision rates a fraction of the permanent dipole moment can
still be induced, especially for the $j=0$ state.

Figure 7: The influence of a static electric field for bosonic and fermionic
NaK with $j+j^{\prime}=0+2$ on (a) the loss rate and (b) the induced dipole
moment. The universal loss is given for bosonic NaK. Figure 8: The influence
of linearly polarized microwaves for bosonic NaK with rotational
$j+j^{\prime}=0+2$ states on (a) the loss rate and (b) the induced dipole
moment, both as a function of microwave detuning $\Delta$ at Rabi frequency
$\Omega=2\pi\times 10$ kHz.

In Fig. 8, we show loss rates for bosonic NaK $0+2$ dressed with microwaves.
We use linearly polarized microwaves red detuned from the $j=1\to 2$
transition for both $|2,0\rangle$ and $|2,1\rangle$ rotational states [40]. We
fix the Rabi frequency $\Omega$ to $2\pi\times 10$ kHz and scan the detuning
$\Delta$ from the transition. The induced dipole moment is then defined as the
effective dipole arising from the dipole-dipole interaction [40]. The largest
dipole moments that can be induced for $|0,0\rangle|2,0\rangle$ and
$|0,0\rangle|2,1\rangle$, respectively, are $d/\sqrt{3}$ and $d/\sqrt{6}$. For
red-detuned dressing, larger $\Delta/\Omega$ decreases the induced dipole
moment by

$d_{\mathrm{ind}}\sim\frac{1}{\sqrt{1+\left(\frac{\Delta}{\Omega}+\sqrt{1+\left(\frac{\Delta}{\Omega}\right)^{2}}\right)^{2}}}\,.$
(15)

From Fig. 8 we then find that relatively small loss rates of $10^{-11}$ to
$10^{-10}$ cm3/s can be obtained, while inducing a dipole moment of around 0.1
Debye. Although not shown here, circularly polarized microwaves yield very
similar results.

Static electric and microwave fields can still be used to induce a fraction of
the dipole moment, while collisional loss is kept low.

###  III.5 Interactions at pair degeneracies

Figure 9: Rotational dimer degeneracy $j+j^{\prime}=2+2\equiv 0+3$ with
$m_{j}=m_{j^{\prime}}=0$ for bosonic NaK. (a,b) Interaction potentials with
centrifugal distortion coefficient taken into account, lifting the degeneracy
by $\sim 80$ $\mu$K, referenced to the prepared state at infinite separation,
which are (a) $2+2$ and (b) $0+3$. The main difference is the parity
$(-1)^{j+j^{\prime}+L}$, which is opposite between these figures. (c,d)
Scaling of the loss rates with respect to the dipole moment, quadrupole moment
and centrifugal distortion constant for the (c) $2+2$ and (d) $0+3$ states.

We have discussed in section II.3 that pairs of rotational states contain
degeneracies for certain combinations of monomer rotational states. The
$2+2\equiv 0+3$ degeneracy is the one with the lowest rotational quantum
numbers and we now explore this specific case further. These two pairs of
states are coupled by the dipole-quadrupole interaction. In the absence of
that interaction, the $2+2$ state has an attractive rotational van der Waals
interaction whereas the $0+3$ state has a repulsive one. What makes this
degeneracy particularly interesting is that the $2+2$ state can involve two
molecules prepared in the same $|j=2,m\rangle$ quantum state. Furthermore, the
$0+3$ level is degenerate with $2+2$ for rigid rotors, but in reality $0+3$ is
slightly lower in energy due to centrifugal distortion. Thus, one can expect
level repulsion due to the dipole-quadrupole coupling between the nearly
degenerate states to create a repulsive potential correlating adiabatically to
$2+2$. That is, dipole-quadrupole coupling to the nearly degenerate $0+3$
threshold may cause molecules in $j=2$ to be collisionally stable without the
need for active shielding.

The interaction curves for $2+2$ and $0+3$ are presented in Figs. 9(a) and
9(b), respectively, where $m_{j}=m_{j^{\prime}}=0$. We see the small lifting
of degeneracy by about 80 $\mu$K between the two states due to centrifugal
distortion of the molecules. The $L=0$ initial channels for the states
$j+j^{\prime}\equiv 2+2$ and $3+0$ have opposite parity, and the two sets of
potentials are shown in panels (a) and (b), respectively. The rotational van
der Waals interaction is attractive in the upper threshold and repulsive in
the lower one, leading to crossings between the potential curves. The
crossings are avoided due to the dipole-quadrupole coupling between the
states. However, the crossings occur where the $2+2$ potential is already
attractive and the local kinetic energy makes it unlikely that these crossings
would be traversed adiabatically. For the $0+3$ state, the effect of the
avoided crossings is to greatly reduce the barrier height, which can
negatively impact the collisional loss rates.

We compute loss rate coefficients as a function of a scaling applied to the
dipole and quadrupole moments as well as the distortion constant and calculate
collisional loss for the $2+2$ and $0+3$ initial states, shown in Figs. 9(c)
and 9(d), respectively. For the $2+2$ state we find high losses across the
entire scaling range of the different parameters. This means that the
intrinsic shielding that one could have expected for the $2+2$ state cannot be
effective. The main feature in Fig. 9(c) is the rise in loss as the dipole
rises, which is simply due to the rotational van der Waals potential becoming
more attractive for $2+2$. For the $0+3$ state rate coefficients are shown in
Fig. 9(d), and here we find different effects. For reduced quadrupole moments,
the coupling to the $2+2$ state is reduced, and the loss rate approaches the
lower loss rate expected for the $0+3$ repulsive van der Waals potential. At
increased distortion constants, the loss rate decreases slightly as the
barrier for $0+3$ increases due to the avoided crossing with the $2+2$
potential being moved outwards to where that potential is less attractive.
Varying the dipole moment changes the repulsive interaction of $0+3$, the
attractive interaction of $2+2$ and the coupling between them. The dependence
of the loss rate is therefore not as simple. For all realistic parameters,
coupling between the nearly degenerate states is not strong enough to shield
the upper state, but it does increase the loss rate for the lower one.

##  IV Applications for cold molecules

The biggest limitation of the stability of ultracold molecular systems is
collisional loss. In “shielding”, external fields are used to reduce
collisional losses [20, 21, 22, 23, 24, 25, 26, 27]. We discussed in this
paper that by an appropriate choice of two rotational states $j,j^{\prime}$
such that $\Delta j\geq 2$ repulsive interactions can be realized without
requiring external field control. Here, we consider several possible
applications of these interactions: evaporating a fermi mixture, fermionic
molecules in an optical lattice, and stable impurities in an ultracold gas.

###  IV.1 Evaporation of fermi mixtures

Cooling down to a degenerate quantum gas of fermions has so far been achieved
by assembly from atoms at ultracold temperatures [43, 44, 45], and by forced
evaporation in two-dimensional confinement [46] or while shielding using
static electric fields [22, 23] or microwave fields [27, 28]. Efficient forced
evaporation requires a high ratio of elastic to inelastic collision rates. For
fermions in the same quantum state, collisions are highly suppressed by the
presence of a $p$-wave centrifugal barrier. This leads to elastic and
inelastic collision rates that are suppressed at ultracold temperatures by
$T^{2}$ and $T$, respectively.

Our idea here is that one could create a fermi mixture of molecules in two
rotational states, $j$ and $j^{\prime}$. Collisions between molecules in the
same rotational state would be highly suppressed at ultracold temperatures,
but “inter-species” collisions could occur. If $\Delta j\geq 2$, the inelastic
rate could be small due to the repulsive rotational van der Waals interaction.
This would leave only inter-species elastic collisions, which could then
enable fast thermalization and efficient forced evaporation. After removal of
molecules in one of the $j$ states, which could be implemented as collisional
removal of the minority species, one could end up with a state-pure quantum
degenerate gas without active shielding using external fields.

In Fig. 10 we show elastic and inelastic collision rates for different pairs
of rotational states of fermionic NaK. The $0+4$ pair performs the best out of
the investigated pairs due to the low collisional loss as was seen in Fig. 4.
Compared to the $0+0$ collisions, the rate for elastic $0+3$ or $0+4$
collisions has indeed increased by orders of magnitude, while especially for
the $0+4$ pair the inelastic rate is generally smaller than the elastic one.
However, the increased elastic rate for $0+4$ is generally not much higher
than the inelastic rate of $0+0$ and $4+4$ collisions that also occur in the
mixture. So while the repulsive interaction improves the elastic collision
rate by orders of magnitude while keeping the collisional loss rate low, the
ratio of elastic to inelastic collisions is realistically not high enough to
enable direct evaporative cooling.

Figure 10: Collision rates for thermalization of fermions for various
rotational state pairs of NaK as a function of temperature.

###  IV.2 Fermions in optical lattices

In quantum simulation using ultracold molecules in optical lattices [47, 48,
49], two molecules can tunnel into the same lattice site and undergo
collisional loss. The repulsive rotational van der Waals interaction can
prevent such losses without actively controlling interactions with external
fields, at least for fermionic molecules in two rotational states with $\Delta
j\geq 2$. Fermions in the same quantum state will not occupy the same lattice
site due to the Pauli principle. Fermions in distinct quantum states can meet
on site, but will be protected from collisional loss by the repulsive
rotational van der Waals interaction. As shown in Sec. III.4, a weak microwave
or static electric field can be applied to access dipolar interactions between
the molecules without compromising the collisional stability.

###  IV.3 Impurity physics

An ultracold mixture of molecules in different rotational states is also an
interesting playground for impurity physics [50, 51, 52], and enables studying
polarons with long-range interactions. We envision introducing $j=2$
impurities into a gas of $j=0$ rotational ground state molecules, such that
the repulsive rotational van der Waals interaction would render the impurities
stable with respect to collisions with the bath. One can address the impurity
by microwaves tuned on the $j=1\to 2$ transition, which induces dipole-dipole
interactions between the impurity and the bath molecules selectively, without
inducing dipolar interactions within the bath.

##  V Conclusion

In this work, we described the electrostatic interaction between two dipolar,
linear molecules that are prepared in specific rotational states. When both
molecules are prepared in the same rotational state, we recover the well-known
attractive rotational van der Waals interaction. When the rotational states
differ by one quantum, resonant dipole-dipole interactions lead to first-order
dipole-dipole interactions. When the rotational states differ by two or more
quanta, the dominant interaction is a repulsive van der Waals interaction. The
repulsive interactions follow directly from the rotational energy levels of a
linear molecule that become sparser as one goes up in energy, and are
therefore universal for polar molecules. This is in stark contrast to the
usual electronic van der Waals interaction which is typically attractive as
generally electronically excited states become denser at higher energies.

The repulsive interactions result in a potential barrier that we computed for
NaK and KCs molecules. These barriers can suppress collisional losses at low
temperatures. Using quantum scattering calculations we studied the ultracold
collision dynamics quantitatively and showed that collisional loss can be
suppressed by several orders of magnitude below the universal loss rate.
Moreover, we discussed the cases when pairs of rotational energy levels of two
molecules become generate, which impacts the collision dynamics. Using static
electric fields or microwaves, a fraction of the limiting dipole moment can be
induced while maintaining a low loss rate.

The repulsive interaction between molecules in different rotational states
opens up several applications in ultracold physics. A clear application is
quantum simulation with fermionic molecules in an optical lattice. Here, only
distinguishable fermionic molecules in different rotational states can tunnel
into a single lattice site, and the repulsive van der Waals interactions can
prevent collisional loss and control on site interactions. Another promising
application is impurity physics with rotationally excited molecules immersed
in a bath of rotational ground state molecules, enabling control of dipolar
and repulsive van der Waals interactions between the impurity and bath
molecules while suppressing collisional loss. Further less obvious
applications of the repulsive rotational van der Waals interactions may exist.
In the accompanying paper we show how these interactions can be used to
implement near-deterministic loading of optical tweezers with laser cooled
molecules [53].

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