# Distance sets bounds for polyhedral norms
via effective dimension

Iqra Altaf  Department of Mathematics
University of Chicago
5734 S University Avenue, Room 108
Chicago, IL, 60637-1505<EMAIL_ADDRESS>,  Ryan Bushling  Department of
Mathematics
University of Washington, Box 354350
Seattle, WA 98195-4350<EMAIL_ADDRESS>and  Bobby Wilson  Department of
Mathematics
University of Washington, Box 354350
Seattle, WA 98195-4350<EMAIL_ADDRESS>
###### Abstract.

We prove that, for every norm on $\mathbb{R}^{d}$ and every
$E\subseteq\mathbb{R}^{d}$, the Hausdorff dimension of the distance set of $E$
with respect to that norm is at least $\operatorname{\dim_{\rm H}}E-(d-1)$. An
explicit construction follows, demonstrating that this bound is sharp for
every polyhedral norm on $\mathbb{R}^{d}$. The techniques of algorithmic
complexity theory underlie both the computations and the construction.

###### 2020 Mathematics Subject Classification:

28A80, 03D32

B. Wilson was supported by NSF grant, DMS 1856124, and NSF CAREER Fellowship,
DMS 2142064.

##  1\. Introduction

Given a set $E\subseteq\mathbb{R}^{d}$, let

$\Delta(E):=\\{\|x-y\|\in[0,\infty):~{}x,y\in E\\}$

be the distance set of $E$, where $\|\cdot\|$ is the Euclidean norm. The
Falconer distance problem is to find a lower bound on $\operatorname{\dim_{\rm
H}}\Delta(E)$, the Hausdorff dimension of the distance set, in terms of the
Hausdorff dimension of $E$ itself. A frequent simplification is to fix a point
$x\in\mathbb{R}^{d}$ and instead investigate the pinned distance set

$\Delta_{x}(E):=\\{\|x-y\|\in[0,\infty):~{}y\in E\\}.$

Since $\Delta_{x}(E)\subseteq\Delta(E)$ whenever $x\in E$, any lower bound on
the size of $\Delta_{x}(E)$ immediately implies the same bound on the size of
$\Delta(E)$. An interesting variant of the Falconer distance problem is to
seek dimension bounds on distance sets relative to a different norm
$\|\cdot\|_{*}$. In this case, the corresponding distance and pinned distance
sets are denoted $\Delta^{*}(E)$ and $\Delta_{x}^{*}(E)$, respectively.

Falconer [4] proves a simple estimate in the Euclidean case:

###### Theorem 1.1 (Falconer [4]).

If $E\subseteq\mathbb{R}^{d}$ is any set, then

$\displaystyle\operatorname{\dim_{\rm H}}\Delta(E)\geq\operatorname{\dim_{\rm
H}}E-(d-1).$

The elementary proof presented by Falconer generalizes to other norms with
little modification. However, the techniques of algorithmic complexity
provided in [8] are highly applicable here, and we use this theory to give an
alternate proof of the general case.

###### Theorem 1.2.

Let $\|\cdot\|_{*}$ be any norm on $\mathbb{R}^{d}$ and let
$E\subseteq\mathbb{R}^{d}$. Then

$\operatorname{\dim_{\rm H}}\Delta_{x}^{*}(E)\geq\operatorname{\dim_{\rm
H}}E-(d-1)$

for all $x\in\mathbb{R}^{d}$.

Polyhedral norms provide a class of finite-dimensional normed spaces which can
be considered the “worst-case scenario” for the distance set problem, as the
curvature of the norm ball plays a major role in precluding the possibility of
a large number of points with equal pairwise distances. The relevance of
nonvanishing curvature manifests itself in [7]: the authors provide the best
known bound in the planar Euclidean case, and it is demonstrated that the
essential characteristic of the Euclidean ball required for their argument is
that it is a smooth norm ball with strictly positive curvature. A construction
of Falconer [5] illustrates the marked differences in the context of
polyhedral norms.

###### Proposition 1.3 (Falconer [5]).

Let $\|\cdot\|_{P}$ be a polyhedral norm on $\mathbb{R}^{d}$. Then there
exists a compact set $F\subset\mathbb{R}^{d}$ with $\operatorname{\dim_{\rm
H}}F=d$ such that $\mathcal{L}(\Delta^{P}(F))$ $=0$.

The relevance of curvature to the distance set conjecture is further explored
in [1], in which Falconer’s construction is generalized to the case in which
at most countably many points of the surface of the unit ball do not lie on
any open line segment contained in the surface of the unit ball (all but
finitely many points on the surface of a polyhedral ball satisfy this
condition) or for certain convex sets for which the measure representing the
curvature field is singular with respect to surface measure. For more state-
of-the-art estimates and information on the history of the Euclidean distance
set conjecture, see [3].

The aforementioned results all concern sets of full Hausdorff dimension. Our
second result shows that polyhedral norms witness the sharpness of the
dimension bound in Theorem 1.2 in a more quantitative way. This, too, will be
proven using complexity theoretic tools.

###### Theorem 1.4.

Let $\|\cdot\|_{P}$ be a polyhedral norm on $\mathbb{R}^{d}$. For any
$s\in[d-1,d]$, there exists a compact set $E\subseteq\mathbb{R}^{d}$ with
$\operatorname{\dim_{\rm H}}E=s$ such that

$\operatorname{\dim_{\rm H}}\Delta_{x}^{P}(E)=s-(d-1)$

for some point $x\in E$.

Following a brief synopsis of the background material, we prove Theorem 1.2 in
§3. In §4 we construct the examples described in Theorem 1.4.

##  2\. Preliminaries

This section summarizes the requisite terminology and results from algorithmic
complexity theory. See [8] for a more thorough exposition of this material. We
denote by

$\\{0,1\\}^{*}:=\bigcup_{n\in\mathbb{N}}\\{0,1\\}^{n}$

the set of all (finite) binary strings, including the empty string
$\lambda\in\\{0,1\\}^{0}$.

###### Definition 2.1 (Kolmogorov Complexity of a String).

Let $\sigma,\tau\in\\{0,1\\}^{*}$. The conditional Kolmogorov complexity of
$\sigma$ given $\tau$ is the length of the shortest program $\pi$ that will
output $\sigma$ given $\tau$. More precisely,

$\displaystyle
K(\sigma|\tau):=\min_{\pi\in\\{0,1\\}^{*}}\\{\ell(\pi):~{}U(\pi,\tau)=\sigma\\},$

where $U$ is a fixed universal prefix-free Turing machine and $\ell(\pi)$ is
the length of $\pi$. The Kolmogorov complexity of $\sigma$ is simply the
conditional Kolmogorov complexity of $\sigma$ given the empty string:

$K(\sigma):=K(\sigma|\lambda),$

where $\lambda$ is the empty string.

Throughout, we work with a fixed encoding
$\\{0,1\\}^{*}\to\bigcup_{d\in\mathbb{N}}\mathbb{Q}^{d}$, under which the
definitions above extend from strings to vectors over the rationals.

###### Definition 2.2 (Kolmogorov Complexity of a Point).

Let $x\in\mathbb{R}^{d}$ and $r\in\mathbb{Z}_{+}$. The Kolmogorov complexity
of $x$ at precision $r$ is the length of the shortest program that outputs a
point in $\mathbb{Q}^{d}$ that approximates $x$ to $r$ bits of precision:

$K_{r}(x):=\min\\{K(p):~{}p\in B(x,2^{-r})\cap\mathbb{Q}^{d}\\}.$

If also $y\in\mathbb{R}^{d^{\prime}}$ and $s\in\mathbb{Z}_{+}$, then the
conditional Kolmogorov complexity of $x$ at precision $r$ given $y$ at
precision $s$ is defined by

$K_{r,s}(x|y):=\max\big{\\{}\min\\{K_{r}(p|q):p\in
B(x,2^{-r})\cap\mathbb{Q}^{d}\\}:q\in
B(y,2^{-s})\cap\mathbb{Q}^{d^{\prime}}\big{\\}}.$

Note that, for $x\in\mathbb{R}^{d}$, $K_{r}(x)$ is always less than equal to
$dr+O(\log r)$. To simplify notation, we also denote $K_{r,r}(x|y)$ by
$K_{r}(x|y)$ and, when $r\in(0,\infty)$, $K_{\lceil r\rceil}(x)$ by
$K_{r}(x)$.

###### Lemma 2.1 (Symmetry of Information [9]).

For all $m,n\in\mathbb{N}$, $x\in\mathbb{R}^{m}$, $y\in\mathbb{R}^{n}$, and
$r,s\in\mathbb{N}$ with $r\geq s$:

  1. a.

$\big{|}K_{r}(x|y)+K_{r}(y)-K_{r}(x,y)\big{|}\leq O(\log r)+O(\log\log\|y\|)$.

  2. b.

$\big{|}K_{r,s}(x|x)+K_{s}(x)-K_{r}(x)\big{|}\leq O(\log r)+O(\log\log\|x\|)$.

Using the notion of Kolmogorov complexity, one defines the concept of
effective dimension as the asymptotic Kolmogorov complexity of a given point.

###### Definition 2.3.

The effective Hausdorff dimension of a point $x\in\mathbb{R}^{d}$ is given by

$\dim(x):=\liminf_{r\to\infty}\frac{K_{r}(x)}{r}.$

In computability theory, a set $A\subseteq\mathbb{N}$ is called an oracle.
Heuristically, a Turing machine $T$ can be said to have access to $A$ if, in
addition to its usual operations, it can inquire whether the number currently
printed on the work tape belongs to $A$. This operation takes one step, and
the answer to the question is recorded as a state. The machine derived from
$T$ by allowing it access to $A$ is denoted $T^{A}$.

The concepts of complexity and dimension defined above can be relativized to
an oracle $A$ by replacing the fixed universal Turing machine $U$ with
$U^{A}$. These relativized concepts are denoted $K^{A}$, $\dim^{A}$, etc. For
us, the utility of oracles is that they allow us to compute the Hausdorff
dimension of a set from the effective dimensions the points it contains.

###### Theorem 2.2 (Point-to-Set Principle [8]).

For every set $E\subseteq\mathbb{R}^{d}$,

$\operatorname{\dim_{\rm H}}E=\min_{A\subseteq\mathbb{N}}\sup_{x\in
E}\dim^{A}(x).$

One can infer from the proof of the point-to-set principle that, for any
oracle $A$ and $s\in(0,d]$, the set of points $x$ that satisfy $\dim^{A}(x)<s$
has $s$-dimensional Hausdorff measure zero. In particular, $\dim^{A}(x)=d$ for
all $x$ outside of a Lebesgue-null set.

As in [10], we describe the complexity of a point $x\in\mathbb{R}^{d}$
relative to a point $y\in\mathbb{R}^{d^{\prime}}$ as the complexity of $x$
relative to an oracle set $A_{y}$ that encodes the binary expansion of $y$ in
a standard way. We define

$K_{r}^{y}(x):=K_{r}^{A_{y}}(x).$

Throughout the paper, we will often invoke the notion of “randomness”. A point
$x\in\mathbb{R}^{d}$ is random with respect to an oracle $A$ if

$\big{|}K_{r}^{A}(x)-rd\big{|}\leq O(\log r).$

This also implies that each of the coordinates of $x$ are random with respect
to each other and have complexity $1$ with respect to the oracle $A$. More
quantitatively, for $t<r$, the symmetry of information 2.1 implies that

$\displaystyle rd+O(\log r)=K_{r}(x)$
$\displaystyle=K_{r,t}(x)+K_{t}(x)+O(\log r)$ $\displaystyle\leq
K_{r,t}(x)+td+O(\log r),$

whence

$|K_{r,t}(x)-(r-t)d|\leq O(\log r).$

This means that the first $t$ digits of the binary expansion of $x$ “do not
help much” in the calculating the remaining $r-t$ digits, and that we would
approximately need $(r-t)d$ bits to calculate them. In the proof of Theorem
1.4, we will have occasion to choose collections of strings of digits that are
random with respect to each other, but we can see from this discussion that
such a choice is always possible.

##  3\. Distance set bounds for arbitrary norms

Proof of Theorem 1.2. The unit sphere in the norm $\|\cdot\|_{*}$ has
Hausdorff dimension $d-1$. Therefore, by the point-to-set principle, there
exists an oracle $A\subseteq\mathbb{N}$ such that

$\sup_{z\in\partial B_{*}(0,1)}\dim^{A}(z)=d-1.$

For any $y\in E$, if we write $y=x+\|x-y\|_{*}z$ for some $z\in\partial
B_{*}(0,1)$, then the following holds: For any oracle $B\subseteq\mathbb{N}$,

$\displaystyle K_{r}^{A,B,x}(y)$ $\displaystyle\leq K_{r}^{A,B,x}(y-x)+O(\log
r)$ $\displaystyle=K_{r}^{A,B,x}(\|x-y\|_{*}z)+O(\log r)$ $\displaystyle\leq
K_{r}^{A,B,x}(\|x-y\|_{*})+K_{r}^{A,B,x}(z)+O(\log r).$ $\displaystyle\leq
K_{r}^{A,B,x}(\|x-y\|_{*})+(d-1)r+O(\log r).$

In particular, for every $\varepsilon>0$, the point-to-set principle gives a
point $y\in E$ such that

$\operatorname{\dim_{\rm H}}E\leq\dim^{A,B,x}(y)+\varepsilon.$

Dividing through by $r$ and letting $r\to\infty$ gives us that for every
$\varepsilon$, there exists a point $t=\|x-y\|_{*}\in\Delta_{x}^{*}(E)$ such
that

$\operatorname{\dim_{\rm H}}E-(d-1)-\varepsilon\leq\dim^{A,B,x}(t).$

Taking the minimum over all oracles and letting $\varepsilon\to 0$ then gives
the desired inequality. $\square$

##  4\. Sharp examples

In this section we construct the set $E\subset\mathbb{R}^{d}$ proving Theorem
1.4. Fundamentally, the construction adapts [6] Example 7.8, which in turn
works on the same principles as the “Venetian blinds” constructions that have
been studied widely in geometric measure theory for the extreme features they
exhibit, e.g., in the context of orthogonal projections, see [2]. As such, the
use of complexity in building our set $E$ is secondary to its use in actually
proving that it has the asserted properties.

Proof of Theorem 1.4. Let $s\in[d-1,d]$ and $\alpha:=s-(d-1)$. Since
$\|\cdot\|_{P}$ is a polyhedral norm, there exists a finite set
$\\{v^{1},...,v^{N}\\}\subset\mathbb{R}^{d}$ of vectors—one for each face of
the $\|\cdot\|_{P}$-ball—such that

$\|x\|_{P}=\max\\{|x\cdot v^{\ell}|\\}_{\ell=1}^{N}$ (4.1)

for all $x\in\mathbb{R}^{d}$. Let $c$ be a large constant to be specified
later, and let $\\{m_{k}\\}_{k=1}^{\infty}$ be a sequence of positive integers
satisfying $m_{1}=1$, $2c<m_{2}$, and, for all $k\in\mathbb{Z}_{+}$,

$k\hskip 0.83298ptm_{k}\leq m_{k+1}.$ (4.2)

We also define a sequence $\\{n_{k}\\}_{k=1}^{\infty}$ by

$n_{k}:=m_{k}+\alpha(m_{k+1}-m_{k}).$

Note that, since $\alpha\in[0,1]$ and $m_{2}>2c$, we have $n_{k}+c<m_{k+1}-c$
for all $k$.

We construct $E\subset[0,1]^{d}$ in blocks of $N$ steps. Define

$\begin{array}[]{rl}F_{k}:=&\Big{\\{}\ x\in[0,1]^{d}:~{}\lfloor 2^{j}(x\cdot
v^{\ell})\rfloor=0\mbox{ for all }\\\ &\ j\in(n_{k}+c,m_{k+1}-c],\mbox{ where
}\ \ell\equiv k\text{ (mod $N$) }\Big{\\}}\end{array}$ (4.3)

and

$E:=\bigcap_{k=1}^{\infty}F_{k}.$

These definitions prescribe that the binary expansions of the points in $E$
alternate between long strings of free digits and long strings of zeros, with
the lengths of the strings rapidly approaching infinity. When a real number
has two different binary expansions, we associate to it the expansion
terminating in an infinite string of ones. This makes the sets $F_{k}$ and
thus $E$ to be closed.

Claim: For every oracle $A\subseteq\mathbb{N}$, there exists $x\in E$ such
that

$\liminf_{r\to\infty}\frac{K_{r}^{A}(x)}{r}\geq s$ (4.4)

and, for all $t\in\Delta_{x}^{P}(E)$,

$\liminf_{r\to\infty}\frac{K_{r}^{A}(t)}{r}\leq\alpha.$ (4.5)

By the definition of $\alpha$, this will imply that $\Delta_{x}^{P}(E)\leq
s-(d-1)\leq\operatorname{\dim_{\rm H}}E-(d-1)$, and an application of Theorem
1.2 will entail that these hold with equality.

To obtain this point $x$ and ensure that it belongs to $E$, we iteratively
select the digits in the binary expansions of its coordinates. In particular,
assuming we have already chosen the digits in the places before the $m_{k}$th
place, the digits in the places $[m_{k},m_{k+1})$ will be chosen in two steps.
For each $\ell=1,...,N$, fix any index $1\leq i_{\ell}\leq d$ such that the
$i_{\ell}$th coordinate of $v^{\ell}$ is nonzero: $v_{i_{\ell}}^{\ell}\neq 0$.
Let $\ell\equiv k~{}\text{(mod $N$)}$, $1\leq\ell\leq d$.

  1. 1.

We define the digits in the binary expansion of $x_{i}$ in the places
$[m_{k},m_{k+1})$, for $i\neq i_{\ell}$ and in the places $[m_{k},n_{k})$, for
$i=i_{\ell}$ such that the corresponding $d$ strings (of which $d-1$ have
length $m_{k+1}-m_{k}$ and one has length $n_{k}-m_{k}$) are random with
respect to each other, the oracle $A$ and the first $m_{k}$ digits of $x$.

  2. 2.

We want to define the digits in the binary expansion of $x_{i_{\ell}}$ in the
places $[n_{k},m_{k+1})$ such that

$\displaystyle\left\lfloor 2^{j}\left(\sum_{i=1}^{d}2^{-m_{k+1}}\lfloor
2^{m_{k+1}}x_{i}\rfloor
v_{i}^{\ell}\right)\right\rfloor\equiv\begin{cases}0&\text{if
}j\in(n_{k}+c,m_{k+1}-c],\\\ 1&\text{if }j=m_{k+1}-c+1,\\\ 0&\text{if
}j=m_{k+1}-c+2,\end{cases}$ (4.6)

where the equivalence holds modulo $2$. Before showing that this is possible,
we prove that Equation (4.6) implies

$\big{\lfloor}2^{j}(x\cdot v^{\ell})\big{\rfloor}\equiv 0~{}\text{(mod
$2$)}\quad\text{for every}\quad j\in(n_{k}+c,m_{k+1}-c),$ (4.7)

provided $c$ is chosen large enough. It will then follow from (4.3) that the
point $x$ satisfying (4.6) belongs to $F_{k}$. In order to derive Equation
(4.7) from Equation (4.6), we first observe that

$x\cdot v^{\ell}=\sum_{i=1}^{d}2^{-m_{k+1}}\lfloor 2^{m_{k+1}}x_{i}\rfloor
v_{i}^{\ell}+\sum_{i=1}^{d}\big{(}x_{i}-2^{-m_{k+1}}\lfloor
2^{m_{k+1}}x_{i}\rfloor\big{)}v_{i}^{\ell}.$ (4.8)

Notice that, if (4.6) holds, then for some $M\in\mathbb{Z}$ and
$t\leq\tfrac{1}{4}$, we have

$\sum_{i=1}^{d}2^{-m_{k+1}}\lfloor 2^{m_{k+1}}x_{i}\rfloor
v_{i}^{\ell}=2^{-n_{k}-c}M+2^{-m_{k+1}+c}\big{(}\tfrac{1}{2}+t\big{)}.$ (4.9)

Choose $c$ large enough such that

$\max\left(\sum_{i=1}^{d}|v_{i}^{\ell}|,\,(v_{i_{\ell}}^{\ell})^{-1}\right)\leq
2^{c-3}\quad\text{for all}\quad\ell,$ (4.10)

from which it follows that

$\displaystyle\left|\sum_{i=1}^{d}(x_{i}-2^{-m_{k+1}}\lfloor
2^{m_{k+1}}x_{i}\rfloor)v_{i}^{\ell}\right|$
$\displaystyle\leq\sum_{i=1}^{d}\big{|}x_{i}-2^{-m_{k+1}}\lfloor
2^{m_{k+1}}x_{i}\rfloor\big{|}|v_{i}^{\ell}|$ (4.11) $\displaystyle\leq
2^{-m_{k+1}}\sum_{i=1}^{d}|v_{i}^{\ell}|$ $\displaystyle\leq
2^{-m_{k+1}+c-3}.$

Combining (4.8), (4.9), and (4.11), we see that

$2^{j}(x\cdot
v^{\ell})=2^{j-n_{k}-c}M+2^{j-(m_{k+1}-c)}\big{(}\tfrac{1}{2}+t+t^{\prime}\big{)},$

where $|t^{\prime}|\leq\tfrac{1}{8}$. Hence, for $j\in(n_{k}+c,m_{k+1}-c]$,
the second term above lies in $[0,1)$:

$0<2^{j-(m_{k+1}-c)}\big{(}\tfrac{1}{2}+t+t^{\prime}\big{)}<1.$

Therefore, we have shown that Equation (4.6) implies

$\left\lfloor 2^{j}\left(\sum_{i=1}^{d}2^{-m_{k+1}}\lfloor
2^{m_{k+1}}x_{i}\rfloor
v_{i}^{\ell}\right)\right\rfloor=\big{\lfloor}2^{j}(x\cdot
v^{\ell})\big{\rfloor}$ (4.12)

for $j\in(n_{k}+c,m_{k+1}-c]$, which immediately entails Equation (4.7).

It remains to see that we can choose the digits of $x_{i_{\ell}}$ in the
places $[n_{k},m_{k+1})$ such that (4.6) holds. The condition (4.6) determines
the digits of $x_{i_{\ell}}\cdot v_{i_{\ell}}^{\ell}$ in places
$[n_{k},m_{k+1}-c+2)$, and we see from (4.10) that

$2^{-c+3}\leq v_{i_{\ell}}\leq 2^{c-3}.$

Therefore, we can indeed make such a choice for the digits of $x_{i_{\ell}}$.

Repeating this process for all $k\in\mathbb{Z}_{+}$ produces the desired point
$x\in E$.

Now we prove that $\dim^{A}(x)=s$. Let $r\in[m_{k},m_{k+1}]$ and
$k\equiv\ell~{}\text{(mod $N$)}$. We estimate $K_{r}^{A}(x)$ in two cases.

Case 1: $r\in(m_{k},n_{k}]$
By symmetry of information, we have

$\displaystyle K^{A}_{r}(x)$
$\displaystyle=K^{A}_{r,m_{k}}(x)+K^{A}_{m_{k}}(x)+o(r)$ (4.13)
$\displaystyle\geq(r-m_{k})d+K^{A}_{m_{k}}(x)-o(r).$

The second inequality is true because the digits of $x$ in the $m_{k}$th
through $r$th places are chosen randomly with respect to $A$ and the digits in
the places $0$ through $m_{k}$.

Case 2: $r\in(n_{k},m_{k+1}]$
By the symmetry of information, we have

$\displaystyle K^{A}_{r}(x)=K^{A}_{r,n_{k}}(x)+K^{A}_{n_{k}}(x)-o(r).$

It follows from (4.13) of Case 1 that

$\displaystyle K^{A}_{n_{k}}(x)$
$\displaystyle\geq(n_{k}-m_{k})d+K^{A}_{m_{k}}(x)-o(r)$
$\displaystyle=\alpha(m_{k+1}-m_{k})d+K^{A}_{m_{k}}(x)-o(r).$

Thus we have

$\displaystyle K^{A}_{r}(x)$ $\displaystyle\geq
K^{A}_{r,n_{k}}(x)+\alpha(m_{k+1}-m_{k})d+K^{A}_{m_{k}}(x)-o(r)$ (4.14)
$\displaystyle\geq
K^{A}_{r,n_{k}}(x_{1},...,x_{i_{j}-1},x_{i_{j}+1},...,x_{d})+\alpha(m_{k+1}-m_{k})d+K^{A}_{m_{k}}(x)-o(r)$
$\displaystyle\geq(r-n_{k})(d-1)+\alpha(m_{k+1}-m_{k})d+K^{A}_{m_{k}}(x)-o(r)$
$\displaystyle=(r-{m_{k}})(d-1)+\alpha(m_{k+1}-m_{k})+K^{A}_{m_{k}}(x)-o(r).$

The third inequality above also follows from randomness. We put $r=m_{k+1}$ in
(4.14) to get

$\displaystyle K^{A}_{m_{k+1}}(x)$
$\displaystyle=(m_{k+1}-m_{k})s+K_{m_{k}}(x)-o(r)$ $\displaystyle\geq
m_{k+1}\left(1-\frac{1}{k}\right)s+K_{m_{k}}(x)-o(r),$

where the inequality follows from (4.2). Thus for any arbitrarily small $t>0$,
we have

$\displaystyle\frac{K^{A}_{m_{k+1}}(x)}{m_{k+1}}\geq s-t\quad\text{for large
enough }k.$

We will now justify that the expression $K^{A}_{r}(x)/r$ is minimized when
$r=m_{k}$ for some $k$. By (4.13) of Case 1,

$\displaystyle\frac{K^{A}_{r}(x)}{r}$
$\displaystyle\geq\left(1-\frac{m_{k}}{r}\right)d+\frac{K^{A}_{m_{k}}(x)}{r}-\frac{o(r)}{r}$
(4.15)
$\displaystyle=\left(1-\frac{m_{k}}{r}\right)d+\frac{K^{A}_{m_{k}}(x)}{m_{k}}\left(\frac{m_{k}}{r}\right)-\frac{o(r)}{r}$
$\displaystyle\geq\left(1-\frac{m_{k}}{r}\right)d+(s-t)\frac{m_{k}}{r}-\frac{o(r)}{r}$
$\displaystyle=d-(\alpha-1+t)\frac{m_{k}}{r}-\frac{o(r)}{r}$

and similarly, by (4.14), we have in Case 2

$\displaystyle\frac{K^{A}_{r}(x)}{r}$
$\displaystyle\geq\left(1-\frac{m_{k}}{r}\right)(d-1)+\frac{\alpha(m_{k+1}-m_{k})}{r}-\frac{o(r)}{r}$
(4.16) $\displaystyle\geq
d-1+\alpha\frac{m_{k+1}}{r}+t\frac{m_{k}}{r}-\frac{o(r)}{r}.$

Recalling $\tfrac{m_{k}}{r}\leq 1$ and $\tfrac{m_{k+1}}{r}\geq 1$, we conclude
(4.4) from (4.15) and (4.16) and therefore we have proved that
$\operatorname{\dim_{\rm H}}E\geq s$.

Now let $z=\|x-y\|\in\Delta_{x}^{P}(E)$, where $x,y\in E$. Thus,
$z=|(x-y)\cdot v^{l}|$ for some $\ell$. Thus for a given
$k\equiv\ell~{}\text{(mod $N$)}$, we have

$\lfloor 2^{j}(x\cdot v^{\ell})\rfloor\equiv 0~{}\text{(mod
$2$)}\quad\text{for all}\quad j\in[n_{k}+c,m_{k+1}-c]$

or

$\lfloor 2^{j}(x\cdot v^{\ell})\rfloor\equiv 1~{}\text{(mod
$2$)}\quad\text{for all}\quad j\in[n_{k}+c,m_{k+1}-c].$

Therefore,

$K^{A}_{m_{k+1}-c,n_{k}+c}(z)\leq o(m_{k+1}).$

This is because specifying a string of 000’s or 111’s having length less than
$m_{k+1}$ requires a program of length at most $O(\log m_{k+1})$. We
consequently have

$\displaystyle K_{m_{k+1}-c}^{A}(z)$ $\displaystyle\leq
K_{m_{k+1}-c,n_{k}+c}^{A}(z)+K_{n_{k}}^{A}(z)+o(m_{k+1})$ $\displaystyle\leq
n_{k}+o(m_{k+1})$ $\displaystyle\leq
m_{k+1}\alpha\left(1-\frac{1}{k}\right)+o(m_{k+1}),$

where the last inequality follows from (4.2). Taking $k\to\infty$ gives (4.5),
and, since this holds for all $z\in\Delta_{x}^{P}(E)$, we conclude that
$\Delta_{x}^{P}(E)\leq\alpha$. The desired upper bound on
$\operatorname{\dim_{\rm H}}E$ then follows immediately from Theorem 1.2:

$s\leq\operatorname{\dim_{\rm
H}}(E)\leq\Delta_{x}^{P}(E)+(d-1)\leq\alpha+(d-1)=s.$

As such, all the inequalities above are actually equalities, so the theorem is
proved. $\square$

## References

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# Collisionally Stable Gas of Bosonic Dipolar Ground State Molecules

Niccolò Bigagli  Department of Physics, Columbia University, New York, New
York 10027, USA  Claire Warner  Department of Physics, Columbia University,
New York, New York 10027, USA  Weijun Yuan  Department of Physics, Columbia
University, New York, New York 10027, USA  Siwei Zhang  Department of Physics,
Columbia University, New York, New York 10027, USA  Ian Stevenson  Department
of Physics, Columbia University, New York, New York 10027, USA  Tijs Karman
Institute for Molecules and Materials, Radboud University, 6525 AJ Nijmegen,
Netherlands  Sebastian Will  Corresponding author. Email:
<EMAIL_ADDRESS>Department of Physics, Columbia University, New
York, New York 10027, USA

###### Abstract

Stable ultracold ensembles of dipolar molecules hold great promise for many-
body quantum physics, but high inelastic loss rates have been a long-standing
challenge. Recently, it was shown that gases of fermionic molecules can be
effectively stabilized through external fields. However, many quantum
applications will benefit from molecular ensembles with bosonic statistics.
Here, we stabilize a bosonic gas of strongly dipolar NaCs molecules against
inelastic losses via microwave shielding, decreasing losses by more than a
factor of 200 and reaching lifetimes on the scale of 1 second. We also measure
high elastic scattering rates, a result of strong dipolar interactions, and
observe the anisotropic nature of dipolar collisions. Finally, we demonstrate
evaporative cooling of a bosonic molecular gas to a temperature of 36(5) nK,
increasing its phase-space density by a factor of 20. This work is a critical
step towards the creation of a Bose-Einstein condensate of dipolar molecules.

††preprint: APS/123-QED

##  I Introduction

Ultracold gases of atoms and molecules have revolutionized the experimental
exploration of many-body quantum systems [1, 2, 3, 4, 5]. In recent years,
systems with dipolar long-range interactions have gained rapid traction,
enabling the creation of new types of strongly correlated and highly entangled
quantum matter. Magnetic atoms [6], operating in a regime of relatively weak
dipole-dipole interactions, have allowed the realization of quantum
ferrofluids [7, 8] and the creation of droplet [9, 10] and supersolid phases
[11, 12]. Rydberg atoms [13], operating in a regime of extremely strong
interactions, have given rise to phases with crystalline order [14, 15],
simulation of quantum magnetic models [16, 17, 18], and the controlled
creation of entanglement [19, 20]. Dipolar molecules [21] are expected to
operate in an intermediate regime where kinetic energy and interaction energy
are on a similar scale, giving rise to nontrivial correlations and self-
organization. Predicted phases include strongly interacting superfluids [22,
23, 24], supersolids [25, 26, 27], dipolar crystals [28, 29], and Mott
insulators with fractional filling [30]. However, molecules have been found to
be prone to strong inelastic loss [31, 32, 33]. Suppression of losses has
emerged as a critical prerequisite to realize stable many-body quantum systems
of dipolar molecules.

Quantum statistics plays an important role in molecular loss dynamics [34].
Fermionic molecules are intriniscally less prone to inelastic loss than
bosonic ones. For indistinguishable fermions, the probability of reaching
short range in a two-body collision is suppressed by the $p$-wave centrifugal
barrier [35, 36, 37]. For bosons, such a barrier is absent and the rate of
two-body loss is typically one to two orders of magnitude larger than for
fermions [32]. To reduce loss below the natural rate, shielding techniques
have been proposed that utilize external electric fields to engineer a
repulsive barrier for intermolecular collisions, leveraging the rich internal
state structure of molecules [38, 39, 40, 41, 42]. Microwave shielding was
demonstrated in a proof-of-principle experiment for two bosonic CaF molecules
in an optical tweezer trap [43]. For bulk gases of fermionic molecules,
shielding with d.c. electric fields was shown for KRb [44, 45, 46] and
microwave shielding for NaK [47, 48], suppressing inelastic loss by about an
order of magnitude, sufficient to demonstrate evaporative cooling. Whether
loss in bosonic molecular gases can be sufficiently suppressed to enable
evaporative cooling has remained an open question.

Here, we demonstrate the stabilization of a gas of bosonic sodium-cesium
(NaCs) molecules against inelastic loss via microwave shielding. NaCs is
strongly dipolar, with a permanent dipole moment of $d_{0}=4.75(20)$ D [49],
which enhances losses of unshielded molecules [32], but in turn makes
shielding more effective. We suppress two-body loss by more than a factor of
200, increasing the lifetime of dense ensembles, with an interparticle spacing
of about 1 $\mu$m, from 16(2) ms to 1.0(1) s. The microwave field also induces
a dipole moment of up to 1.3 D, leading to dipolar interactions that enhance
elastic collisions. In a cross-thermalization experiment, we measure strong
elastic interactions and obtain a ratio of elastic-to-inelastic collisions of
up to $\gamma=4(1)\times 10^{3}$. Under these conditions, we demonstrate the
evaporative cooling of the bosonic molecular gas, increasing its phase-space
density (PSD) by a factor of 20 and reaching a temperature of $36(5)$ nK.

Figure 1: Microwave shielding of NaCs molecules. (A) Illustration of the
trapped molecular gas. Molecular dipoles are set into rotation by the electric
field of a $\sigma^{+}$-polarized microwave field generated by an antenna
array. Vertical beams for stimulated Raman adiabatic passage (STIRAP) allow
for time-of-flight expansion of the microwave shielded gas up to 50 $\mu$m
cloud waist, enabling precise measurement of temperature. (B) Rotational
levels of NaCs at 864 G. The states $|J,m_{J}\rangle=|0,0\rangle$ and
$|1,1\rangle$ are split by an energy $\hbar\omega_{\mathrm{res}}$ with
$\omega_{\mathrm{res}}=2\pi\times 3.471323(2)$ GHz. The microwave field is
blue-detuned with respect to the resonance by an amount $\Delta$. (C) Energy
of the single-molecule dressed states, $|+\rangle$ and $|-\rangle$, as a
function of Rabi frequency. The states are split by an energy
$\hbar\Omega_{\mathrm{eff}}=\hbar\sqrt{\Omega^{2}+\Delta^{2}}$. (D) Effective
dipole moment in the lab frame of the $|+\rangle$ state as a function of
$\Delta/\Omega$. (E) Potential energy curves of a pair of microwave dressed
molecules approaching in the $s$-wave channel for $\Omega/(2\pi)=4$ MHz and
$\Delta/(2\pi)=6$ MHz. The adiabatic potentials for $|++\rangle$ (blue solid
line), $|+0\rangle$ (grey solid line), and $|+-\rangle$ (red solid line) are
shown. Molecules are either (1) reflected by the repulsive potential, (2) lost
to non-shielded states, or (3) reach short range. The inset shows the
difference between bosonic NaCs ($s$-wave scattering, solid line) and a
hypothetical fermionic NaCs molecule ($p$-wave scattering with $m_{l}=\pm 1$,
dashed line) for which the $p$-wave barrier provides further shielding.

##  II Shielding

Our experiment begins with a gas of $3.0(5)\times 10^{4}$ NaCs molecules
prepared in their rovibrational ground state at a temperature of 750(50) nK.
The molecules are held in an optical dipole trap (ODT) with trap frequencies
$(\omega_{x},\omega_{y},\omega_{z})/(2\pi)=(60,\,65,\,140)$ Hz at a magnetic
field of about 864 G (Fig. 1A) which sets the quantization axis. Details on
the sample preparation can be found in [50, 51, 52]. Initially, the molecules
are in the rotational ground state $|J,\ m_{J}\rangle=|0,\ 0\rangle$, where
$J$ denotes the total angular momentum excluding nuclear spin and $m_{J}$ its
projection onto the quantization axis. Then, a circularly polarized microwave
field is applied at a frequency that is blue-detuned by an amount $\Delta$
from the resonance $\omega_{\mathrm{res}}$ with the excited state $|J,\
m_{J}\rangle=|1,\ 1\rangle$ (Fig. 1B). The field is created by a phased-array
antenna [53], ensuring a high purity of polarization. The intensity of the
microwave field is adiabatically increased within 40 $\mu$s, transferring each
molecule into the state $|+\rangle=\cos(\phi)|0,\ 0\rangle+\sin(\phi)|1,\
1\rangle$, where the mixing angle $\phi$ is defined by
$\sin(2\phi)=1/\sqrt{1+(\Delta/\Omega)^{2}}$ and $\Omega$ denotes the Rabi
frequency. The orthogonal dressed state,
$|-\rangle=\sin(\phi)|0,\,0\rangle-\cos(\phi)|1,\,1\rangle$, remains
unpopulated. Fig. 1C shows the energy splitting between the dressed states as
a function of Rabi coupling.

In a semiclassical picture, the dressed states represent dipoles rotating in
the $xy$-plane at a frequency $\omega_{\mathrm{res}}+\Delta$. Due to the
superposition of opposite parity states $|0,0\rangle$ and $|1,1\rangle$, the
dressed states feature an induced dipole moment. The effective dipole moment,
$d_{\mathrm{eff}}$, as a function of $\Delta/\Omega$ is shown in Fig. 1D. At
large intermolecular distances, shielded molecules interact through the long-
range dipole-dipole interaction $V_{\rm
dd}=d_{\mathrm{eff}}^{2}(3\cos^{2}\theta-1)/(4\pi\epsilon_{0}R^{3})$ [54, 55],
where $\theta$ denotes the angle between the rotation axis of the dipole and
the intermolecular axis, $\epsilon_{0}$ the vacuum permittivity, and $R$ the
intermolecular distance. When approaching each other, molecules in the
$|+\rangle$ state mutually align the orientation of their dipoles and repel
each other [41]. This is illustrated by the dressed intermolecular potentials
shown in Fig. 1E. The repulsion prevents the molecules from reaching short
range and suppresses loss from inelastic collisions [41]. The shielding
efficiency is limited by residual loss channels, such as tunneling of the
molecule pair through the microwave barrier, reaching short range, as well as
non-adiabatic transitions to other scattering channels between $|+\rangle$ and
$|-\rangle$ and the spectator states $|J,\ m_{J}\rangle=|1,\ 0\rangle$ and
$|1,\ -1\rangle$, collectively labeled $|0\rangle$.

For optimal parameters of microwave shielding, we observe an enhancement of
the lifetime from 16(2) ms to 1.0(1) s for ensembles with an initial peak
density of $1.0(2)\times 10^{12}$ cm-3. Fig. 2A displays lifetime data of
unshielded and shielded molecules, illustrating this dramatic change. As a
function of hold time in the dipole trap, $t_{\mathrm{hold}}$, we track both
the molecule number and temperature, as shown in Fig. 2A and B, and fit the
data with a kinetic model that includes one-body, two-body and evaporative
losses [52].

Figure 2: Lifetime and inelastic collisions of microwave shielded NaCs
molecules. (A) Lifetime of molecular ensembles with (blue) and without (grey)
shielding. The dashed lines indicate the respective $1/e$ lifetimes. Error
bars show 1$\sigma$ standard-error-of-the-mean. The shielded data is taken at
$\Omega/(2\pi)=4$ MHz and $\Delta/\Omega=1$. (B) Temperature evolution of
shielded (blue) and unshielded (grey) samples, corresponding to the data in
panel (A). The solid curves in (A) and (B) are fits of the solutions of a
kinetic model for molecule number and temperature. (C) Measured inelastic rate
coefficient as a function of $\Delta/\Omega$ at $\Omega/(2\pi)=4$ MHz. Circles
represent data points taken at $750(50)$ nK and squares are taken at $160(10)$
nK. The black dotted line corresponds to the measured two-body loss rate
coefficient in the absence of microwave shielding at $750(50)$ nK. The orange
shaded area shows a coupled-channel calculation for microwave ellipticities
between $1^{\circ}$ and $5^{\circ}$ at 750 nK. The calculation is scaled by a
factor of 2 to highlight the matching trend between experiment and theory.
Insets show the relevant experimental sequence. When shielding is ramped on,
the ODT power is adjusted to compensate for the change in AC polarizability of
the dressed state, ensuring that trap frequencies remain constant [56].
Microwave shielding is kept on during time-of-flight to prevent inelastic
losses in the initial phase of time-of-flight.

##  III Inelastic Collisions

We record lifetime data under different microwave parameters, which allows us
to extract the two-body loss rate coefficient, $\beta_{\mathrm{2B}}$, as a
function of $\Delta/\Omega$. Our study is conducted at $\Omega/(2\pi)=4.0(4)$
MHz. This value was chosen after we measured a plateau in the shielding
quality as a function of Rabi frequency between $\Omega/(2\pi)=4$ MHz and
$\Omega/(2\pi)=10$ MHz, with loss rates increasing on either side of this
range [52]. The measured loss rate coefficients are shown in Fig. 2C. For
$\Delta/\Omega>1$, the trend of the data agrees well with the results of a
coupled-channel calculation that takes into account our measured microwave
ellipticity of $\xi=3(2)^{\circ}$ [52, 53]. At $\Delta/\Omega=1$, the data
shows a 225-fold reduction in the loss rate coefficient compared to the
unshielded case, going from $\beta_{\mathrm{2B}}=450(50)\times 10^{-12}$ cm3/s
to $\beta_{\mathrm{2B}}=2.0(5)\times 10^{-12}$ cm3/s. For $\Delta/\Omega<1$,
the data deviates from the theoretical expectations. We compare the data taken
at 750(50) nK to a second run at 160(10) nK. While for $\Delta/\Omega>2$ the
two runs are practically indistinguishable, there is a marked difference for
$\Delta/\Omega<2$ with a significant uptick of the inelastic rate coefficient
for colder temperatures. Based on the coupled-channels calculation, which only
takes into account two-body physics, such a temperature dependence is not
expected. Three-body effects, not accounted for in the calculation, may be a
driver of this physics, potentially in conjunction with heating caused by
microwave-induced loss. A detailed understanding of these effects in future
work may allow reaching even lower loss rates, as predicted by theory.

Figure 3: Elastic collisions of microwave shielded NaCs molecules. (A) Cross-
thermalization experiment at $\Omega/(2\pi)=4$ MHz and $\Delta/(2\pi)=6$ MHz.
First, shielding is turned off and two-body loss in the unshielded gas is used
to heat the sample to between 650 and 800 nK; then shielding is turned back
on, triggering evaporation along the $z$-axis and cross-thermalization in the
$xy$-plane. Grey data points show the temperature evolution of the unshielded
molecules, blue data points show the temperature in the $xy$-plane after
shielding is turned back on. The blue solid line shows a fit to the data using
the kinetic model. (B) Theoretical calculation of $N_{\mathrm{col}}$ as a
function of $\Delta/\Omega$. The dark grey (light grey) band is the calculated
number of collisions for $xz$ ($xy$)-thermalization for a temperature range
between 650 nK and 800 nK. The dashed line indicates $N_{\mathrm{col}}=2.5$ as
expected for an isotropically interacting system. (C) Measured elastic cross
section as a function of $\Omega/(2\pi)=4$ MHz. The orange shaded band is the
integral scattering cross section from the coupled-channel calculation for a
temperature range between 650 nK and 800 nK [52]. The grey shaded area
corresponds to the hydrodynamic limit [47], which we do not enter at the
densities used for this measurement. The dashed vertical line marks the
transition between the semiclassical ($kT_{\mathrm{B}}>E_{\mathrm{d}}$) and
threshold ($kT_{\mathrm{B}}<E_{\mathrm{d}}$) regimes of elastic scattering.
(D) Ratio of elastic-to-inelastic collisions, $\gamma$, as a function of
shielding parameter $\Delta/\Omega$.

##  IV Elastic Collisions

In addition to the suppression of inelastic collisions, the dipole moment
induced by the microwave field enhances elastic collisions. The effective
dipole moment depends on the microwave parameters as
$d_{\mathrm{eff}}=d_{0}/\sqrt{12(1+(\Delta/\Omega)^{2})}$ (see Fig. 1D), and
the resulting dipole-dipole interactions vary as a function of the shielding
parameter, $\Delta/\Omega$. For example, for small $\Delta/\Omega$ the dressed
state approaches an equal superposition of $|0,0\rangle$ and $|1,1\rangle$,
leading to a maximal induced dipole moment and an enhancement of the dipolar
contribution to the elastic collisions.

As the shielding parameter $\Delta/\Omega$ is varied, the molecular gas probes
two different regimes of dipolar scattering depending on the relative
magnitude of the thermal energy, $k_{\mathrm{B}}T$, and the dipolar energy,
$E_{\mathrm{d}}=d_{\mathrm{eff}}^{2}/(4\pi\epsilon_{0}\,a_{\mathrm{d}}^{3})$
[57], where $a_{\mathrm{d}}=Md_{\mathrm{eff}}^{2}/(8\pi\hbar^{2}\epsilon_{0})$
is the dipolar length and $M$ the molecular mass. The elastic scattering cross
section, $\sigma_{\mathrm{el}}$, varies in magnitude, temperature dependence,
and anisotropy depending on which of these energies is dominant. For
$k_{\mathrm{B}}T\gg E_{\mathrm{d}}$, collisions are semiclassical and
$\sigma_{\mathrm{sc}}=8\pi\,a_{\mathrm{d}}/(3k)$, where
$k=\sqrt{\pi\,M\,k_{\mathrm{B}}T}/\hbar$ is the thermally-averaged collisional
$k$-vector. For $k_{\mathrm{B}}T\leq E_{\mathrm{d}}$, i.e. as the thermal
deBroglie wavelength, $\lambda_{\mathrm{th}}=h/\sqrt{2\pi Mk_{\mathrm{B}}T}$,
approaches or exceeds the length scale of dipole-dipole interactions, the
collisional properties are modified and the cross section enters the threshold
regime, becoming $\sigma_{\mathrm{th}}=32\pi\,a_{\mathrm{d}}^{2}/45+8\pi
a_{s}^{2}$, where $a_{s}$ is the $s$-wave scattering length of the molecules
[57]. Since $d_{\mathrm{eff}}$ is a function of $\Delta/\Omega$, our
experiment accesses both regimes, with $k_{\mathrm{B}}T\approx E_{\mathrm{d}}$
at large microwave detunings and $k_{\mathrm{B}}T\gg E_{\mathrm{d}}$ close to
resonance.

We measure the elastic collision cross section via a cross-thermalization
experiment (Fig. 3). For these measurements we keep the peak density below
$0.2\times 10^{12}$ cm-3 to avoid entering the hydrodynamic regime, where the
thermalization rate would be limited to the mean trap frequency [47]. At
constant ODT depth, fast heating of the cloud to a set temperature is induced,
followed by evaporative cooling along the vertical $z$-axis and cross-
thermalization in the $xy$-plane. During the sequence, we follow the
temperature evolution of the cloud in $x$-direction, as shown in Fig. 3A. Via
the kinetic model [52], we extract the thermalization rate
$\Gamma_{\mathrm{th}}=\sigma_{\mathrm{el}}\,\bar{n}\,v_{\mathrm{th}}/N_{\mathrm{col}}$
[47], where $v_{\mathrm{th}}=4\sqrt{k_{\mathrm{B}}T/(\pi M)}$ is the
molecules’ mean thermal velocity, $\bar{n}$ the mean density of the cloud, and
$N_{\mathrm{col}}$ the average number of collisions required for cross-
thermalization between the $z$-axis and the $xy$-plane. Smaller
$N_{\mathrm{col}}$ means more efficient energy transfer. From
$\Gamma_{\mathrm{th}}$ we obtain the elastic scattering cross section
$\sigma_{\mathrm{el}}$.

The anisotropic nature of dipolar interactions has a profound impact on the
thermalization dynamics via the value of $N_{\mathrm{col}}$. Close to
resonance, where the molecular gas is in the semiclassical regime and
$E_{\mathrm{d}}$ is small, forward collisions that do not deflect the
molecules’ trajectories by a large angle are favored, thus limiting the
transverse energy transfer. More efficient energy redistribution is achieved
in the threshold regime at larger detunings, when the induced dipole moment is
lower and $E_{\mathrm{d}}$ becomes larger. This is shown in Fig. 3B, in which
a calculation of $N_{\mathrm{col}}$ for cross-thermalization between the $x$
and $z$ (or $x$ and $y$) axes is provided. Unlike experiments on fermionic
dipoles [58, 46] where there is only the dipole-dipole interaction, bosonic
systems also have an $s$-wave van der Waals contribution to elastic
scattering. The scattering length, $a_{s}$, is not known for NaCs and we
obtain a value of $a_{s}=1200$ $a_{0}$ from a fit to our data [52]. With
$a_{s}$ being the only free fitting parameter, we find excellent agreement for
$\sigma_{\mathrm{el}}$ between the experiment and a coupled channel
calculation, as shown in Fig. 3C.

From the measured elastic and inelastic collision rates, we calculate the
ratio of elastic-to-inelastic collisions, $\gamma$, as shown in Fig. 3D. We
observe a peak value of $\gamma\approx 4(1)\times 10^{3}$ at
$\Delta/\Omega=1$. The quantity $\gamma$ is typically used as a key parameter
to characterize the efficacy of forced evaporative cooling [59, 60]. However,
evaporative cooling with dipolar elastic collisions in the semiclassical
regime is qualitatively different from evaporation in systems with $s$-wave or
threshold dipolar interactions, as is typically the case in atomic and
molecular systems, including the recent demonstrations of evaporative cooling
in fermionic dipolar molecules [44, 47]. In our case the reduced quantity
$\gamma/N_{\mathrm{col}}$, rather than $\gamma$, sets the thermalization rate,
and thus the evaporation efficiency. Our highest value of
$\gamma/N_{\mathrm{col}}\approx 250$ is still favorable for efficient
evaporation [43, 47].

##  V Evaporation

We demonstrate evaporative cooling in the stabilized ultracold gas of NaCs
molecules. We start with a gas at a temperature of $750(50)$ nK and a PSD of
$5(1)\times 10^{-3}$. Then, the depth of the ODT is continuously reduced over
1.5 s, while the molecular cloud is shielded at $\Omega/(2\pi)=4$ MHz and
$\Delta/(2\pi)=6$ MHz. At different stages of the evaporation, we measure the
molecule number and temperature, and extract the phase-space density of the
cloud, as shown in Fig. 4. At the end of the cooling sequence, we reach a
temperature of $36(5)$ nK and a corresponding PSD of $0.10(3)$. For small
molecule numbers, the measured phase-space density seems to show a plateau,
likely the result of the limited signal-to-noise of our detection system. The
extracted evaporation efficiency, $-d{\rm ln}({\rm PSD})/d{\rm ln}(N)$, is
$1.0(1)$. This efficiency is similar to those found in recent work on
evaporative cooling of fermionic ground state molecules [44, 46, 47]. We note
that, besides the prospect of cooling the molecular gas to degeneracy,
evaporative cooling allows the preparation of molecular samples at well-
defined temperatures over a wide dynamic range, which will facilitate studies
of quantum chemistry and collisional physics in bosonic molecular gases [33].

Figure 4: Evaporation of microwave dressed molecules at $\Omega/(2\pi)=4$ MHz
and $\Delta/(2\pi)=6$ MHz. (A) Evolution of phase-space density during
evaporation. The insets show images of molecular gases after 3 ms of time-of-
flight. The low temperature cloud is an average of five images; the high
temperature cloud is a single image. (B) Temperature evolution corresponding
to panel (A).

##  VI Outlook

In this work, we have demonstrated that ultracold gases of bosonic ground
state molecules can be effectively stabilized via microwave shielding,
reaching low inelastic loss rates similar to shielded fermionic molecules
despite the absence of a $p$-wave barrier. Our data on anisotropic cross-
thermalization shows that, even above quantum degeneracy, the strongly dipolar
character of our gas leads to nontrivial thermodynamic behavior. Dipolar
liquids similar to our system have recently been predicted to show anisotropic
thermal conductivity [61] and viscosity [62]. Thanks to the rapid tunability
of microwave parameters $\Omega$ and $\Delta$, allowing the quasi-
instantaneous tuning of inelastic and elastic scattering properties, novel
non-equilibrium measurement protocols can be envisioned to probe such
thermodynamics.

A key question that emerges from this work is whether Bose-Einstein
condensation of microwave-shielded dipolar molecules can be achieved. The
measured gain in phase-space density brings us to the brink of Bose-Einstein
condensation. Following the observed trend in evaporation, a BEC with 100
molecules would be expected. This is currently below the signal-to-noise level
of our imaging system, but straightforward upgrades will provide the necessary
improvements. In addition, further reduction of microwave noise and a better
understanding of inelastic loss at small detunings may allow us to reach even
lower loss levels, as predicted by theory. Field-linked resonances [38, 48],
which should be accessible for NaCs at low microwave ellipticity and moderate
microwave intensity [48], allow for independent tuning of $a_{\mathrm{d}}$ and
$a_{\mathrm{s}}$ [42] and offer a tuning knob to further improve on the
scattering properties of our molecules. Using field-linked resonances to
tailor interactions may also tune $N_{\mathrm{col}}$ close to or below the
$s$-wave value of 2.5 [63], increasing the thermalization rate by an order of
magnitude, enhancing our evaporation.

Starting from shielded three-dimensional bulk samples of bosonic molecules,
efficient transfer with minimal loss to lower dimensional systems comes within
reach. In two-dimensional systems, microwave shielding and d.c. electric
fields should enable the realization of strongly correlated phases [28], such
as supersolidity and self-organized crystallization of dipoles, both in single
layers [27] and multilayers [23]. NaCs molecules are a highly promising
platform for such explorations due their large dipole moment, which allows a
characteristic range of dipolar interactions, $a_{\mathrm{d}}$, of tens of
micrometers. Shielded loading of optical lattices, in particular from a BEC of
molecules, may offer a way to reach unity filling, necessary to realize
extended Hubbard models [30, 64], and enabling studies of many-body spin
models in defect-free molecular arrays [65, 66, 67, 68].

##  VII Acknowledgements

We thank Andreas Schindewolf, Goulven Quéméner, and Timon Hilker for helpful
discussions, Michal Lipson and Javad Shabani for the loan of equipment, and
Tarik Yefsah for critical reading of the manuscript. This work was supported
by an NSF CAREER Award (Award No. 1848466), an ONR DURIP Award (Award No.
N00014-21-1-2721), and a Lenfest Junior Faculty Development Grant from
Columbia University. C.W. acknowledges support from the Natural Sciences and
Engineering Research Council of Canada (NSERC). W.Y. acknowledges support from
the Croucher Foundation. I.S. was supported by the Ernest Kempton Adams Fund.
S.W. acknowledges additional support from the Alfred P. Sloan Foundation.

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