# Non-almost regular quaternary $m$-gonal forms

Dayoon Park  Department of Mathematical Sciences, Ulsan National Institute of
Science and Technology, Ulsan, Korea<EMAIL_ADDRESS>
###### Abstract.

The maximal $n$ which admits a non-almost regular $m$-gonal form of rank $n$
is $4$. In this article, we consider the $m$-gonal forms of rank $4$ which is
not almost regular.

##  1\. Introduction

The $m$-gonal number which is defined as a total number of dots to constitute
a regular $m$-gon is one of the classical objects in the number theory. More
precisely, we call the total number of dots of regular $m$-gon with $x$ dots
for each side

(1.1) $P_{m}(x):=\frac{m-2}{2}(x^{2}-x)+x$

as $x$-th $m$-gonal number. One of the most famous stories about $m$-gonal
number is a Fermat’s Conjecture which states that every positive integer may
be written as a sum of at most $m$ $m$-gonal numbers. This conjecture was
resolved by Lagrange, Gauss, and Cauchy for $m=4$, $m=3$, and $m\geq 3$ ,
respectively. As a generalization of Gauss’s work, Liouvelle classified the
$3$-tuples $(a_{1},a_{2},a_{3})\in{\mathbb{N}}^{3}$ for which every positive
integer can be written as a sum
$a_{1}P_{3}(x_{1})+a_{2}P_{3}(x_{2})+a_{3}P_{3}(x_{3})$ for some
$(x_{1},x_{2},x_{3})\in{\mathbb{N}}_{0}^{3}$. And as a generalization of
Lagrange’s work, Ramanujan classified the $4$-tuples
$(a_{1},a_{2},a_{3},a_{4})\in{\mathbb{N}}^{4}$ for which every positive
integer can be written as a sum
$a_{1}P_{4}(x_{1})+a_{2}P_{4}(x_{2})+a_{3}P_{4}(x_{3})+a_{4}P_{4}(x_{4})$ for
some $(x_{1},x_{2},x_{3},x_{4})\in{\mathbb{N}}_{0}^{4}$, but one mistake was
found later in the Ramanujan’s list.

We call a weighted sum of $m$-gonal numbers

(1.2) $F_{m}(\mathbf{x})=a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$

where $(a_{1},\cdots,a_{n})\in{\mathbb{N}}^{n}$ as $m$-gonal form. We simply
write the $m$-gonal form $a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$ as
$\left<a_{1},\cdots,a_{n}\right>_{m}$. But exceptionally we adopt a notation
$\left<a_{1},\cdots,a_{n}\right>$ for the square ($4$-gonal) form (i.e.,
diagonal quadratic form) instead of $\left<a_{1},\cdots,a_{n}\right>_{4}$ true
to the tradition of the theory of quadratic form. By the definition of
original $m$-gonal number, only positive rational integer $x$ would be
admitted in (1.1). But as a generalized $m$-gonal number version, we admit
every rational integer $x\in{\mathbb{Z}}$ in (1.1) in this article. For
$N\in{\mathbb{N}}$, if the diophantine equation

(1.3) $F_{m}(\mathbf{x})=N$

has an integer soulution $\mathbf{x}\in{\mathbb{Z}}^{n}$, then we say that the
$m$-gonal form $F_{m}(\mathbf{x})$ (globally) represents $N$. As is well
known, the problem of determining the solvability of non-linear diophantine
equation is not easy in general. In an effort to understand the
representability of $N$ by $F_{m}(\mathbf{x})$, we consider the congruence
equation

(1.4) $F_{m}(\mathbf{x})\equiv N\pmod{r}.$

When the congruence equation (2.1) has an integer solution
$\mathbf{x}\in{\mathbb{Z}}^{n}$ for every $r\in{\mathbb{Z}}$, we say that
$F_{m}(\mathbf{x})$ locally represents $N$. The solvability of congruence
equation

$F_{m}(\mathbf{x})\equiv N\pmod{p^{r}}$

for every prime $p$ and $r\in{\mathbb{N}}$ would be enough to derive the local
representability of $N$ by $F_{m}(\mathbf{x})$ in virtue of the Chinese
Remainder Theorem. Note that for some prime $p$, the congruence equation
$F_{m}(\mathbf{x})\equiv N\pmod{p^{r}}$ has an integer solution
$\mathbf{x}\in{\mathbb{Z}}^{n}$ for every $r\in{\mathbb{N}}$ if and only if
$F_{m}(\mathbf{x})=N$ has an $p$-adic integer solution
$\mathbf{x}\in{\mathbb{Z}}_{p}^{n}$. When $F_{m}(\mathbf{x})=N$ has a $p$-adic
integer solution $\mathbf{x}\in{\mathbb{Z}}_{p}^{n}$, we say that
$F_{m}(\mathbf{x})$ represents $N$ over ${\mathbb{Z}}_{p}$. Obviously, the
representability implies the local representability, but the converse does not
hold in general. Especially, when the converse also holds, i.e., an $m$-goanl
form $F_{m}(\mathbf{x})$ represents every positive integer $N$ which is
locally represented by $F_{m}(\mathbf{x})$, we say that $F_{m}(\mathbf{x})$ is
regular. When $F_{m}(\mathbf{x})$ represents every positive integer $N$ which
is locally represented by $F_{m}(\mathbf{x})$ but finitely many, we say that
$F_{m}(\mathbf{x})$ is almost regular.

The concept regular was firstly introduced by Dickson in [3]. In [11], Watson
proved that there are only finitely many primitive positive definite regular
ternary quadratic forms by introducing very nice Watson’s $\Lambda$\-
transformation which is widely applied in the study of regular form. In [5],
Jagy, Kaplansky and Schiemann who used the Watson’s $\Lambda$-transformation
classified 913 candidates for primitive positive definite regular ternary
quadratic forms. And they claimed the regularity of all of them but 22. Oh [8]
showed the regularity of $8$ forms of unconfirmed $22$ forms. And Lemke Oliver
[7] proved the regularity of the remaining $14$ candidates under the
assumption of generalized Riemann Hypothesis.

The finiteness of primitive regular ternary triangular forms was shown by Chan
and Oh [1] and all the finitely many primitive regular ternary triangular
forms was determined by Kim and Oh. As an improvement of the result in [1],
Chan and Ricci [2] proved the finiteness of primitive regualr ternary
quadratic polynomials with a fixed conductor in [2]. As a another version’s
improvement of the work in [1], He and Kane showed the finiteness of primitive
regular ternary polygonal forms. Which implies that there are only finitely
many $m$ which admits a regular $m$-gonal form of rank $3$. In [6], Kim and
the author showed that for each $n\geq 4$, there are only finitely many $m$
which admits a regular $m$-gonal form of rank $n$. Moreover in [6], they
completely determined the type of regualr $m$-gonal forms of rank $n\geq 4$
for all $m\geq 14$ with $m\not\equiv 0\pmod{4}$ and $m\geq 28$ with $m\equiv
0\pmod{4}$ by introducing $\Lambda_{p}$-transformation which is well designed
to trace the regular $m$-gonal forms.

On the other hand, by Theorem 4.9 (1) in [1], every $m$-gonal form of rank
$n\geq 5$ is almost regular. On the other words, every $m$-gonal form of rank
$n\geq 5$ represents all the sufficiently large integers which are locally
represented by the form. In [13], the author considered about the sufficiently
large integers. And then one may notice that the largest $n$ which admits a
non-almost regular $m$-gonal form of rank $n$ would be at most $4$. In this
short article, we consider non-almost regular $m$-gonal forms of rank $4$. The
following two theorems are the main goal in this article.

######  Theorem 1.1.

Let $a_{1},a_{2},a_{3},a_{4}\in{\mathbb{N}}$ with
$(a_{1},a_{2},a_{3},a_{4})=1$.

  * (1)

If the quaternary quadratic form
$\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{p}$ is isotropic for
every prime $p$, then the quaternary $m$-gonal form
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ is almost regular for every $m\geq
3$.

  * (2)

Otherwise, there are infinitely many $m\geq 3$ for which
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ is not almost regular.

######  Theorem 1.2.

For any $m\geq 7$, there are infinitely many primitive quaternary $m$-gonal
forms which are not almost regular.

In order to to avoid the appearance of unneecessary duplications, we always
assume that an m-gonal form $\left<a_{1},\cdots,a_{n}\right>_{m}$ is
primitive, i.e., $(a_{1},\cdots,a_{n})=1$ in this article.

######  Remark 1.3.

By Theorem 1.1, we have that for each $4$-tuple
$(a_{1},a_{2},a_{3},a_{4})\in{\mathbb{N}}^{4}$, either

$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ is almost regular for every $m\geq
3$

or

$\text{$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ are not almost regular for
infinitely many $m\geq 3$}.$

Throughout this article, we adopt the arithmetic theory of quadratic forms.
Before we move on, we briefly set our languages and terminologies which are
used throughout this article.

If a quadratic form (or $m$-gonal form) represents every $p$-adic integer in
$N{\mathbb{Z}}_{p}$ over ${\mathbb{Z}}_{p}$, then we say that the quadratic
form (or $m$-gonal form) is $N{\mathbb{Z}}_{p}$-universal and especially, when
$N{\mathbb{Z}}_{p}={\mathbb{Z}}_{p}$, we say that universal over
${\mathbb{Z}}_{p}$. If a quadratic form (or $m$-gonal form) is universal over
${\mathbb{Z}}_{p}$ for every prime $p$, then we say that the quadratic form
(or $m$-gonal form) is locally universal.

We say that a quadratic form $Q(\mathbf{x})$ is isotropic if the equation
$Q(\mathbf{x})=0$ has a non-trivial solution $\mathbf{x}\not\equiv\mathbf{0}$
and $\it anisotropic$ otherwise. It is well known that any quadratic form
$Q(\mathbf{x})$ of rank $n\geq 5$ is isotropic over local ring
${\mathbb{Z}}_{p}$. This property is critical in the regularity of arbitrary
quadratic form of rank $n\geq 5$.

Any unexplained notation and terminology can be found in [10].

##  2\. Preliminaries

We may get an easy and important observation that

$N=a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$

if and only if

$8(m-2)N+(m-4)^{2}(a_{1}+\cdots+a_{n})=\sum_{i=1}^{n}a_{i}(2(m-2)x_{i}-(m-4))^{2}.$

Which allows transfer the issue of the representability by $m$-gonal form to
the issue of the representability by diagonal quadratic form with a congruence
condition. By using the following proposition, one may effectively determine
the representability of integer by an $m$-gonal form over local ring with help
of known results in the quadratic form theory.

######  Proposition 2.1.

Let $F_{m}(\mathbf{x})=a_{1}P_{m}(x_{1})+\cdots+a_{n}P_{m}(x_{n})$ be a
primitive $m$-gonal form.

  * (1)

When $p$ is an odd prime with $p|m-2$, $F_{m}(\mathbf{x})$ is universal over
${\mathbb{Z}}_{p}$.

  * (2)

When $m\not\equiv 0\pmod{4}$, $F_{m}(\mathbf{x})$ is universal over
${\mathbb{Z}}_{2}$.

  * (3)

When $p$ is an odd prime with $(p,m-2)=1$, an integer $N$ is represented by
$F_{m}(\mathbf{x})$ over ${\mathbb{Z}}_{p}$ if and only if the integer
$8(m-2)N+(a_{1}+\cdots+a_{n})(m-4)^{2}$ is represented by the diagonal
quadratic form $\left<a_{1},\cdots,a_{n}\right>$ over ${\mathbb{Z}}_{p}$.

  * (4)

When $m\equiv 0\pmod{4}$, an integer $N$ is represented by $F_{m}(\mathbf{x})$
over ${\mathbb{Z}}_{2}$ if and only if the integer
$\frac{m-2}{2}N+(a_{1}+\cdots+a_{n})\left(\frac{m-4}{4}\right)^{2}$ is
represented by the diagonal quadratic form $\left<a_{1},\cdots,a_{n}\right>$
over ${\mathbb{Z}}_{2}$.

###### Proof.

See Proposition 3.1 in [13]. ∎

######  Remark 2.2.

Now, we may need understanding the behavior of representation by a quadratic
form which is anisotropic over a local ring.

  * (1)

First, we consider a quadratic form which is anisotropic over a non-dyadic
local ring ${\mathbb{Z}}_{p}$. A quaternary anisotropic diagonal qudaratic
form over a non-dyadic local ring ${\mathbb{Z}}_{p}$ is the form of

$\left<ap^{2r_{1}+1},aup^{2r_{2}+1},bp^{2r_{3}},bu^{\prime}p^{2r_{4}}\right>$

where $a,b\in{\mathbb{Z}}_{p}^{\times}$, $r_{i}\in{\mathbb{N}}_{0}$, and
$-u,-u^{\prime}\in{\mathbb{Z}}_{p}^{\times}$ are non-residues. So one may get
that for a quadratic form $\left<a_{1},a_{2},a_{3},a_{4}\right>$ which is
anisotropic over ${\mathbb{Z}}_{p}$, when

$\text{ord}_{p}(a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+a_{3}x_{3}^{2}+a_{4}x_{4}^{2})\geq
r$

for some $(x_{1},x_{2},x_{3},x_{4})\in{\mathbb{Z}}_{p}^{4}$ (so also
$\in{\mathbb{Z}}^{4}$), $\text{ord}_{p}(a_{i}x_{i}^{2})\geq r$ for each $i$.

  * (2)

A primitive quaterary anisotropic diagonal quadratic form over
${\mathbb{Z}}_{2}$ is the form of either

$\left<a_{1}2^{2r_{1}},a_{2}2^{2r_{2}},a_{3}2^{2r_{3}+1},a_{4}2^{2r_{4}+1}\right>$

where $r_{1},r_{2},r_{3},r_{4}\in{\mathbb{N}}_{0}$ and
$a_{1},a_{2},a_{3},a_{4}\in{\mathbb{Z}}_{2}^{\times}$ with $a_{1}\equiv
a_{2}\equiv\frac{a_{3}+a_{4}}{2}\pmod{4}$ and $a_{3}\equiv
a_{4}\equiv\frac{a_{3}+a_{4}}{2}\pmod{4}$ or

$\left<a_{1}2^{2r_{1}},a_{2}2^{2r_{2}},a_{3}2^{2r_{3}},a_{4}2^{2r_{4}}\right>$

where $r_{1},r_{2},r_{3},r_{4}\in{\mathbb{N}}_{0}$ and
$a_{1},a_{2},a_{3},a_{4}\in{\mathbb{Z}}_{2}^{\times}$ with $a_{1}\equiv
a_{2}\equiv a_{3}\equiv a_{4}\pmod{4}$ and $a_{1}+a_{2}+a_{3}+a_{4}\not\equiv
0\pmod{8}$. So one may get that for a quadratic form
$\left<a_{1},a_{2},a_{3},a_{4}\right>$ which is anisotropic over
${\mathbb{Z}}_{2}$, when

$\text{ord}_{p}(a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+a_{3}x_{3}^{2}+a_{4}x_{4}^{2})\geq
r+2$

for some $(x_{1},x_{2},x_{3},x_{4})\in{\mathbb{Z}}_{2}^{4}$ (so also
$\in{\mathbb{Z}}^{4}$), $\text{ord}_{2}(a_{i}x_{i}^{2})\geq r$ for each $i$.

######  Proposition 2.3.

A quaternary quadratic form

$\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{p}$

is $p^{\max_{1\leq i\leq 4}\text{ord}_{p}(a_{i})}{\mathbb{Z}}_{p}$-universal
for every prime $p$.

###### Proof.

First, assume that $p$ is an odd prime. By 92:1b in [10], a ternary unimodular
(i.e., $u_{1},u_{2},u_{3}\in{\mathbb{Z}}_{p}^{\times}$) quadratic form
$\left<u_{1},u_{2},u_{3}\right>\otimes{\mathbb{Z}}_{p}$ is
${\mathbb{Z}}_{p}$-universal and a binary unimodular (i.e.,
$u_{1},u_{2}\in{\mathbb{Z}}_{p}^{\times}$) quadratic form
$\left<u_{1},u_{2}\right>\otimes{\mathbb{Z}}_{p}$ represents every $p$-adic
unit ${\mathbb{Z}}_{p}^{\times}$. Which may yield the claim.

Now assume that $p=2$. One may show that for arbitrary $2$-adic units
$u_{i}\in{\mathbb{Z}}_{2}^{\times}$, the quadratic forms

$\left<u_{1},2u_{2},,2u_{3},2u_{4}\right>,\
\left<u_{1},u_{2},,2u_{3},2u_{4}\right>,\
\left<u_{1},u_{2},,u_{3},2u_{4}\right>,\
\left<u_{1},u_{2},,u_{3},u_{4}\right>$

are ${\mathbb{Z}}_{2}$-universal over ${\mathbb{Z}}_{2}$. For two $2$-adic
units $u,u^{\prime}\in{\mathbb{Z}}_{2}^{\times}$, since

$u\equiv u^{\prime}\pmod{8}\iff u\in
u^{\prime}({\mathbb{Z}}_{2}^{\times})^{2},$

it would be enough to show the claim only for $u_{i}\in\\{1,3,5,7\\}$. The
tedious processing is omitted in this article. ∎

##  3\. Non-almost regular quaternary $m$-gonal forms

Now we are ready to prove our main goals. At the end, in this last section, we
prove the Theorem 1.1 by assigning a suitable $m$ and Theorem 1.2 by
concretely constructing a non-almost regular $m$-gonal form of rank $4$.

###### proof of Theorem 1.1.

(1) Theorem 4.9 (2) yields the claim.
(2) Let $T$ be a finite set (by Proposition 2.3) of all odd primes $q$ for
which the quaternary quadratic form
$\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{p}$ is not universal
($T$ could be empty set).

First, assume that there is an odd prime $p$ for which
$\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{p}$ is anisotropic.
Now take an odd $m\geq a_{1}+a_{2}+a_{3}+a_{4}+p^{\max_{1\leq i\leq
i}\\{\text{ord}_{p}(a_{i})\\}}+4$ such that

$\begin{cases}m-2\equiv 0\pmod{q}&\text{ for every }q\in T\setminus\\{p\\}\\\
m-4\equiv 0\pmod{p}\\\ \end{cases}$

(note that there are infinitely many such $m$). The $m$-gonal form
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ would not represent anyting between
$a_{1}+a_{2}+a_{3}+a_{4}+1$ and $a_{1}+a_{2}+a_{3}+a_{4}+p^{\max_{1\leq i\leq
4}\\{\text{ord}_{p}(a_{i})\\}}(<m-3)$ since the smallest $m$-gonal number is
$m-3$ except $0$ and $1$. So we may take
$N_{0}\in[a_{1}+a_{2}+a_{3}+a_{4}+1,a_{1}+a_{2}+a_{3}+a_{4}+p^{\max_{1\leq
i\leq 4}\\{\text{ord}_{p}(a_{i})\\}}]$ for which

$\begin{cases}N_{0}\not\rightarrow\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}\\\
8(m-2)N_{0}+(m-4)^{2}(a_{1}+a_{2}+a_{3}+a_{4})\equiv 0\pmod{p^{\max_{1\leq
i\leq 4}\\{\text{ord}_{p}(a_{i})\\}}}.\end{cases}$

For $k\in{\mathbb{N}}$ satisfying

$p^{k}\equiv 1\pmod{8(m-2)}$

(the existence such $k\in{\mathbb{N}}$ comes from $(p,8(m-2))=1$), the
positive integers

$N_{n}:=\frac{p^{2kn}\\{8(m-2)N_{0}+(m-4)^{2}(a_{1}+\cdots+a_{4})\\}-(m-4)^{2}(a_{1}+\cdots+a_{4})}{8(m-2)}$

are locally represented by the $m$-gonal form
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ but not globally represented by
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ for all $n\geq 0$. For recall that

$N=a_{1}P_{m}(x_{1})+a_{2}P_{m}(x_{2})+a_{3}P_{m}(x_{3})+a_{4}P_{m}(x_{4})$

if and only if

$8(m-2)N+(m-4)^{2}(a_{1}+a_{2}+a_{3}+a_{4})=\sum_{i=1}^{4}a_{i}(2(m-2)x_{i}-(m-4))^{2}.$

By Proposition 2.1, the $m$-gonal form would be ${\mathbb{Z}}_{q}$-universal
for all primes $q\not=p$. And since the quadratic form
$\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{p}$ is
$p^{\max_{1\leq i\leq
4}\\{\text{ord}_{p}(a_{i})\\}}{\mathbb{Z}}_{p}$-univresal by Proposition 2.3,
for every $n\geq 0$, the integer $N_{n}$ satisfying
$8(m-2)N_{n}+(m-4)^{2}(a_{1}+\cdots+a_{n})\equiv 0\pmod{p^{\max_{1\leq i\leq
4}\\{\text{ord}_{p}(a_{i})\\}}}$ would be represented by
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ over ${\mathbb{Z}}_{p}$ for
$2(m-2)\in{\mathbb{Z}}_{p}^{\times}$. As the above arguments, we have that the
positive integer $N_{n}$ is locally represented by the $m$-goanl form
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ for every $n\geq 0$.

Now we show that the integer $N_{n}$ is not (globally) represented by
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ for every $n\geq 0$. For a
contradiction, we suppose that $N_{n}$ is represented by
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ over ${\mathbb{Z}}$, i.e.,

$8(m-2)N_{n}+(m-4)^{2}(a_{1}+a_{2}+a_{3}+a_{4})=\sum_{i=1}^{4}a_{i}(2(m-2)x_{i}-(m-4))^{2}$

for some $x_{i}\in{\mathbb{Z}}$. Since

$8(m-2)N_{n}+(m-4)^{2}(a_{1}+a_{2}+a_{3}+a_{4})\equiv
0\pmod{p^{2kn+{\max_{1\leq i\leq 4}\\{\text{ord}_{p}(a_{i})\\}}}}$

and $\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{p}$ is
anisotropic, we have that

$2(m-2)x_{i}-(m-4)\equiv 0\pmod{p^{kn}}$

by Remark 2.2. From our choice of $k\in{\mathbb{N}}$ satisfying $p^{k}\equiv
1\pmod{8(m-2)}$, we have that

$\frac{2(m-2)x_{i}-(m-4)}{p^{kn}}=2(m-2)y_{i}-(m-4)$

for some $y_{i}\in{\mathbb{Z}}$. And then
$N_{0}=a_{1}P_{m}(y_{1})+a_{2}P_{m}(y_{2})+a_{3}P_{m}(y_{3})+a_{4}P_{m}(y_{4})$
which is a contradition. This completes the proof.

Secondly, assume that
$\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{p}$ is isotropic for
every odd prime $p$ and
$\left<a_{1},a_{2},a_{3},a_{4}\right>\otimes{\mathbb{Z}}_{2}$ is anisotropic.
Now take an $m\geq a_{1}+a_{2}+a_{3}+a_{4}+2^{\max_{1\leq i\leq
i}\\{\text{ord}_{2}(a_{i})\\}+1}+4$ such that

$\begin{cases}m-2\equiv 0\pmod{q}&\text{ for every }q\in T\\\ m\equiv
0\pmod{4}\\\ \end{cases}$

(note that there are infinitely many such $m$). By the same token with the
above, we may take
$N_{0}\in[a_{1}+a_{2}+a_{3}+a_{4}+1,a_{1}+a_{2}+a_{3}+a_{4}+2^{\max_{1\leq
i\leq 4}\\{\text{ord}_{2}(a_{i})\\}+1}]$ for which

$\begin{cases}N_{0}\not\rightarrow\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}\\\
\frac{m-2}{2}N_{0}+\left(\frac{m-4}{4}\right)^{2}(a_{1}+a_{2}+a_{3}+a_{4})\equiv
0\pmod{2^{\max_{1\leq i\leq 4}\\{\text{ord}_{2}(a_{i})\\}+1}}.\end{cases}$

One may show that for $k\in{\mathbb{N}}$ satisfying

$2^{k}\equiv 1\pmod{\frac{m-2}{2}},$

the positive integers

$N_{n}:=\frac{4^{kn}\\{\frac{m-2}{2}N_{0}+\left(\frac{m-4}{4}\right)^{2}(a_{1}+\cdots+a_{4})\\}-\left(\frac{m-4}{4}\right)^{2}(a_{1}+\cdots+a_{4})}{\frac{m-2}{2}}$

are locally represented by the $m$-gonal form
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ but not globally represented by
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ for all $n\geq 0$ through similar
arguments with the above. ∎

###### proof of Theorem 1.2.

Let $T$ be a finite set of all primes $p$ for which $p|m-2$.

First, assume that there is an odd prime $q$ for which $q|m-4$. For some
suitable two different odd primes
$p^{\prime}(\not=q),p^{\prime\prime}(\not=q)$ and $r(p)\geq 0$, we may
construct a quadratic form

$\left<\prod_{p\in T}p^{r(p)},p^{\prime}\prod_{p\in
T}p^{r(p)},2q,p^{\prime\prime}q\right>:=\left<a_{1},a_{2},a_{3},a_{4}\right>$

with $\prod\limits_{p\in T}p^{r(p)}>q$ which is anisotropic over
${\mathbb{Z}}_{q}$ (note that there are infinitely many such
$\left<a_{1},a_{2},a_{3},a_{4}\right>$). And then we have that the $m$-gonal
form

$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$

is locally universal from Proposition 2.1.

By our construction, $a_{i}>q$ for each $i=1,2,3,4$. So there is an
$N_{0}\in[1,q]$ which is not represented by
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ and

$8(m-2)N_{0}+(m-4)^{2}\left(\prod\limits_{p\in
T}p^{r(p)}+p^{\prime}\prod\limits_{p\in
T}p^{r(p)}+2q+p^{\prime\prime}q\right)\equiv 0\pmod{q}$

holds. Through similar processings with the proof of Theorem 1.1, one may see
that for $k\in{\mathbb{N}}$ satisfying $q^{k}\equiv 1\pmod{8(m-2)}$, the
positive integer

$N_{n}=\frac{q^{2kn}\\{8(m-2)N_{0}+(m-4)^{2}(a_{1}+\cdots+a_{4})\\}-(m-4)^{2}(a_{1}+\cdots+a_{4})}{8(m-2)}$

is not represented by the $m$-gonal form
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ for every $n\geq 0$. As a result,
we conclude that the $m$-gonal form $\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$
is locally universal, but there are infinitely many positive integers $N_{n}$
which are not (globally) represented by the $m$-gonal form
$\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ and we could construct infinitely
many such $m$-gonal forms $\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ through
the above processings. Which completes our claim.

Secondly, assume that $m-4$ is a power of $2$. In this case, we have that

$N=a_{1}P_{m}(x_{1})+a_{2}P_{m}(x_{2})+a_{3}P_{m}(x_{3})+a_{4}P_{m}(x_{4})$

if and only if

$\frac{m-2}{2}N+\left(\frac{m-4}{4}\right)^{2}(a_{1}+a_{2}+a_{3}+a_{4})=\sum_{i=1}^{4}a_{i}\left(\frac{m-2}{2}x_{i}-\frac{m-4}{4}\right)^{2}.$

Take four different odd primes $a_{1},a_{2},a_{3},a_{4}\equiv 1\pmod{8}$ (note
that such primes are infinitely many). Then the $m$-gonal form
$\left<a_{1},a_{2},a_{3},a_{4}\right>$ is locally universal by Proposition
2.1. Through similar processings with the proof of Theorem 1.1, one may show
that for $N_{0}\in[1,4]$ with
$\frac{m-2}{2}N_{0}+\left(\frac{m-4}{4}\right)^{2}(a_{1}+a_{2}+a_{3}+a_{4})\equiv
0\pmod{4}$, every positive integer

$N_{n}:=\frac{4^{kn}\\{\frac{m-2}{2}N_{0}+\left(\frac{m-4}{4}\right)^{2}(a_{1}+a_{2}+a_{3}+a_{4})\\}-\left(\frac{m-4}{4}\right)^{2}(a_{1}+a_{2}+a_{3}+a_{4})}{\frac{m-2}{2}}$

is not represented by $\left<a_{1},a_{2},a_{3},a_{4}\right>_{m}$ for
$k\in{\mathbb{N}}$ satisfying $2^{k}\equiv 1\pmod{\frac{m-2}{2}}$ (for the
existence of $k$, note that $\frac{m-2}{2}$ is odd). Which completes the
proof. ∎

## References

  * [1] W. K. Chan, B.-K. Oh, Representations of integral quadratic polynomials, Contemp. Math. 587 (2013), 31–46.
  * [2] W. K. Chan and J. Ricci, The representation of integers by positive ternary quadratic polynomials, J. Number Theory. 156 (2015), 75–94.
  * [3] L. E. Dickson, Ternary quadratic forms and congruences, Ann. Math. 28(1926), 333-341.
  * [4] Z. He and B. Kane, Regular Ternary Polygonal Forms, Ramanujan Journal. 55 (2021), 1039–1062.
  * [5] W. Jagy, I. Kaplansky, and A. Schiemann, There are 913 regular ternary quadratic forms, Mathematika 44 (1997), 332-341.
  * [6] B. M. Kim, D. Park, Regular $m$-gonal forms, preprint.
  * [7] R. Lemke Oliver, Representation by ternary quadratic forms, Bull. London Math. Soc. 46 (2014), 1237-1247.
  * [8] B.-K. Oh, Regular positive ternary quadratic forms, Acta Arith. 147 (2011), 233-243.
  * [9] O. T. O’Meara, The Integral representations of quadratic forms over local fields, Amer. J. of Math, 80(1958), 843-878.
  * [10] O. T. O’Meara, Introduction to quadratic forms, Springer-Verlag, New York, (1963).
  * [11] G. L. Watson, Some problems in the theory of numbers, Ph.D. Thesis, University of London, 1953.
  * [12] G. L. Watson, Regular positive ternary quadratic forms, J. Lond. Math. Soc. 13 (1976), 97-102.
  * [13] D. Park, Representations of almost regular $m$-gonal forms $II$, preprint.


# Preparation of ultra-cold atomic-ensemble arrays using time-multiplexed
optical tweezers

Katja Gosar<EMAIL_ADDRESS>Jožef Stefan Institute, Jamova 39, SI-1000
Ljubljana, Slovenia  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia  Vesna Pirc Jevšenak
Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia  Faculty of
Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000
Ljubljana, Slovenia  Tadej Mežnaršič  Jožef Stefan Institute, Jamova 39,
SI-1000 Ljubljana, Slovenia  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia  Dušan Babič  Aresis Ltd,
Ulica Franca Mlakarja 1a, SI-1000 Ljubljana, Slovenia  Igor Poberaj  Aresis
Ltd, Ulica Franca Mlakarja 1a, SI-1000 Ljubljana, Slovenia  Erik Zupanič
Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia  Peter Jeglič
<EMAIL_ADDRESS>Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana,
Slovenia

###### Abstract

We use optical tweezers based on time-multiplexed acousto-optic deflectors to
trap ultra-cold cesium atoms in one-dimensional arrays of atomic ensembles.
For temperatures between $2.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$
and $50\text{\,}\mathrm{n}\mathrm{K}$ we study the maximal time between
optical tweezer pulses that retains the number of atoms in a single trap. This
time provides an estimate on the maximal number of sites in an array of time-
multiplexed optical tweezers. We demonstrate evaporative cooling of atoms in
arrays of up to 25 optical tweezer traps and the preparation of atoms in a box
potential. Additionally, we demonstrate three different protocols for the
preparation of atomic-ensemble arrays by transfer from an expanding ultra-cold
atomic cloud. These result in the preparation of arrays of up to 74 atomic
ensembles consisting of $\sim$100 atoms on average.

##  I Introduction

Optical tweezers is a common name for steerable optical dipole traps. In cold
atom experiments optical tweezers are used for painting arbitrary static and
dynamic potentials. This includes creating arrays of optical tweezers at
arbitrary positions [1, 2, 3, 4], moving of such optical traps [5, 6, 7, 8]
and painting box, ring [9] and other shapes of optical potentials. The
positional control of the traps can be achieved with several different
methods: using acousto-optic deflectors (AODs), spatial-light modulators
(SLMs) [10, 11] or digital micromirror devices (DMDs) [1, 12, 13]. In this
work we focus on optical tweezers based on AODs.

Deflection of a laser beam on an AOD is controlled by the frequency of the
drive signal. When creating multiple optical traps or painting an arbitrary
potential, AODs can be used in two different ways (i) by using continuous
multi-frequency driving, where the AODs are simultaneously driven with all
frequencies corresponding to trap positions [14, 2, 15], or (ii) by using the
time-multiplexed approach, where only a single frequency is applied at a time,
but the frequencies are rapidly switching to create a time-averaged optical
potential [16, 9, 17, 18]. The use of the latter approach is studied in this
article.

In our experiments the optical traps are created at the intersection of
optical tweezers with a dimple beam. In principle, a single-axis AOD would
allow the control of the position of the tweezers along the dimple, however
that would call for perfect alignment of the AOD axis with the dimple beam
[2]. We use a two-axis AOD which allows us to control the position of optical
tweezers in two dimensions. In the case of two-axial AODs the multi-frequency
approach creates “ghost” traps, which lie outside the dimple beam, causing a
loss of optical power into the unused traps [19]. Because of the ghost traps
multi-frequency driving cannot produce an arbitrary potential in two-
dimensions, but with time-multiplexing an arbitrary potential can be painted
[20, 21]. For potentials with a suitable symmetry, a combination of the multi-
frequency and time-multiplexed approach can be implemented [22].

Figure 1: Experimental setup schematically showing the optical tweezers setup.
The position of the optical tweezers focal plane is controlled by the position
of a lens before the AODs. The two-dimensional position of the optical
tweezers is controlled by the AODs. The optical tweezers cross the horizontal
dimple beam inside the experimental vacuum chamber. The guide beam (not shown)
is perpendicular to the page.  Figure 2: Characterisation of multiplexing. (a)
Temperature of the atomic cloud as a function of the evaporation time in a
single optical tweezer trap. The vertical dashed line marks the start of the
magnetic-field gradient rampdown. (b) An example of a $t_{\mathrm{max}}$
measurement ($t_{\mathrm{evap}}=$ $2700\text{\,}\mathrm{m}\mathrm{s}$),
showing the number of atoms retained in the trap for different $t_{\mathrm{on-
off}}$. $t_{\mathrm{max}}$ is determined as the time, for which the number of
retained atoms rapidly drops. (c) Maximal time $t_{\mathrm{max}}$ between
optical tweezer pulses that maintains the number of atoms in the trap, as a
function of the temperature and the linear fit to the measured data.

Preparation of ultra-cold atomic-ensemble arrays is motivated by the
possibility of using them as quantum simulators using Rydberg atoms.
Previously, this type of quantum simulation was performed using arrays of
single atoms [3, 2, 23, 24], but it is also possible to use collective
excitations in ensembles of a few hundred atoms [25, 1, 26]. With ensembles
the preparation and detection of Rydberg states can be significantly faster
than with single atoms [27].

We demonstrate four different methods of preparing a one-dimensional array of
atomic ensembles in time-multiplexed optical tweezers. The first method is
based on loading the traps directly from the large dipole trap and performing
simultaneous evaporative cooling with them. In the other three methods the
first step is evaporative cooling of an atomic cloud in the dimple trap. It is
followed by different forms of expansion of the atomic cloud along the dimple
beam and trapping into an array of optical tweezers.

In addition to optical tweezer arrays, optical tweezers are also used for the
preparation of arbitrary time-averaged and dynamic potentials. These can be
used to perform experiments with solitons [28, 5] or to prepare condensates in
different shaped traps [29]. In this work we present examples of painting a
box potential and a one-dimensional harmonic potential. As an example of a
dynamic time-averaged potential, we present splitting of atomic ensembles.

##  II Experiment

We prepare cold 133Cs atoms by laser cooling with the standard procedure
described in detail in Ref. 30. All experiments presented in this article
start with $3\text{\times}{10}^{6}\text{\,}$ atoms in a large dipole trap,
cooled to $2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$. For evaporative
cooling we use a small dipole trap inside the large dipole trap. It is created
by crossing two laser beams that we call the dimple beam and the guide beam.
The dimple beam has a $1/\mathrm{e}^{2}$ radius of
$32\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and the guide
$60\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. Instead of the guide beam,
we can use optical tweezers. Our experimental setup setup with commercial
optical tweezers (Aresis, CALM 1064) and the dimple beam is illustrated in
Fig. 1. The radius of the optical tweezers beam waist is
$w_{0}\approx$2.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$, and the
Rayleigh length is
$z_{0}\approx$13.0\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. To create
a trap that holds the atoms against gravity the tweezers have to be combined
with another beam. In all our experiments with optical tweezers the traps are
created by crossing a tweezer beam with the dimple beam. The position of the
optical tweezers is optimized so that they are centered with the dimple beam.
This is important because misalignment of the two beams leads to leakage of
atoms from the trap [19]. The accessible area of our optical tweezers is
$1.7\text{\,}\mathrm{m}\mathrm{m}$ $\times$ $1.7\text{\,}\mathrm{m}\mathrm{m}$
and we calibrate the AOD drive amplitude so that the intensity of the tweezers
is independent of the trap position. Our optical tweezers setup features an
additional lens that can move the tweezers’ beamwaist, as shown in Fig. 1. For
experiments in section A and B the crossing of optical tweezers with the
dimple beam is approximately
$14\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ from the tweezers’
beamwaist, where the radius of the optical tweezers beam is
$3.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. For experiments in
section C, the tweezers are used at the beamwaist.

###  II.1 Temperature dependence of the maximal trap switching time

We prepare an array of optical tweezers traps by time multiplexing with the
switching frequency $\nu=$ $100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, which
is the upper limit for our optical tweezers. This means that each trap is
active for only $10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ in each
cycle and then it is off for $t_{\mathrm{off}}=\frac{N-1}{\nu}$, where $N$ is
the number of traps in the array. That is why we first characterise the
dependence of $t_{\mathrm{off}}$ on the temperature. This gives us an estimate
on the number of traps in an array that would efficiently trap an atomic
ensemble at a given temperature.

We start by transferring atoms from the large dipole trap into a single
optical tweezer trap in which we then evaporatively cool the atoms. First, the
power of the tweezer and dimple beam is linearly ramped up for
$1500\text{\,}\mathrm{m}\mathrm{s}$. Then, we hold the power for
$400\text{\,}\mathrm{m}\mathrm{s}$ to let the atoms thermalize. Afterwards, we
turn off the large dipole trap and start the evaporation cooling. The power of
the beams is ramped down exponentially ($\tau_{\mathrm{dimple}}=$
$0.625\text{\,}\mathrm{s}$, $\tau_{\mathrm{tweezers}}=$
$1\text{\,}\mathrm{s}$) for the evaporation time $t_{\mathrm{evap}}$,
asymptotically approaching 6.8% (dimple) and 5% (tweezers) of the maximum
power. After $2\text{\,}\mathrm{s}$ of evaporation, we start with an
additional exponential rampdown of the magnetic field gradient, which
contributes to more efficient evaporative cooling [30]. We measure the
temperature of the atoms as a function of the evaporation time
$t_{\mathrm{evap}}$ (Fig. 2(a)). After $3\text{\,}\mathrm{s}$ of evaporation,
the atoms form a Bose-Einstein condensate (BEC), which can be seen from the
standard time-of-flight measurement.

To estimate the maximal possible number of traps in an array, we measure the
maximal time between optical tweezer pulses that retains the number of trapped
atoms and its dependence on the temperature of the atoms. During the
evaporation, the optical trap is switched on and off with
$100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$; the trap is on for
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ and then off for
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$. After a given evaporation
time $t_{\mathrm{evap}}$, we stop the evaporation by keeping the trap powers
constant. The tweezer trap is then switched on and off with pulses of length
$t_{\mathrm{on-off}}$ and a 50% duty cycle. This way the time-averaged power
of the trap is independent of the $t_{\mathrm{on-off}}$ when using the same
amplitude. We measure the number of atoms that are retained in a pulsing trap
after $100\text{\,}\mathrm{m}\mathrm{s}$ as a function of $t_{\mathrm{on-
off}}$. An example of this measurement is shown in Fig 2(b). The number of
atoms is constant for sufficiently short $t_{\mathrm{on-off}}$ and it drops
off to zero at time $t_{\mathrm{max}}$. The colder the atom cloud is, the
longer the $t_{\mathrm{max}}$.

$t_{\mathrm{max}}$ is determined by the velocity of the atoms and the size of
the optical tweezer trap. For the atoms to be recaptured with the next pulse,
the average distance $d$ an atom travels in the time $t_{\mathrm{on-off}}$ has
to be smaller than the effective radius of the trap $w$ [31, 32]. We calculate
$d$ as

$d=\sqrt{\frac{2k_{B}T}{m}}t_{\mathrm{on-off}},$ (1)

where $\sqrt{2k_{B}T/m}$ is the average (rms) velocity of an atom of an ideal
gas in two dimensions at temperature $T$. $k_{B}$ is the Boltzmann constant
and $m$ is the mass of a cesium atom. We use the two dimensional average
velocity because atoms escape the optical tweezers only in the radial
direction. Since the dimple beam is approximately 10-times wider then the
tweezers we can neglect its effect on this process. From Eq. (1) we can
conclude that $t_{\mathrm{max}}$ is proportional to $T^{-1/2}$ as

$t_{\mathrm{max}}=\sqrt{\frac{m}{2k_{B}T}}w.$ (2)

Here we assume that the effective trap radius $w$ is constant. We plot
$t_{\mathrm{max}}$ as a function of $T^{-1/2}$ in Fig. 2(c). The linear fit to
the data gives us an estimate of the effective radius of the optical tweezers
$w=$ $1.8(1)\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. The deviation of
the measured $t_{\mathrm{max}}$ from the model could be explained by the
dependence on the trapping width on the optical power, the effect of the
dimple beam and the magnetic field gradient, which additionally deforms the
trap geometry.

Figure 3: Arrays loaded directly from a dipole trap. (a) An absorption image
of atoms in an array of 20 tweezers of equal intensity. (b) Atoms in an array
of 20 sites with a V-shaped profile of tweezer intensities. All other
parameters are the same as in image (a). The red bars illustrate the
intensities of the optical tweezers. (c) Absorption image of a 25 site array
of atomic ensembles in optical tweezers, loaded directly from the large dipole
trap. The image is an average of 10 realisations. Below, we plot
$\sigma_{N_{a}}/\sqrt{N_{a}}$ and $N_{a}$, where $\sigma_{N_{a}}$ is the
variance of the number of atoms within each trap determined from 10
measurements and $N_{a}$ is the average number of atoms in each trap. The
dashed line marks $\sigma_{N_{a}}/\sqrt{N_{a}}=1$ which represents the
Poissonian distribution of the number of atoms in each trap. (d) Splitting of
5 atomic ensembles into 10. Absorption images taken at
$10\text{\,}\mathrm{m}\mathrm{s}$ intervals of the
$100\text{\,}\mathrm{m}\mathrm{s}$ splitting. All images use the same
colorbar, except (c) where the same colors correspond to optical densities
from 0 to 0.2. The optical tweezers are time-multiplexed with a frequency of
$100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. This means that each trap is
active for $10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ and its duty
cycle is $1/N$, where $N$ is the number of traps forming the array.

###  II.2 Loading of an array and a box potential directly from the dipole
trap

Figure 4: Characterisation of the number of traps forming the potential for
preparation of cold atoms in a line box-shaped potential. (a) The number of
atoms after $3000\text{\,}\mathrm{m}\mathrm{s}$ of evaporation in a box-shaped
potential as a function of the number of traps $N$ forming the box. (b) The
number of atoms in a box potential after $2800\text{\,}\mathrm{m}\mathrm{s}$
of evaporation in an array of 5 tweezers and additional
$400\text{\,}\mathrm{m}\mathrm{s}$ of evaporation in a box potential formed by
$N$ tweezer traps. (c) Absorption image (colorbar in Fig. 3) and the linear
density profile $p$ of atoms in a box-shaped potential, prepared with a
transfer from 5 to 40 sites. Average of 10 measurements. Figure 5: Arrays
loaded from a dimple trap using one-dimensional time-of-flight or expansion in
a harmonic potential. Protocol A: loading the tweezers from an expanding
cloud. (a) shows the linear density of the atomic ensembles in an array of 10
optical tweezers as a function of time after the optical tweezers are turned
on. A breathing mode oscillation can be seen. Protocol B: expansion of the
atomic cloud in a painted harmonic potential and switching to an array of
optical tweezers after half the breathing oscillation. (b) shows the
oscillation of the atomic cloud in the painted harmonic potential and (c) the
resulting array (optical density image and the normalized variance
$\sigma_{N_{a}}/\sqrt{N_{a}}$). Protocol C: ramping the optical tweezer array
power and position with the expansion of the atomic cloud. (d) and (e) show
the results of protocol C with 60 and 80 traps (74 filled), respectively. (c),
(d) and (e) show an average of 10 realisations of the experiment and the
variance is determined from the same 10 measurements. The colorbar is the same
as in Fig. 3.

The simplest method of preparing an array of atomic ensembles with optical
tweezers is simultaneous evaporation in the optical tweezers. The protocol for
evaporative cooling is identical to the one presented in Section II.1, except
for the larger number of optical tweezer traps. We observe that the
distribution of atoms in the array reflects the atom density profile of the
dipole trap. We were able to achieve more uniform filling by adjusting the
power in the tweezers to a V-profile; traps on the edge of the array have more
power than those in the middle. Examples of arrays created with a uniform
strength profile and a V-profile are shown are Fig. 3(a) and 3(b). The largest
array we are able to prepare by trapping directly from the large dipole trap
has 25 atomic ensembles. As shown in Fig. 3(c), the variance of the number of
atoms per trap is relatively large. This method is similar to experiments in
Ref. 18, where evaporative cooling to BEC in four time-multiplexed optical
traps was achieved. In contrast to our method, the atomic clouds for
evaporation were prepared with splitting.

Splitting of the atomic ensembles can also be done after the evaporative
cooling, as a way to achieve a higher number of traps in an array. The atoms
are cooled in a smaller array and then the number of array sites is doubled by
splitting the atomic ensembles [19]. Fig. 3(d) shows the splitting of an array
of 5 traps into 10. The splitting starts after
$2800\text{\,}\mathrm{m}\mathrm{s}$ of evaporation and is finished in
$100\text{\,}\mathrm{m}\mathrm{s}$. The tweezers’ trajectories are linear. The
challenge with splitting is ensuring even distribution of atoms in the split
traps, which can be achieved by precise tuning of the traps’ position and
intensity. Additionally, the splitting could be improved by using a minimum
jerk cost trajectory of the tweezers, as it was done in Ref. 19.

If the distance between the traps in the array is smaller than the width of a
trap, they form a box potential. We measured the number of atoms in an array
of $N$ traps that span $185\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$,
after $3000\text{\,}\mathrm{m}\mathrm{s}$ of evaporation. The measured number
of the atoms as a function of the number of traps is shown in Fig. 4. For
$N<15$ the number of atoms increases with the number of traps. This is because
the traps are separate and an additional trap can trap additional atoms. For
$N>20$, we see a drop in the number of atoms. Here the traps still do not form
a box potential, but the number of trapped atoms lowers due to larger
$t_{\mathrm{off}}$.

If we wanted to prepare a larger box potential, we would need a larger number
of traps, but that also leads to higher $t_{\mathrm{off}}$ that has an adverse
effect on trapping. However, the lower the temperature of the atoms, the
larger number of traps we can use. This gives the idea of cooling the atoms in
an array with a lower number of traps and then, when the atoms are
sufficiently cooled, switching to a larger number of traps, that cover the
same area, forming a box potential. For example, we start by evaporatively
cooling atoms in an array of 5 optical tweezer traps. After
$2800\text{\,}\mathrm{m}\mathrm{s}$ of evaporation the temperature is
$48\text{\,}\mathrm{n}\mathrm{K}$. We then switch to an array of $N$ tweezers,
and keep the power of the beams constant for
$400\text{\,}\mathrm{m}\mathrm{s}$ to let the atoms cover the
$185\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ box potential. Fig. 4(c)
shows an example profile of a filled box potential. Fig. 4(b) shows the final
number of atoms as a function of $N$. For small $N$ we see the same effect as
with loading directly into a box potential: the traps are separated and the
number of atoms increases with the number of traps. For more than 100 traps,
the number of atoms starts decreasing. This is the regime were the trap
separation is the same as the trap radius and the traps truly form a box
potential. The number of atoms decreases with $N$ due to the larger
$t_{\mathrm{off}}$. With more than 400 traps no atoms are trapped in the box
potential. This is due to atom losses at the edges of the box potential, if
the traps at the edges are not repeated with a high enough frequency.

###  II.3 Loading from the dimple trap

We see that for creating larger atomic-ensemble arrays lower temperatures of
atoms are needed. This is why we developed alternative methods of preparing
atomic-ensemble arrays. In these methods, the atoms are first cooled in the
crossed dimple and guide trap and then transferred into an array of optical
tweezers. However, the transfer can lead to unwanted oscillations of the
atomic ensembles inside the optical traps. We present three different
protocols that tackle this problem in different ways.

The first protocol starts after $4500\text{\,}\mathrm{m}\mathrm{s}$ of
evaporative cooling (atoms in BEC), when the guide beam is slowly turned off,
which allows the atomic cloud to expand along the dimple beam. The power of
the guide beam is decreased as a hyperbolic tangens from the initial power to
zero in $1500\text{\,}\mathrm{m}\mathrm{s}$. This ensures slower expansion of
the atomic cloud as opposed to a sudden switching off of the guide beam. We
allow the cloud to expand for additional $100\text{\,}\mathrm{m}\mathrm{s}$
and then turn on an array of 10 optical tweezer traps that span
$215\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. We will refer to this
method as protocol A. The resulting array of 10 atomic ensembles is shown in
Fig. 5(a). In the image, 11 bunches of atoms can be seen. The 10 bunches on
the right side correspond to atoms trapped by the optical tweezers and the
remaining one in the left-most position is the untrapped atoms.

The drawback of protocol A is that the filling of the tweezers is
intrinsically non-uniform, because it is determined by the distribution of
atom velocities that is mapped into their positions by the expansion. This
also means that the atoms, that we attempt to trap at the edges of the array
have a higher velocity. This additionally decreases the number of trapped
atoms and induces oscillations of atoms inside the traps. We observed that the
atomic clouds oscillate inside all the tweezer traps. This is shown in Fig.
5(a), where we plot the linear density of the atoms as a function of time.

We aim to reduce the motion of atoms due to expansion with protocol B. Here,
we instantly turn off the guide beam after $3000\text{\,}\mathrm{m}\mathrm{s}$
of evaporation, and then paint a harmonic potential with optical tweezers. The
harmonic potential is painted in 1000 points on a span of
$900\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ along the dimple beam and
it is centred around the position of the crossed dimple trap. The breathing
oscillation of the atomic cloud in the painted harmonic potential is shown in
fig. 5(b). The frequency of the oscillation is
$2.8\text{\,}\mathrm{H}\mathrm{z}$. After $180\text{\,}\mathrm{m}\mathrm{s}$,
which is a half of the oscillation, the atoms are at the maximal amplitude and
have, in principle, zero velocity (if the center of the harmonic potential is
perfectly aligned with the initial position of the atoms). This is when we
stop painting the harmonic potential and instead paint an array of 40 optical
tweezers to capture the atoms. The strength of the tweezers is ramped for
$100\text{\,}\mathrm{m}\mathrm{s}$, after which we image the atomic-ensemble
array. Ramping of the optical tweezer power results in less oscillations in
the tweezers compared to a sudden switch to traps at full power. The result is
shown in Fig. 5(c). All 40 traps are occupied, but the occupation of the edge
traps varies more than of the middle traps, where it even reaches the
Poissonian limit $\sigma_{N_{a}}=\sqrt{N_{a}}$.

In protocol C, we instantly turn off the guide beam after
$3000\text{\,}\mathrm{m}\mathrm{s}$ of evaporation (the temperature of the
atomic cloud is $\sim$$70\text{\,}\mathrm{n}\mathrm{K}$) and start the
$200\text{\,}\mathrm{m}\mathrm{s}$ ramp up of the optical tweezer power. We
start with all the traps in the center and then, during the ramp-up, linearly
expand the array to the final span of
$1100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. This is approximately
the same rate as the expansion of the atomic cloud in absence of tweezer
traps. This way, we follow the motion of the atoms and therefore avoid
exciting oscillations in the trap. An example of an array of 59 atomic
ensembles prepared by this protocol is shown in Fig. 5(d). The image is taken
at the end of the expansion. We were able to achieve larger arrays with
shorter evaporation times. Because of the higher temperature, the rate of
expansion is higher, allowing for a larger array. For example, Fig. 5(e) shows
an array of 74 filled traps, out of 80 optical tweezers, where the time of
evaporation was shortened to $2900\text{\,}\mathrm{m}\mathrm{s}$,
corresponding to $\sim$$100\text{\,}\mathrm{n}\mathrm{K}$ at the end of
evaporation). In this case, the number of filled traps in the array is not
limited by temperature but by the rate of expansion, resulting in a larger
number of filled traps at the higher temperature. Similarly to protocol B, the
variance is larger at the edges of the array and is about
$\sigma_{N_{a}}/\sqrt{N_{a}}=1$ in the middle of the array.

##  III Discussion and Conclusions

Comparing the $t_{\mathrm{off}}=\frac{N-1}{\nu}$ to the measured
$t_{\mathrm{max}}$ at a given temperature gives us an estimate for the maximal
number $N$ of traps in an array that would still efficiently trap atoms at
that temperature. The observed maximal number of traps is higher. This can by
understood through the effect of neighbouring traps; if an atom escapes from a
certain trap it is not necessarily “lost” as it can be captured by a
neighbouring trap. We note that the temperature of the atoms is not the only
limiting factor on the number of traps. The size of the array is also limited
by the available trap power and the maximal possible switching frequency. For
very large arrays the available optical power would need to be increased to
ensure sufficient effective trap depths. The trap frequency sets a constraint
on the number of traps in array at a given switching frequency. This is due to
parametric heating that occurs when the frequency of the multiplexing seen by
the atoms is close to the trap frequency [17]. In our case, the trap
frequencies used in Section II.1 are at most
$215\text{\,}\mathrm{H}\mathrm{z}$ at the start of evaporation and
$100\text{\,}\mathrm{H}\mathrm{z}$ after $3\text{\,}\mathrm{s}$ of
evaporation, which is much lower that the frequencies of on-off switching at
which we observed the loss of atoms.

The arrays presented in this work have a high variation in the number of atoms
per trap inside an array. In Section II.2 we demonstrate that the homogeneity
of the trap filling can be improved by adjusting the power of the traps. We
used a simple V-profile of trap intensities. To achieve even more uniform
filling, we propose fine tuning of individual trap power. In Ref. 9, atom
density images were used to correct the intensity profile of a time-averaged
ring trap to achieve homogeneous filling of the trap. We plan to implement a
similar technique to prepare arrays of atomic ensembles with an even
distribution of atoms in the array sites.

V-profiles and tilted profiles were used in measurements in section II.2 and
for protocol C, to achieve a more uniform filling of the arrays or the box
potential. In some cases, the sequence of multiplexing was also important: the
atoms distributed over the array differently if the optical tweezer traps were
multiplexed from left to right or from right to left. For protocol A and B it
was best to use a random permutation of the array sites.

In addition to the uniform filling of the array sites, the variance of atom
numbers inside individual traps is also important for applications of atomic-
ensemble arrays such as quantum simulation. In Ref. 1 a sub-Poissonian
variance of the occupation number of a single trap inside an optical tweezer
array was observed. Our measurements show above Poissonian distribution for
direct loading into optical tweezers, and close to Poissonian distribution for
protocol C, but only for the central part of the array.

The repeatability of loading directly from the dipole trap is worse than the
repeatability of protocols A, B and C. This is probably mostly due to the
higher initial temperature of the atoms, which also limits the observed
maximal number of traps in the array to only 25. The advantage of direct
loading is that the intrinsic distribution of atoms within the array is
determined only by the density of the atoms in the dipole trap and the mean
velocity of the atoms is zero across the whole array. This is in contrast to
protocols A, B and C, where the uneven distribution of atoms is caused by the
expansion determined by the initial velocity distribution. In protocols A and
C the mean velocity of atoms at the positions of the traps is in general non-
zero, because we are trapping an expanding atomic cloud. Therefore, protocol A
causes oscillations within the trap that can be avoided by stopping the
expansion as it is done in protocol B. Protocol C is the most promising in
terms of the achieved size of the array. However, for this protocol we only
show the distribution of atoms during the expansion, but the atoms will
oscillate like in protocol A, if we suddenly stop the expansion of the optical
tweezer array. Avoiding inducing the oscillations is possible by slowly
stopping the expansion.

This article provides an analysis of time-multiplexed optical tweezers and
presents the limitations that the temperature of the atoms sets on the number
of array sites. We conclude that, close to the BEC critical temperature, an
array of about a hundred atomic-ensembles could be prepared with the proposed
improvements regarding trap intensities. The arrays can be of an arbitrary
geometry in a plane, not only in one dimension, if combined with a light-sheet
instead of a dimple beam. Furthermore, time-multiplexing can be used in
combination with multi-frequency driving or even DMDs or SLMs to achieve
hundreds of traps that could be used for quantum simulators.

###### Acknowledgements.

We thank Rok Žitko for his comments and discussions. This work was supported
by the Slovenian Research Agency (Research Core Fundings No. P1-0125, No.
P1-0099 and No. P1-0416).

## References

  * Wang  _et al._ [2020] Y. Wang, S. Shevate, T. M. Wintermantel, M. Morgado, G. Lochead, and S. Whitlock, Preparation of hundreds of microscopic atomic ensembles in optical tweezer arrays, npj Quantum Information 6, 1 (2020).
  * Endres  _et al._ [2016] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletić, M. Greiner, and M. D. Lukin, Atom-by-atom assembly of defect-free one-dimensional cold atom arrays, Science 354, 1024 (2016).
  * Bernien  _et al._ [2017] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Probing many-body dynamics on a 51-atom quantum simulator, Nature 551, 579 (2017).
  * Pu  _et al._ [2018] Y. Pu, Y. Wu, N. Jiang, W. Chang, C. Li, S. Zhang, and L. Duan, Experimental entanglement of 25 individually accessible atomic quantum interfaces, Science advances 4, eaar3931 (2018).
  * Carpentier  _et al._ [2008] A. V. Carpentier, J. Belmonte-Beitia, H. Michinel, and M. I. Rodas-Verde, Laser tweezers for atomic solitons, Journal of Modern Optics 55, 2819 (2008).
  * Beugnon  _et al._ [2007] J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. Sortais, A. M. Lance, M. P. Jones, G. Messin, A. Browaeys, and P. Grangier, Two-dimensional transport and transfer of a single atomic qubit in optical tweezers, Nature Physics 3, 696 (2007).
  * Gustavson  _et al._ [2001] T. L. Gustavson, A. P. Chikkatur, A. E. Leanhardt, A. Görlitz, S. Gupta, D. E. Pritchard, and W. Ketterle, Transport of Bose-Einstein condensates with optical tweezers, Physical Review Letters 88, 020401 (2001).
  * Rakonjac  _et al._ [2012] A. Rakonjac, A. B. Deb, S. Hoinka, D. Hudson, B. J. Sawyer, and N. Kjærgaard, Laser based accelerator for ultracold atoms, Optics Letters 37, 1085 (2012).
  * Bell  _et al._ [2016] T. A. Bell, J. A. P. Glidden, L. Humbert, M. W. J. Bromley, S. A. Haine, M. J. Davis, T. W. Neely, M. A. Baker, and H. Rubinsztein-Dunlop, Bose-Einstein condensation in large time-averaged optical ring potentials, New journal of Physics 18, 035003 (2016).
  * Boyer  _et al._ [2006] V. Boyer, R. M. Godun, G. Smirne, D. Cassettari, C. M. Chandrashekar, A. B. Deb, Z. J. Laczik, and C. J. Foot, Dynamic manipulation of Bose-Einstein condensates with a spatial light modulator, Physical Review A 73, 031402(R) (2006).
  * Muldoon  _et al._ [2012] C. Muldoon, L. Brandt, J. Dong, D. Stuart, E. Brainis, M. Himsworth, and A. Kuhn, Control and manipulation of cold atoms in optical tweezers, New Journal of Physics 14, 073051 (2012).
  * Gauthier  _et al._ [2016] G. Gauthier, I. Lenton, N. M. Parry, M. Baker, M. J. Davis, H. Rubinsztein-Dunlop, and T. W. Neely, Direct imaging of a digital-micromirror device for configurable microscopic optical potentials, Optica 3, 1136 (2016).
  * Stuart and Kuhn [2018] D. Stuart and A. Kuhn, Single-atom trapping and transport in DMD-controlled optical tweezers, New Journal of Physics 20, 023013 (2018).
  * Shin  _et al._ [2004] Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, Atom interferometry with Bose-Einstein condensates in a double-well potential, Physical Review Letters 92, 050405 (2004).
  * Trypogeorgos  _et al._ [2013] D. Trypogeorgos, T. Harte, A. Bonnin, and C. Foot, Precise shaping of laser light by an acousto-optic deflector, Optics Express 21, 24837 (2013).
  * Schnelle  _et al._ [2008] S. K. Schnelle, E. D. Van Ooijen, M. J. Davis, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Versatile two-dimensional potentials for ultra-cold atoms, Optics Express 16, 1405 (2008).
  * Chisholm  _et al._ [2018] C. S. Chisholm, R. Thomas, A. B. Deb, and N. Kjærgaard, A three-dimensional steerable optical tweezer system for ultracold atoms, Review of Scientific Instruments 89, 103105 (2018).
  * Deb  _et al._ [2014] A. B. Deb, T. McKellar, and N. Kjærgaard, Optical runaway evaporation for the parallel production of multiple Bose-Einstein condensates, Physical Review A 90, 051401(R) (2014).
  * Roberts  _et al._ [2014] K. O. Roberts, T. McKellar, J. Fekete, A. Rakonjac, A. B. Deb, and N. Kjærgaard, Steerable optical tweezers for ultracold atom studies, Optics letters 39, 2012 (2014).
  * Onofrio  _et al._ [2000] R. Onofrio, D. S. Durfee, C. Raman, M. Köhl, C. E. Kuklewicz, and W. Ketterle, Surface excitations of a Bose-Einstein condensate, Physical Review Letters 84, 810 (2000).
  * Henderson  _et al._ [2009] K. Henderson, C. Ryu, C. MacCormick, and M. G. Boshier, Experimental demonstration of painting arbitrary and dynamic potentials for Bose-Einstein condensates, New Journal of Physics 11, 043030 (2009).
  * Zimmermann  _et al._ [2011] B. Zimmermann, T. Mueller, J. Meineke, T. Esslinger, and H. Moritz, High-resolution imaging of ultracold fermions in microscopically tailored optical potentials, New Journal of Physics 13, 043007 (2011).
  * Wu  _et al._ [2021] X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, A concise review of Rydberg atom based quantum computation and quantum simulation, Chinese Physics B 30, 020305 (2021).
  * Ebadi  _et al._ [2021] S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletić, and M. D. Lukin, Quantum phases of matter on a 256-atom programmable quantum simulator, Nature 595, 227 (2021).
  * Ebert  _et al._ [2015] M. Ebert, M. Kwon, T. G. Walker, and M. Saffman, Coherence and Rydberg blockade of atomic ensemble qubits, Physical Review Letters 115, 093601 (2015).
  * Spong  _et al._ [2021] N. L. R. Spong, Y. Jiao, O. D. W. Hughes, K. J. Weatherill, I. Lesanovsky, and C. S. Adams, Collectively encoded Rydberg qubit, Physical Review Letters 127, 063604 (2021).
  * Xu  _et al._ [2021] W. Xu, A. V. Venkatramani, S. H. Cantú, T. Šumarac, V. Klüsener, M. D. Lukin, and V. Vuletić, Fast preparation and detection of a Rydberg qubit using atomic ensembles, Physical Reviev Letters 127, 050501 (2021).
  * Ismailov  _et al._ [2021] K. K. Ismailov, B. B. Baizakov, and F. K. Abdullaev, Confinement of bright matter-wave solitons on top of a pedestal-shaped potential, Physics Letters A , 127831 (2021).
  * Navon  _et al._ [2021] N. Navon, R. P. Smith, and Z. Hadzibabic, Quantum gases in optical boxes, Nature Physics 17, 1334 (2021).
  * Mežnaršič  _et al._ [2019] T. Mežnaršič, T. Arh, J. Brence, J. Pišljar, K. Gosar, Ž. Gosar, R. Žitko, E. Zupanič, and P. Jeglič, Cesium bright matter-wave solitons and soliton trains, Physical Review A 99, 033625 (2019).
  * Tuchendler  _et al._ [2008] C. Tuchendler, A. M. Lance, A. Browaeys, Y. R. P. Sortais, and P. Grangier, Energy distribution and cooling of a single atom in an optical tweezer, Physical Review A 78, 033425 (2008).
  * He  _et al._ [2011] J. He, B.-D. Yang, T.-C. Zhang, and J.-M. Wang, Extending a release-and-recapture scheme to single atom optical tweezer for effective temperature evaluation, Chinese Physics B 20, 073701 (2011).