englishen-US

# Even quantum advice is unlikely to solve $\mathsf{PP}$

Justin Yirka
The University of Texas at Austin
<EMAIL_ADDRESS>
(March 2024)

###### Abstract

We give a corrected proof that if
$\mathsf{PP}\subseteq\mathsf{BQP\\!/\\!qpoly}$, then the Counting Hierarchy
collapses, as originally claimed by [Aaronson, CCC 2006]. This recovers the
related unconditional claim that $\mathsf{PP}$ does not have circuits of any
fixed size $n^{k}$ even with quantum advice. We do so by proving that
$\mathsf{YQP^{*}}$, an oblivious version of $\mathsf{QMA}\cap\mathsf{coQMA}$,
is contained in $\mathsf{APP}$, and so is $\mathsf{PP}$-low.

##  1 Introduction

In [Aar06], Aaronson proved new quantum circuit lower bounds for
$\mathsf{PP}$, among other results. First, Aaronson proved that
$\mathsf{P}^{\mathsf{PP}}$ does not have circuits of size $n^{k}$ for any
fixed constant $k$ even if the circuits use quantum advice states. Second, he
gave an analogue of the Karp-Lipton theorem for quantum circuits, showing that
if $\mathsf{PP}\subseteq\mathsf{BQP\\!/\\!qpoly}$, then the Counting Hierarchy
collapses to $\mathsf{QMA}$, where the Counting Hierarchy is the infinite
sequence of classes $\mathsf{C_{1}P}=\mathsf{PP}$ and
$\mathsf{C_{i}P}=\left(\mathsf{C_{i-1}P}\right)^{\mathsf{PP}}$. Finally,
Aaronson combined these two results to give the unconditional bound that
$\mathsf{PP}$ does not have circuits of size $n^{k}$ for any fixed constant
$k$ even with quantum advice.111Slightly earlier, Vinochandran [Vin05] gave a
proof that $\mathsf{PP}$ does not have _classical_ circuits of fixed
polynomial size.

However, Aaronson later noted there was an error in one of the proofs [Aar17].
The first of the above results was unaffected, but the proofs of the second
and third results only held for quantum circuits with classical advice.
Fortunately, no other results in [Aar06] were affected, but no fix for this
bug was forthcoming. Very briefly, the error was a claim that for some oracle
class $\mathsf{C}^{\mathsf{BQP\\!/\\!qpoly}}$ in which a machine for the base
class $\mathsf{C}$ is able to find the quantum advice state that will be used
by the oracle machine, the base machine could “hard-code” the advice state
into its oracle queries so that the oracle no longer needs the power to find
its own advice, thus reducing $\mathsf{C}^{\mathsf{BQP\\!/\\!qpoly}}$ to
$\mathsf{C}^{\mathsf{BQP}}$. This approach works for classes with classical
advice, like $\mathsf{C}^{\mathsf{BQP\\!/\\!poly}}$. But, because complexity
classes such as $\mathsf{BQP}$ and their associated oracles are defined as
maps from $\\{0,1\\}^{*}$ to $\\{0,1\\}$, there is no way to hard-code a
quantum advice state into a query.

In this note, we give a corrected proof of Aaronson’s full claims. We show
that if $\mathsf{PP}\subseteq\mathsf{BQP\\!/\\!qpoly}$, then the Counting
Hierarchy collapses to $\mathsf{QMA}$ and in fact to $\mathsf{YQP^{*}}$. Given
this correction, Aaronson’s proof for the third claim, that $\mathsf{PP}$ does
not have circuits of size $n^{k}$ for any fixed constant $k$ even with quantum
advice, now goes through.

Our corrected proof relies on the known equality
$\mathsf{BQP\\!/\\!qpoly}=\mathsf{\mathsf{YQP}^{*}\\!/\\!poly}$,
serendipitously proven by Aaronson with Drucker [AD14], where
$\mathsf{YQP^{*}}$ is an oblivious version of
$\mathsf{QMA}\cap\mathsf{coQMA}$. Our primary technical contribution is to
show $\mathsf{YQP^{*}}\subseteq\mathsf{APP}$, which is known to be
$\mathsf{PP}$-low, where $\mathsf{APP}$ is a subclass of $\mathsf{PP}$ with an
arbitrarily small but nonzero promise gap.

For comparison, other Karp-Lipton style bounds on quantum complexity classes
include that if $\mathsf{QCMA}\subseteq\mathsf{BQP\\!/\\!poly}$, then
$\mathsf{QCPH}$ collapses [AGKR24] and that if
$\mathsf{NP}\subseteq\mathsf{BQP\\!/\\!qpoly}$, then
$\mathsf{\Pi^{p}_{2}}\subseteq\mathsf{QMA}^{\mathsf{PromiseQMA}}$ [AD14]. As
for unconditional bounds, following Aaronson’s unaffected result that
$\mathsf{P}^{\mathsf{PP}}$ does not have quantum circuits with quantum advice
of any fixed polynomial size, our result for $\mathsf{PP}$ is the first
improved bound on fixed-size circuits with quantum advice. Our primary lemma
establishes $\mathsf{YQP^{*}}$ as the largest natural quantum complexity class
known to be $\mathsf{PP}$-low, improving on the fact that $\mathsf{BQP}$ is
$\mathsf{PP}$-low [FR99].222Morimae and Nishimura [MN16] gave definitions
involving quantum postselection chosen to equal $\mathsf{AWPP}$ and
$\mathsf{APP}$. Additionally, while the largest witness-based class previously
known to be contained in $\mathsf{APP}$ was $\mathsf{FewP}$ [Li93], our result
shows that $\mathsf{APP}$ in fact contains oblivious-witness classes including
$\mathsf{YQP^{*}}\supseteq\mathsf{YMA^{*}}\supseteq\mathsf{YP^{*}}\supseteq\mathsf{FewP}$.

##  2 Preliminaries

As this note builds directly on [Aar06] and [AD14], we intentionally give only
a concise background. For further background, motivation, and technical
details concerning the complexity classes discussed, see these earlier works.

Note that our definitions of $\mathsf{BQP\\!/\\!poly}$ and
$\mathsf{BQP\\!/\\!qpoly}$ are the standard ones in which a circuit is only
required to satisfy the promise gap when the correct advice is provided. The
same notation has sometimes been used to refer to other definitions, see e.g.
[Zoo].

###

The class $\mathsf{YQP}$ was first described in [Aar07], but the definition
was later corrected by Aaronson and Drucker [AD14]. Informally, it is the
oblivious version of $\mathsf{QMA}{}\cap\mathsf{coQMA}$, so that the witness
sent by Merlin depends only on the length of the input. In contrast to the
“advice” of $\mathsf{P\\!/\\!poly}$, this has been described as “untrusted
advice”. Oblivious proofs can also be thought of as restricting non-uniform
classes, like $\mathsf{P\\!/\\!poly}$ or $\mathsf{BQP\\!/\\!qpoly}$, to advice
which is verifiable [GM15].

######  Definition 2.1.

A language $L$ is in $\mathsf{YQP}$ if there exists a polynomial-time uniform
family of quantum circuits $\\{Y_{n}\\}_{n\in\mathbb{N}}$ that satisfy the
following. Circuit $Y_{n}$ is of size $\operatorname{poly}(n)$ and takes as
input $x\in\\{0,1\\}^{n}$, a $p(n)$-qubit state $\rho$ for some
$p(n)\leq\operatorname{poly}(n)$, and an ancilla register initialized to the
all-zero state, and has two designated 1-qubit “advice-testing” and “output”
qubits. $Y_{n}(x,\rho)$ acts as follows:

  1. 1.

First $Y_{n}$ applies a subcircuit $A_{n}$ to all registers, after which the
advice-testing qubit is measured, producing a value
$b_{\text{adv}}\in\\{0,1\\}$.

  2. 2.

Next, $Y_{n}$ applies a second subcircuit $B_{n}$ to all registers, then
measures the output qubit, producing a value $b_{\text{out}}\in\\{0,1\\}$.

These output bits satisfy the following:

  * •

For all $n$, there exists a $\rho_{n}$ such that for all $x$, the advice bit
satisfies $\operatorname{E}[b_{\text{adv}}]\geq 9/10$.

  * •

If for any $x,\rho$ we have $\operatorname{E}[b_{\text{adv}}]\geq 1/10$, then
on input $x,\rho$ we have

$\operatorname{Pr}\left[b_{\text{out}}=L(x)\mid b_{\text{adv}}=1\right]\geq
9/10.$

$L$ is in the subclass $\mathsf{YQP^{*}}$ if the family can be chosen such
that $b_{\text{adv}}$ is independent of $x$.

Just as $\mathsf{Oblivious\mathchar 45\relax{}NP}$ is unlikely to contain
$\mathsf{NP}$ [FSW09], it also seems unlikely that $\mathsf{QMA}$ is contained
in $\mathsf{YQP}$. On the other hand, it is straightforward to show that any
sparse language can be verified obliviously, so
$\mathsf{FewP}\subseteq\mathsf{YP^{*}}$ and
$\mathsf{FewQMA}\subseteq\mathsf{YQP^{*}}$. We also have the trivial bounds
$\mathsf{BQP}\subseteq\mathsf{YQP^{*}}\subseteq\mathsf{YQP}\subseteq\mathsf{QMA}$
and $\mathsf{YQP}\subseteq\mathsf{BQP\\!/\\!qpoly}$. Studying $\mathsf{YQP}$
may be motivated by the use of oblivious complexity classes in constructing
circuit lower bounds [FSW09, GLV24], by the fact that
$\mathsf{BQP\\!/\\!qpoly}=\mathsf{\mathsf{YQP}^{*}\\!/\\!poly}=\mathsf{\mathsf{YQP}\\!/\\!poly}$
shown by [AD14], or by the results shown in this work.

###

The class $\mathsf{APP}$ was introduced by Li [Li93] in pursuit of a large
class of $\mathsf{PP}$-low languages. We use the equivalent definition given
by Fenner [Fen03, Corollary 3.7].

######  Definition 2.2.

$L\in\mathsf{APP}$ if and only if there exist functions $f,g\in\mathsf{GapP}$
and constants $0\leq\lambda<\upsilon\leq 1$ such that for all $n$ and
$x\in\\{0,1\\}^{n}$, we have $g(1^{n})>0$ and

  * •

If $x\in L$ then $\upsilon g(1^{n})\leq f(x)\leq g(1^{n})$;

  * •

If $x\notin L$ then $0\leq f(x)\leq\lambda g(1^{n})$.

In the above definition, recall that $\mathsf{GapP}$ is the closure of
$\mathsf{\\#P}$ under subtraction. In other words, while every function
$f\in\mathsf{\\#P}$ corresponds to a nondeterministic polynomial-time Turing
Machine $N$ such that $f(x)$ equals the number of accepting paths of $N(x)$, a
$\mathsf{GapP}$ function equals the number of accepting paths minus the number
of rejecting paths.

$\mathsf{APP}$ is a subclass of $\mathsf{PP}$ and is $\mathsf{PP}$-low,
meaning $\mathsf{PP}^{\mathsf{APP}}=\mathsf{PP}$. Recall that $\mathsf{PP}$
can be thought of as comparing a $\mathsf{\\#P}$ function to a threshold
exactly, with no promise gap. The class in fact remains unchanged if it is
defined as comparing a $\mathsf{GapP}$ function to a threshold as simple as
one-half of the possible paths or as complex as a $\mathsf{GapP}$ function.
Then, $\mathsf{APP}$ can be thought of as comparing a $\mathsf{GapP}$ function
(here $f(x)$) to some threshold (here $g(1^{n})$), where the complexity of the
threshold is limited to a $\mathsf{GapP}$ function which may depend on the
input size but not the input, and where there is some arbitrarily small but
nonzero promise gap (from $\lambda g(1^{n})$ to $\upsilon g(1^{n})$).

The best known upper bound on $\mathsf{APP}$ is $\mathsf{PP}$. Compared with
the class $\mathsf{A_{0}PP}=\mathsf{SBQP}\subseteq\mathsf{PP}$ [Kup15],
$\mathsf{A_{0}PP}$ contains $\mathsf{QMA}$ and is _not_ known to be
$\mathsf{PP}$-low, while $\mathsf{APP}$ is not known to contain even
$\mathsf{NP}$ but is $\mathsf{PP}$-low.

###

We will use the following fact shown for uniform circuit families by Watrous
[Wat08, Section IV.5], and shown earlier for QTMs by Fortnow and Rogers
[FR99].

######  Lemma 2.3.

For any polynomial-time uniformly generated family of quantum circuits
$\\{Q_{n}\\}_{n\in\mathbb{N}}$ each of size bounded by a polynomial $t(n)$,
there is a $\mathsf{GapP}$ function $f$ such that for all $n$-bit $x$,

$\operatorname{Pr}\left[Q_{n}(x)\text{ accepts}\right]=\frac{f(x)}{5^{t(n)}}.$

##  3 Results

We first prove our main technical result which will later allow us to prove
Theorems 3.4 and 3.5.

######  Lemma 3.1.

$\mathsf{YQP^{*}}\subseteq\mathsf{APP}$.

###### Proof.

Consider any language $L\in\mathsf{YQP^{*}}$. Let
$\\{Y_{n},A_{n},B_{n}\\}_{n\in\mathbb{N}}$ be the associated family of
circuits and subcircuits, in which $Y_{n}$ takes string $x$ and a supposed
witness or advice state as input, in which subcircuit $A_{n}$ validates the
advice and produces output bit $b_{\text{adv}}$, and in which, given $A_{n}$
accepted, $B_{n}$ uses the advice to verify the particular input $x$ and
outputs bit $b_{\text{adv}}$. Note that because we consider
$\mathsf{YQP^{*}}$, the circuit $A_{n}$ only takes the witness state, not $x$,
as input. Let $k$ and $m$ be polynomials in $n$ denoting the respective sizes
of the ancilla and proof registers.

We use the technique of strong, or in-place, error reduction of Marriott and
Watrous [MW05] on the circuits $A_{n}$ with a polynomial $q$ in $n$ of our
choosing to produce a new circuit family $\\{A^{\prime}\\}_{n\in\mathbb{N}}$
such that for any proof $\rho$,

  * •

$\operatorname{Pr}\left[A_{n}(\rho)\right]\geq\frac{9}{10}\Rightarrow\operatorname{Pr}\left[A^{\prime}_{n}(\rho)\right]\geq
1-2^{-q}$;

  * •

$\operatorname{Pr}\left[A_{n}(\rho)\right]\leq\frac{1}{10}\Rightarrow\operatorname{Pr}\left[A^{\prime}_{n}(\rho)\right]\leq
2^{-q}$.

Recall the error reduction algorithm of [MW05] involves, given some quantum
input or witness state, applying a circuit $C$, recording whether the output
is $\ket{0}$ or $\ket{1}$ in a variable $y_{i}$, applying $C^{\dagger}$,
recording whether the circuit’s ancilla register is in the all-zero state or
not in a variable $y_{i+1}$, and repeating these steps for some number of
iterations $M$. Call the full, amplified circuit $C^{\prime}$.

######  Remark 3.2.

Studying the proof of [MW05], if the final recorded bit $y_{2M+1}=1$, then the
final state of the ancilla register was projected into the all-zero state.
Additionally, suppose the circuit $C^{\prime}$ is applied to an $m$-qubit
proof state, so there are $2^{m}$ eigenstate
$\\{\ket{\lambda_{i}}\\}_{i\in[2^{n}]}$ of $C^{\prime}$. Further studying the
proof of [MW05], if the initial state given to $C^{\prime}$ was an eigenstate
$\ket{\lambda_{i}}$ and after applying $C^{\prime}$ the final two recorded
bits were $y_{2M}=y_{2M+1}=1$, then the final state of the proof register is
the same as the initial state, $\ket{\lambda_{i}}$. If an eigenstate
$\ket{\lambda_{i}}$ was accepted by the original circuit $C$ with probability
$p$, then when $C^{\prime}$ is run on $\ket{\lambda_{i}}$, the probability
that of $y_{2M}=y_{2M+1}=1$ is at least $p\times\min\left\\{p,1-p\right\\}$.

Combining all of the above, we define
$\\{A^{\prime\prime}_{n}\\}_{n\in\mathbb{N}}$ to be the amplified circuits
$\\{A^{\prime}_{n}\\}_{n\in\mathbb{N}}$ with the additional rule that the
circuit accepts iff both $b_{\text{adv}}=1$ and the final two recorded
variables $y_{2M}=y_{2M+1}=1$. Further, define
$\\{A^{\prime\prime\prime}_{n}\\}_{n\in\mathbb{N}}$ so that
$A^{\prime\prime\prime}_{n}=A^{\prime\prime}_{n}(\frac{\mathbb{I}}{2^{m}})$,
with the maximally mixed state hard-wired into the proof register. Similarly,
we define $\\{Y^{\prime}_{n}\\}_{n\in\mathbb{N}}$ to apply the amplified
subcircuit $A^{\prime}_{n}$ and $B_{n}$, we define
$\\{Y^{\prime\prime}_{n}\\}_{n\in\mathbb{N}}$ to apply $A^{\prime\prime}_{n}$
and $B_{n}$ and thus accept iff
$b_{\text{adv}},b_{\text{out}},y_{2M},y_{2M+1}$ all equal 1, and we define
$\\{Y^{\prime\prime\prime}_{n}\\}_{n\in\mathbb{N}}$ so that
$Y^{\prime\prime\prime}_{n}(x)=Y^{\prime\prime}_{n}(x,\frac{\mathbb{I}}{2^{m}})$
with the maximally mixed state hard-wired into the proof register, meaning
that it uses $A^{\prime\prime\prime}_{n}$ as a subcircuit.

Applying Lemma 2.3, there exist $\mathsf{GapP}$ functions $f,g$ and
polynomials $r,t$ such that for all $n$-bit $x$,

$\displaystyle\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]=\frac{f(1^{n})}{5^{r(n)}}\quad\text{and}\quad\operatorname{Pr}\left[Y^{\prime\prime\prime}_{n}(x)\text{
accepts}\right]=\frac{g(x)}{5^{t(n)}}.$

The function $f$ depends only on the input length $n$, not $x$, because the
circuit $A^{\prime\prime\prime}_{n}$ is independent of $x$. Next, we define
$F(1^{n})=f(1^{n})5^{t(n)-r(n)}$, which is a $\mathsf{GapP}$ function since
$5^{t(n)-r(n)}\in\mathsf{FP}\subseteq\mathsf{GapP}$ and $\mathsf{GapP}$ is
closed under multiplication. We have

$\displaystyle\frac{g(x)}{F(1^{n})}=\frac{\operatorname{Pr}\left[Y^{\prime\prime\prime}_{n}(x)\text{
accepts}\right]}{\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]}$

We will show bounds on the ratio $g(x)/F(1^{n})$ based on whether $x$ is in
$L$ or not in $L$ in order to prove $L$ is in $\mathsf{APP}$. First, note that
the ratio is upper-bounded by 1 since $Y^{\prime\prime\prime}_{n}$ only
accepts if the subcircuit $A^{\prime\prime\prime}_{n}$ accepts, and it is
lower-bounded by 0 since probabilities are non-negative. Next, let
$\left\\{\ket{\lambda_{i}}\right\\}_{i\in[2^{m}]}$ be the set of eigenvectors
$\ket{\lambda_{i}}$ of the circuit $A_{n}$. By writing the maximally mixed
state, which is hard-wired into the proof register of
$Y^{\prime\prime\prime}_{n}$, in terms of this eigenbasis, we find

$\displaystyle\frac{\operatorname{Pr}\left[Y^{\prime\prime\prime}_{n}(x)\text{
accepts}\right]}{\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]}=\frac{\operatorname{Pr}\left[Y^{\prime\prime}_{n}(x,\frac{\mathbb{I}}{2^{m}})\text{
accepts}\right]}{\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]}=\frac{\sum\limits_{i=1}^{2^{m}}\operatorname{Pr}\left[Y^{\prime\prime}_{n}(x,\ket{\lambda_{i}})\text{
accepts}\right]}{2^{m}\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]}$ $\displaystyle=$
$\displaystyle\frac{\sum\limits_{i=1}^{2^{m}}\operatorname{Pr}\left[Y^{\prime\prime}_{n}(x,\ket{\lambda_{i}})\text{
accepts}\mid A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]}{2^{m}\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]}$ $\displaystyle=$
$\displaystyle\frac{\sum\limits_{i=1}^{2^{m}}\operatorname{Pr}\left[B_{n}(x,\ket{\lambda_{i}})\text{
accepts}\right]\cdot\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]}{2^{m}\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]}$

where we have used the facts that $Y^{\prime\prime}_{n}$ accepting requires
$A^{\prime\prime}_{n}$ to accept and that $A^{\prime\prime}_{n}$ accepting
guarantees the initial eigenstate $\ket{\lambda_{i}}$ is sent on to the
subcircuit $B_{n}$ within $Y^{\prime\prime}_{n}$. Define

$\mathcal{B}=\left\\{i\in[2^{m}]\mid\operatorname{Pr}\left[A_{n}(\ket{\lambda_{i}})\right]\leq
0.1\right\\}.$

Then we can rewrite both the numerator and denominator in the above ratio to
give

$\frac{\text{\small$\sum\nolimits_{i\in\mathcal{B}}$}\text{\footnotesize$\operatorname{Pr}\left[B_{n}(x,\ket{\lambda_{i}})\text{
accepts}\right]\cdot\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]+$}\text{\small$\sum\nolimits_{i\in\overline{\mathcal{B}}}$}\text{\footnotesize$\operatorname{Pr}\left[B_{n}(x,\ket{\lambda_{i}})\text{
accepts}\right]\cdot\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]$}}{\sum\nolimits_{i\in\mathcal{B}}\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]+\sum\nolimits_{i\in\overline{\mathcal{B}}}\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]}.$

We will use this expression as the starting point for our analysis of the YES
and NO cases.

Now, suppose we have a YES instance with $x\in L$. As this is
$\mathsf{YQP^{*}}$, we are guaranteed at least one proof is accepted by $A$
with high probability, and denote it by $\ket{\lambda^{*}}$. Then, we may
calculate that $g(x)/F(1^{n})$ is at least

$\displaystyle\frac{\sum\nolimits_{i\in\mathcal{B}}0+\sum\nolimits_{i\in\overline{\mathcal{B}}}\frac{9}{10}\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]}{\sum\nolimits_{i\in\mathcal{B}}2^{-q}+\sum\nolimits_{i\in\overline{\mathcal{B}}}\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda_{i}})\text{
accepts}\right]}\geq\frac{\frac{9}{10}\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda^{*}})\text{
accepts}\right]}{\left\lvert\mathcal{B}\right\rvert
2^{-q}+\operatorname{Pr}\left[A^{\prime\prime}_{n}(\ket{\lambda^{*}})\text{
accepts}\right]}$

which, using the fact that $x/(c+x)$ decreases as $x$ decreases as well as the
bound stated in Remark 3.2 on the probability the error-reduction variables
$y_{2M},y_{2M+1}$ are 1, is at least

$\displaystyle\frac{\frac{9}{10}(1-2^{-q})(0.9)(0.1)}{\left\lvert\mathcal{B}\right\rvert
2^{-q}+(1-2^{-q})(0.9)(0.1)}$
$\displaystyle=\frac{0.081(1-2^{-q})}{2^{m-q}+0.09(1-2^{-q})}$
$\displaystyle\geq\frac{0.081(1-2^{-q})}{2^{-q/2}+0.09(1-2^{-q})}\geq\frac{0.081(1-2^{-10})}{2^{-5}+0.09(1-2^{-10})}>0.66,$

where in the last line we used our freedom to choose the polynomial $q$.

On the other hand, in a NO instance, we have that $g(x)/F(1^{n})$ is at most

$\displaystyle\frac{\left\lvert\mathcal{B}\right\rvert\times 1\times
2^{-q}+\left\lvert\overline{\mathcal{B}}\right\rvert\times\frac{1}{10}\times
1}{2^{m}\operatorname{Pr}\left[A^{\prime\prime\prime}_{n}\text{
accepts}\right]}\leq\frac{2^{m}2^{-q}+2^{m}\frac{1}{10}}{2^{m}}=2^{-q}+\frac{1}{10}\leq
0.2.$

We have shown a constant separation of $g(x)/F(1^{n})$ in YES and NO
instances. This satisfies the definition of $\mathsf{APP}$ in Definition 2.2
of $\mathsf{APP}$, so we conclude $\mathsf{YQP^{*}}\subseteq\mathsf{APP}$. ∎

Next, the fact $\mathsf{APP}$ is known to be $\mathsf{PP}$-low [Li93, Theorem
6.4.14] gives us the following corollary.

######  Corollary 3.3.

$\mathsf{YQP^{*}}$ is $\mathsf{PP}$-low, i.e.
$\mathsf{PP}^{\mathsf{YQP^{*}}}=\mathsf{PP}$.

We are now able to give a corrected proof of the result originally claimed for
$\mathsf{BQP\\!/\\!qpoly}$ but only proved for $\mathsf{BQP\\!/\\!poly}$ by
Aaronson [Aar06].

######  Theorem 3.4.

If $\mathsf{PP}\subseteq\mathsf{BQP\\!/\\!qpoly}$, then the Counting Hierarchy
collapses to $\mathsf{CH}=\mathsf{QMA}=\mathsf{YQP^{*}}$.

###### Proof.

We repeat Aaronson’s original claimed proof [Aar06], but substitute
$\mathsf{YQP^{*}}$ where he relied on $\mathsf{QMA}$.

Suppose $\mathsf{PP}\subseteq\mathsf{BQP\\!/\\!qpoly}$. From [AD14], we know
that $\mathsf{BQP\\!/\\!qpoly}=\mathsf{\mathsf{YQP}^{*}\\!/\\!poly}$. Then in
$\mathsf{YQP^{*}}$, without trusted classical advice, Arthur can request
Merlin sends many copies of the quantum advice $\ket{\psi}$ and a description
of the circuit $C$ such that $C,\ket{\psi}$ compute Permanent, a
$\mathsf{PP}$-complete problem. Of course, this advice is now untrusted.
Arthur verifies that $C,\ket{\psi}$ in fact work on a large fraction of inputs
by simulating the interactive protocol for $\mathsf{\\#P}$ due to [LFKN92],
which also works for $\mathsf{PP}$, using $C,\ket{\psi}$ in place of the
prover. If the protocol accepts, then Arthur can use the random self-
reducibility of Permanent to generate a circuit $C^{\prime}$ which is correct
on _all_ inputs (see e.g. [AB09, Sec. 8.6.2]). Thus, we have
$\mathsf{PP}=\mathsf{YQP^{*}}$.

In this way, any level of the Counting Hierarchy
$\mathsf{C_{i}P}=\left(\mathsf{C_{i-1}P}\right)^{\mathsf{PP}}$ with $i>1$ is
reducible to $\left(\mathsf{C_{i-1}P}\right)^{\mathsf{YQP^{*}}}$ which by
Corollary 3.3 equals $\mathsf{C_{i-1}P}$. This works recursively for all
levels, collapsing $\mathsf{C_{i}P}$ to $\mathsf{C_{1}P}=\mathsf{PP}$, so that
all of $\mathsf{CH}=\mathsf{PP}=\mathsf{YQP^{*}}$. ∎

Given the above result, we can also fully recover the following result
originally claimed by Aaronson [Aar06].

######  Theorem 3.5.

$\mathsf{PP}$ does not have quantum circuits of size $n^{k}$ for any fixed
$k$. Furthermore, this holds even if the circuits can use quantum advice.

###### Proof.

Suppose $\mathsf{PP}$ does have circuits of size $n^{k}$. This implies
$\mathsf{PP}\subseteq\mathsf{BQP\\!/\\!qpoly}$, which by Theorem 3.4 implies
$\mathsf{CH}=\mathsf{YQP^{*}}$, which includes
$\mathsf{P}^{\mathsf{PP}}=\mathsf{PP}=\mathsf{YQP^{*}}$. Together, there are
circuits of size $n^{k}$ for $\mathsf{P}^{\mathsf{PP}}$, which contradicts the
result of [Aar06] (unaffected by the bug) that $\mathsf{P}^{\mathsf{PP}}$ does
not have such circuits even with quantum advice. ∎

In fact, as [Aar06] observes, because his proof that
$\mathsf{P}^{\mathsf{PP}}$ does not have circuits of size $n^{k}$ for fixed
$k$ can be strengthened, we have that Theorem 3.5 can be strengthened to show
for all functions $f(n)\leq 2^{n}$, the class $\mathsf{PTIME(f(f(n)))}$, which
is like $\mathsf{PP}$ but for machines of running time $f(f(n))$, requires
quantum circuits using quantum advice of size at least $f(n)/n^{2}$. In
particular, this implies $\mathsf{PEXP}$, the exponential-time version of
$\mathsf{PP}$, requires quantum circuits with quantum advice of “half-
exponential” size (meaning a function that becomes exponential when composed
with itself [MVW99]).

## References

  * [Aar06] Scott Aaronson.  Oracles are subtle but not malicious.  In Proceedings of the 21st Annual IEEE Conference on Computational Complexity, pages 340––354. IEEE Computer Society, 2006.  doi:10.1109/CCC.2006.32.
  * [Aar07] Scott Aaronson.  The learnability of quantum states.  Proc. R. Soc. A., 463(2088):3089–3114, 2007.  doi:10.1098/rspa.2007.0113.
  * [Aar17] Scott Aaronson.  Yet more errors in papers, May 2017.  Accessed 14 Jan. 2024.  URL: https://scottaaronson.blog/?p=3256.
  * [AB09] Sanjeev Arora and Boaz Barak.  Computational complexity: a modern approach.  Cambridge University Press, 2009.
  * [AD14] Scott Aaronson and Andrew Drucker.  A full characterization of quantum advice.  SIAM Journal on Computing, 43(3):1131–1183, 2014.  doi:10.1137/110856939.
  * [AGKR24] Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph.  Quantum polynomial hierarchies: Karp-Lipton, error reduction, and lower bounds, 2024.  arXiv:2401.01633.
  * [Fen03] Stephen A. Fenner.  PP-lowness and a simple definition of AWPP.  Theory of Computing Systems, 36:199–212, 2003.  doi:10.1007/s00224-002-1089-8.
  * [FR99] Lance Fortnow and John Rogers.  Complexity limitations on quantum computation.  Journal of Computer and System Sciences, 59(2):240–252, 1999.  doi:10.1006/jcss.1999.1651.
  * [FSW09] Lance Fortnow, Rahul Santhanam, and Ryan Williams.  Fixed-polynomial size circuit bounds.  In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, pages 19–26. IEEE, 2009.  doi:10.1109/CCC.2009.21.
  * [GLV24] Karthik Gajulapalli, Zeyong Li, and Ilya Volkovich.  Oblivious classes revisited: Lower bounds and hierarchies.  ECCC: TR24-049, 2024.  URL: https://eccc.weizmann.ac.il/report/2024/049/.
  * [GM15] Oded Goldreich and Or Meir.  Input-oblivious proof systems and a uniform complexity perspective on P/poly.  ACM Transactions on Computation Theory, 7(4):1–13, 2015.  doi:10.1145/2799645.
  * [Kup15] Greg Kuperberg.  How hard is it to approximate the Jones polynomial?  Theory of Computing, 11(1):183–219, 2015.
  * [LFKN92] Carsten Lund, Lance Fortnow, Howard Karloff, and Noam Nisan.  Algebraic methods for interactive proof systems.  Journal of the ACM (JACM), 39(4):859–868, 1992.  doi:10.1145/146585.146605.
  * [Li93] Lide Li.  On the counting functions.  PhD thesis, The University of Chicago, 1993.  URL: https://www.proquest.com/dissertations-theses/on-counting-functions/docview/304080357/se-2.
  * [MN16] Tomoyuk Morimae and Harumichi Nishimura.  Quantum interpretations of AWPP and APP.  Quantum Info. Comput., 16(5–-6):498––514, 2016.  doi:10.26421/QIC16.5-6-6.
  * [MVW99] Peter Bro Miltersen, N. V. Vinodchandran, and Osamu Watanabe.  Super-polynomial versus half-exponential circuit size in the Exponential Hierarchy.  In International Computing and Combinatorics Conference, pages 210–220. Springer, 1999.  doi:10.1007/3-540-48686-0_21.
  * [MW05] Chris Marriott and John Watrous.  Quantum Arthur–Merlin games.  Computational Complexity, 14:122–152, 2005.  doi:10.1007/s00037-005-0194-x.
  * [Vin05] N. V. Vinodchandran.  A note on the circuit complexity of PP.  Theoretical Computer Science, 347(1):415–418, 2005.  doi:10.1016/j.tcs.2005.07.032.
  * [Wat08] John Watrous.  Quantum computational complexity, 2008.  arXiv:0804.3401v1.
  * [Zoo] Complexity Zoo: BQP/poly, BQP/mpoly, BQP/qpoly, BQP.  Accessed 13 Mar. 2024.  URL: https://complexityzoo.net/Complexity_Zoo:B.


# Thermal Phase Fluctuations in Narrow Superfluid Rings

Parth Sabharwal  Department of Physics and Astronomy, Dartmouth College, 6127
Wilder Laboratory, Hanover NH 03766, USA  Daniel G. Allman  Pradipta Debnath
Department of Physics and Astronomy, Dartmouth College, 6127 Wilder
Laboratory, Hanover NH 03766, USA  Kevin C. Wright  Department of Physics and
Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover NH 03766, USA

###### Abstract

Using matter-wave interference, we have investigated thermal phase
fluctuations in narrow coplanar, concentric rings of ultracold fermionic
superfluids. We found that the correlation length decreases with number
density, consistent with theoretical expectations. We also observed that
increasing the coupling between the rings leads to greater overall coherence
in the system. The phase fluctuations increased with a change from periodic to
closed boundary conditions as we applied a potential barrier at one point in a
ring. These results are relevant for the implementation of proposals to
utilize ultracold quantum gases in large and elongated circuit-like
geometries, especially those that require deterministic preparation and
control of quantized circulation states.

††preprint: APS/123-QED

##  I Introduction

Coherence is a fundamental property of superfluids and superconductors that is
essential to many of their applications in circuits used for sensing and
information processing. However, phase fluctuations can degrade coherence and
even destroy long-range order in sufficiently elongated systems, even when the
particle-like excitations are three-dimensional and temperatures are well
below the critical temperature Petrov  _et al._ (2001). Under these
conditions, local macroscopic order can still be maintained over distances
shorter than the correlation length, but not globally, and if one or more of
the system dimensions exceeds this correlation length the system is described
as quasi-condensed Mora and Castin (2003). It also becomes challenging to
reliably initialize and control supercurrents in circuits, especially since
coherence over longer paths can be disrupted by changes in the system’s
topology and boundary conditions. These challenges are less acute in systems
with large phase stiffness such as conventional superconductors Raychaudhuri
and Dutta (2021), but can be important in low-carrier density superconductors
Emery and Kivelson (1995) and in ultracold quantum gases such as those
considered in this work.

The first experiments to show the impact of thermal phase fluctuations on
elongated three-dimensional (3D) quantum gases were conducted with Bose-
Einstein condensates (BECs) of 87Rb in a prolate harmonic trap with a variable
aspect ratio of up to 51. Modulations of the density profile were observed
after ballistic expansion, and the power spectrum of these modulations was
shown to depend on the aspect ratio of the trapping potential, the atom
number, and the temperature Dettmer  _et al._ (2001); Hellweg  _et al._
(2001). Ballistic expansion techniques have been used to characterize phase
fluctuations in two-dimensional Bose gases Choi  _et al._ (2012); Seo  _et
al._ (2014) and to measure the change in coherence in the crossover from three
dimensions to one dimension Shah  _et al._ (2023). Other experiments have used
Bragg interferometry to measure spatial correlation lengths and correlation
functions more directly Hellweg  _et al._ (2003); Richard  _et al._ (2003);
Cacciapuoti  _et al._ (2003); Hugbart  _et al._ (2005). Phase fluctuations
have also been observed using matter-wave interference of two adjacent,
independent quantum gases in elongated, linear double-well configurations Shin
_et al._ (2004); Gati  _et al._ (2006); Jo  _et al._ (2007); Scott  _et al._
(2008, 2009).

This experiment is the first to examine phase fluctuations in an ultracold gas
that can be varied between closed (hard-walled) and periodic boundary
conditions. The first theoretical analysis of how boundary conditions affect
phase fluctuations in a ring geometry was conducted by Mathey et al., who
considered the scenario of a weakly-interacting BEC in the hydrodynamic limit
where phase fluctuations arise mainly due to excitation of long-wavelength
sound modes Mathey  _et al._ (2010). The phase-fluctuating regime was
predicted to arise in the limit of low one-dimensional (1D) density, with a
simple criterion involving the trap aspect ratio, particle mass, and number of
particles.

Although phase variations have previously been measured in ring-shaped quantum
gases using matter-wave interference techniques Eckel  _et al._ (2014); Corman
_et al._ (2014); Cai  _et al._ (2022); Del Pace  _et al._ (2022); Allman  _et
al._ (2024), in those experiments the combination of aspect ratio, 1D density,
and temperature were not suitable for studying the phase fluctuating regime,
where the correlation length is smaller than half the ring circumference. Our
experimental setup allows us to create narrow, concentric superfluid rings of
fermionic 6Li with aspect ratios $>$10 and uniform 1D densities as low as 20
pairs/$\upmu$m, which is sufficient to realize the phase fluctuating regime.
In the hydrodynamic limit of strong interactions, the collective phononic
excitations of the superfluid dominate quasiparticle (pair-breaking)
excitations, and phase fluctuations should be adequately described by the same
Bogoliubov de Gennes (BdG) framework previously applied to weakly-interacting
BECs.

Figure 1: Individual absorption images of an equal spin mixture of 6Li atoms
in narrow concentric rings, showing how the ring geometry and inter-ring
coupling is altered for different versions of the experimental procedures. A
localized “cutting” barrier is applied to both rings during the evaporative
cooling process as shown in (a), to ensure the average azimuthal current
density is negligible. After cooling, we adiabatically remove the cutting
barrier from both rings (b) or just the outer ring (c). These are the
‘Connected’ and ‘Cut’ ring cases described in the text. The smaller inner ring
is less affected by fluctuations, and is used as a phase reference for matter-
wave interference. We also examined the effects of reducing the strength of
the “inter-ring” barrier by up to 50%, which can be seen as a slight increase
in the average optical depth between the rings in (d). The field of view is 42
$\upmu$m$\times$42 $\upmu$m.

This work focuses on two distinct physical configurations: a pair of smooth
and fully connected concentric rings, and a connected inner ring surrounded by
an outer ring that is cut at a fixed point by a narrow potential barrier, as
shown in Fig. 1. Our motivation for studying configurations with and without
localized potential barriers comes from our experiences with using such
barriers to initialize narrow superfluid rings in a selected persistent
current state. Evaporative cooling through the superfluid phase transition in
a smooth, connected ring often leads to the appearance of a persistent current
around the ring. This can occur spontaneously via the Kibble-Zurek mechanism
Allman  _et al._ (2024), or due to bulk flow generated by changes in the
trapping potential during the evaporation process. Our most reliable procedure
for initializing two concentric rings each in a zero-current state involves
evaporative cooling with a barrier breaking each ring, and reconnecting them
only after the temperature is low enough for the superfluid to be robust. The
success rate of this initialization procedure is $>99\%$ for sufficiently high
final densities and strong interactions, but we also observed that this
success rate decreases at lower densities. This work confirms that increased
phase fluctuations were responsible for difficulties with initialization, and
to determine what other factors might also impact our ability to control the
quantized current around the rings.

One essential technical development in this experiment is a robust method for
using matter-wave interference to measure the angular variations in the phase
difference between adjacent regions of two concentric superfluid rings. The
procedure involves using ballistic expansion to generate a suitable matter-
wave interference pattern from the two concentric rings, and employing a phase
extraction algorithm to infer the phase difference between them. We have
validated this experimental measurement procedure using simulated data that
combines theoretical predictions of the phase fluctuations with numerical
modeling of the matter-wave interference patterns that should appear after
ballistic expansion. Sec. II contains a detailed description of the
experimental setup, the phase extraction algorithm, and a comparison of its
effectiveness against the model described in Sec. III. In Sec. IV, we describe
our results in detail and compare them with theoretical predictions, and
discuss the limitations of our study.

##  II Experimental Techniques/Methods

The goal of the experiment is to study phase fluctuations of a 6Li superfluid
confined in a double ring configuration, as we vary the pair densities and
inter-ring coupling. We prepare the 6Li atoms in the trap in an equal spin
mixture of the lowest two hyperfine states, which have a broad Feshbach
resonance occurring at $83.4$ mT. This work is focused on the phase
fluctuating properties of the fermionic pair superfluid in the unitary limit
where the superfluid critical temperature is relatively high, and it is
reasonable to treat the system in the hydrodynamic limit.

Core technical details of the cooling, trapping and state preparation
procedure are described in detail in Refs. Cai  _et al._ (2022); Allman  _et
al._ (2023, 2024), but the final part involves evaporative cooling of the 6Li
atoms in a dynamically reconfigurable optical dipole trap generated by two
red-detuned laser fields. The main contribution to the trap potential is from
a sheet-like trapping beam with a wavelength of 1064 nm, propagating in the
horizontal direction. The broad, oblate harmonic potential near the focus of
this beam is modified by a second beam propagating upward, perpendicular to
the sheet beam, with a wavelength of 780 nm and a typical power of 50-100 mW.
The spatial intensity profile of this beam is shaped by a digital micromirror
device (DMD, Texas Instruments DLP Lightcrafter 6500) with a feature
resolution limit of 1.5 $\upmu$m, after which 1.5-3 mW of power reaches the
atoms. In this experiment we use the DMD to generate the concentric-ring
patterns, with the resultant density profiles shown schematically in Fig. 1,
where the radii of the inner and outer rings are 7.5 $\upmu$m and 12.5
$\upmu$m, respectively, and their Gaussian widths are 3 $\upmu$m each. In the
cases where we cut the rings with a localized azimuthal barrier, the width of
the cutting barrier is 3 $\upmu$m. The final vertical trapping frequency of
our sheet-like trap after evaporative cooling is 1.4(1) kHz, and the
oscillation frequency for radial motion about the minima of the rings is
5.1(2) kHz. The peak Fermi energy Fer  lies within the range $h\times
8.7-13.4$ kHz ($h$ is Planck’s constant) for the atom numbers observed in this
experiment.

Previous experiments have measured spatial variations of the superfluid phase
in a ring configuration using interferometric techniques with bosonic Eckel
_et al._ (2014); Corman  _et al._ (2014) and fermionic Del Pace  _et al._
(2022) condensates in a “bulls-eye” configuration with a disk surrounded by a
ring. Instead, we use a double ring configuration, similar to that employed in
Aidelsburger  _et al._ (2017); Allman  _et al._ (2024), replacing the disk
with an inner ring. The goal is to maximize the sensitivity and accuracy of
the interferometric measurement by achieving the best fringe contrast in the
shortest possible expansion time after release from the trap. The total atom
number in the inner ring matches that of the outer ring more closely than a
disc of a similar radius and density, which has been empirically determined to
provide much better fringe contrast. This is especially true for fringes
formed in the interior region of the disk, where the presence of a large
background of atoms reduces the fringe contrast greatly. Using a smaller disk
to address this atom number mismatch causes a large increase in the time-of-
flight (ToF) before the fringes begin to form. A larger ToF increases the
degree of self-interference of neighboring sections of the ring by the end of
the expansion, effectively blurring the azimuthal phase information. An inner
ring may provide a less uniform phase reference than a disk, but we account
for these phase fluctuations in our analysis, as described in Section III. The
dimension of the outer ring is itself set by the desired aspect ratio, and the
estimated correlation length in our experiment.

We now discuss how we vary the density and inter-ring coupling, which is
essential for obtaining the results presented in later sections. We vary the
final number of atoms in the trap between $4$-$12\times 10^{3}$ by changing
the number of atoms initially loaded into the dipole trap before forced
evaporative cooling, which is achieved by lowering the sheet beam power from
$3.0$ W to $42$ mW over 2 seconds. We vary the amount of inter-ring coupling,
by weakening what we will refer to here as the “inter-ring barrier”. This is
achieved by programming the DMD to alter the amount of light in the inter-ring
region during the evaporation stage and thereafter, which in turn varies the
number of non-condensed atoms in that region and effectively varies the amount
of coupling between the two ensembles. We have validated that the column
density in the inter-ring region grows linearly with the weakening of the
inter-ring barrier. We note that the possible existence of pre-formed pairs in
the inter-ring region could lead to non-trivial modifications of the coupling
dynamics. A barrier strength of 100$\%$ corresponds to no red-detuned light in
the inter-ring region from the DMD-controlled beam, which is reduced to 50$\%$
in increments of 10$\%$, by increasing the amount of red-detuned light. After
evaporation and a hold time of 100 ms, we reconnect one or both rings by
adiabatically removing the cutting barrier(s) over another 100 ms, by
dynamically updating the pattern on the DMD, resulting in a configuration with
a ‘Cut’ or ‘Connected’ outer ring, respectively.

After preparation of the superfluid in the concentric trap potentials
described above, we measure the superfluid phase difference profile via
matter-wave interference (See Sec. II.1). To do this, we shut off the optical
trapping beams and rapidly ramp the magnetic field down to $61$ mT, which
converts most of the weakly-bound pairs at resonance to tightly bound
molecules before they are allowed to expand Regal  _et al._ (2004). This
ensures that the pairs do not break and their coherence is preserved after
they are released, which is essential for the initial phase variations to lead
to observable density variations in the pattern that appears when the
concentric expanding rings begin to overlap and interfere with each other. All
imaging of the density of the molecules after expansion is performed by
resonant absorption imaging of atoms in the $|1\rangle$ state on a closed
transition at $61$ mT.

###  II.1 Phase Measurement Procedure

As demonstrated by Eckel et al. Eckel  _et al._ (2014) and Corman et al.
Corman  _et al._ (2014), when two sufficiently independent ring condensates
expand and overlap, observable interference fringes can form under certain
conditions. These fringes contain information about the in-trap state of each
superfluid, with the assumption that initial local phase variations map
linearly onto observable deviations in the fringe positions. For two non-
circulating, concentric ring condensates in equilibrium, this resultant
pattern should look like a series of concentric closed loops, with the polar
profile $r(\theta)$ of each loop dependent on the local angular phase
difference between the condensates before they were released from the trap. If
there is a non-zero winding number difference $\Delta w$ between the two
condensates, spirals are seen in the resultant interferogram, with the number
of arms equal to $|\Delta w|$. The deviations of the fringes about the
expected radial positions indicate fluctuations in the “relative” phase about
the overall winding value. In-trap variations in the local chemical potential
also give rise to deviations in fringe positions, although this effect is
typically stable shot-to-shot, and small compared to the effect of phase
fluctuations on the fringe positions. We account for these deviations by
subtracting the offset apparent in the average phase extracted for all the
experimental runs considered in our experiment. Our entire analysis is
predicated on accurate extraction of the azimuthal variation in phase from an
interferogram and relating it back to the phase around the experiment ring.

Figure 2: Interferograms (left) with a Gaussian background subtracted off
(center), accompanied by their polar projection (right) over a range of radii
(enclosed by black circles), about the center (yellow dot). This polar image
is used for extracting relative phase (black line), which is scaled to match
2$\pi$ to the typical fringe width (right axis), with an arbitrary offset
applied. The interferograms shown are from a pair of connected rings with
(top), 4.2 $\times 10^{3}$ (middle) 1.9 $\times 10^{3}$ and (bottom) 2.1
$\times 10^{3}$ pairs, with the last case showing an overall relative winding
number of 1. The color scale for the interferograms (left) has been normalized
to the respective peaks. The field of view is 42 $\upmu$m$\times$42 $\upmu$m.

We automate extraction of the in-situ relative phase from the interferograms
by using a Fourier technique. The first step is the accurate determination of
the centre of symmetry, which is determined by locating the central
interference peak in the interferogram. To improve contrast and facilitate
detection of the interference fringes, we perform a Gaussian background
subtraction with a filter radius of 1.7 $\upmu$m, which is on the order of the
fringe width 2.1 $\upmu$m, but smaller than the fringe spacing 4.4 $\upmu$m.
This radius was chosen empirically to remove the lowest spatial frequency
components from the Fourier spectrum while preserving enough of the desired
signal at the spatial frequency of the fringes to ensure a robust measurement.

An annular region of this background subtracted image containing the fringes
and excluding the central peak is then mapped from the $x$-$y$ plane to the
$\theta$-$r$ plane, effectively “unwrapping” the fringes present in the
original interferogram, as can be seen in Fig. 2. The radial extent of the
annular region is chosen to be 3.8 $\upmu$m to 13.0 $\upmu$m from the center,
and is twice the typical radial fringe spacing. The value of this spacing
depends primarily on the chosen ToF and the radii of the inner and outer
rings, and is otherwise very stable against changes in experimental
conditions. At the optimal ToF, any radial cut of this “unwrapped” image is
close to sinusoidal, and a Fourier transform of this 2D image along the radial
axis shows a spectral peak at the second harmonic. For each angular position
$\theta$, we extract the phase argument of this Fourier component, which we
treat as an estimate of the in-situ relative azimuthal phase of the rings,
plus a constant phase related to the overall radial offset. We subtract this
average offset under the assumption that the average relative azimuthal phase
is zero. This routine effectively generates a single composite fringe from the
interferogram that can be sampled on the interval $\theta\in[0,2\pi)$. Fig. 2
shows the extracted phase for a few experimental runs starting with two
connected rings for various atom numbers and relative winding numbers ($\Delta
w$). It can be seen that the algorithm works well across various atom numbers,
and can be effectively used to extract phase and winding number information
for large sets of data.

Figure 3: Comparison between the phase imprinted on a simulated absorption
image (solid line), and extracted using the method described above (stars).
Both plots represent the phase difference between the inner and outer ring at
each azimuthal position. The correlation length assumed to simulate this
instance of imprinted phase is 0.3 times the circumference of the outer ring.
Figure 4: Comparison between experimental observations of ballistic expansion
of atoms after release from a double ring (top row) against numerical
simulations (bottom row), for an overall winding number difference $\Delta
w=0$. Each image is an average of 5 absorption images. The radii of the inner
and outer ring in these images is 7.5 and 13.5 $\upmu$m, with Gaussian half-
widths of 1.5 $\upmu$m. The most optimal fringes for phase extraction are seen
here at 1400 $\upmu$s. For the rest of the experimental data, a smaller radius
for outer ring (12.5 $\upmu$m) is used, which reveals similar optimal fringes
at 1100 $\upmu$s. The field of view is 50 $\upmu$m$\times$50 $\upmu$m.

We validated the phase extraction procedure outlined above by extracting the
phase from interferograms simulated by ballistically propagating two
independent superfluids initially confined to two concentric rings and
imprinted with a known, controllable phase difference. We find that the
assumption of ballistic expansion is sufficient to describe the main features
of the interference pattern throughout the relevant times of flight, as can be
seen in Fig. 4. A more thorough treatment would account for interactions to
capture additional complexities in the expansion dynamics, but this was not
essential for validating our phase extraction routine. We perform simulations
on a spatial grid of size 50 $\upmu$m$\times$50 $\upmu$m, with a grid spacing
of 0.42 $\upmu$m, chosen because a single camera pixel in our EMCCD (Andor
Ixon Ultra) represents an extent of 0.42 $\upmu$m in the atomic plane. The
dimensions of the rings are identical to those chosen in our experiment.
Azimuthal phase variations for each ring are simulated using the model
described in the Sec. III for various correlation lengths, which are then
imprinted on the superfluid wavefunctions. After propagating the initial
wavefunctions with imprinted phases using standard Fourier techniques, the
resulting density profile is convolved with an Airy disk of radius 1.5
$\upmu$m to account for the finite resolution of our imaging objective. As
exemplified in Fig. 3, the algorithm substantially recovers the imprinted
phase, with better accuracy for the long wavelength components. Crucially,
phase variations on length scales comparable to the correlation length are
preserved and effectively extracted. The root-mean-square deviations from the
imprinted phase are typically less than 0.3 radians for various correlation
lengths.

The evolution of the matter-wave interference pattern after the atoms are
released from the trap can be quite complex, as can be seen in the images
shown in Fig. 4. There are many factors involved in determining the optimum
expansion time for obtaining information about the superfluid phase from the
concentric rings of fringes appearing in the patterns. This optimum time is
generally expected to occur when the two rings have expanded enough to
interfere substantially over a large region, but not so much that the inner
ring has expanded through the center. For a ring width $d=3$ $\upmu$m, a
radial separation of $\Delta r=5$ $\upmu$m, and a pair mass $m_{p}$ we can
estimate that the optimum ToF will be around $t_{\text{ToF}}=m_{p}d\Delta
r/\hbar\approx$ 1.4 ms, which is close to our chosen ToF of 1.1 ms. The
characteristic radial fringe spacing for the interference of molecular matter-
wave components with relative velocity $\delta v=\Delta r/t_{\text{ToF}}$ is
set by the de Broglie wavelength $\Lambda=h/m_{p}\delta v\approx 7.3$
$\upmu$m. For the conditions of our experiment, the fringe spacing is around 5
$\upmu$m, which is easily resolvable using absorption imaging. Finally, for
the sake of reliable automated phase-extraction, it is important to employ a
ToF such that one observes roughly equal-spaced interference fringes (at fixed
polar angles) such that radial slices of the density profile have a single
prominent sinusoidal component. For the conditions used in the experiment, we
found this to always be true.

##  III Phase fluctuations in a superfluid ring

In this section, we present a simple model for phase fluctuations arising from
thermal excitations of Bogoliubov-Anderson (BA) sound modes for a superfluid
in a ring geometry. Our main goal is to obtain expressions for the auto-
correlation function that explicitly account for the effects of periodic or
hard-wall boundary conditions in our ring-shaped channel. We adopt the same
Bogoliubov-de Gennes (BdG) approach applied to a weakly-interacting BEC in Ref
Mathey  _et al._ (2010), under the assumptions that the hydrodynamic
approximation is still valid for the elongated unitary Fermi gas in our
experiment, the superfluid fraction is not small, and the temperature is high
enough compared to the phonon energy that quantum fluctuations are negligible.
We also assume that the interaction energy is large enough that density
fluctuations are of negligible importance at distances much larger than the
healing length. A more accurate and comprehensive treatment of fluctuations is
possible using the framework of a low-temperature effective field theory for
fermions Klimin  _et al._ (2015), but the simpler BdG approach should suffice
for present purposes. We use it to make predictions about the correlation
function, and to generate simulated experimental data for validation of our
overall phase extraction and analysis procedures.

Under our current experimental conditions, it is clear that the correlation
length is always large compared to the width of the ring-shaped channels, and
it is reasonable to assume that excitation of transverse sound modes is
negligible. In this case a one-dimensional description is valid, and the long-
wavelength BA sound modes along the channel may be treated as independent
Gaussian-distributed variables with a linear phononic dispersion. For a
connected ring-shaped channel of radius $R$, we may expand the phase
$\phi_{\text{con}}(\theta)$ of the pair order parameter in a Fourier series in
the angular coordinate $\theta$

$\displaystyle\phi_{\text{con}}(\theta)$
$\displaystyle=\sqrt{\frac{R}{l_{\phi}}}\sum_{m\neq
0}\frac{\xi_{m}}{|m|}\frac{e^{im\theta}}{\sqrt{2\pi}}+c.c.,$ (1)

where the $\xi_{m}$ are complex-valued Gaussian normal and independently
distributed variables, which satisfy $\langle\xi_{m}\rangle=0$ and
$\langle\xi_{m}^{*}\xi_{m^{\prime}}\rangle=\delta_{mm^{\prime}}$ with the
brackets denoting an ensemble average. The correlation length $l_{\phi}$
describes the length scale over which the correlation function
$g_{1}(\theta)\propto\langle\exp\\{i[\phi(\theta)-\phi(0)]\\}\rangle$ decays.
We explicitly compute the closely-related auto-correlation function later in
this section and show that $l_{\phi}$ indeed represents a correlation length.

This formalism can be adapted to the scenario of the cut ring, keeping in mind
the constraint from the boundary condition that $\bm{\nabla}\phi$ must vanish
at the break in the ring. Physically, this means that there is no superflow
into or out of the break in the ring, which is required by mass conservation.
Therefore, the phonon spatial mode functions in this case are cosinusoidal,
and the phase function for the cut ring can be expressed as

$\displaystyle\phi_{\text{cut}}(\theta)$
$\displaystyle=\sqrt{\frac{R}{l_{\phi}}}\sum_{m>0}\frac{\xi_{m}}{m/2}\frac{\cos(m\theta/2)}{\sqrt{\pi}}+c.c.,$
(2)

where $\theta\in[0,2\pi)$ and there is a break in the ring with zero angular
width at $\theta=0$.

In a narrow torus with harmonic transverse confinement, and under the
condition that the correlation length is larger than the transverse widths of
the toroidal channel, the thermal correlation length of a weakly interacting
molecular BEC is given by Mathey  _et al._ (2010)

$l_{\phi}=\frac{2\hbar^{2}}{m_{p}k_{B}T}\frac{N_{0}}{L},\\\ $ (3)

where $T$ is the temperature, $L$ is the channel length, $m_{p}$ is the
molecule pair mass and $N_{0}$ is the condensed molecule number. Notably, in
this approximation the correlation length is expected to depend on temperature
and the 1D density $n_{0}\equiv N_{0}/L$, but not explicitly on the
interaction strength.

Assuming density fluctuations are negligible, we can write general expressions
for the angular auto-correlation function in terms of a generic Gaussian-
random phase $\phi(\theta)$, defined for a system with either periodic or
closed (hard-walled) boundary conditions. For the periodic case, the angles
$\theta$ and $\theta+2\pi$ represent physically indistinguishable points, and
we define the periodic normalized angular auto-correlation function

$\displaystyle g_{\text{periodic}}[\phi](\theta^{\prime})$
$\displaystyle\equiv\frac{\int_{0}^{2\pi}\langle
e^{i[\phi(\theta)-\phi(\theta+\theta^{\prime})]}\rangle
d\theta}{\int_{0}^{2\pi}d\theta}$ (4)
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-\langle[\delta\phi(\theta,\theta^{\prime})]^{2}\rangle/2}d\theta,$
(5)

where
$\delta\phi(\theta,\theta^{\prime})\equiv\phi(\theta)-\phi(\theta+\theta^{\prime})$
and the form in Eq. (5) is obtained using Wick’s theorem. For the case of
closed boundary conditions, the phase is only defined over the extent of the
channel, and so we define the closed normalized angular auto-correlation
function

$\displaystyle g_{\text{closed}}[\phi](\theta^{\prime})$
$\displaystyle\equiv\frac{\int_{0}^{2\pi-\theta^{\prime}}\langle
e^{i[\phi(\theta)-\phi(\theta+\theta^{\prime})]}\rangle
d\theta}{\int_{0}^{2\pi-\theta^{\prime}}d\theta}$ (6)
$\displaystyle=\frac{1}{2\pi-\theta^{\prime}}\int_{0}^{2\pi-\theta^{\prime}}e^{-\langle[\delta\phi(\theta,\theta^{\prime})]^{2}\rangle/2}d\theta.$
(7)

With this definition $\delta\phi(\theta,\theta^{\prime})$ is unique and well-
defined on the interval $\theta\in[0,2\pi-\theta^{\prime}]$ for
$\theta^{\prime}\in[0,2\pi)$.

Explicit expressions for auto-correlation functions can be calculated for the
connected and cut ring (with a barrier of negligible angular extent) using
Eqs. (1) and (2) for the connected and cut cases, respectively:

$\displaystyle g_{\text{con}}(\theta^{\prime};l_{\phi},R)$
$\displaystyle\equiv
g_{\text{periodic}}[\phi_{\text{con}}](\theta^{\prime})=e^{-|x|(1-|x|)/\tilde{l}_{\phi}}$
(8) $\displaystyle g_{\text{cut}}(\theta^{\prime};l_{\phi},R)$
$\displaystyle\equiv
g_{\text{closed}}[\phi_{\text{cut}}](\theta^{\prime})=e^{-|x|/\tilde{l}_{\phi}},$
(9)

where $x\equiv\theta^{\prime}/2\pi$ and $\tilde{l}_{\phi}\equiv
l_{\phi}/2\pi$R are the scaled angular offset and scaled correlation length
respectively, and the dependence on the correlation length has been made
explicit. While the final expression for the connected ring is the same for
both Eqs. (5) and (7) due to translational invariance, the Eq. (7) is used for
the cut ring case.

Having calculated the correlation functions for a single ring, we now use
these expressions to determine the auto-correlation function of the
experimentally-measurable relative phase
$\phi_{\text{rel}}(\theta)\equiv\phi_{1}(\theta)-\phi_{2}(\theta)$ between two
concentric rings (labelled 1 and 2) of radii $R_{1}$ and $R_{2}$ that may
either be cut or connected. We take, without loss of generality, $R_{1}<R_{2}$
and refer to ring 1 as the inner “reference” ring and ring 2 as the outer
“experiment” ring. Under the assumption of uncorrelated inter-ring phases,
i.e. $\langle\phi_{1}(\theta_{1})\phi_{2}(\theta_{2})\rangle=0$, for two rings
of the same type, this function takes a form similar to the single-ring
expressions, with an apparent scaled correlation length satisfying
$\tilde{l}_{\phi,\text{rel}}^{-1}=\tilde{l}_{\phi,1}^{-1}+\tilde{l}_{\phi,2}^{-1}$.
For the case of a double ring consisting of a connected reference ring and cut
experiment ring, we compute the auto-correlation function using Eq. (7) since
the relative phase is ill-defined at the cut. Due to the independence of the
rings, it is easy to show that the auto-correlation function is simply the
product of the single-ring auto-correlation functions:

$\displaystyle g_{\text{rel}}(\theta^{\prime})$
$\displaystyle=g_{\text{con}}(\theta^{\prime},l_{\phi,1},R_{1})g_{\text{cut}}(\theta^{\prime},l_{\phi,2},R_{2}).$
(10)

Accounting for a finite barrier of angular extent $2\pi f$ located at
$\theta=0$, the expression is modified as follows:

$\displaystyle\begin{split}g_{\text{rel}}(\theta^{\prime})&=\exp\left\\{-\frac{|x|(1-|x|)}{\tilde{l}_{\phi,1}}-\frac{|x|}{(1-f)\tilde{l}_{\phi,2}}\right\\}.\end{split}$
(11)

In our experiment, $2\pi f=0.24$ radians. Furthermore, the domain of the auto-
correlation function is restricted to the angular extent of the cut ring,
i.e., $\theta^{\prime}\in[0,2\pi(1-f))$.

For all of the data taken in this experiment, the average 1D densities of each
sub-ring are not exactly equal, and this must be included in the model to make
a quantitative comparison against the data. In the case where the reference
ring is connected and the experiment ring is cut, the 1D density of the
reference ring, $n_{0,1}$, is 1.2 times that of the experiment ring,
$n_{0,2}$. Eq. (3) then implies that the correlation length of the inner ring
is a factor $\eta=1.2$ times that of the outer ring, with the assumption that
the temperature is uniform throughout the system. With this information, we
express our final fitting function as

$\displaystyle\begin{split}g_{\text{rel}}(\theta^{\prime})&=\exp\left\\{-\frac{1}{\tilde{l}_{\phi,2}}\left(\frac{|x|(1-|x|)R_{1}}{\eta
R_{2}}+\frac{|x|}{1-f}\right)\right\\}.\end{split}$ (12)

We can further substitute the measured scaled correlation lengths into the
expression given in Eq. 3, to obtain a temperature estimate $T$, where
$N_{\text{tot}}$ is the total number of pairs observed in the interferogram:

$\displaystyle T$ $\displaystyle=\frac{2\hbar^{2}}{m_{p}k_{B}(2\pi
R_{2}\tilde{l}_{\phi,2})}\frac{N_{\text{tot}}}{(2\pi\eta R_{1}+2\pi R_{2})}.$
(13)

In the above, we have implicitly assumed that $N_{\text{tot}}$ is the same as
the number of pairs initially in the rings. We have confirmed this to be true
to within approximately $10\%$ in our experiments by comparing the average in-
situ pair numbers to those observed in the interferograms. This assumption is
important to be able to associate a definite pair number to a particular
realization of an observed phase profile without precise knowledge of the in-
situ state. Only the section of the interferograms containing the fringes,
which is a circle of radius 21 $\upmu$m, is considered for these quantitative
comparisons.

##  IV Results

Figure 5: Standard deviation of phase difference across an angular region of
total width 0.78 radians for an outer ring with (green, ‘Cut’) or without
(blue, ‘Connected’), the cutting barrier. The inter-ring barrier is set to
100$\%$ for all these experimental runs. We observe a dramatic increase in the
variance for pair numbers below 2500. Runs with a non-zero overall winding
number for the connected ring have been excluded here. The error bars
represent the standard error of the sample standard deviation shown here std .

The first investigation we performed was to measure the difference between the
relative phase of the rings at two points separated by an angular region of
width 0.78 radians, located symmetrically on either side of the cutting
barrier, using the procedure described in section II.1. Figure 5 shows a
comparison of the results with a cutting barrier in the outer ring versus the
case where both rings were smooth and continuous, as the number of pairs in
both rings is varied. The mean of the measured phase difference for both the
cases remains close to zero, for all pair numbers. An important observation is
that for the case where no cutting barrier is present, the standard deviation
of the phase difference across the test region is consistently measured to be
much less than $\pi$, even as the number of pairs is reduced to less than
1500. While our phase measurement procedure can become unreliable when the
signal to noise ratio in the absorption images falls too low, this consistent
result for the connected ring indicates that it is still sufficiently reliable
throughout our experimental conditions.

In both cases, the standard deviation of phase difference across the cutting
barrier increases as the number of pairs in the system is reduced, with the
fluctuations in the cut ring significantly exceeding those of the connected
ring. The most interesting feature of Fig. 5 is the dramatic increase in
fluctuations when the number of pairs falls below approximately 2500. This
increased likelihood of observing a phase difference on the order of $\pi$
points to the onset of the phase-fluctuating regime in the cut ring. The
impact of these fluctuations is still evident in the data for the connected
ring case, however, because our experimental procedure for creating a
connected ring involves first evaporating in a cut (and potentially
fluctuating) ring, and reconnecting it before release. We found that the
probability for observing a non-zero winding number in a (re)connected outer
ring under these same conditions was $0.16^{+0.28}_{-0.01}$ for pair numbers
below 2500, and $0.00^{+0.01}_{-0.00}$ above that number. Taken together,
these results strongly suggest that phase fluctuations are responsible for
difficulties in reliably initializing the system in a selected current state
in the low-density limit.

We can obtain a more comprehensive picture of the effect of phase fluctuations
at different length scales by calculating the angular auto-correlation of the
relative phase of the rings for an individual realization as defined by Eq. 7,
then averaging over many realizations. Fig. 6 shows an example set of data
obtained for the case of a cut outer ring, as the total number of pairs in the
system is varied. As we decrease the pair number, we see a clear reduction in
the coherence and the correlation length, as predicted by Eq. 3. We fit these
auto-correlations to Eq. 12 to extract the scaled correlation length, which we
plot in Fig. 8. A section of the phase profile around the barrier, with a
total angular width of 0.78 radians, is excluded in the computation since we
do not expect the in-situ phase to be well-defined for the cut ring in that
region. We see that the measured auto-correlations are well fit by the model
over most of the angular domain. At angular offsets on the order of $\pi$
radians, we see deviation from our fit function due to the declining
reliability of the auto-correlation owing to the limited overlap between the
two copies of the phase profile that are being correlated. Another notable
feature of Fig. 6 is the “plateau” observed for the lower pair number cases at
offsets larger than $\pi/2$. We suspect this plateau is an effect of coupling
both between the rings and between the endpoints of the cut ring, but further
investigation will be needed to confirm this conjecture.

Figure 6: Change in the auto-correlation of the phase difference between a
connected inner ring and a cut outer ring, as the total number of pairs in
both rings is varied. The curves are averaged results for approximately 60
experimental runs each. The average pair number for each bin is indicated in
the legend, and the bin width of 1000 pairs is similar to the shot-to-shot
number variation for the experiment. Data for another bin centered at 1000
pairs follows the overall trend, but have been omitted for visual clarity. The
error bars on each curve are the standard error of the mean. The curves are
fit using Eq. 12, and the measured scaled correlation lengths are given in
Fig. 8. The x-axis is curtailed at roughly half the domain of the auto-
correlation function, where the data is not reliable due to limited overlap
between the two copies of the phase profile that are being correlated. Figure
7: Change in auto-correlation of the measured phase difference between a
connected inner ring, and a cut outer ring, as the strength of the inter-ring
barrier is reduced from its maximal value to the percentage indicated in the
legend, for comparable values of total pair number. The figure shows
increasing coherence as the inter-ring barrier is successively weakened. The
data for the 90$\%$,70$\%$ and 50$\%$ cases follow this overall trend, but
have been omitted for visual clarity. The curves are the averages of results
from 15,69 and 63 experimental runs respectively, with the total number of
pairs for each run falling within a bin width of 1000, centered around 2000.
The error bars are the standard error of the mean of the observed statistical
variations. The x-axis terminates at roughly half the domain of the auto-
correlation function, similar to Fig. 6.

Since the 6Li pairs are confined with an attractive red-detuned potential, the
inter-ring region can become populated by paired or unpaired atoms if the
chemical potential exceeds the ring trap depth. This in turn results in a non-
negligible amount of coupling between the two rings. We observe the effect of
this inter-ring coupling on the phase fluctuations, by repeating our
experiment with the inter-ring barrier weakened by successively larger
amounts. Fig. 7 shows the auto-correlation where the total number of pairs in
the system is held fixed at approximately 2000, but the inter-ring barrier is
incrementally weakened by a factor of up to 50%. This pair number corresponds
to the regime where the density in the inter-ring region is effectively zero
for the 100$\%$ barrier case, making it ideal for comparing the effects of the
inter-ring coupling. Our expectation is supported by the observation of
increasing coherence as the inter-ring barrier is weakened. This implies that
for higher pair numbers, where inter-ring coupling is more pronounced, even
for the 100$\%$ barrier, our correlation length measurement exceeds any
prediction based on the assumption of independent rings. It is difficult for
us to separate the effects of increasing pair density and inter-ring coupling
at present, and a more sophisticated model is required to understand their
individual contributions.

The auto-correlations can be fitted to the explicit expression Eq. 12 for
interference between a connected and a cut ring, allowing us to measure the
correlation length for the outer ring. The scaled correlation lengths for the
outer ring for various configurations are plotted against the number of pairs
in Fig. 8. For similar pair numbers, the apparent correlation length is
significantly higher for a weaker inter-ring barrier. As the correlation
length begins to exceed the circumference of the rings, the precision of our
estimation of the correlation length reduces as superfluid coherence extends
around the ring and a true condensate is realized.

Figure 8: Measured scaled correlation lengths $\tilde{l}_{\phi,2}$ for the
outer ring, for various barrier strengths and pair numbers. The data for the
90$\%$, 70$\%$ and 50$\%$ cases follow the overall trend, but have been
omitted for visual clarity. The error bars are 1-$\sigma$ confidence intervals
obtained from the fit covariance matrix.

One aspect of this work that may be of practical significance in the future is
the possibility of estimating the temperature of the system from measurements
of the phase fluctuations. The model outlined in section III is only a valid
description for a single ring, and we would expect it to be most accurate in
describing our experiment in the limit of low numbers and maximal height of
the inter-ring barrier. Using Eq. (13), the value of the best-fit temperature
was 52(2) nK, yielding $T/T_{c}\approx 0.9$ given an estimate of the unitary
superfluid critical temperature $T_{c}\equiv 0.167T_{F}=60(5)$ nK Ku  _et al._
(2012). We are encouraged by the result still being below the critical
temperature, in spite of the fact that noise and other effects would tend to
result in a higher temperature estimate. While we do not presently have a
reliable way of independently measuring the temperature to confirm this
result, this will be possible in future experiments that combine support for
using other thermometry techniques with the capability of controllably
increasing the system temperature by a known amount.

##  V Conclusions

We have performed the first systematic analysis of phase fluctuations in a
superfluid under periodic boundary conditions, doing so under conditions where
the correlation length is similar to or smaller than the system size.
Employing a Fourier analysis-based algorithm, we successfully extract
information about the azimuthally varying in-situ phase of a superfluid in a
ring potential. This was achieved through matter-wave interferometry with a
reference ring and analyzing the interference pattern after time-of-flight
expansion. Increasing the pair density and inter-ring coupling increased the
coherence in the ring, as determined from the auto-correlation of the angular
relative phase between the two rings. By measuring the effective coherence
length for the outer ring under varying conditions, we observed a transition
where the correlation length surpasses the system size. We also found our
procedure for initializing a zero-current state in a superfluid ring becomes
more successful as the magnitude of phase fluctuations decreases.

Using a simple model for phase fluctuations in independent rings, we obtained
a temperature estimate of 52(2) nK for the system in the limit of minimal
inter-ring coupling, which is reasonable in comparison to the estimated
critical temperature of 60(5) nK. A more precise analysis of the temperature
dependence of the phase fluctuations could provide a useful means of
thermometry for elongated superfluid systems. This work also raises the
possibility of investigating the effect of inter-ring coupling, with a
potentially unique opportunity to study Josephson vortex dynamics.
Alternatively, the influence of inter-ring coupling could be entirely removed
by employing a blue-detuned repulsive trap. In future experiments it should be
possible to realize even narrower rings with lower atom numbers and reach the
quasi-1D limit. In this regime thermal and even quantum phase fluctuations can
become significant, and their measurements may provide valuable insight into
the behavior of low-dimensional quantum systems.

##  VI Acknowledgments

We thank Roberto Onofrio and Danelle Akanova for helpful discussions and
careful reading of the manuscript. This work was supported by the National
Science Foundation (Grant No. PHY-2046097).

## References

  * Petrov  _et al._ (2001) D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 87, 050404 (2001).
  * Mora and Castin (2003) C. Mora and Y. Castin, Phys. Rev. A 67, 053615 (2003).
  * Raychaudhuri and Dutta (2021) P. Raychaudhuri and S. Dutta, Journal of Physics: Condensed Matter 34, 083001 (2021).
  * Emery and Kivelson (1995) V. J. Emery and S. A. Kivelson, Nature 374, 434 (1995).
  * Dettmer  _et al._ (2001) S. Dettmer, D. Hellweg, P. Ryytty, J. J. Arlt, W. Ertmer, K. Sengstock, D. S. Petrov, G. V. Shlyapnikov, H. Kreutzmann, L. Santos, and M. Lewenstein, Phys. Rev. Lett. 87, 160406 (2001).
  * Hellweg  _et al._ (2001) D. Hellweg, S. Dettmer, P. Ryytty, J. Arlt, W. Ertmer, K. Sengstock, D. Petrov, G. Shlyapnikov, H. Kreutzmann, L. Santos, and M. Lewenstein, Applied Physics B 73, 781–789 (2001).
  * Choi  _et al._ (2012) J. Choi, S. W. Seo, W. J. Kwon, and Y. Shin, Phys. Rev. Lett. 109, 125301 (2012).
  * Seo  _et al._ (2014) S. W. Seo, J. Choi, and Y. Shin, Phys. Rev. A 89, 043606 (2014).
  * Shah  _et al._ (2023) R. Shah, T. J. Barrett, A. Colcelli, F. Oručević, A. Trombettoni, and P. Krüger, Phys. Rev. Lett. 130, 123401 (2023).
  * Hellweg  _et al._ (2003) D. Hellweg, L. Cacciapuoti, M. Kottke, T. Schulte, K. Sengstock, W. Ertmer, and J. J. Arlt, Phys. Rev. Lett. 91, 010406 (2003).
  * Richard  _et al._ (2003) S. Richard, F. Gerbier, J. H. Thywissen, M. Hugbart, P. Bouyer, and A. Aspect, Phys. Rev. Lett. 91, 010405 (2003).
  * Cacciapuoti  _et al._ (2003) L. Cacciapuoti, D. Hellweg, M. Kottke, T. Schulte, W. Ertmer, J. J. Arlt, K. Sengstock, L. Santos, and M. Lewenstein, Phys. Rev. A 68, 053612 (2003).
  * Hugbart  _et al._ (2005) M. Hugbart, J. A. Retter, F. Gerbier, A. F. Varón, S. Richard, J. H. Thywissen, D. Clément, P. Bouyer, and A. Aspect, The European Physical Journal D 35, 155–163 (2005).
  * Shin  _et al._ (2004) Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, Phys. Rev. Lett. 92, 050405 (2004).
  * Gati  _et al._ (2006) R. Gati, J. Esteve, B. Hemmerling, T. B. Ottenstein, J. Appmeier, A. Weller, and M. K. Oberthaler, New Journal of Physics 8, 189 (2006).
  * Jo  _et al._ (2007) G.-B. Jo, J.-H. Choi, C. A. Christensen, Y.-R. Lee, T. A. Pasquini, W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 99, 240406 (2007).
  * Scott  _et al._ (2008) R. G. Scott, T. E. Judd, and T. M. Fromhold, Phys. Rev. Lett. 100, 100402 (2008).
  * Scott  _et al._ (2009) R. G. Scott, D. A. W. Hutchinson, T. E. Judd, and T. M. Fromhold, Phys. Rev. A 79, 063624 (2009).
  * Mathey  _et al._ (2010) L. Mathey, A. Ramanathan, K. C. Wright, S. R. Muniz, W. D. Phillips, and C. W. Clark, Phys. Rev. A 82, 033607 (2010).
  * Eckel  _et al._ (2014) S. Eckel, F. Jendrzejewski, A. Kumar, C. J. Lobb, and G. K. Campbell, Phys. Rev. X 4, 031052 (2014).
  * Corman  _et al._ (2014) L. Corman, L. Chomaz, T. Bienaimé, R. Desbuquois, C. Weitenberg, S. Nascimbène, J. Dalibard, and J. Beugnon, Phys. Rev. Lett. 113, 135302 (2014).
  * Cai  _et al._ (2022) Y. Cai, D. G. Allman, P. Sabharwal, and K. C. Wright, Phys. Rev. Lett. 128, 150401 (2022).
  * Del Pace  _et al._ (2022) G. Del Pace, K. Xhani, A. Muzi Falconi, M. Fedrizzi, N. Grani, D. Hernandez Rajkov, M. Inguscio, F. Scazza, W. J. Kwon, and G. Roati, Phys. Rev. X 12, 041037 (2022).
  * Allman  _et al._ (2024) D. G. Allman, P. Sabharwal, and K. C. Wright, Phys. Rev. A 109, 053320 (2024).
  * Allman  _et al._ (2023) D. G. Allman, P. Sabharwal, and K. C. Wright, Phys. Rev. A 107, 043322 (2023).
  * (26) The Fermi energy of a trapped unitary Fermi gas can be estimated as $E_{F}=\beta^{-1/4}\hbar\omega_{z}\sqrt{2\pi a_{z}^{2}n_{2}}$, with $\beta\approx 0.38$ the Bertsch parameter, $\omega_{z}$ the axial trap frequency, $a_{z}^{2}\equiv\hbar/M\omega_{z}$ and $n_{2}$ the peak column density.
  * Aidelsburger  _et al._ (2017) M. Aidelsburger, J. L. Ville, R. Saint-Jalm, S. Nascimbène, J. Dalibard, and J. Beugnon, Phys. Rev. Lett. 119, 190403 (2017).
  * Regal  _et al._ (2004) C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004).
  * Klimin  _et al._ (2015) S. N. Klimin, J. Tempere, G. Lombardi, and J. T. Devreese, The European Physical Journal B 88, 1 (2015).
  * (30) The standard error of the sample standard deviation approaches 1/$\sqrt{2(n-1)}$ for large $n$, and is a reasonable approximation for $n>10$.
  * Ku  _et al._ (2012) M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Science 335, 563–567 (2012).