$\displaystyle\underline{v}_{3}(2)=[0.3158,0.1072,0.019,0.558]$

which yields the following comparison for the incurred estimation errors:

$\displaystyle
0.4342=\sum_{i=1}^{3}||\underline{u}_{i}(2)-\underline{v}_{i}(2)||<\sum_{i=1}^{3}||\underline{u}^{\prime}_{i}(2)-\underline{v}_{i}(2)||=0.5018.$

The left hand side shows the estimation error under our proposed backward-
forward update and the right hand side shows the estimation error under the
naive approach.

##  Appendix I Delay of Testing Results

One can extend the framework to a more realistic case where testing results
are not able to be obtained on the same day, but will be obtained after a
delay $a$. In other words, if nodes are tested on day $t-a$, the test results
are provided on day $t$. The extended framework is summarized as follows.

On day $t$, before getting the test results of day $t-a$, the algorithm knows
the following information: (i) the network topology from day $t-a-1$ to day
$t$, i.e., $\mathcal{G}(t-a-1),\cdots,\mathcal{G}(t)$ (it is affected by the
past actions); (ii) the posterior probability of nodes on day $t-a-1$,
$\big{\\{}\underline{w}_{i}(t-a-1)\big{\\}}_{i\in\mathcal{G}(t-a-1)}$; and
(iii) and the prior probability vectors of nodes from day $t-a$ to day $t$,
i.e.,
$\big{\\{}\underline{u}_{i}(t-a)\big{\\}}_{i\in\mathcal{G}(t-a)},\cdots,\big{\\{}\underline{u}_{i}(t)\big{\\}}_{i\in\mathcal{G}(t)}$.

After getting the test results on day $t-a$, we can obtain the updated
posterior probability vectors on day $t-a-1$, and the posterior probability
vectors on day $t-a$, i.e.,
$\big{\\{}\underline{e}_{i}(t-a-1)\big{\\}}_{i\in\mathcal{G}(t-a-1)}$, and
$\big{\\{}\underline{w}_{i}(t-a)\big{\\}}_{i\in\mathcal{G}(t-a)}$, by Step 1
in Algorithm 4.

Based on $\\{\underline{w}(t-a)\\}_{i\in\mathcal{G}(t-a)}$, by Step 2 in
Algorithm 4, we update the prior probability from day $t-a+1$ to day $t$, and
obtain the prior probability on day $t+1$, i.e.,
$\\{\underline{u}_{i}(t+1)\\}_{i\in\mathcal{G}(t+1)}$.

Repeating the process, we can compute the estimated probability vectors of
nodes and apply the exploration and exploitation policies.

##  Appendix J Proof of Theorem 2

Step 1: Preliminaries.

We divide the distributions of initial infectious nodes into two complementary
events:

$\displaystyle\mathcal{I}_{1}=$ $\displaystyle\\{\text{No node is
infectious}\\}$ $\displaystyle\mathcal{I}_{2}=$
$\displaystyle\mathcal{I}_{1}^{c}.$

Let $N$ be sufficiently large,

$\displaystyle\Pr\\{\mathcal{I}_{1}\\}=(1-1/N)^{N}\approx 1/e$
$\displaystyle\Pr\\{\mathcal{I}_{2}\\}\approx 1-1/e.$

In event $\mathcal{I}_{1}$, since there is no infection on the initial day,
then no node is infectious in the future, i.e., the true probability of nodes
$v_{I}^{(i)}(t)=0$ for all $i\in\mathcal{V}(t)$ and $t\geq 1$.

Note that in Example 1, each node can be in one of two states, $S$ and $I$.
The transmission probability $\beta=1$. So, on day $t$, the probability of
node $i$ being in state $I$ includes the infection of node $i$ on day $t-1$,
and the infection from its neighbors. Then, based on (15), we have

$\displaystyle u_{I}^{(i)}(t)=$ $\displaystyle
w_{I}^{(i)}(t-1)+\\{1-w_{I}^{(i)}(t-1)\\}\Big{\\{}1-\big{(}1-w_{I}^{(i-1)}(t-1)\big{)}\big{(}1-w_{I}^{(i+1)}(t-1)\big{)}\Big{\\}}$
(49) $\displaystyle=$ $\displaystyle
1-\\{1-w_{I}^{(i-1)}(t-1)\\}\\{1-w_{I}^{(i)}(t-1)\\}\\{1-w_{I}^{(i+1)}(t-1)\\}.$

For convention, we assume that nodes $0$ and $N+1$ are two virtual nodes with
no probability of infection, i.e., $u_{I}^{(0)}(t)=u_{I}^{(N+1)}(t)=0$ for all
$t$, and no tests are applied to these two nodes all the time.

Since $w_{I}^{(i)}(t+1),w_{I}^{(i-1)}(t+1)$, $w_{I}^{(i+1)}(t+1)\in[0,1]$,
then from (49),

$\displaystyle u_{I}^{(i)}(t)\geq 1-1\times(1-w_{I}^{(i)}(t-1))\times
1=w_{I}^{(i)}(t-1).$ (50)

Thus, by symmetry over $w_{I}^{(i-1)}(t-1)$, $w_{I}^{(i)}(t-1)$, and
$w_{I}^{(i+1)}(t-1)$ we get the inequality

$\displaystyle
u_{I}^{(i)}(t)\geq\max\\{w_{I}^{(i-1)}(t-1),w_{I}^{(i)}(t-1),w_{I}^{(i+1)}(t-1)\\}.$
(51)

Step 2: Consider the computation of $\\{\underline{u}_{i}(t)\\}_{i}$ based on
(15) (equivalently (49)) under event $\mathcal{I}_{1}$.

Recall that $B(t)=1$ for all $t$. On any day $t$, if node $i_{0}$ is tested,
then the result is negative, and $w_{I}^{(i_{0})}(t)=0$, and

$\displaystyle w_{I}^{(i)}(t)=u_{I}^{(i)}(t)\,\,\text{for all}\,\,i\neq
i_{0}.$ (52)

In (51), at most one of $w_{I}^{(i-1)}(t-1)$, $w_{I}^{(i)}(t-1)$, and
$w_{I}^{(i+1)}(t-1)$ is updated to $0$. We first prove the following facts.

Fact 1. $u_{I}^{(i)}(t)\geq\frac{1}{N}$ for all $t$. On any day $t$,
$w_{I}^{(i)}(t)\geq\frac{1}{N}$ with $i\neq i_{0}$, where $i_{0}$ is the index
of node tested on day $t$.

Proof. We prove Fact 1 by mathematical induction. On the initial day, by model
assumption in Example 1, $u_{I}^{(i)}(0)=\frac{1}{N}$ for all $i$. Then, if
node $i_{0}$ is tested, then as mentioned above, $w_{I}^{(i_{0})}(0)=0$, and
by (52) $w_{I}^{(i)}(0)=u_{I}^{(i)}(0)=1/N$ for all $i\neq i_{0}$.

Suppose Fact 1 holds for all $\tau\leq t-1$. Now, we consider $\tau=t$. From
(51), we have
$u_{I}^{(i)}(t)\geq\max\\{w_{I}^{(i-1)}(t-1),w_{I}^{(i)}(t-1),w_{I}^{(i+1)}(t-1)\\}\geq
1/N$. Then, if node $i_{0}$ is tested, we have $w_{I}^{(i_{0})}(t)=0$, and
then by (52), $w_{I}^{(i)}(t)=u_{I}^{(i)}(t)\geq 1/N$ for all $i\neq i_{0}$. ∎

Fact 2. If node $i$ has not been tested up to day $t$, then $u_{I}^{(i)}(t)$
tends to $1$ as $t\rightarrow\infty$.

Proof. Since node $i$ is not tested from the initial day to day $t$, then

$\displaystyle w_{I}^{(i)}(\tau)=u_{I}^{(i)}(\tau),\,\,\tau\leq t.$ (53)

Note that at most one of its neighbors is tested on day $t$. By (49) and Fact
1,

$\displaystyle u_{I}^{(i)}(t)\geq
1-(1-1/N)(1-w_{I}^{(i)}(t-1))=1-(1-1/N)(1-u_{I}^{(i)}(t-1)),,$

which implies

$\displaystyle(1-1/N)(1-u_{I}^{(i)}(t-1))\geq 1-u_{I}^{(i)}(t),$

which implies

$\displaystyle
1-u_{I}^{(i)}(t)\leq(1-1/N)^{t}(1-u_{I}^{(i)}(0))=(1-1/N)^{t+1}.$

Letting $t\to\infty$ completes the proof.∎

Fact 3. If node $i$ is not tested on day $t-1$, then

$\displaystyle u_{I}^{(i)}(t)\geq
w_{I}^{(i)}(t-1)+\frac{1}{N}(1-w_{I}^{(i)}(t-1))w_{I}^{(i)}(t-1).$

Proof. By Fact 3, if node $i$ is not tested on day $t-1$, then
$w_{I}^{(i)}(t-1)>0$. From (49), by some algebra,

$\displaystyle u_{I}^{(i)}(t)=$ $\displaystyle
w_{I}^{(i)}(t-1)+(1-w_{I}^{(i)}(t-1))(w_{I}^{(i-1)}(t-1)+w_{I}^{(i+1)}(t-1)-w_{I}^{(i-1)}(t-1)w_{I}^{(i+1)}(t-1))$
$\displaystyle=$ $\displaystyle(1+\epsilon)w_{I}^{(i)}(t-1)$

where

$\displaystyle\epsilon=\frac{1-w_{I}^{(i)}(t-1)}{w_{I}^{(i)}(t-1)}\times(w_{I}^{(i-1)}(t-1)+w_{I}^{(i+1)}(t-1)-w_{I}^{(i-1)}(t-1)w_{I}^{(i+1)}(t-1)).$

Note that at most one of the neighbors of node $i$ is tested on day $t-1$,
then

$\displaystyle\frac{1-w_{I}^{(i)}(t-1)}{w_{I}^{(i)}(t-1)}\geq$ $\displaystyle
1-w_{I}^{(i)}(t-1)$ $\displaystyle
w_{I}^{(i-1)}(t-1)+w_{I}^{(i+1)}(t-1)-w_{I}^{(i-1)}(t-1)w_{I}^{(i+1)}(t-1)\geq$
$\displaystyle\max\\{w_{I}^{(i-1)}(t-1),w_{I}^{(i+1)}(t-1)\\}.$

From Fact 1, $\max\\{w_{I}^{(i-1)}(t-1),w_{I}^{(i+1)}(t-1)\\}\geq 1/N$. Thus,
$\epsilon\geq(1-w_{I}^{(i)}(t-1))\times 1/N$. Hence, $u_{I}^{(i)}(t)\geq
w_{I}^{(i)}(t-1)+\frac{1}{N}(1-w_{I}^{(i)}(t-1))w_{I}^{(i)}(t-1)$.∎

Since we consider all possible sequential testing policies, then we divide all
nodes into two sets

$\displaystyle\mathcal{S}_{1}(t)=$ $\displaystyle\\{\text{nodes that have not
been tested up to day $t$}\\}$ $\displaystyle\mathcal{S}_{2}(t)=$
$\displaystyle\mathcal{S}_{1}^{c}(t).$

In the following proof, let $t\to\infty$. By Fact 2, $u_{I}^{(i)}(t)\to 1$ if
$i\in\mathcal{S}_{1}(t)$. Next, we focus on the set $\mathcal{S}_{2}(t)$.
Denote the index of node which is tested on day $t-1$ as $i_{0}(t)$. By Fact
1, $w_{I}^{(i)}(t-1)\geq 1/N$ for all $i\neq i_{0}(t)$. Then, we define

$\displaystyle\mathcal{S}_{21}(t)=$ $\displaystyle\\{i|1/N\leq
w_{I}^{(i)}(t-1)<1-1/N\\}$ $\displaystyle\mathcal{S}_{22}(t)=$
$\displaystyle\\{i|1-1/N\leq w_{I}^{(i)}(t-1)\\}.$

Thus, we have
$\mathcal{S}_{2}(t)=\mathcal{S}_{21}(t)\cup\mathcal{S}_{22}(t)\cup\\{i_{0}(t)\\}$.
Due to the equivalence of norms, without loss of generality, we consider
$L_{1}$ norm in the rest of the proof.

  * (i)

If $i\in\mathcal{S}_{1}(t)$, then $u_{I}^{(i)}(t)\to 1$. Thus
$||\underline{u}_{i}(t)-\underline{v}_{i}(t)||_{1}\to 2$.

  * (ii)

If $i\in\mathcal{S}_{21}(t)$, then
$||\underline{u}_{i}(t)-\underline{v}_{i}(t)||_{1}\geq||\underline{u}_{i}(t-1)-\underline{v}_{i}(t-1)||_{1}+\frac{2(N-1)}{N^{3}}$.
In fact, since $i\in\mathcal{S}_{21}(t)$, then $i\neq i_{0}(t)$, thus by (52)
and Fact 3,

$\displaystyle u_{I}^{(i)}(t)\geq
u_{I}^{(i)}(t-1)+\frac{1}{N}(1-w_{I}^{(i)}(t-1))w_{I}^{(i)}(t-1).$

Note that $N$ is sufficiently large, so $1/N<1/2<1-1/N$. If $x\in[1/N,1-1/N)$,
then the fuction $f(x)=x(1-x)$ has the minimum value $\frac{N-1}{N^{2}}$ when
$x=1/N$. Thus,

$\displaystyle u_{I}^{(i)}(t)\geq u_{I}^{(i)}(t-1)+\frac{N-1}{N^{3}}.$ (54)

Recall that $v_{I}^{(i)}(t)=0$ and $v_{S}^{(i)}(t)=1$ for all $t$, and
$u_{I}^{(i)}(t)+u_{S}^{(i)}(t)=1$, then

$\displaystyle||\underline{u}_{i}(t)-\underline{v}_{i}(t)||_{1}=|u_{I}^{(i)}(t)-v_{I}^{(i)}(t)|+|u_{S}^{(i)}(t)-v_{S}^{(i)}(t)|=2|u_{I}^{(i)}(t)-v_{I}^{(i)}(t)|.$
(55)

From (54),

$\displaystyle||\underline{u}_{i}(t)-\underline{v}_{i}(t)||_{1}=2|u_{I}^{(i)}(t)-v_{I}^{(i)}(t)|\geq
2|u_{I}^{(i)}(t-1)+\frac{N-1}{N^{3}}-v_{I}^{(i)}(t-1)|$ $\displaystyle\geq
2|u_{I}^{(i)}(t-1)-v_{I}^{(i)}(t-1)|+\frac{2(N-1)}{N^{3}}=||\underline{u}_{i}(t-1)-\underline{v}_{i}(t-1)||_{1}+\frac{2(N-1)}{N^{3}}.$

  * (iii)

If $i\in\mathcal{S}_{22}(t)$, then node $i$ is not tested on day $t$, thus
from (51), $u_{I}^{(i)}(t)\geq w_{I}^{(i)}(t-1)=1-1/N$. Thus, by (55),
$||\underline{u}_{i}(t)-\underline{v}_{i}(t)||_{1}\geq\frac{2(N-1)}{N}$.

Since we consider $N$ sufficiently large, then we can prove the following
lemma.

######  Lemma 4.

$\lim_{t\to\infty}\mathcal{S}_{21}(t)=\varnothing$.

Proof. We first prove the following Claims.

Claim 1. If (i) $u_{I}^{(i-1)}(t)\geq 1-1/N$ and node $i-1$ is not tested on
day $t$, or (ii) $u_{I}^{(i+1)}(t)\geq 1-1/N$ and node $i+1$ is not tested on
day $t$, or (iii) $u_{I}^{(i-1)}(t)\geq 1-1/N$ and $u_{I}^{(i+1)}(t)\geq
1-1/N$, then $u_{I}^{(i)}(t+1)\geq 1-1/N$.

Proof. By (49) and (52), we can derive $u_{I}^{(i)}(t+1)\geq 1-1/N$ directly.∎

Claim 2. No node can stay in $\mathcal{S}_{21}(t)$ for successive
$\left\lceil{N^{3}/(N-1)}\right\rceil$ days.

Proof. if node $i$ stays in $\mathcal{S}_{21}(t)$ for successive
$\left\lceil{N^{3}/(N-1)}\right\rceil$ days, i.e., from day $\tau$ to day
$\tau+\left\lceil{N^{3}/(N-1)}\right\rceil$, then by (54),
$u_{I}^{(i)}(\tau+\left\lceil{N^{3}/(N-1)}\right\rceil)>1$, which contradicts
with $u_{I}^{(i)}(t)\leq 1$ for all $t$.∎

Now, we prove the lemma by contradiction. Based on Claim 2, assume there
exists at least one $j$ and an increasing sequence
$\\{t_{i}\\}_{i=0}^{\infty}$ with $\lim_{n\to\infty}t_{n}=\infty$, such that
$j\in\mathcal{S}_{21}(t_{i})$ for all $\\{t_{i}\\}_{i=0}^{\infty}$.

For some $i$, node $j$ is in $\mathcal{S}_{22}(t_{i}-1)$ on day $t_{i}-1$, and
node $j$ is in $\mathcal{S}_{21}(t_{i})$ on day $t_{i}$. In other words,
$u_{I}^{(j)}(t_{i})<1-1/N\leq u_{I}^{(j)}(t_{i}-1)$. From (49) and Calim 1,
$u_{I}^{(j)}(t_{i})<1-1/N\leq u_{I}^{(j)}(t_{i}-1)$ holds only because node
$j$ is tested on day $t_{i}-1$, and all of its neighbors (i.e., nodes $j-1$,
$j+1$) have $u_{I}^{(j-1)}(t_{i}-1)<1-1/N$ and $u_{I}^{(j+1)}(t_{i}-1)<1-1/N$.
However, since $u_{I}^{(j)}(t_{i}-1)\geq 1-1/N$ and node $j$ is tested on day
$t_{i}-1$, then by Claim 1, $u_{I}^{(j-1)}(t_{i})\geq 1-1/N$ and
$u_{I}^{(j+1)}(t_{i})\geq 1-1/N$. Subsequently, by Claim 1, we have
$u_{I}^{(j)}(t_{i}+1)\geq 1-1/N$. Thus, on day $t_{i}+1$, at least one of its
neighbors, say $j-1$, has $u_{I}^{(j-1)}(t_{i}+1)\geq 1-1/N$. By Claim 1, node
$j$ never fall into $\mathcal{S}_{21}(t)$ for
$t\in\\{t_{i+1},t_{i+2},\cdots\\}$, which contradicts with the assumption.∎

From Lemma 4, when $t\to\infty$, we have $|\mathcal{S}_{1}(t)|=\Theta(N)$ or
$|\mathcal{S}_{22}(t)|=\Theta(N)$. Thus,
$\sum_{i=1}^{N}||\underline{u}_{i}(t)-\underline{v}_{i}(t)||_{1}=\Theta(N)$.

Step 3: Consider the computation of $\\{\underline{u}_{i}(t)\\}_{i}$ based on
Algorithm 4.

In this step, we consider a specific testing policy: We test node $i$ on day
$k$, where $k\equiv i-1(mod\,\,M)$ for all $1\leq i\leq M$.

In event $\mathcal{I}_{2}$, since the transmission probability $\beta=1$, then
all nodes are infected at most $N$ days because there is no recovery. Thus, no
node with positive testing result is repeatedly tested. So in at most $2N$
days, all nodes are infectious, and the algorithm finds all infected nodes, so
$\underline{u}_{i}(t)=\underline{v}_{i}(t)$, $t\geq 2N$.

In event $\mathcal{I}_{1}$, whenever a node is tested, it is negative. Node
$1$ is tested on day $0$, the result is negative. On day $1$, node $2$ is
tested and the result is negative. By backward updating, since $\beta=1$ and
no recovery, then nodes $1\&3$ are inferred to be in state $S$ on day $0$.
Since node $2$ is in state $S$ on day $1$. Then, node $1$ is inferred in state
$S$ on days $0$ and $1$.

Assume that nodes $1,2,\cdots,k-2$ are inferred to be in state $S$ by day
$k-1$. Now, we day $k$, where $k\leq N$. On day $k$, node $k-1$ is tested
negative, hence by backward updating, nodes $k-2$ and $k$ are inferred to be
in state $S$ on day $k-1$. By the testing result of node $k-1$ on day $k$,
nodes $1,2,\cdots,k-1$ are inferred in state $S$ by day $k$. By induction,
after $N$ days, it clears every node, so
$\underline{u}_{i}(t)=\underline{v}_{i}(t)$, $t\geq N$.

From Steps 1$\sim$3, we complete the proof.

##  Appendix K $\alpha$-linking Backward Updating

###  K-A Complexity Reduction

Let $\\{x_{j}\\}_{j\in\mathcal{O}(t)}$ be a realization of
$\\{\theta_{j}(t)\\}_{j\in\mathcal{O}(t)}$ and
$\\{y_{l}\\}_{l\in\Theta_{i}(t)}$ be a realization of
$\\{\zeta_{l}(t-1)\\}_{l\in\Theta_{i}(t)}$. Let node $i$ have state $x$ in day
$t-1$. Consider one node $k\in\partial^{+}_{j}(t-1)\backslash\\{i\\}$ and the
probability

$\Pr\big{(}x_{j}|\\{y_{l}\\}_{l\in\partial^{+}_{j}(t-1)\backslash\\{i\\}},x\big{)},j\in\Psi_{i}(t).$

Since node $k$ is not infectious if $y_{k}=L$, $y_{k}=R$ or $y_{k}=S$, then
the probability above remians the same no matter whether $y_{k}=L$, $y_{k}=R$
or $y_{k}=S$.

Thus, we introduce a new state, denoted by $E$, to be a replacement of
$\\{L,R,S\\}$, and

$\displaystyle\Pr\big{(}y_{k}=E\big{)}=\sum_{x\in\\{L,R,S\\}}\Pr\big{(}y_{k}=x\big{)}.$

Next, denote $\mathcal{X}^{\prime}=\\{I,E\\}$. Equation (19) can be re-written
as follows:

$\displaystyle\Pr\big{(}\\{Y_{j}(t)\\}_{j\in\Psi_{i}(t)}|\zeta_{i}(t-1)=x\big{)}$
(56)
$\displaystyle=\sum_{\\{x_{j}\\}_{j\in\Psi_{i}(t)}}\prod_{j\in\Psi_{i}(t)}\Pr\big{(}Y_{j}(t)|\theta_{j}(t)\big{)}\times\sum_{\\{y_{l}\\}_{l\in\Theta_{i}(t)}}\prod_{j\in\Psi_{i}(t)}{\tt
P}_{j}\big{(}x_{j}|\\{y_{l}\\}_{l\in\partial^{+}_{j}(t-1)\backslash\\{i\\}},x\big{)}$
$\displaystyle\times\prod_{z_{l}\in\mathcal{X}^{\prime},l\in\Theta_{i}(t)}\Pr\big{(}\zeta_{l}(t-1)=z_{l}\big{)},$

with reduces the computation complexity. Subsequently,
$\underline{e}_{i}(t-1)$ in (18) can be calculated by (56) directly.

###  K-B $\alpha$-linking Backward Updating

In the backward step, the computation complexity is large even in (56). To
further reduce the complexity in (56), one way is to update the posterior
probability $\underline{e}_{i}(t)$ in a sparser network. Now, we define
$\alpha$-linking Backward Updating as follows:

  * (i)

We generate a subgraph $\mathcal{G}_{\alpha}(t)$ based on the pre-determined
graph $\mathcal{G}(t)$: Suppose that each edge (in $\mathcal{G}(t)$) exists
with probability $\alpha$, $0\leq\alpha\leq 1$. If $\alpha=1$, then
$\mathcal{G}_{\alpha}(t)=\mathcal{G}(t)$; if $\alpha=0$, then
$\mathcal{G}_{\alpha}(t)$ is a graph with no edges.

  * (ii)

Backward updating in $\mathcal{G}_{\alpha}(t)$: Similar with
$\partial_{i}(t)$, $\Psi_{i}(t)$, $\Phi_{i}(t)$ and $\Theta_{i}(t)$, we define
$\partial_{i,\alpha}(t)$, $\Psi_{i,\alpha}(t)$, $\Phi_{i,\alpha}(t)$ and
$\Theta_{i,\alpha}(t)$ on graph $\mathcal{G}_{\alpha}(t)$, respectively.
Subsequently, replace $\partial_{i}(t)$, $\Psi_{i}$, $\Phi_{i}(t)$ and
$\Theta_{i}(t)$ by $\partial_{i,\alpha}(t)$, $\Psi_{i,\alpha}(t)$,
$\Phi_{i,\alpha}(t)$ and $\Theta_{i,\alpha}(t)$ in (56), respectively.

##  Appendix L Proof of Theorem 3

Step 1. Preliminaries.

In Example 2, $\beta=1$, $\lambda=0$, and $\gamma=0$, there is no recovery and
we assume no latent state. Based on (11), the expression of rewards
$\hat{r}_{i}(t)$ for every node is given as follows. If node $i$ has two
neighbors (without quarantine)

$\displaystyle\hat{r}_{i}(t)=$ $\displaystyle
u_{S}^{(i-1)}(t)(1-u_{I}^{(i-2)}(t))u_{I}^{(i)}(t)+u_{S}^{(i+1)}(t)(1-u_{I}^{(i+2)}(t))u_{I}^{(i)}(t).$
(57)

If node $i$ only has one neighbor, then

$\displaystyle\hat{r}_{i}(t)=$ $\displaystyle
u_{S}^{(i+d)}(t)\big{(}1-u_{I}^{(i+2d)}(t)\big{)}u_{I}^{(i)}(t),\,d\in\\{-1,1\\}.$
(58)

For simplicity, we introduce artificial nodes $-1,0,N+1,N+2$ with
$u_{I}^{(-1)}(t)=u_{I}^{(0)}(t)=u_{I}^{(N+1)}(t)=u_{I}^{(N+2)}(t)=0$ for all
$t$, and these $4$ nodes are never tested.

Step 2. The RbEx policy.

Under the RbEx policy, the algorithm always tests the nodes with maximum
rewards. Let an infectious node be found, for the first time, on day $aN$,
where $a$ is a positive real number. Note that until the first infected node
is found, in any application of the RbEx policy, $u_{I}^{(i)}(t)$ is the same
for any given $i$, and hence $\hat{r}_{i}(t)$ is also the same. So, $a$ is the
same for any application of the RbEx policy. Recall that in Example 2, nodes
that are tested positive will be isolated. The cumulative infections is at
least $\min\\{aN,N\\}$ in the end.

Step 3. Consider the exploration process of the specific exploration policy.

Recall that from Step 2, an infectious node is found, for the first time, by
the RbEx policy with budget $10$ tests on day $aN$. Under the specific defined
exploration policy, we can choose a specific $b^{\prime}$ with $b^{\prime}<a$,
such that no infectious node is tested by the RbEx policy with budget $9$
tests before and including day $t=b^{\prime}N$.

We know that on day $\tau$, nodes $1,2,\cdots,\tau$ are infectious since
$\beta=1$. Note that one test is applied to exploration (randomly choice) on
every day, so with probability

$\displaystyle\prod_{\tau^{\prime}=1}^{\tau}(1-\frac{\tau^{\prime}}{N}),$ (59)

no infectious node is explored from the initial day to day $\tau$. Then, with
probability

$\displaystyle\prod_{\tau^{\prime}=1}^{\tau-1}(1-\frac{\tau^{\prime}}{N})\cdot\frac{\tau}{N},$

one infectious node is detected on day $\tau$. Thus, with probability

$\displaystyle\sum_{\tau=1}^{t}\prod_{\tau^{\prime}=1}^{\tau-1}(1-\frac{\tau^{\prime}}{N})\cdot\frac{\tau-1}{N},$
(60)

one infectious node is tested by exploration process on day $\tau$ ($\tau\leq
t$), and this node is not the new infectious one on day $\tau$, i.e., has
index $\tau$. The probability defined in (60) increases with $t$ when $N$ is
fixed, and it can be close to $1$ when $t$ close to $N$. Therefore, We can
choose proper parameters $b^{\prime}$ and $N$ such that the probability
defined in (60) is larger than or equal to $p_{0}$. In particular, if $N$ is
large, we can choose a relatively small $b^{\prime}$. In Theorem 3, we set
$p_{0}\geq 99/100$.

Let the infectious node detected (for the first time) by the exploration
process have index $j$ on day $t^{\prime}$, where $t^{\prime}\leq t$. As
discussed above, node $j$ is not the new infectious node on day $t^{\prime}$,
so we have $j<t^{\prime}$. In other words, node $j+1$ must be infecitous on
day $t^{\prime}$ with a positive test result, i.e., $Y_{j}(t^{\prime})=1$. By
Step 1 in Algorithm 4, the updated posterior probability of node $j$

$\displaystyle e_{I}^{(j)}(t^{\prime}-1)=1,\quad e_{S}^{(j)}(t^{\prime}-1)=0.$
(61)

Again, by Step 1 in Algorithm 4,

$\displaystyle w_{I}^{(j-1)}(t^{\prime})=w_{I}^{(j+1)}(t^{\prime})=1.$ (62)

Then, by Step 2 in Algorithm 4,

$\displaystyle
u_{I}^{(j-2)}(t^{\prime}+1)=u_{I}^{(j-1)}(t^{\prime}+1)=u_{I}^{(j)}(t^{\prime}+1)=u_{I}^{(j+1)}(t^{\prime}+1)=u_{I}^{(j+2)}(t^{\prime}+1)=1.$
(63)

Since $j$ is detected and isolated on day $t^{\prime}$, then,

$\displaystyle\hat{r}_{j}(t^{\prime}+1)=0.$ (64)

By (58) and (63),

$\displaystyle\hat{r}_{j-1}(t^{\prime}+1)=u_{S}^{(j-2)}(t^{\prime}+1)\big{(}1-u_{I}^{(j-3)}(t^{\prime}+1)\big{)}=0$
(65)
$\displaystyle\hat{r}_{j+1}(t^{\prime}+1)=u_{S}^{(j+2)}(t^{\prime}+1)\big{(}1-u_{I}^{(j+3)}(t^{\prime}+1)\big{)}=0.$

By (57) and (63),

$\displaystyle\hat{r}_{j-2}(t^{\prime}+1)=u_{S}^{(j-3)}(t^{\prime}+1)\big{(}1-u_{I}^{(j-4)}(t^{\prime}+1)\big{)}$
(66)
$\displaystyle\hat{r}_{j+2}(t^{\prime}+1)=u_{S}^{(j+3)}(t^{\prime}+1)\big{(}1-u_{I}^{(j+4)}(t^{\prime}+1)\big{)}.$

Step 4.The exploitation process of the specific exploration policy.

We first study an extreme case where no tests are applied. In this case,
denote the prior probability of node $i$ on day $\tau$ as $U_{I}^{(i)}(\tau)$,
which can be calculated by the following recursion:

$\displaystyle
U_{I}^{(i)}(\tau+1)=U_{I}^{(i)}(\tau)+U_{S}^{(i)}(\tau)\big{(}1-(1-U_{I}^{(i-1)}(\tau))(1-U_{I}^{(i+1)}(\tau))\big{)}.$
(67)

Based on (67), recall that $U_{I}^{(i)}(0)=0$ if $i\leq\frac{9N}{10}$, and
$U_{I}^{(i)}(0)=\frac{10\epsilon}{N}$ if $\frac{9N}{10}<i\leq N$, then
$U_{I}^{(i)}(\tau)$ increases over $\tau$ and is a function of $\epsilon$.
Then, given $b^{\prime}$, $N$ and $t=b^{\prime}N$, we can choose a small
enough $\epsilon$, denoted by $\epsilon(b^{\prime},N)$, such that
$U_{I}^{(i)}(2t)<\frac{1}{2}$ for all $i$. Since $U_{I}^{(i)}(\tau)$ increases
over $\tau$, then $U_{I}^{(i)}(\tau)<\frac{1}{2},\,\tau\leq 2t$.

Now, we introduce the exploitation process. Let
$t=b^{\prime}N<\min\\{\frac{9}{40},a\\}N$. There are at most $2t$ infectious
nodes on day $2t$, i.e., nodes $1,2,\cdots,2t$. Since
$t<\min\\{\frac{9}{40},a\\}N$, then nodes with index from $9N/10-2t$ to $N$
are in state $S$, which implies nodes with index from $9N/10-2t$ to $N$ can
never be tested positive before day $2t$. Thus, on any day $\tau\leq 2t$, for
$9N/10-2t\leq i\leq N$, if node $i$ is tested, and the testing result is
negative. Recall that $U_{I}^{(i)}(\tau)$ in (67) is calculated without any
negative testing results. Hence, $u_{I}^{(i)}(\tau)\leq U_{I}^{(i)}(\tau)$.
Furthermore, with the condition $t=b^{\prime}N<\min\\{\frac{9}{40},a\\}N$, we
can find a small enough $\epsilon(b^{\prime},N)$, such that under the specific
exploration policy,

$\displaystyle u_{I}^{(i)}(\tau)<\frac{1}{2},\quad\tau\leq 2t,\,\,9N/10-2t\leq
i\leq N.$ (68)

In the rest, we divide the nodes in to $3$ sets: $\mathcal{Q}_{1}=\\{i|i\leq
2t\\}$, $\mathcal{Q}_{2}=\\{i|2t<i<9N/10-2t\\}$, and
$\mathcal{Q}_{3}=\\{i|9N/10-2t\leq i\leq N\\}$.

Fact 1. For $i\in\mathcal{Q}_{1}$ and $\tau\leq 2t$, $u_{I}^{(i)}(\tau)=1$ or
$u_{I}^{(i)}(\tau)=0$.

Proof. If no test is applied to $\mathcal{Q}_{1}$, then $u_{I}^{(i)}(\tau)=0$
for all $i\in\mathcal{Q}_{1}$.

On some day $\tau\leq 2t$, if one node with index $j\in\mathcal{Q}_{1}$ is
tested positive on day $\tau-1$, then by (63),
$u_{I}^{(j-2)}(\tau)=u_{I}^{(j-1)}(\tau)=u_{I}^{(j)}(\tau)=u_{I}^{(j+1)}(\tau)=u_{I}^{(j+2)}(\tau)=1$.
In other words, if node $j$ is tested positive on day $\tau-1$, then node $j$,
its neighbors and neighbors of neighbors have probability of infection equal
to $1$ on day $\tau$.

On some day $\tau$, if node $j$ is not tested positive on day $\tau-1$, and
neither of its neighbors and neighbors of neighbors are is not tested
positive, then $u_{I}^{(j)}(\tau)=1$ only when $u_{I}^{(j)}(\tau-1)=1$, or
$u_{I}^{(j-1)}(\tau-1)=1$ or $u_{I}^{(j+1)}(\tau-1)=1$ since $\beta=1$.
Otherwise $u_{I}^{(j)}(\tau)=0$.∎

Fact 2. For $i\in\mathcal{Q}_{1}$ and $\tau\leq 2t$, $\hat{r}_{i}(\tau)=1$ or
$\hat{r}_{i}(\tau)=0$.

Proof. If $u_{I}^{(i)}(\tau)=0$, then $\hat{r}_{i}(\tau)=0$ by (57) and (58).

Now, we consider $u_{I}^{(i)}(\tau)=1$ in the following cases: (i) If both
neighbors of node $i$ are isolated, then $\hat{r}_{i}(\tau)=0$. (ii) If one of
neighbors of node $i$ (for example, node $i-1$) is isolated, then by (58),
$\hat{r}_{i}(\tau)=0$ when $u_{I}^{(i+1)}(\tau)=1$, and $\hat{r}_{i}(\tau)=1$
when $u_{I}^{(i+1)}(\tau)=0$. (iii) If both neighbors are not isolated, then
$u_{I}^{(i-1)}(\tau-1)=1$ or $u_{I}^{(i+1)}(\tau-1)=1$, otherwise,
$u_{I}^{(i)}(\tau)=0$. Since there is no recovery, then
$u_{I}^{(i-1)}(\tau)=1$ or $u_{I}^{(i+1)}(\tau)=1$. By Fact 1,
$u_{I}^{(j)}(\tau)=1$ or $u_{I}^{(j)}(\tau)=0$ when $j\in\mathcal{Q}_{1}$. If
$u_{I}^{(i+1)}(\tau)=u_{I}^{(i-1)}(\tau)=1$, then $\hat{r}_{i}(\tau)=0$. If
$u_{I}^{(i+1)}(\tau)=1$, then by (57), $\hat{r}_{i}(\tau)=0$ when
$u_{I}^{(i-2)}(\tau)=1$, $\hat{r}_{i}(\tau)=1$ when $u_{I}^{(i-2)}(\tau)=0$.
If $u_{I}^{(i-1)}(\tau)=1$, then by (57), $\hat{r}_{i}(\tau)=0$ when
$u_{I}^{(i+2)}(\tau)=1$, $\hat{r}_{i}(\tau)=1$ when $u_{I}^{(i+2)}(\tau)=0$. ∎

From (68), for all $\tau\leq 2t$ and $i\in\mathcal{Q}_{3}$, we have

$\displaystyle\hat{r}_{i}(\tau)\leq 2u_{I}^{(i)}(\tau)<1.$ (69)

Note that only nodes in $\mathcal{Q}_{1}$ and $\mathcal{Q}_{3}$ may have
positive probability of infection. For all $\tau\leq 2t$ and
$i\in\mathcal{Q}_{2}$, since $u_{I}^{(i)}(\tau)=0$, then
$\hat{r}_{i}(\tau)=0$. Therefore, a node with reward equal to $1$ has the
largest reward.

Recall that on day $t^{\prime}$, node $j$ is tested positive. From (66), nodes
$j-2$ and $j+2$ have largest rewards ($=1$) on day $t^{\prime}+1$, which are
exploited on day $t^{\prime}+1$, and all other nodes in $\mathcal{Q}_{1}$ have
rewards $0$. This is because $t^{\prime}$ is the first day when a positive
node is found. Since node $j$ is tested positive and isolated on day
$t^{\prime}$, then all infectious nodes with index less than $j$ can no longer
infect other nodes in the line network. Now, we consider the nodes with index
larger than $j$. Recall that $j<t^{\prime}$, so node $j+1$ must be infectious
on day $t^{\prime}$, and node $j+2$ must be infectious on day $t^{\prime}+1$
since $\beta=1$. Thus, node $j+2$ is tested positive and is isolated. Since
the network is a line, both nodes $j+1$ and $j+2$ can no longer infect other
nodes once node $j+2$ is isolated. Note that nodes in $\mathcal{Q}_{3}$ have
positive rewards. When $N$ is sufficiently large, in the rest of the
exploitation process, nodes in $\mathcal{Q}_{3}$ are tested. Recall that we
have one test for exploration, and we can isolate at least $2$ infectious
nodes with index larger than $j$.

Repeat the process, we exploit nodes $j+4,j+6,\cdots$ on day
$t^{\prime}+2,t^{\prime}+3,\cdots$, respectively. Consider the direction from
node $1$ to node $N$. On every day, there is at most one new infectious node,
but at least two infectious nodes can be isolated. On some day, denote as day
$t^{\prime}+x$, the exploitation process can progress beyond the infections
(exceeding by one node) for the first time. In other words, node $j+2x$ is
tested negative on day $t^{\prime}+x$. By Step 2 in Algorithm 4,
$e_{I}^{(j+2x)}(t^{\prime}+x-1)=0$. However, since
$w_{I}^{(j+2x-1)}(t^{\prime}+x)=1$ becuase node $j+2x-2$ is tested positive on
day $t^{\prime}+x-1$. By Step 1 in Algorithm 4,
$u_{I}^{(j+2x)}(t^{\prime}+x+1)=1$, hence by (66),
$\hat{r}_{j+2x}(t^{\prime}+x+1)=1$, which implies node $j+2x$ has the largest
reward and is exploited on day $t^{\prime}+x+1$, and it will be tested
positive. On day $t^{\prime}+x+1$, all infectious nodes are isolated.

Finally, we can calculate the total number of infections to be

$\displaystyle j+2(t^{\prime}-j)=2t^{\prime}-j\leq 2t^{\prime}\leq
2t=2b^{\prime}N.$

Let $b=2b^{\prime}$. This is an improvement by a factor of at least
$\frac{a}{b}$ in comparison to the RbEx strategy, where $\frac{a}{b}$ can be
as large as desired by increasing the value of $N$ or decreasing $p_{0}$.

##  Appendix M Construction of Networks and Further Results

###  M-A Constructions of SBM and V-SBM

In this section, we construct SBMs and its variants.

##### SBM

The SBM is a generative model for random graphs. The graph is divided into
several communities, and subsets of nodes are characterized by being connected
with one another with particular edge densities.888Here, we assume that $M$ is
an exact divisor of $N$. The intra-connection probability is $p_{1}$, and
inter-connection probability is $p_{2}$. We denote the SBM as
SBM$(N,M,p_{1},p_{2})$. Note that the (expected) number of edges, denoted by
$|\mathcal{E}|$, is

$\displaystyle|\mathcal{E}|=\frac{p_{1}}{2}N(\frac{N}{M}-1)+\frac{p_{2}}{2}\frac{N^{2}}{M}(M-1).$
(70)

Now, we fix $|\mathcal{E}|$, and choose the pair $(p_{1},p_{2})$ under a fixed
$|\mathcal{E}|$ in (70). The aim of fixing $|\mathcal{E}|$ is to guarantee
that the transmission of the disease would not be affected by edges.

##### V-SBM

Now, we consider a variant of SBM, denoted by V-SBM. Different from SBM, we
only allow nodes in cluster $i$ to connect to nodes in successive clusters
(the neighbor clusters). Denote the V-SBM as V-SBM$(N,M,p_{1},p_{2})$.
Similarly, the expected number of edges, denoted by $|\mathcal{E}|$, is

$\displaystyle|\mathcal{E}|=\frac{p_{1}}{2}N(\frac{N}{M}-1)+p_{2}\frac{N^{2}}{M}.$
(71)

Now, we fix $|\mathcal{E}|$, and choose the pair $(p_{1},p_{2})$ under a fixed
$|\mathcal{E}|$ in (71). The aim of fixing $|\mathcal{E}|$ is to guarantee
that the transmission of the disease would not be affected by edges.

###  M-B The impact of $\gamma_{c}$ and $L_{p}$ individually

In this subsection, we investigate the role of $\gamma_{c}$ and $L_{p}$
individually, not through the common factor $\delta$. We consider different WS
networks with degrees $d=4,6$, and then adjust the rewiring probability
$\delta$, such that one of $(\gamma_{c},L_{p})$ is almost constant, and the
other is varying. We can see that the trend is similar to what we observed by
varying $\delta$ in Table VI.

WS, $(d,\delta)$ | $\gamma_{c}$ | $L_{p}$ | ${\tt Ratio}$
---|---|---|---
$(6,0.05)$ | 0.504 | $4.952$ | $.0003$
$(4,0)$ | 0.500 | $62.876$ | $0.191$
$(6,0.1)$ | 0.456 | $5.718$ | $-0.027$
$(4,0.03)$ | 0.456 | $10.810$ | $0.097$
TABLE XII: Clustering coefficients of WS networks WS, $(d,\delta)$ | $\gamma_{c}$ | $L_{p}$ | ${\tt Ratio}$
---|---|---|---
$(6,.001)$ | $0.599$ | 21.188 | 0.209
$(4,.0075)$ | $0.489$ | 21.264 | $0.182$
$(6,.005)$ | $0.592$ | 14.310 | $0.211$
$(4,.015)$ | $0.473$ | 14.253 | $0.174$
$(6,.009)$ | $0.585$ | 12.081 | $0.137$
$(4,.0225)$ | $0.467$ | 12.171 | $0.125$
TABLE XIII: Clustering coefficients of WS networks


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# Longitudinal spin-relaxation optimization for miniaturized optically pumped
magnetometers

A. P. McWilliam<EMAIL_ADDRESS>S. Dyer  D. Hunter  M. Mrozowski
S. J. Ingleby  O. Sharp  D. P. Burt  P. F. Griffin  J. P. McGilligan  E. Riis
Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United
Kingdom  2Kelvin Nanotechnology, University of Glasgow, Glasgow G12 8LS,
United Kingdom

###### Abstract

The microfabrication of cesium vapor cells for optically pumped magnetometry
relies on optimization of buffer gas pressure in order to maximize atomic
coherence time and sensitivity to external magnetic signals. We demonstrate
post-bond nitrogen buffer gas pressure tuning through localized heating of an
integrated micro-pill dispenser. We characterize the variation in the
intrinsic longitudinal relaxation rate, $\gamma_{10}$, and magnetic
sensitivity, as a function of the resulting nitrogen buffer gas pressure.
Measurements are conducted through employing an optically pumped magnetometer
operating in a free-induction-decay configuration. $\gamma_{10}$ is extracted
across a range of nitrogen pressures between $\sim$ 60 - 700 Torr, measuring a
minimum of 140 Hz at 115 Torr. Additionally, we achieve sensitivities as low
as 130 fT/$\sqrt{\text{Hz}}$ at a bias field amplitude of $\sim 50~{}$µT. With
the optimal nitrogen buffer gas pressure now quantified and achievable post-
fabrication, these mass-producible cells can be tailored to suit a variety of
sensing applications, ensuring peak magnetometer performance.

††preprint: APS/123-QED

##  I Introduction

Precise magnetic field sensing in finite field conditions is crucial in
several areas including geophysical surveying [1], land ordinance [2] and
archaeology [3]. Additionally, continuous research has revealed promising
applications in the medical field including magnetocardiography (MCG) and
magnetoencephalography (MEG) [4]. These applications typically require a
network of sensors to adequately resolve both spatial and temporal
information, and often necessitate gradiometric cancellation when operated in
unshielded conditions [5]. Therefore, precise tuning of the buffer gas
pressure offers significant advantages by providing consistent sensor
performance, improved common-mode noise suppression, and a uniform response
across the sensor array.

Optically pumped magnetometers (OPMs) have emerged as a versatile solution for
satisfying the numerous sensing requirements across these avenues. OPMs
measure the precession frequency of spin-polarized atoms, providing extremely
accurate and precise magnetic field readings [6]. Importantly, the
incorporation of thermal atomic ensembles in OPMs has enabled miniaturization
into portable and chip-scale sensor heads to address real-world
implementations [7].

At the heart of the OPM comprises an atomic vapor cell, where atom-light
interactions occur. Such vapor cells contain the atomic spin medium, in
conjunction with a buffer gas or paraffin coating, required to extend the
atomic spin-coherence time. A core component in the miniaturization of OPMs is
the micro-machined vapor cell, typically formed of a glass-silicon-glass stack
containing buffer gas to slow the diffusion of the alkali atoms towards the
cell walls. Such cells are commonly manufactured with a uniform target buffer
gas pressure intended across the entire wafer [8]. The reproducibility and
achievable yield is on the order of 100’s of cells per wafer making this
approach advantageous for development of sensor arrays and mass-producible
quantum technologies [9], with custom geometries and multi-chamber designs
realizable [10, 11, 12]. Additionally, precise buffer gas control enables the
optimization between spatial resolution and sensor precision in OPM magnetic
imaging applications [13]. This refinement enables an optimal balance between
two factors for peak performance: spin-destruction collisions between alkali
and buffer gas atoms, and the atomic diffusion rate.

We employ an OPM based on the free-induction-decay (FID) measurement protocol
[14]. One significant advantage of FID-based sensors is their insensitivity to
systematics, achieved by temporally separating the optical pumping and
detection processes using a pulsed measurement scheme. This separation allows
the polarized spins to precess freely at the Larmor frequency without
interference from intense pumping light, significantly reducing light shifts
and power broadening effects compared to continuous-wave pumping schemes.
Additionally, an enhanced optical pumping mode is exploited to facilitate
suppression of spin-exchange collisions, reducing the overall spin-relaxation
contribution [15]. This vastly extends the sensor’s dynamic range to bias
fields surpassing the Earth’s field ($\sim~{}50~{}$µT).

In this paper, we demonstrate the ability to optimize the longitudinal spin-
relaxation rate in a micro-machined vapor cell by controllable gettering of
the nitrogen (N2) buffer gas pressure from localized heating of an alkali
micro-pill source. The intrinsic spin-relaxation properties for a range of
vapor cells is ascertained using the FID approach [14, 16, 15, 13]. Power
broadening caused by residual optical pumping from the probe light is
mitigated through extrapolation to zero-light power [17]. Furthermore, the
impact of the buffer gas pressure on the measured sensitivity of the OPM is
shown for cell pressures measured between $60-700$ Torr. The ability to
optimize the OPM sensitivity and longitudinal relaxation rate by reducing the
vapor cell buffer gas pressure in a single vapor cell gives critical insight
to the operational characteristics of the sensor. Our findings will guide
future mass production of cells with the optimal N2 pressure content to be
applied via back-filling now quantified. Moreover, the N2 content can be
equalized across a population of cells achieving uniform response in magnetic
gradiometers and OPM sensor arrays.

##  II Methodology

###  II.1 Experimental setup

Figure 1: (a) OPM experimental setup containing 852 nm and 895 nm pump and
probe beams that interact with cesium (Cs) atoms in a thermal vapor cell
placed in a magnetic shield with a controlled bias field applied along the
z-axis. The transmitted probe was detected with a balanced polarimeter while
the residual pump was blocked with a spectral filter. GT: Glan-Thompson
polarizer; NPBS: Non-polarizing beam-splitter; PBS: Polarizing beam-splitter;
DWP: Dual-wavelength waveplate; BPF: Band-pass filter; WP: Wollaston prism;
B-P: Balanced polarimeter; PD: Photodiode used in conjunction with an AOM (not
pictured) for probe beam intensity stabilization. (b) Absorption spectroscopy
(Cs $\text{D}_{1}$ line) was used to extract (N2) pressure readings with
spectra shown for 700 Torr (green) and 135 Torr (blue) compared against a
reference cell (black). (c) Corresponding FID signal data relating to a probe
power of 1.3 mW along with relaxation envelopes (black). The probe beam
induces a broadening effect, $\gamma_{\text{P}}$, leading to an increase in
the relaxation rate.

Figure 1(a) displays a simplified schematic of the experimental setup used in
this work. We employed three separate vapor cells, which were anodically
bonded with different N2 pressures (P${}_{\text{N}_{2}}$). Each cell had
dimensions of ($6~{}\times~{}6~{}\times~{}6$) mm within the science chamber.
Two cells were cylindrically shaped whilst the third was a cuboid geometry. We
conducted sequential measurements for each using this OPM setup. The N2
content was decreased by laser heating the alkali pill source to activate the
non-evaporable getter material inside as discussed in [18, 19].
P${}_{\text{N}_{2}}$ within the cells was determined after each depletion step
by performing absorption spectroscopy on the Cs $\text{D}_{1}$ line with the
cell heated to 70 ∘C. The resulting spectrum was then fitted with an
absorption profile to extract the N2 induced pressure broadening and shift,
which were used in conjunction with the coefficients characterized by Andalkar
et al. [20] to determine the P${}_{\text{N}_{2}}$. P${}_{\text{N}_{2}}$ scales
as a function of cell temperature according to the ideal gas law. The cell was
operated at 30 ∘C for spin-relaxation measurements to suppress the spin-
exchange contributions observed at higher vapor densities, although an
appreciable signal could still be observed. The sensitivity was measured at 70
∘C which is optimal for the vapor cell configurations considered here. For
each measurement throughout this study the cell was placed within a three-
layer µ-metal shield to attenuate the Earth’s ambient field amplitude and
sources of environmental noise.
A set of coils were thermally connected to both sides of the cell using
printed circuit boards (PCBs), as depicted in Fig. 1(a). These coils provided
resistive heating and established a well-defined optical pumping
(quantization) axis, ensuring high spin-polarization even in the presence of
substantial bias fields. This is due to the strong magnetic field,
$|\vec{\text{B}}_{\text{Pol}}|>1~{}$mT, applied along the quantization axis by
the coils, which greatly exceeds the fields within the measurement range of
interest [15]. Surrounding the cell were a set of Helmholtz coils which
provided static transverse magnetic field control. This coil set was driven by
a programmable low-noise current driver that does not contribute significantly
to the overall noise budget of the sensor [21].
We operated a two-color pump-probe scheme as described in previous works [13,
15]. Strong pulsed optical pumping was applied ($\sim 120$ mW peak power) with
circularly polarized light on the pressure broadened F = 3$~{}\rightarrow$ F’
hyperfine transition of the Cs $\text{D}_{2}$ line. The pump light was
extinguished from peak optical power to $<$ 5 µW using an acousto-optic
modulator (AOM). The pump light recycles atomic population present in the F =
3 manifold which are subsequently redistributed to the F = 4 ground state,
inducing an orientation in the atomic ensemble. This is akin to the hyperfine-
repump technique employed in [22], albeit without an additional laser source.
Moreover, the continuous absorption and emission process transfers a large net
spin-polarization into a stretched Zeeman state with maximum angular momentum
$m_{F}=4$. Collision of atoms occupying the same hyperfine ground state do not
result in decoherence through spin-exchange, therefore efficient optical
pumping to a highly stretched state allows effective suppression of spin-
exchange dephasing [23].

Detection was performed on the Cs $\text{D}_{1}$ line with a weaker probe
beam, less than 1.2 % of the pump beam’s peak optical power. The probe
frequency was set $\approx$ 21 GHz blue-detuned from the F = 3$~{}\rightarrow$
F’ transition. This was close enough to resonance to yield an appreciable
signal amplitude, whilst avoiding excessive power broadening due to residual
optical pumping. To obtain the intrinsic spin-relaxation property,
$\gamma_{10}$, it was necessary to negate these operational systematics
further. This was achieved by applying a range of probe powers from $\sim$ 0.1
mW to 1.3 mW such that $\gamma_{10}$ could be ascertained by extrapolating to
zero-light power. The probe power was measured before transmission through the
cell which was $\geq$ 70 %. The pump and probe beams were ensured to overlap
at the center of the cell, where they both possessed a 3.1 mm beam diameter
($1/\text{e}^{2}$). Therefore, both were sufficiently well contained inside
the cell volume to avoid beam clipping.

###  II.2 Signal analysis

A balanced polarimeter (Thorlabs PDB210A), which rejects common-mode laser
intensity noise, was used to capture the signal. A data acquisition device
(Picoscope Model 5444D) sampled at a base rate of f${}_{\text{s}}$ = 125 MHz
and functioned with 15-bit voltage resolution. Over-sampling was performed to
average 25 successive data points resulting in a decimated sample rate of
$\text{f}_{\text{s}}=1/\Delta\text{t}=$ 5 MHz. This was due to
f${}_{\text{s}}$ greatly exceeding the signals of interest, which oscillated
at either 3.5 kHz (Section III.1) or 175 kHz (Section III.2). Furthermore, for
the former in post-process a low pass Butterworth filter (2nd order) was
applied to suppress any spurious higher frequency noise. This data was also
further down-sampled by a factor of 100 to reduce the effective sample rate to
f${}_{\text{s}}$ = 50 kHz (equating to approximately 14 points per cycle).
This served only to expedite the fitting analysis and did not corrupt the
extracted fit parameters.

The FID signal manifests as a result of optical rotation of the probe beam
polarization, which maps the induced spin-polarization precessing around the
magnetic field at the Larmor frequency. The digitized FID signal can be
represented as follows,

$\text{S}_{\text{n}}=\text{A}\,\mathrm{sin}\big{(}\omega_{\text{L}}\,\text{n}\,\Delta{\text{t}}+\phi_{0}\big{)}\,e^{-\gamma_{2}\,\text{n}\,\Delta{\text{t}}}+\epsilon_{\text{n}}.$
(1)

where A is the signal amplitude, relating to the spin-polarization generated
during the optical pumping phase, $\omega_{\text{L}}$ is the Larmor precession
frequency, and $\phi_{0}$ is the initial phase. n relates to the sample data
point under consideration and $\epsilon_{\text{n}}$ represents noise present
in the signal. Nonlinear fitting was applied with the Levenberg-Marquardt
algorithm [24] using a damped sinusoidal model to extract the relevant
parameters. After optical pumping, the precessing atomic spin-polarization
decays as the polarization returns to its equilibrium (unpolarized) state,
with a total spin-relaxation rate denoted $\gamma_{2}$. Section II.3 describes
methods of lowering $\gamma_{2}$ such that this equates to the intrinsic
longitudinal rate of the vapor cell. This was enabled by reducing the cell
temperature and probe power with the results presented in Section III.1. In
Section III.2 our chosen operating regime involved the vapor density being
raised to increase atom-light interactions, thus elevating $\gamma_{2}$.
However, this ultimately enhanced the sensitivity and provided a better analog
for a practical sensor with optimized performance.

###  II.3 Relaxation rate measurements

The spin-relaxation rate limits the total possible measurement time of the
OPM. The various intrinsic relaxation mechanisms inherent to vapor cells are
extensively described in [25]. The intrinsic longitudinal relaxation rate,
$\gamma_{10}$, is the ultimate spin-relaxation limit. The main contributions
to this in this study arise from alkali-wall collisions ($\gamma_{\text{WC}}$)
and alkali-buffer gas collisions ($\gamma_{\text{BG}}$). A negligible
contribution to the spin-decoherence arises through Cs-Cs spin-destruction
collisions ($\gamma_{\text{SD}}$). The cross section for this is a factor of
$\sim$ 100 smaller than Cs-Cs spin-exchange collisions [26, 27]. Therefore,
$\gamma_{10}$ is defined as,

$\gamma_{10}=\gamma_{\text{WC}}+\gamma_{\text{BG}}+\gamma_{\text{SD}}.$ (2)

The aggregated term representing the total transverse relaxation rate,
$\gamma_{2}$, including operational systematics, is given by,

$\gamma_{\text{2}}=\gamma_{10}+\gamma_{\text{SE}}+\gamma_{\text{P}}+\gamma_{\Delta\text{B}}.$
(3)

Therefore, to successfully extract the intrinsic relaxation rate,
$\gamma_{10}$, the additional relaxation contributions including optical power
broadening ($\gamma_{\text{P}}$), magnetic gradients across the cell
($\gamma_{\Delta\text{B}}$), and spin-exchange collisions
($\gamma_{\text{SE}}$) have to be suppressed.

Operating the system at a low bias magnetic field strength of
$|\vec{\text{B}_{\text{z}}}|\approx$ 1 µT lowered magnetic gradients to
$\gamma_{\Delta\text{B}}<1$ Hz based on calculations considering the coil
geometry and vapor cell dimensions. Furthermore, spin-exchange collisions are
negligible at the low cell temperatures (T = 30 ∘C) considered here.
Temperature was monitored using three non-magnetic T-type thermocouple sensors
attached to separate positions of the cell. All agreed to within 0.1 ∘C,
confirming a uniform temperature across the cell. The cell was not found to
drift by more than 0.1 ∘C over the course of each set of measurements. It
should be noted that an insignificant increase in $\gamma_{10}$ occurs for
increasing temperatures.

The pump-probe cycle was applied at a repetition rate, $\text{R}_{\text{r}}$,
of 10 Hz with a pump duration, $\text{T}_{\text{OP}}$, of 5 ms and total
probing duration, $\text{T}_{\text{PR}}$, of 95 ms. This measurement period
ensured the atoms fully decohered and re-thermalized before subsequent optical
pumping cycles occurred, thus avoiding fitting systematics from signal
truncation. To determine the intrinsic relaxation rate under these conditions,
measurements were taken at various probe powers, starting from a maximum of
1.3 mW and decreasing incrementally to 140 µW. Nonlinear fitting using a
damped sinusoidal model (Eq. 1) was applied to the FID datasets to extract
$\gamma_{\text{2}}$ in each case. A linear extrapolation to zero-light power
accounts for the contribution from $\gamma_{\text{P}}$ such that
$\gamma_{\text{10}}$ can be determined as seen in Fig. 2 (a).

##  III Results

###  III.1 Relaxation rate results

Figure 2: (a) Measured values for $\gamma_{\text{2}}$ as a function of probe
power for P${}_{\text{N}_{2}}=115$ Torr, where $\gamma_{10}$ is minimized.
Extrapolation to zero-light power at each N2 pressure provides the respective
$\gamma_{10}$ values shown in (b). The spin-exchange and magnetic field
broadening contributions to $\gamma_{\text{2}}$ were minimal throughout these
spin-relaxation measurements. (b) Blue data points relate to measurements with
the cuboid cell. Black (green) data points refer to two separate cylindrical
shaped cells. The various lines relate to the theoretical spin-relaxation
mechanisms: $\gamma_{10}$ (green solid line); $\gamma_{\text{WC}}$ (blue dot-
dashed line); $\gamma_{\text{BG}}$ (red dashed line). Error-bars are smaller
than the markers.

Figure 2 (b) displays the results over the full buffer gas pressure range.
Each pressure iteration produced small statistical errors for each data point
on the order of 1 Hz. The data demonstrates a continuous optimization of
$\gamma_{10}$ when lowering $\text{P}_{\text{N}_{2}}$ to 115 Torr, which is
where it is minimized. $\gamma_{10}$ then begins to increase as
$\text{P}_{\text{N}_{2}}$ is reduced further due to Cs-wall collisions
($\gamma_{\text{WC}}$) becoming the dominant depolarizing mechanism and the
contribution from Cs-$\text{N}_{2}$ collisions ($\gamma_{\text{BG}}$)
diminishing. Across the pressure range $\text{P}_{\text{N}_{2}}\approx 57-212$
Torr, the intrinsic relaxation rate extends from $\gamma_{10}\approx 140-184$
Hz. This equates to intrinsic spin-coherence times of between 5.4 ms and 7.1
ms. Moreover, $\gamma_{10}$ maintains a consistency below $150$ Hz between a
range of $108-162$ Torr, showcasing a fairly wide pressure range that permits
extensive coherence times. This is a significant time period in which magnetic
sensing measurements can be made before spin-relaxation occurs providing
enhanced sensitivity performance.
The data aligns with the theoretical model outlined in [25] when constraining
all known parameters, with the exception of two critical parameters; namely,
the Cs-N2 spin-destruction rate, $\sigma_{\text{Cs}-\text{N}_{2}}$, and the Cs
diffusion coefficient in N2, $\text{D}_{0:\text{Cs}-\text{N}_{2}}$. The data
indicates that these values are $\sigma_{\text{Cs}-\text{N}_{2}}$ = 2.9 x
10-26 m2 and $\text{D}_{0:\text{Cs}-\text{N}_{2}}$ = 0.12
$\text{cm}^{2}\text{s}^{-1}$ respectively. This value for
$\text{D}_{0:\text{Cs}-\text{N}_{2}}$ matches well with the works from [28,
29, 30, 31] which have published values for each.
$\sigma_{\text{Cs}-\text{N}_{2}}$ resides within a fairly wide spectrum of
these published results. More recent data for these parameters are not readily
available. The disparity of measured values from the literature makes it
reasonable to incorporate these values in the model as they concur with the
observed data. This alignment is further supported by the reliability of our
measurements, involving multiple data points with three distinct cells which
agree for similar $\text{P}_{\text{N}_{2}}$ values.
Previously, we have demonstrated that our heating and enhanced optical pumping
strategy does not cause additional systematics to $\gamma_{10}$ by perturbing
the precessing ensemble [15]. This was further verified in this work by
collecting data at room temperature (T $\approx 21$ ∘C), without current being
applied through the resistive heating coils. We obtained a value of
$\gamma_{10}\approx 139$ Hz at $\text{P}_{\text{N}_{2}}\approx 125$ Torr which
is 2 Hz lower than the 141 Hz extracted when the cell was heated to 30 ∘C.
This agrees with theoretical predictions for $\Delta\gamma_{10}$ for this
temperature variation.

###  III.2 Sensitivity results

Figure 3: (a) OPM sensitivity measured across a range of N2 pressures at
$|\vec{\text{B}_{\text{z}}}|\sim 50~{}$µT bias field. The error-bars were
calculated from the standard deviation in the noise densities from each
associated ASD over a 70 - 250 Hz frequency range, excluding technical noise
peaks. Negligible change occurs between $\mathrm{P_{N_{2}}}\approx 57-340$
Torr providing an extensive buffer gas target range to aim for in future
fabrication for optimal sensitivity performance. (b) ASD at
$\mathrm{P_{N_{2}}}\approx 130$ Torr, $\text{T}\approx 70~{}^{\circ}\text{C}$,
and a probe power of 1.2 mW. A sensitivity of 147 $\pm$ 11
fT$/\sqrt{\text{Hz}}$ was calculated.

Assessment of the achievable sensitivity performance of the sensor required
raising of the vapor density. Therefore, the cell was heated to 70 ∘C for each
$\mathrm{P_{N_{2}}}$ iteration. This elevated temperature results in an
increase in the rate of spin-exchange collisions. Demonstrating the system’s
candidacy for unshielded applications also required increasing the bias field
to $|\vec{\text{B}_{\text{z}}}|\approx 50$ µT. To minimize technical magnetic
noise contributions, e.g. power line noise, the current through the Helmholtz
coils was driven by connecting a 12 V battery to a resistor in series.
Consistent pump and probe powers of 120 mW and 1.2 mW were respectively
employed throughout. $\text{R}_{\text{r}}$, was set to 500 Hz with an optical
pumping duty cycle of 10 %, equating to $\text{T}_{\text{OP}}$ = 0.2 ms and
$\text{T}_{\text{PR}}$ = 1.8 ms for each FID cycle. A faster
$\text{R}_{\text{r}}$ here aids the sensitivity performance, which can be
optimized based on the decoherence rate and the CRLB condition [32, 33]. In
contrast to Section III.1, subsequent optical pumping cycles were commenced
before the atoms fully decohered or re-thermalized to maximize sensor
bandwidth and sensitivity.
Each evaluation of the magnetic sensitivity involved capturing a sequence of
FID signal trains. 40 independent FID trains were recorded over 1 s time
periods and subsequently processed to provide a time series of the magnetic
field magnitude. The corresponding amplitude spectral densities (ASDs) were
averaged using Welch’s method to provide a smoother spectrum [34]. Figure 3
(a) displays the sensitivity results across the full buffer gas range for each
cell. An example of the ASD for $\mathrm{P_{N_{2}}}$ = 130 Torr is displayed
in (b).
The experimental sensitivities improved as $\mathrm{P_{N_{2}}}$ was reduced
from 780 - 340 Torr. Between 57 - 340 Torr a consistent performance was found
with measured experimental sensitivities between 130 - 160
fT$/\sqrt{\text{Hz}}$. The best sensitivity of 130$~{}\pm~{}$11
fT$/\sqrt{\text{Hz}}$ was found to occur at a buffer gas pressure of 180 Torr.
Notably, line and technical noise peaks are observable, indicating coupling
through the electronics and magnetic environment. Variations in the amplitude
and frequencies of these peaks are also evident.
These results equate to around 3 ppb in fractional sensitivity terms [35]
representing a state-of-the-art resolution using a micro-machined vapor cell.
These results owe much to the manufacturing, with the 6 mm thickness reducing
the relaxation rates whilst also improving the signal amplitude in comparison
to our previous works with 3 mm thick cells [13, 15]. Additionally, the
intense pump light combined with applying $\vec{\text{B}}_{\text{Pol}}$
greatly assists in maintaining the spin-polarization during the atomic state
preparation. Micro-machined cells have previously shown higher sensitivities
than the work here, however, this was at a lower magnetic field amplitude
[36]. In that work, magnetic field nulling was used during the optical pumping
stage, ultimately producing a sensitivity of around 100 fT$/\sqrt{\text{Hz}}$.
While demonstrating excellent sensitivity, the advantage of our approach is
that it circumvents the need for measuring the field and subsequently nulling
it. This presents an advantage for unshielded sensing as it reduces dead-time
and bypasses potential issues incurred in dynamic magnetic environments, e.g.,
sensing on board moving vehicles or vessels.

##  IV Conclusion

Our micro-machining capabilities have led to successful production of vapor
cells with 6 mm optical path lengths (single pass). Extensive measurements
were made to extract the intrinsic longitudinal spin-relaxation rate
$\gamma_{10}$, which sets the ultimate limit on spin-relaxation, across a
range of buffer gas pressures using three distinct vapor cells. We have
demonstrated a successful application of buffer gas pressure tuning to reach
the optimal value where $\gamma_{10}$ is minimized. A minimum of 140 Hz was
found, measured at a cell temperature of 30 ∘C and $\mathrm{P_{N_{2}}=115}$
Torr. Further reduction of $\gamma_{10}$ will be challenging without further
increasing the cell size or thickness. Pivoting to fabrication approaches
which do not include a buffer gas would circumvent these spin-destructive
alkali-buffer gas collisions, however, the anti-relaxation coatings required
to preserve spin-coherence are not compatible with the temperatures required
for anodic bonding. Overall, our results indicate that the differing geometry
of cells does not markedly impact the cell performance characteristics. This
is an encouraging result which bodes well for mass cell fabrication as this
indicates a level of consistency. A noteworthy finding is the large buffer gas
pressure range at which prolonged relaxation times are achieved.
We have demonstrated sensitivities reaching state-of-the-art levels for
microfabricated vapor cells in bias fields emulating Earth’s field conditions
($\sim 50~{}$µT). We obtained a minimum of 130 $\pm$ 11 fT$/\sqrt{\text{Hz}}$.
Clearly sensitivities at these levels in geophysical fields is of value in
several finite-field OPM applications. Additionally, our results validate the
fabrication and activation approach employed for these cells. This study has
demonstrated the large buffer gas pressure range at which exemplary sensor
performance is attainable. One of the main advantages of the buffer gas tuning
approach is the ability to decrease the pressure post-fabrication. Recent
studies have demonstrated technical capabilities to sequentially increase the
buffer gas pressure within microfabricated cells via a break-seal technique
that well compliments the methods demonstrated in this work [37]. Magnetic
gradiometers would undoubtedly benefit from this, as the capacity to finely
adjust the pressure to achieve uniformity across identical cell geometries is
desirable [38, 39].

## acknowledgments

APM was supported by a Ph.D. studentship from the Defence Science and
Technology Laboratory (DSTL). JPM gratefully acknowledges funding from a Royal
Academy of Engineering Research Fellowship.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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| 0.2 | RMSE | 1.3173 | 1.5442 | 1.2882 | 1.2840 | 1.2215 | 1.2202 | 1.1768
|  | MAPE | 0.0898 | 0.1040 | 0.0930 | 0.0922 | 0.0950 | 0.0892 | 0.0865
|  | MAE | 0.6333 | 0.6421 | 0.6223 | 0.6161 | 0.5879 | 0.5803 | 0.5609
| 0.3 | RMSE | 1.0650 | 1.1240 | 1.0641 | 1.0697 | 1.0241 | 1.0148 | 0.9876
|  | MAPE | 0.0670 | 0.0653 | 0.0645 | 0.0638 | 0.0575 | 0.0566 | 0.0557
|  | MAE | 0.4536 | 0.4520 | 0.4023 | 0.4018 | 0.3905 | 0.3876 | 0.3770
| 0.4 | RMSE | 0.8274 | 0.8495 | 0.7676 | 0.7645 | 0.7366 | 0.7345 | 0.7211
|  | MAPE | 0.0509 | 0.0467 | 0.0419 | 0.0419 | 0.0407 | 0.0403 | 0.0385
|  | MAE | 0.2663 | 0.3063 | 0.2831 | 0.2825 | 0.2605 | 0.2615 | 0.2560
| 0.5 | RMSE | 0.5450 | 0.5934 | 0.5912 | 0.5855 | 0.5580 | 0.5525 | 0.5435
|  | MAPE | 0.0263 | 0.0350 | 0.0308 | 0.0306 | 0.0306 | 0.0303 | 0.0292
PM2.5 Concentration |  | MAE | 3.5170 | 3.7224 | 3.2365 | 3.2133 | 3.2120 | 3.1685 | 3.0717
0.1 | RMSE | 5.7922 | 6.5506 | 5.4857 | 5.4166 | 5.3981 | 5.3666 | 5.2669
| MAPE | 0.5125 | 0.5987 | 0.5828 | 0.4831 | 0.4811 | 0.5309 | 0.5278
|  | MAE | 2.5809 | 2.7509 | 2.5363 | 2.5412 | 2.4906 | 2.4782 | 2.4226
| 0.2 | RMSE | 4.4577 | 4.8055 | 4.5982 | 4.6150 | 4.6008 | 4.5416 | 4.2820
|  | MAPE | 0.4588 | 0.5110 | 0.4598 | 0.4614 | 0.4834 | 0.4863 | 0.4717
|  | MAE | 2.0887 | 2.1562 | 2.0231 | 2.0161 | 1.9740 | 1.9534 | 1.9405
| 0.3 | RMSE | 4.1175 | 4.0359 | 3.9335 | 3.9242 | 3.8544 | 3.8286 | 3.8262
|  | MAPE | 0.3420 | 0.3872 | 0.3613 | 0.3574 | 0.3583 | 0.3541 | 0.3452
|  | MAE | 1.7716 | 1.7880 | 1.5922 | 1.5841 | 1.5715 | 1.5508 | 1.5285
| 0.4 | RMSE | 3.6206 | 3.5443 | 3.3331 | 3.3171 | 3.3405 | 3.3078 | 3.1535
|  | MAPE | 0.2983 | 0.3133 | 0.2796 | 0.2760 | 0.2809 | 0.2786 | 0.2737
|  | MAE | 1.4244 | 1.5043 | 1.2903 | 1.2876 | 1.2904 | 1.2867 | 1.2645
| 0.5 | RMSE | 2.9851 | 3.1396 | 2.9521 | 2.9361 | 2.9533 | 2.9315 | 2.8521
|  | MAPE | 0.2506 | 0.2260 | 0.2477 | 0.2375 | 0.2303 | 0.2283 | 0.2504
Sea Level Pressure |  | MAE | 452.3376 | 469.8179 | 460.6198 | 459.5086 | 457.6376 | 452.2161 | 444.3823
0.1 | RMSE | 673.5309 | 704.6300 | 722.8781 | 715.2535 | 714.0710 | 707.0870 | 696.8868
| MAPE | 0.0045 | 0.0046 | 0.0048 | 0.0047 | 0.0047 | 0.0045 | 0.0044
|  | MAE | 395.4735 | 404.9930 | 376.2283 | 374.5962 | 365.2090 | 360.5889 | 355.2410
| 0.2 | RMSE | 602.4211 | 609.5108 | 641.7902 | 606.5960 | 582.4397 | 581.1579 | 578.3478
|  | MAPE | 0.0039 | 0.0040 | 0.0038 | 0.0038 | 0.0037 | 0.0036 | 0.0035
|  | MAE | 319.3464 | 328.3074 | 309.2513 | 308.1569 | 298.5832 | 296.9049 | 293.2719
| 0.3 | RMSE | 535.5085 | 547.2752 | 547.0275 | 537.0053 | 525.3399 | 521.1263 | 513.2530
|  | MAPE | 0.0034 | 0.0034 | 0.0030 | 0.0030 | 0.0030 | 0.0030 | 0.0029
|  | MAE | 265.9806 | 274.8237 | 251.6239 | 251.0470 | 249.6740 | 246.4856 | 243.6812
| 0.4 | RMSE | 483.8443 | 476.1510 | 479.6474 | 480.5829 | 469.6758 | 461.5433 | 456.8314
|  | MAPE | 0.0028 | 0.0027 | 0.0026 | 0.0026 | 0.0025 | 0.0025 | 0.0025
|  | MAE | 222.3163 | 224.0090 | 206.3858 | 205.5674 | 200.1395 | 198.1765 | 195.2719
| 0.5 | RMSE | 408.2798 | 413.9014 | 428.3704 | 429.2268 | 411.1974 | 370.1874 | 398.6022
|  | MAPE | 0.0022 | 0.0022 | 0.0021 | 0.0021 | 0.0020 | 0.0020 | 0.0020

TABLE XIII: Performance comparisons for Entire Snapshots Sampling (P2). Red-
colored numbers are lowest errors.

##  Appendix F More results on dMRI reconstruction

For demonstration, Table X examines different sizes of TDDIP by varying the
number of layers and hidden dimensions, where TDDIP1 is the setting scoring
the lowest NRMSE, TDDIP2 and TDDIP3 are close to MultiL-KRIM[$M=7,Q=3$] and
MultiL-KRIM[$M=1,Q=3$] in number of parameters, respectively, while TDDIP4
matches the computational times of [$M=7$]. Specifically, performance of these
variations of TDDIP are reported in Table XI. Note that if the complexity of
TDDIP decreases, its NRMSE and SSIM approach those of MultiL-KRIM.
Nevertheless, increasing the number of unknowns of MultiL-KRIM, whether by
increasing the number of kernels $M$ or adopting larger inner dimensions, does
not significantly improve its performance. These findings suggest a research
direction of developing a “deep” architecture for MultiL-KRIM, e.g., by adding
non-linear activation function layers into its factorization, inspired by
results showing that deep matrix factorization outperforms the standard MMF in
matrix completion tasks [15, 16]. More results on this research front will be
reported at other publication venues.