# Adaptive Learning-Based Detection for
One-Bit Quantized Massive MIMO Systems

Yunseong Cho, Jinseok Choi†, and Brian L. Evans
†Dept. of Electrical Engineering, Ulsan National Institute of Science and
Technology
Email<EMAIL_ADDRESS>6G@UT Research Center, Wireless Networking and
Communications Group, The University of Texas at Austin
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose an adaptive learning-based framework for uplink massive multiple-
input multiple-output (MIMO) systems with one-bit analog-to-digital
converters. Learning-based detection does not need to estimate channels, which
overcomes a key drawback in one-bit quantized systems. During training,
learning-based detection suffers at high signal-to-noise ratio (SNR) because
observations will be biased to $+1$ or $-1$ which leads to many zero-valued
empirical likelihood functions. At low SNR, observations vary frequently in
value but the high noise power makes capturing the effect of the channel
difficult. To address these drawbacks, we propose an adaptive dithering-and-
learning method. During training, received values are mixed with dithering
noise whose statistics are known to the base station, and the dithering noise
power is updated for each antenna element depending on the observed pattern of
the output. We then use the refined probabilities in the one-bit maximum
likelihood detection rule. Simulation results validate the detection
performance of the proposed method vs. our previous method using fixed
dithering noise power as well as zero-forcing and optimal ML detection both of
which assume perfect channel knowledge.

###### Index Terms:

Massive MIMO, one-bit ADC, dithering, ML detection, Machine Learning, Deep
Neural Network.

##  I Introduction

Positioning large antenna arrays has been considered as one of the emerging
technologies for future communications such as massive multiple-input-
multiple-output MIMO for sub-6GHz systems [1, 2] and millimeter wave
communications [3, 4, 5]. Due to the small wavelength of mmWave signals and
small antenna spacing, the mmWave system allows the installation of more
antennas per unit area, and hence a large number of high-precision analog-to-
digital converters (ADCs) at receivers causes a prohibitively huge power
consumption, which becomes the main bottleneck in the realistic deployment
because a high-resolution ADC is particularly power-hungry as the power
consumption of an ADC is scaled exponentially with the number of quantization
bits. To overcome this issue, deploying low-precision ADCs has been evaluated
as a low-power solution over recent years [6, 7, 8, 9, 10]. The one-bit data
converter is particularly attractive due to its ability to enhance power
efficiency, lower hardware cost, and simplify analog processing in receivers
[11, 12, 13, 14, 15, 16, 17, 18, 19].

Figure 1: A receiver architecture for the training phase with a dithering
signal (in red). The BS decides whether to add intensity to the dither power
based on the observation patterns (in blue).

Several modern one-bit channel estimation, detection, and beamforming
techniques have been introduced [13, 14, 15, 16, 17, 18]. Low-complexity
symbol-level beamforming methods for one-bit quantized systems were developed
for quadrature-amplitude-modulation (QAM) constellations [18]. Taking into
account the heavily quantized signals and antenna correlation, the authors in
[13] devised an iterative multiuser detection by using a message-passing de-
quantization algorithm. For MIMO systems, the optimal maximum likelihood (ML)
detector was introduced and a near-optimal ML detector was also proposed by
transforming the ML detection problem into a tractable convex optimization
problem [15]. In [16], the authors presented successive-interference-
cancellation one-bit receiver which can be applied to modern channel coding
techniques. However, such detection methods require perfect channel state
information (CSI), which is unrealistic with one-bit quantized signals.
Various channel estimation methods were developed such as least-squares (LS),
ML, zero-forcing (ZF), and Bussgang decomposition-based methods [15, 20];
however, channel estimation with one-bit quantized signals still suffers
degradation in estimation accuracy compared to high-precision ADC systems. In
this context, we investigate a learning-based detection that replaces one-bit
channel estimation with an ML probability learning process.

Several learning-based techniques have recently been investigated [21, 22,
23]. Support vector machines [21] were used for efficient channel estimation
and data detection with one-bit quantized observations. The authors in [17]
applied sphere decoding to the one-bit quantized system to reduce complexity
while achieving near-optimal detection performance. Viewing the one-bit ADC
systems as a classification problem, various supervised-learning-based data
detection techniques were provided by estimating effective channels and
learning the non-linear system response [22]. However, such methods are
heavily influenced by channel estimation accuracy. Recently, Unlike previous
learning-based approaches that focused on developing detection mechanisms
based on channel estimation, we rather focus on applying one-bit ML detection
and learning likelihood functions to overcome the problem of the learning
process with the limited training length.

In this work, we explore an adaptive learning-based ML detection approach that
replaces one-bit channel estimation with the adaptive probability learning
process in Fig. 1. We use a dithering technique to infer likelihood functions
from the dithered signals, which significantly reduces the number of zero-
valued likelihood functions. To this end, we first add artificial dithering
noise before quantizing the received signal, and then use de-noising to
retrieve the artificial-noise-free likelihood probability. [24] We extend our
work in [24] to adapt the dithering power for each antenna element using
feedback during training. During training, we estimate the likelihood
probability with an acceptable training length as the extra randomness is
likely to trigger a change in sign in the sequence of the quantized signals.
Since the dithering-based algorithm is affected by the dithering power, we
propose a heuristic algorithm to adjust the dithering noise power depending on
the pattern of the one-bit quantized vectors. Simulation results demonstrate
that, in contrast to the conventional learning-based one-bit ML detection, the
proposed adaptive learning-based detection technique exhibits more reliable
detection performance and achieves comparable performance to the optimal one-
bit ML detection that unrealistically requires perfect CSI.

Notation: $\bf{A}$ is a matrix and $\bf{a}$ is a column vector.
$\mathbf{A}^{T}$ denotes transpose operation. We denote $a_{i}$ as the $i$th
element of $\bf a$. With mean $\mu$ and variance $\sigma^{2}$, we generate a
real Gaussian distribution and a complex Gaussian distribution using
$\mathcal{N}(\mu,\sigma^{2})$ and $\mathcal{CN}(\mu,\sigma^{2})$,
respectively. $\rm diag(\bf a)$ creates a diagonal matrix that has $a_{i}$’s
as its diagonal entries. ${\bf I}_{N}$ denotes a $N\times N$ identity matrix.
${\bf 1}_{N}$ and ${\bf 0}_{N}$ are a $N\times 1$ one vector and zero vector,
respectively. ${\rm Re}\\{{\bf A}\\}$ and ${\rm Im}\\{{\bf A}\\}$ take the
real and imaginary part of ${\bf A}$, respectively. $\mathbbm{1}{\\{A\\}}$ is
the indicator function which outputs 1 if $A$ is true, and 0 otherwise.
$\mbox{$\mathbb{E}$}[\cdot]$ is the expectation operator. For a complex-based
vector and matrix, we define real-valued expansions as $\omega({\bf
a})=\begin{bmatrix}{\rm Re}\\{{\bf a}\\}\\\ {\rm Im}\\{{\bf
a}\\}\end{bmatrix}$ and $\omega({\bf A})=\begin{bmatrix}{\rm Re}\\{{\bf
A}\\}&-{\rm Im}\\{{\bf A}\\}\\\ {\rm Im}\\{{\bf A}\\}&{\rm Re}\\{{\bf
A}\\}\end{bmatrix}$, respectively.

##  II Preliminaries

###  II-A System Model

We consider uplink multiuser MIMO communication systems where the BS has
$N_{r}$ receive antennas and concurrently communicates with $N_{u}$ single-
antenna user devices. For our massive MIMO systems, we assume $N_{r}\gg
N_{u}$. A block fading channel is assumed where a channel matrix is invariant
during $N_{c}$ time slots. We then split the uplink transmission into training
with $N_{t}$ time slots and data transmission with $N_{d}$ slots, i.e.,
$N_{c}=N_{t}+N_{d}$. During training, users transmit pilot symbols up to
$N_{t}$ symbol times in total. We use $K$ to denote the number of possible
symbol combinations that users can send together and $N_{tr}$ to represent the
number of transmissions of each signal, which implies $N_{t}\geq KN_{tr}$.

Let $\bar{\bf s}[t]\in\mbox{$\mathbb{C}$}^{N_{u}}$, $t=1,\dots,N_{c}$, denote
a symbol vector at time $t$. Then, the received signal vector at time $t$ is

$\displaystyle\bar{\bf r}[t]=\sqrt{\rho}\bar{\bf H}\bar{\bf s}[t]+\bar{\bf
z}[t],$ (1)

where $\bar{\bf H}\in\mbox{$\mathbb{C}$}^{N_{r}\times N_{u}}$ is the channel
matrix between the BS and $N_{u}$ users, whose row vector $\bar{\bf
h}_{i}^{T}$ indicates the channel vector between all users and the $i$th BS
antenna element. Let $\mbox{$\mathbb{Q}$}_{M}$ denote the set of $M$-ary QAM
constellation points from which $\bar{s}_{u}[t]$ is generated where
$\bar{s}_{u}[t]$ denotes the $u$th element of $\bar{\bf s}[t]$ and symbol of
the $u$th user. $\bar{s}_{u}[t]\in\mbox{$\mathbb{Q}$}_{M}$ has zero mean and
unit variance, i.e., $\mbox{$\mathbb{E}$}[\bar{s}_{u}]=0$ and
$\mbox{$\mathbb{E}$}[|\bar{s}_{u}[t]|^{2}]=1$. $\bar{\bf z}[t]$ is the
additive noise vector at time $n$ that follows $\mathcal{CN}\left({\bf
0}_{N_{r}},{N_{0}}{\bf I}_{N_{r}}\right)$ with noise variance $N_{0}$. We
define SNR as $\gamma=\rho/N_{0}$.

The real and imaginary components of the received signal are each quantized
with one-bit ADCs to capture the sign, i.e., $+1$ or $-1$. The quantized
signal can be represented as

$\displaystyle\bar{\bf y}[t]=\mbox{$\mathcal{Q}$}({\rm Re}\\{\bar{\bf
r}[t]\\})+j\mbox{$\mathcal{Q}$}({\rm Im}\\{\bar{\bf r}[t]\\})$ (2)

where $\mbox{$\mathcal{Q}$}(\cdot)$ is a element-wise one-bit quantizer. The
received signal in the complex-vector form $\bar{\bf r}$ can be rewritten in a
real-valued vector representation as

$\displaystyle{\bf r}[t]=\omega(\bar{\bf r}[t])=\sqrt{\rho}{\bf H}{\bf
s}[t]+{\bf z}[t]$ (3)

where ${\bf H}=\omega(\bar{\bf H})$, ${\bf s}[t]=\omega(\bar{\bf s}[t])$, and
${\bf z}[t]=\omega(\bar{\bf z}[t])$. We rewrite the quantized signal in a
real-vector form as

$\displaystyle{\bf y}[t]$ $\displaystyle=\mbox{$\mathcal{Q}$}({\bf
r}[t])=\mbox{$\mathcal{Q}$}(\sqrt{\rho}{\bf H}{\bf s}[t]+{\bf z}[t]),$ (4)

and each value $r_{i}[t]$ is quantized to be $y_{i}[t]=+1$ if $r_{i}[t]\geq 0$
or $y_{i}[t]=-1$ otherwise.

###  II-B One-Bit ML Detection with CSI

We first introduce the conventional one-bit ML detection with the full CSI.
Since each user sends one of $M$ possible signals, the number of possible
symbol vectors from $N_{u}$ users is $K=M^{N_{u}}$. We define the index set of
all possible symbol vectors as $\mbox{$\mathcal{K}$}=\\{1,\ldots,K\\}$ and use
${\bf s}_{k}$ to denote the $k$th pilot symbol vector in a real-vector form.
Let $p_{k}(\beta)$ where $\beta\in\\{-1,+1\\}$ denote the probability that
$i$th antenna component receives $\beta$ when users transmit $k$th symbol
vector. Assuming uncorrelated antennas, the likelihood probability of the
quantized signal vector ${\bf y}[t]$ for a given channel ${\bf H}$ and
transmit symbol vector ${\bf s}_{k}$ is given as

$\displaystyle\mbox{$\mathbb{P}$}({\bf y}[t]|{\bf H},{\bf
s}_{k})=\prod_{i=1}^{2N_{r}}p_{k}(y_{i}[t])$ (5)

where the likelihood function for $i$th antenna element for an observation
$y_{i}[t]\in\\{-1,+1\\}$ with the full CSI is defined as

$\displaystyle p_{k}(y_{i}[t])\\!$
$\displaystyle=\\!\mbox{$\mathbb{P}$}(y_{i}[t]|{{\bf h}}_{i},{\bf s}_{k}\\!)$
(6) $\displaystyle=\Phi\left(y_{i}[t]\psi_{k,i}\right),$ (7)

where $\psi_{k,i}=\sqrt{\frac{\rho}{N_{0}/2}}{\bf h}^{T}_{i}{\bf s}_{k}$ is
the effective channel when transmitting $k$th symbol vector and
$\Phi(x)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-{\tau^{2}}/{2}}d\tau$ is
the cumulative distribution function of a standard Gaussian distribution.
Based on (5), the one-bit ML detection rule is given as

$\displaystyle
k^{\star}[t]=\mathop{\rm\arg\\!\max}_{k\in\mbox{$\mathcal{K}$}}\prod_{i=1}^{2N_{r}}p_{k}(y_{i}[t])$
(8)

The detected symbol vector is defined as $\hat{\bf s}[t]$ = ${\bf
s}_{k^{\star}[t]}$. However, the detection rule in (8) requires full CSI to
compute (6), which is intractable when employing one-bit ADCs.

###  II-C One-bit ML Detection without CSI (Naive Approach)

Now, we outline a straightforward learning strategy that does not require
channel estimation, however, does require $N_{tr}$ training sequences. Each
pilot symbol vector ${\bf s}_{k}\in\mbox{$\mathbb{Q}$}_{M}^{Nu}$ is
transmitted $N_{tr}$ times throughout the pilot transmission of length
$N_{t}$. The BS learns likelihood functions by observing the frequency of
$y_{i}[t]=1$ and $y_{i}[t]=-1$ as

$\displaystyle\hat{p}^{(\beta)}_{k,i}\\!=\\!\begin{cases}\hat{p}^{(+1)}_{k,i}=\frac{1}{N_{tr}}\sum_{t=1}^{N_{tr}}\mathbbm{1}{\\{y_{i}[(k-1)N_{tr}+t]=1\\}}\\\
\hat{p}^{(-1)}_{k,i}=1-\hat{p}^{(+1)}_{k,i}\end{cases}$ (9)

where $\beta\in\\{+1,-1\\}$. The operation in (9) measures the number of
$+1$’s at $i$th antenna element out of the $N_{tr}$ observations triggered by
${\bf s}_{k}$. After learning the likelihood functions, the BS obtains the
estimate of the likelihood probability for a given data signal ${\bf y}[t]$ as

$\displaystyle\mbox{$\mathbb{P}$}({\bf y}[t]|{\bf H},{\bf
s}_{k})\\!\approx\\!\prod_{i=1}^{2N_{r}}\\!\\!\Big{(}\hat{p}^{(+1)}_{k,i}\mathbbm{1}\\{y_{i}[t]\\!=\\!1\\}\\!+\\!\hat{p}^{(-1)}_{k,i}\mathbbm{1}\\{y_{i}[t]\\!=\\!-1\\}\\!\Big{)},$
(10)

and the receiver can perform the ML detection in (8) by searching the best
index that maximizes (10) over the $K$ possible symbol vectors.

Although such a one-bit ML approach can provide a near-optimal detection
performance with the simple function learning, it may suffer from critical
performance degradation with a limited length of training; in the high SNR,
the $N_{tr}$ observations of each antenna repeatedly observe either $+1$ or
$-1$ due to low variance aggregate noise. This phenomenon further leads to a
number of zero empirical likelihood functions in (9), e.g.,
$\hat{p}^{(\beta)}_{k,i}=0$. This is because the one-bit quantized observation
in the high SNR becomes so deterministic that it is hard to observe a change
in the sign of the quantized output sequences during the $N_{tr}$
transmissions of the symbol vector ${\bf s}_{k}$. We name it an undertrained
likelihood function, which completely ruins the ML detection rule since any
zero probability can cancel out the whole product in (5). To overcome such a
limitation, we propose a novel learning-based one-bit ML detection method that
does not require explicit CSI and is robust to the length of the training
sequences in the following section.

##  III Adaptive Statistical Learning without CSI

We present an adaptive learning-based ML detection method for one-bit ADC
systems in order to achieve optimal ML detection performance without requiring
explicit estimation of wireless channels.

###  III-A Incremental Dither-and-Learning with $N$-steps

To resolve the problem caused by the undertrained likelihood functions, we
propose the incremental dither-and-learning (iDL) method that can learn the
likelihood functions with a reasonable $N_{tr}$ without CSI. As shown in Fig.
1, the BS appends dithering signals $d_{i}[t]$ to $r_{i}[t]$ during the
training phase. After placing the additional noise, the quantization input in
the real-vector form becomes

$\displaystyle{\bf r}_{d,k}[t]$ $\displaystyle={\bf r}_{k}[t]+{\bf d}[t]$ (11)
$\displaystyle=\sqrt{\rho}{\bf H}{\bf s}_{k}+{\bf z}[t]+{\bf d}[t].$ (12)

We use $\sigma_{d,i}^{2}$ to denote the dithering power at $i$th antenna and
assume ${\bf d}[t]\sim\mathcal{N}({\bf 0}_{2N_{r}},\boldsymbol{\Sigma})$ where
$\boldsymbol{\Sigma}={\rm
diag}(\sigma_{d,1}^{2},\ldots,\sigma_{d,2N_{r}}^{2})$ represents the
collection of dithering powers. Note that a small $\sigma_{d,i}^{2}$ still
triggers undertrained likelihood functions while a larger $\sigma_{d,i}^{2}$
hinders extracting the symbol information as noise term becomes dominant.
Then, the dithered and quantized signal associated with $k$ symbol vector
becomes

$\displaystyle{\bf y}_{d,k}[t]=\mbox{$\mathcal{Q}$}(\sqrt{\rho}{\bf H}{\bf
s}_{k}+{\bf z}[t]+{\bf d}[t])\in\\{+1,-1\\}^{2N_{r}}.$ (13)

The BS computes the estimated likelihood function for the dithered signals
$\hat{p}_{k,i}^{(\beta)}$ as in (9) for $\beta\in\\{+1,-1\\}$. Without loss of
generality, let us fix $\beta=+1$ for simplicity. Then, as shown in (6),
$\hat{p}_{k,i}^{(+1)}$ is theoretically derived as

$\displaystyle\hat{p}_{k,i}^{(+1)}\approx\Phi\left(\sqrt{\frac{2\rho}{N_{0}+\sigma^{2}}}{\bf
h}_{i}^{T}{\bf s}_{k}\right),$ (14)

Since $N_{0}$ and $\sigma_{d}^{2}$ are known to the BS and
$\tilde{p}^{(+1)}_{k,i}$ is learned from (9), the BS can find the estimate of
$\psi_{k,i}\triangleq\sqrt{\frac{\rho}{N_{0}/2}}{\bf h}_{i}^{T}{\bf s}_{k}$ by
using (14) and the de-nosing phase defined as

$\displaystyle\hat{\psi}_{k,i}$
$\displaystyle=\sqrt{\frac{\rho}{N_{0}/2}}\tilde{{\bf h}}_{i}^{T}{\bf s}_{k}$
(15)
$\displaystyle=\sqrt{1+\frac{\sigma^{2}}{N_{0}}}\Phi^{-1}\left(\hat{p}_{k,i}^{(+1)}\right).$
(16)

Finally, the BS exploits the estimated effective channel $\hat{\psi}_{k,i}$
and known or estimated $N_{0}$ to approximate the true (non-dithered)
likelihood function $p_{k,i}^{(+1)}$ by using (6) as
$\Phi\left(\hat{\psi}_{k,i}\right)$. Since the likelihood function of the
dithered signal $\hat{p}_{k,i}^{(+1)}$ in (14) is much less likely to have
zero probability compared with the non-dithered case, the BS can learn the
majority of the likelihood functions $\hat{p}^{(+1)}_{k,i}$ with a reasonable
training length.

The undertrained likelihood functions are undesirable because we lead to
$\mbox{$\mathbb{P}$}({\bf y}[t]|{\bf H},{\bf s}_{k})=0$ for many candidate
symbols ${\bf s}_{k}$, which may exclude the desired symbol from the ML
detection. However, the fixed dithering variance does not suitably adjust the
dithering power and this behavior can cause two fundamental problems: 1) When
the dithering power is low and the SNR is equivalently high, there still
exists lots of undertrained likelihood functions in spite of the increased
variance, however, the BS cannot do further; and 2) In addition, with a high
dithering power, even if the training framework rarely shows zero
probabilities, it is demanding to capture the effect of the channel because
the noise power becomes dominant. Therefore, the BS has to support an
acceptable dithering power based on the behavior of received observations. To
prevent the problems, we propose an incremental dither-and-learning (iDL)
method that fits the dithering power into a proper range.

Figure 2: Illustration of communication frame composed of a training phase
with $N$ sub-blocks and a data transmission phase. Figure 3: Illustration of
the SNR offline training via deep neural network.

As shown in Fig. 2, we first divide the $N_{tr}$ observations of ${\bf s}_{k}$
into $N$ disjoint sub-blocks in which each sub-block accommodates
$N_{tr}^{sub}=N_{tr}/N$ training samples where $N_{tr}$ has to be a multiple
of $N$. Then, the $n$th dithered and quantized sub-block observed at $i$th
antenna when transmitting ${\bf s}_{k}$ can be represented as

$\displaystyle\tilde{\bf y}_{d,k,i,n}$
$\displaystyle=\\{y_{d,k,i}\big{[}1+(k-1)N_{tr}+\left(n-1\right)N_{tr}^{sub}\big{]},$
$\displaystyle\;\ldots,y_{d,k,i}\big{[}(k-1)N_{tr}+nN_{tr}^{sub}\big{]}\\}^{T}\in\\{+1,-1\\}^{N_{tr}^{sub}},$
(17)

where $n\in\\{1,\ldots,N\\}$ and $y_{d,k,i}[t]$ denotes the dithered
observation at the $i$th antenna at time $t$ when users send $k$th symbol
vector. The ML detection is not able to have an attractive performance if any
of $p_{k,i}^{(+1)}$ and $p_{k,i}^{(-1)}$ is trained as zero, which means the
received training sequence has the pattern of all zeros or all ones.

To prohibit the same problem at $(n+1)$st sub-block, we intentionally increase
the dithering noise variance of $i$th antenna by $\Delta\sigma_{d}^{2}$ if
$n$th sub-block outputs either $+{\bf 1}_{N_{tr}^{sub}}$ or $-{\bf
1}_{N_{tr}^{sub}}$, so that the subsequent training sequence is more likely to
observe any sign changes than the previous sequence. However, if $n$th sub-
block of $i$th antenna already reports any sign transition due to the current
dithering power, it indicates that the current SNR is sufficient not to
observe zero probabilities, the BS hence no longer updates the dithering
noise. The described work can be addressed by introducing the indicator
function $\mbox{$\mathcal{I}$}_{i}$ defined for $i$th antenna and the
corresponding update operation, i.e.,
$\mbox{$\mathcal{I}$}_{i}\Delta\sigma_{d}^{2}$. The indicator
$\mbox{$\mathcal{I}$}_{i}$ is initialized and maintained as 1 and the
indicator is finally set to be zero to interrupt the further update of the
dithering level if the sub-block starts to observe the sign changes. Even
though $\mbox{$\mathcal{I}$}_{i}$ is set to zero, the indicator can revert to
one if sign flips are not observed again.

During each $n$th sub-block, we add intensity to the dithering noise by
$\mbox{$\mathcal{I}$}_{i}\Delta\sigma_{d}^{2}$. Therefore, the BS can maintain
the proper SNR region of $i$th antenna by using the update and stopping
criterion based on received observations. Upon completing all sub-blocks, the
likelihood probability of $k$-th symbol vector is determined as the mean of
the likelihood probabilities among all $N$ sub-blocks associated with symbol
vector $k$. The whole process is summarized in Algorithm 1.

1: Initialize $p_{k,i}^{(+1)}=0\;\;\forall k,i$

2: Fix the increase of the dithering variance, $\Delta\sigma_{d}^{2}$.

3: for $k=1$ to $K$ do

4: Initialize dithering variance as $\sigma_{d,i}^{2}=0$.

5: for $n=1$ to $N$ do

6: for $i=1$ to $2N_{r}$ do

7: Observe $\tilde{\bf y}_{d,k,i,n}$ for $n$th sub-block in (17)

8: Compute the $\hat{p}^{(+1)}_{k,i}$ of $\tilde{\bf y}_{d,k,i,n}$ using (9)

9: Derive $\hat{p}_{k,i}^{(+1)}$ in (14) and compute $\hat{\psi}_{k,i}$ in
(15)

10: $p_{k,i}^{(+1)}\leftarrow
p_{k,i}^{(+1)}+\Phi\left(\hat{\psi}_{k,i}\right)/N$

11: if $\tilde{\bf y}_{d,i}$ is either $+{\bf 1}_{N_{tr}^{sub}}$ or $-{\bf
1}_{N_{tr}^{sub}}$ then

12: $\mbox{$\mathcal{I}$}_{i}\leftarrow 1$

13: else

14: $\mbox{$\mathcal{I}$}_{i}\leftarrow 0$

115:
$\sigma_{d,i}^{2}\leftarrow\sigma_{d,i}^{2}+\mbox{$\mathcal{I}$}_{i}\Delta\sigma_{d}^{2}$

16: return $p_{k,i}^{(+1)}$ and $p_{k,i}^{(-1)}=1-p_{k,i}^{(+1)}\;\;\forall
k,i$

Algorithm 1 Incremental Dither-and-Learning (iDL)

As a result, the effective SNR is decreased until we meet a proper sequence to
capture the sign changes. The wild fluctuations in the output values can be
also prevented since the BS has control over the variance and the update is
supervised by the BS to fit into the appropriate SNR region. Even though the
BS keeps changing the dithering variance, we are still able to compute the
refined likelihood function using (14) and (15) since the updated variance is
also perfectly known to the BS.

###  III-B SNR Estimation

Despite of the properly controlled dithering power, the computation of
likelihood functions using 15 requires the estimated SNR $\gamma$ or noise
variance $N_{0}$ equivalently. In this work, we perform the SNR estimation
task by offline supervised learning using the deep neural network as shown in
Fig. 3. The offline training first collects training data points $\\{{\bf
y}[j];\gamma[j]\\}$ where ${\bf y}[j]\in\\{+1,-1\\}^{2N_{r}}$ is the $j$th
one-bit quantized observation and $\gamma[j]$ is the true SNR at time $j$.
Upon collecting enough samples, the BS selects a few training samples and
performs the supervised offline learning that considers ${\bf y}[j]$’s as
input and $\gamma[j]$’s as output to estimate. Assuming that there exist $L$
layers, the estimated SNR is represented as the scalar output of the neural
network expressed as

$\hat{\gamma}[j]={\bf w}_{L}^{T}{\bf x}_{L-1}+b_{L},$ (18)

where each intermediate vector is defined as ${\bf x}_{\ell}=\phi\left({\bf
W}_{\ell}{\bf x}_{\ell-1}+{\bf b}_{\ell}\right)$ for
$\ell\in\\{1,\ldots,L-1\\}$ with the initial point ${\bf x}_{0}={\bf y}[j]$
when $\phi(\cdot)$ is the element-wise activation function such as rectified
linear unit or sigmoid function. The deep neural network is updated by
minimizing the estimation error, e.g., $(\gamma[j]-\hat{\gamma}[j])^{2}$, and
hence estimates the SNR by extracting meaningful information of the one-bit
observation such as statistical pattern and the number of zeros.

##  IV Simulation Results

We evaluate the performance of the proposed learning-based method in terms of
the number of undertrained probabilities and symbol error probability (SER).
We consider $N_{r}=32$, $N_{u}=4$ with $4$-QAM modulation, Rayleigh channels
$\bar{\bf H}$ in which each element follows
$\mbox{$\mathcal{C}$}\mbox{$\mathcal{N}$}(0,1)$. We fix the dithering variance
as $\sigma_{d}^{2}=\rho/2$ and increment as $\Delta\sigma_{d}^{2}=\rho/2$.

###  IV-A Undertrained Likelihood Functions

Fig. 4 shows the average number of undertrained likelihood functions, i.e.,
$\hat{p}_{k,i}^{(b)}=0$, over the wide range of the SNR levels for the non-
dithering, dithering, and adaptive dithering cases with $N_{tr}=30$. We note
that the DL method is a special case of the iDL method when all antennas
exploit identical and fixed dithering power, i.e., $N=1$. As the SNR
increases, the number of undertrained likelihood functions for the non-
dithering case approaches $2N_{r}$. For the dithering case, however, the
number of undertrained likelihood functions slowly increases with the SNR and
converges to about $20$ due to the dithering effect. Furthermore, for the
incremental dithering case, the number of undertrained likelihood functions
approaches $17$ and $9$ when $N$ is 3 and 5, respectively. Since the iDL
method decides whether to increase the dithering noise depending on the
realization of each sub-block, we can further optimize the learning procedure
in terms of the number of undertrained likelihood functions. If we properly
increase $N$, each antenna is more likely to avoid zero-valued likelihood
probabilities. As a result, with dithering and slicing, the proposed algorithm
can estimate much more non-zero likelihood functions, thereby increasing the
detection accuracy.

Figure 4: The number of undertrained likelihood functions out of the $2N_{r}$
receivers for $N_{u}=4$ users, 4-QAM, $N_{r}=32$ antennas, and $N_{tr}=30$
with Rayleigh channels ($\sigma_{d}^{2}=\rho/2$ for DL and
$\Delta\sigma_{d}^{2}=\rho/2$ for iDL). Figure 5: Simulation results
$N_{u}=4$, 4-QAM, $N_{r}=32$, and $N_{tr}=30$ with Rayleigh channels. Figure
6: Simulation results $N_{u}=4$, 4-QAM, $N_{r}=32$, and $N_{tr}=50$ with
Rayleigh channels.

###  IV-B Data Detection

In the massive multiuser MIMO system, we compare the following one-bit
detection methods:

  1. 1.

Learning one-bit ML: naive learning-based ML

  2. 2.

Dithered learning (DL) one-bit ML [24]

  3. 3.

Incremental dithered learning (iDL) one-bit ML (proposed)

  4. 4.

Incremental dithered learning (iDL) one-bit ML with estimated SNR (proposed)

  5. 5.

One-bit zero-forcing (ZF) [14]

  6. 6.

Optimal one-bit ML

The first four methods do not require channel estimation, however, the last
two methods assume that perfect CSI is known to the BS.

Fig. 5 illustrates the SER curves for $N_{tr}=30$. The one-bit ZF detection
shows the large performance degradation in the medium to high SNR, and the
proposed method outperforms the one-bit ZF detection. We note that the
proposed dithered case closely follows the SER performance of the optimal one-
bit ML case. The performance improvement is achieved because the proposed
method provides the data-driven adaptive likelihood function learning with the
same $N_{tr}$. In addition, the proposed iDL method has around 1.0 dB gain
over the DL method by splitting the training sequence into three sub-blocks,
thereby getting closer to the optimal case. Even though the optimal ML is
implausible without explicit channel estimation, the proposed method can
achieve an impressive capability without the fundamental component. We can
also notice from both simulations that the performance gap between the iDL
method and the iDL with estimated SNR is marginal. It validates the fact that
the offline supervised learning can successfully capture the observation
pattern to estimate the SNR required for the de-noising phase.

Fig. 6 illustrates the SER curves for $N_{tr}=50$. Compared to Fig. 5, we can
observe that the learning-based algorithms are improved and the iDL methods
further approach the performance of optimal one-bit ML decoding. The naive-
learning-based one-bit ML bounces up at the higher SNR than Fig. 5, however,
the reverse trend in the high SNR still exists because $N_{tr}=50$ is not
enough to avoid undertrained likelihood probabilities.

##  V Conclusion

In this paper, we proposed a statistical learning-based one-bit ML detection
method for uplink massive MIMO communications. Since the performance of a
learning-based one-bit detection approach can be severely degraded when the
number of training samples is insufficient, the proposed method handled such
challenges by adopting the adaptive dithering craft. Without requiring channel
knowledge, the dithering-and-learning method perform ML detection through
learning likelihood functions at each antenna. The proposed method is robust
to the number of training symbols because the adaptive randomness triggers
moderate fluctuation in the change of signs of the training sequence, thereby
successfully extracting statistical pattern of one-bit quantized signals.
Simulation results demonstrate the detection performance of the proposed
method in terms of SER.

## References

  * [1] H. Q. Ngo, E. G. Larsson, and T. L. Marzetta, “Energy and spectral efficiency of very large multiuser MIMO systems,” _IEEE Trans. Commun._ , vol. 61, no. 4, pp. 1436–49, 2013.
  * [2] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” _IEEE Commun. Mag._ , vol. 52, no. 2, pp. 186–95, 2014.
  * [3] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband systems,” _IEEE Commun. Mag._ , vol. 49, no. 6, 2011.
  * [4] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. Soong, and J. C. Zhang, “What will 5G be?” _IEEE Journal Sel. Areas in Commun._ , vol. 32, no. 6, pp. 1065–82, 2014.
  * [5] R. W. Heath Jr and A. Lozano, _Foundations of MIMO Communication_. Cambridge University Press, 2018.
  * [6] C.-K. Wen, C.-J. Wang, S. Jin, K.-K. Wong, and P. Ting, “Bayes-optimal joint channel-and-data estimation for massive MIMO with low-precision ADCs,” _IEEE Trans. Sig. Process._ , vol. 64, no. 10, pp. 2541–56, 2016.
  * [7] C. Studer and G. Durisi, “Quantized massive MU-MIMO-OFDM uplink,” _IEEE Trans. Commun._ , vol. 64, no. 6, pp. 2387–99, 2016.
  * [8] J. Choi, B. L. Evans, and A. Gatherer, “Resolution-adaptive hybrid MIMO architectures for millimeter wave communications,” _IEEE Trans. Sig. Process._ , vol. 65, no. 23, pp. 6201–16, 2017.
  * [9] J. Choi, J. Sung, B. L. Evans, and A. Gatherer, “Antenna Selection for Large-Scale MIMO Systems with Low-Resolution ADCs,” _IEEE Int. Conf. Acoustics, Speech, and Signal Process._ , 2018.
  * [10] J. Choi, Y. Cho, and B. L. Evans, “Quantized massive MIMO systems with multicell coordinated beamforming and power control,” _IEEE Trans. Commun._ , vol. 69, no. 2, pp. 946–61, 2021.
  * [11] A. Mezghani and J. A. Nossek, “On ultra-wideband MIMO systems with 1-bit quantized outputs: Performance analysis and input optimization,” in _IEEE Int. Symp. Info. Theory_. IEEE, 2007, pp. 1286–89.
  * [12] J. Mo and R. W. Heath, “Capacity analysis of one-bit quantized MIMO systems with transmitter channel state information,” _IEEE Trans. Sig. Process._ , vol. 63, no. 20, pp. 5498–5512, 2015.
  * [13] S. Wang, Y. Li, and J. Wang, “Multiuser detection in massive spatial modulation MIMO with low-resolution ADCs,” _IEEE Trans. Wireless Commun._ , vol. 14, no. 4, pp. 2156–68, 2015.
  * [14] J. Choi, D. J. Love, D. R. Brown III, and M. Boutin, “Quantized Distributed Reception for MIMO Wireless Systems Using Spatial Multiplexing,” _IEEE Trans. Sig. Process._ , vol. 63, no. 13, pp. 3537–48, 2015.
  * [15] J. Choi, J. Mo, and R. W. Heath, “Near maximum-likelihood detector and channel estimator for uplink multiuser massive MIMO systems with one-bit ADCs,” _IEEE Trans. Commun._ , vol. 64, no. 5, pp. 2005–18, 2016.
  * [16] Y. Cho and S.-N. Hong, “One-Bit Successive-Cancellation Soft-Output (OSS) Detector for Uplink MU-MIMO Systems with One-Bit ADCs,” _IEEE Access_ , 2019\.
  * [17] Y.-S. Jeon, N. Lee, S.-N. Hong, and R. W. Heath, “One-bit sphere decoding for uplink massive MIMO systems with one-bit ADCs,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 7, pp. 4509–21, 2018.
  * [18] S. Park, Y. Cho, and S. Hong, “Construction of 1-bit transmit signal vectors for downlink mu-miso systems: Qam constellations,” _IEEE Trans. Veh. Techn._ , vol. 70, no. 10, pp. 10 065–76, 2021.
  * [19] C. Mollén, J. Choi, E. G. Larsson, and R. W. Heath Jr, “Uplink Performance of Wideband Massive MIMO With One-Bit ADCs,” _IEEE Trans. Wireless Commun._ , vol. 16, no. 1, pp. 87–100, 2017.
  * [20] Y. Li, C. Tao, G. Seco-Granados, A. Mezghani, A. L. Swindlehurst, and L. Liu, “Channel estimation and performance analysis of one-bit massive MIMO systems,” _IEEE Trans. Sig. Process._ , vol. 65, no. 15, pp. 4075–89, 2017\.
  * [21] L. V. Nguyen, A. L. Swindlehurst, and D. H. Nguyen, “SVM-based channel estimation and data detection for one-bit massive MIMO systems,” _IEEE Trans. Sig. Proces._ , vol. 69, pp. 2086–99, 2021.
  * [22] Y.-S. Jeon, S.-N. Hong, and N. Lee, “Supervised-Learning-Aided Communication Framework for MIMO Systems with Low-Resolution ADCs,” _IEEE Trans. Veh. Technol._ , 2018.
  * [23] Y.-S. Jeon, M. So, and N. Lee, “Reinforcement-learning-aided ML detector for uplink massive MIMO systems with low-precision ADCs,” in _IEEE Wireless Commun. and Networking Conf._ , 2018.
  * [24] J. Choi, Y. Cho, B. L. Evans, and A. Gatherer, “Robust learning-based ML detection for massive MIMO systems with one-bit quantized signals,” in _IEEE Global Commun. Conf._ , 2019, pp. 1–6.


# Alignment-based optically pumped magnetometer using a buffer gas cell

L. M. Rushton ID Electronic mail<EMAIL_ADDRESS>L. Elson ID A.
Meraki ID K. Jensen ID Electronic mail<EMAIL_ADDRESS>School
of Physics and Astronomy, University of Nottingham, University Park,
Nottingham, NG7 2RD, UK

###### Abstract

Alignment-based optically pumped magnetometers (OPMs) are capable of measuring
oscillating magnetic fields with high sensitivity in the fT/$\sqrt{\text{Hz}}$
range. Until now, alignment-based magnetometers have only used paraffin-coated
vapour cells to extend the spin relaxation lifetimes of the alkali vapour. The
drawback of these cells is that they are hand-blown and are therefore time-
intensive, and somewhat unreliable, to produce. Buffer gas cells, on the other
hand, can be manufactured on a mass scale using microfabrication techniques.
We present the first demonstration of an alignment-based magnetometer using a
buffer gas vapour cell containing caesium (Cs) alkali vapour and nitrogen
(N${}_{\text{2}}$) buffer gas. The OPM is operated at $55\degree$C and we
achieve a $325$ fT/$\sqrt{\text{Hz}}$ sensitivity to 10 kHz oscillating
magnetic fields with an 800 Hz bandwidth. The alignment-based magnetometer
uses a single laser beam for optical pumping and probing and could potentially
allow for more rapid commercialisation of radio-frequency OPMs, due to the
robustness of the one-beam geometry and the potential for mass-scale
microfabrication of buffer gas cells.

††preprint: AIP/123-QED

##  I Introduction

Optically pumped magnetometers (OPMs) labyt_sander_tilmann_wakai_2022 ;
auzinsh_budker_rochester_2014 ; budker_romalis_2007  based on spin-polarized
atoms (e.g. alkali atoms such as caesium (Cs) or rubidium) can measure
magnetic fields with high sensitivity in the fT/$\sqrt{\text{Hz}}$ range
kominis_kornack_allred_romalis_2003 ; Wasilewski2010 ; Chalupczak2012 ;
yao_maddox_renzoni_2022 . Current commercial OPMs quspin ; fieldline ;
twinleaf  are operated close to zero magnetic field in the spin-exchange
relaxation-free (SERF) regime measuring one, two or three components of the
magnetic field, or in the Earth’s field as scalar magnetometers measuring the
total magnetic field amplitude. These OPMs use one or two beams of circularly
polarized light generated from a single laser diode inside the OPM, making the
sensors compact and robust. The circularly polarized light effectively
generates spin-orientation along the light propagation (i.e., the atomic spins
point in a certain direction) which responds to magnetic fields and can be
measured by detecting the transmitted light. When detecting oscillating
magnetic fields in the kHz-MHz frequency range, radio-frequency (RF) OPMs
savukov_seltzer_romalis_sauer_2005 ; auzinsh_budker_rochester_2014 ;
rochester_thesis ; deans_marmugi_renzoni_2018_subpT_unshielded_tabletop ;
deans_cohen_yao_maddox_vigilante_renzoni_2021 ;
dhombridge_claussen_iivanainen_schwindt_2022  must be used. One type of RF OPM
using only a single laser beam is the alignment-based magnetometer
ledbetter_acosta_rochester_budker_pustelny_yashchuk_2007 ; zigdon_wilson-
gordon_guttikonda_bahr_neitzke_rochester_budker_2010 ;
atomicdensitymatrix_nonlinear , which uses linearly polarized light capable of
effectively aligning the atoms in the direction perpendicular to its
propagation. As a result, as the RF field affects such alignment being
created, its presence can be sensed directly by measuring properties of the
same beam.

High sensitivity optical magnetometry requires a long atomic spin-coherence
time. This can be achieved using vapour cells coated on the inside with an
anti-relaxation coating (e.g. paraffin), such that the moving alkali atoms can
bounce off the inner glass walls of the vapour cell many times without losing
their spin-coherence Balabas2010prl ; Li2017 . Alternatively, a long coherence
time can be achieved by filling the vapour cell with buffer gas (e.g.
N${}_{\text{2}}$). Rapid collisions between the buffer gas atoms and the
alkali atoms make the alkali atoms diffuse slowly, which mitigates the effects
of spin-destroying wall collisions. Alkali vapour cells for magnetometry are
typically hand-blown, however buffer gas cells for magnetometry can be
produced on a mass scale using microfabrication techniques Shah2007natphot ;
Dyer_2022 . Such microfabrication techniques have not, as of yet, been
compatible with anti-relaxation coating.

So far, alignment-based optical magnetometry has been demonstrated using hand-
blown, anti-relaxation coated cells
ledbetter_acosta_rochester_budker_pustelny_yashchuk_2007 ; zigdon_wilson-
gordon_guttikonda_bahr_neitzke_rochester_budker_2010 . The presence of buffer
gas leads to pressure broadening of the alkali vapour absorption spectrum,
reducing the light-atom coupling and affecting the optical pumping preparing
the aligned state. The buffer gas N${}_{\text{2}}$ is also a quenching gas
seltzer_thesis  which causes the alkali atoms not to de-excite via spontaneous
emission. Rapid collisional mixing in the excited state seltzer_thesis  also
occurs in buffer gas cells, but not in paraffin-coated cells. We show here
that, despite these complexities, it is possible to realise an alignment-based
magnetometer using a buffer gas cell. We experimentally demonstrate an
alignment-based magnetometer using a Cs alkali vapour and 65 Torr
N${}_{\text{2}}$ buffer gas cell with a sensitivity of $325$
fT/$\sqrt{\text{Hz}}$ to oscillating magnetic fields at 10 kHz. We also
demonstrate an alignment-based magnetometer with a paraffin-coated cell placed
in the same experimental setup to verify the methods and for comparison. Our
results open up the possibility for miniaturisation
dhombridge_claussen_iivanainen_schwindt_2022 ; rushton_2022  and
commercialisation of RF OPMs, with potential impact in areas such as medical
physics deans_marmugi_renzoni_2020 ;
jensen_zugenmaier_arnbak_staerkind_balabas_polzik_2019 ; marmugi_renzoni_2016
, remote sensing rushton_2022 ; deans_marmugi_renzoni_2018  and non-
destructive testing bevington_gartman_chalupczak_2021 ;
bevington_gartman_chalupczak_2019 .

##  II Alignment-based optical magnetometry

The theory underpinning the alignment-based magnetometer
auzinsh_budker_rochester_2014 ; zigdon_wilson-
gordon_guttikonda_bahr_neitzke_rochester_budker_2010 ; rochester_thesis ;
atomicdensitymatrix_nonlinear  will now be revised and discussed. Consider
atoms with a $F=1\rightarrow F^{\prime}=0$ optical transition with groundstate
sublevels $|F,m\rangle=\left\\{|1,1\rangle,|1,0\rangle,|1,-1\rangle\right\\}$
and an excited state
$|F^{\prime},m^{\prime}\rangle=|0^{\prime},0^{\prime}\rangle$ as shown in Fig.
1(a). A single laser beam is used for optical pumping and probing of the
atoms. Assume the light is linearly ($\pi$) polarized along the direction of a
static magnetic field $B_{0}\hat{\textbf{z}}$. In this case, the atoms will be
optically pumped into the $m=\pm 1$ sublevels with equal probability, creating
a so-called “spin-aligned state”. This is a dark state, such that with perfect
optical pumping, the light will be fully transmitted through the atomic
vapour. Now assume further that there is a transverse oscillating (RF)
magnetic field which we would like to detect. That RF field will affect the
optical pumping and thereby the transmitted light which can be detected by
measuring its intensity or polarization.

Figure 1:  (a) Energy level diagram with a $F=1$ ground state and
$F^{\prime}=0$ excited state. The neighbouring groundstate sublevels are each
split by the Larmor frequency $\Omega_{L}$. Linearly polarised ($\pi$) light
drives transitions between $m=0$ and $m^{\prime}=0$. (b) D1 line of Cs with
$F=3,4$ ground states and $F^{\prime}=3,4$ excited states. Linearly polarised
($\pi$) light drives transitions between $F=4$ and $F^{\prime}=3$. Small
corrections ($\delta$) for the non-linear Zeeman splitting are shown.

The total Hamiltonian $\hat{H}$ which describes the system is given by

$\hat{H}=\hat{H}_{0}+\hat{H}_{l}+\hat{H}_{B},$ (1)

where $\hat{H}_{0}$, $\hat{H}_{l}$ and $\hat{H}_{B}$ are the unperturbed,
light-atom interaction and magnetic field-atom interaction Hamiltonians,
respectively. The unperturbed Hamiltonian $\hat{H}_{0}$ written in the overall
basis
$\\{\ket{1,1}\equiv\ket{1},\ket{1,0}\equiv\ket{0},\ket{1,-1}\equiv\ket{-1},\ket{0^{\prime},0^{\prime}}\equiv\ket{0^{\prime}}\\}$
is

$\begin{split}\hat{H}_{0}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&\hbar\omega_{0}\end{pmatrix}\end{split}$ (2)

where $\hbar$ is the reduced Planck constant, $\omega_{0}=2\pi c/\lambda$ is
the optical transition frequency, $\lambda$ its wavelength and $c$ the speed
of light. The light-atom interaction is governed by

$\hat{H}_{l}=-\textbf{E}\cdot\hat{\textbf{d}},$ (3)

where $\hat{\textbf{d}}$ is the dipole operator and
$\textbf{E}=E_{0}\cos(\omega t)\hat{\textbf{z}}$ is the electric field of the
light. The light-atom interaction Hamiltonian $\hat{H}_{l}$ is

$\hat{H}_{l}=\hbar\Omega_{R}\cos\omega t\begin{pmatrix}0&0&0&0\\\ 0&0&0&-1\\\
0&0&0&0\\\ 0&-1&0&0\end{pmatrix},$ (4)

where $\Omega_{R}=\braket{1}{d}{0^{\prime}}E_{0}/(\sqrt{3}\hbar)$ is the Rabi
frequency and $\braket{1}{d}{0^{\prime}}$ is the transition dipole matrix
element. Assuming $B_{x}=B_{\text{RF}}\cos\omega_{\text{RF}}t$, $B_{y}=0$ and
$B_{z}=B_{0}$, the magnetic field-atom interaction is given by

$\begin{split}\hat{H}_{B}&=-\hat{\bm{\mu}}\cdot\textbf{B}=\frac{g_{F}\mu_{B}}{\hbar}(\hat{F}_{x}B_{x}+\hat{F}_{z}B_{z})\\\
&=g_{F}\mu_{B}\begin{pmatrix}B_{0}&\frac{B_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&0&0\\\
\frac{B_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&0&\frac{B_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&0\\\
0&\frac{B_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&-B_{0}&0\\\
0&0&0&0\end{pmatrix}.\end{split}$ (5)

Here
$\hat{\bm{\mu}}=g_{F}\mu_{B}(\hat{F}_{x}\hat{\textbf{x}}+\hat{F}_{y}\hat{\textbf{y}}+\hat{F}_{z}\hat{\textbf{z}})/\hbar$
is the Cs atom’s magnetic dipole operator, $g_{F}$ is the hyperfine Lande
g-factor steck , and $\mu_{B}$ is the Bohr magneton. Defining the Larmor
frequency $\Omega_{L}=g_{F}\mu_{B}B_{0}/\hbar$ and letting the strength of the
RF field be represented by
$\Omega_{\text{RF}}=g_{F}\mu_{B}B_{\text{RF}}/\hbar$, the total Hamiltonian
$\hat{H}=\hat{H}_{0}+\hat{H}_{B}+\hat{H}_{l}$ is

$\begin{split}\hat{H}=\hbar\begin{pmatrix}\Omega_{L}&\frac{\Omega_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&0&0\\\
\frac{\Omega_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&0&\frac{\Omega_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&-\frac{\Omega_{R}\cos\omega
t}{\sqrt{3}}\\\
0&\frac{\Omega_{\text{RF}}\cos\omega_{\text{RF}}t}{\sqrt{2}}&-\Omega_{L}&0\\\
0&-\frac{\Omega_{R}\cos\omega
t}{\sqrt{3}}&0&\omega_{0}\end{pmatrix}.\end{split}$ (6)

Going to a rotating frame at the optical frequency $\omega$, followed by going
to another rotating frame at the RF frequency $\omega_{\text{RF}}$, then
neglecting the fast oscillating terms using the rotating wave approximation
and setting $\Delta=\omega-\omega_{0}$,
$\Delta_{\text{RF}}=\omega_{\text{RF}}-\Omega_{L}$, the Hamiltonian
$\hat{\tilde{H}}$ in the rotating frame is

$\begin{split}\hat{\tilde{H}}=\hbar\begin{pmatrix}-\Delta_{\text{RF}}&\frac{\Omega_{\text{RF}}}{2\sqrt{2}}&0&0\\\
\frac{\Omega_{\text{RF}}}{2\sqrt{2}}&0&\frac{\Omega_{\text{RF}}}{2\sqrt{2}}&-\frac{\Omega_{R}}{2\sqrt{3}}\\\
0&\frac{\Omega_{\text{RF}}}{2\sqrt{2}}&\Delta_{\text{RF}}&0\\\
0&-\frac{\Omega_{R}}{2\sqrt{3}}&0&-\Delta\end{pmatrix}.\end{split}$ (7)

Next the relaxation $\hat{\Gamma}$ and repopulation $\hat{\Lambda}$ matrices
must be taken into account and are given by

$\hat{\Gamma}=\begin{pmatrix}\gamma&0&0&0\\\ 0&\gamma&0&0\\\ 0&0&\gamma&0\\\
0&0&0&\gamma+\Gamma\end{pmatrix},$ (8)

$\hat{\Lambda}=\begin{pmatrix}\frac{\gamma+\Gamma\rho_{0^{\prime},0^{\prime}}}{3}&0&0&0\\\
0&\frac{\gamma+\Gamma\rho_{0^{\prime},0^{\prime}}}{3}&0&0\\\
0&0&\frac{\gamma+\Gamma\rho_{0^{\prime},0^{\prime}}}{3}&0\\\
0&0&0&0\end{pmatrix}.$ (9)

The atoms in the excited state decay to the ground state sublevels at a rate
$\Gamma$ and are assumed to repopulate the three ground state sublevels in
equal proportion $\Gamma\rho_{0^{\prime},0^{\prime}}/3$, where
$\rho_{0^{\prime},0^{\prime}}$ is the population of the excited state. For a
paraffin-coated cell, the excited atoms decay via spontaneous emission (at the
rate $2\pi(4.56~{}\text{MHz})$ steck  for Cs atoms in the first excited
state). For a buffer gas cell, the excited atoms mainly decay via quenching,
as discussed in the next section. Note that the model above only includes one
excited state and therefore does not describe collisional mixing (between
multiple excited states). The atoms also have a spin-coherence (or transverse
relaxation) time $T_{2}=1/\gamma$. In a buffer gas cell, the alkali atoms
diffuse slowly due to collisions with the buffer gas, increasing $T_{2}$. The
spins relax when the alkali atoms hit the glass walls due to electron
randomisation collisions or via spin-exchange or spin-destruction collisions
between two alkali atoms
graf_kimball_rochester_kerner_wong_budker_alexandrov_balabas_yashchuk_2005 ;
ledbetter_acosta_rochester_budker_pustelny_yashchuk_2007 ;
labyt_sander_tilmann_wakai_2022 . In a paraffin-coated cell the alkali atoms
can bounce off the walls thousands of times before spin relaxation occurs
Balabas2010prl . Power broadening due to laser light also reduces $T_{2}$.

The Liouville equation for the density matrix $\hat{\tilde{\rho}}$ in the
rotating frame is given by

$i\hbar\frac{\partial\hat{\tilde{\rho}}}{\partial
t}=[\hat{\tilde{H}},\hat{\tilde{\rho}}]-i\hbar\frac{1}{2}(\hat{\Gamma}\hat{\tilde{\rho}}+\hat{\tilde{\rho}}\hat{\Gamma})+i\hbar\hat{\Lambda}.$
(10)

In the steady state $d\hat{\tilde{\rho}}/dt=0$ and the right-hand-side of the
equation can be solved to determine $\hat{\tilde{\rho}}$ in the rotating
frame. The density matrix is returned to the lab frame $\hat{\rho}$ by using a
transformation matrix. The polarisation
$\textbf{P}=n\text{Tr}(\hat{\rho}\hat{\textbf{d}})$ of the atomic vapour can
then be calculated, where $n$ is the alkali atom number density. The formalism
in Ref. auzinsh_budker_rochester_2014  allows for the in-phase and out-of-
phase rotations of a linearly polarised beam to be extracted zigdon_wilson-
gordon_guttikonda_bahr_neitzke_rochester_budker_2010 ;
atomicdensitymatrix_nonlinear . The expressions for the in-phase
$\partial\phi^{\text{in}}/\partial l$ and quadrature
$\partial\phi^{\text{out}}/\partial l$ values are zigdon_wilson-
gordon_guttikonda_bahr_neitzke_rochester_budker_2010

$\displaystyle\frac{\partial\phi^{\text{in}}}{\partial l}$
$\displaystyle=\frac{n\Delta_{\text{RF}}\lambda^{2}\Omega_{\text{RF}}(2\gamma^{2}+8\Delta_{\text{RF}}^{2}-\Omega_{\text{RF}}^{2})\Omega_{R}^{2}}{36\pi\Gamma\gamma(\gamma^{2}+4\Delta_{\text{RF}}^{2}+\Omega_{\text{RF}}^{2})[4(\gamma^{2}+\Delta_{\text{RF}}^{2})+\Omega_{\text{RF}}^{2}]}$
$\displaystyle\approx\frac{n\lambda^{2}}{72\pi}\cdot\frac{\Omega_{R}^{2}}{\Gamma}\cdot\Omega_{\text{RF}}\cdot\frac{\Delta_{\text{RF}}/\gamma}{\Delta_{\text{RF}}^{2}+\gamma^{2}}\quad\mathrm{for}\;\;\Omega_{\text{RF}}^{2}\ll\gamma^{2},$
(11) $\displaystyle\frac{\partial\phi^{\text{out}}}{\partial l}$
$\displaystyle=\frac{n\lambda^{2}\Omega_{\text{RF}}(4\gamma^{2}+16\Delta_{\text{RF}}^{2}+\Omega_{\text{RF}}^{2})\Omega_{R}^{2}}{72\pi\Gamma(\gamma^{2}+4\Delta_{\text{RF}}^{2}+\Omega_{\text{RF}}^{2})[4(\gamma^{2}+\Delta_{\text{RF}}^{2})+\Omega_{\text{RF}}^{2}]}$
$\displaystyle\approx\frac{n\lambda^{2}}{72\pi}\cdot\frac{\Omega_{R}^{2}}{\Gamma}\cdot\Omega_{\text{RF}}\cdot\frac{1}{\Delta_{\text{RF}}^{2}+\gamma^{2}}\quad\mathrm{for}\;\;\Omega_{\text{RF}}^{2}\ll\gamma^{2},$
(12)

where $l$ is the length of the vapour cell. In the limit when
$\Omega_{\text{RF}}^{2}\ll\gamma^{2}$, as is the case throughout this paper,
$\partial\phi^{\text{in}}/\partial l$ and $\partial\phi^{\text{out}}/\partial
l$ are proportional to the RF magnetic field amplitude
$B_{\text{RF}}\propto\Omega_{\text{RF}}$ and have dispersive- and absorptive-
Lorentzian lineshapes, respectively, when varying the RF detuning
$\Delta_{\text{RF}}$. The light polarization rotation is measured using a
balanced photodetector and lock-in detection (at the RF frequency), yielding
the lock-in outputs which can be written as

$\displaystyle X\propto$
$\displaystyle\frac{\partial\phi^{\text{out}}}{\partial l}\propto
B_{\text{RF}}\cdot\frac{1}{\left(\omega_{\text{RF}}-\Omega_{L}\right)^{2}+\gamma^{2}},$
(13) $\displaystyle Y\propto$
$\displaystyle\frac{\partial\phi^{\text{in}}}{\partial l}\propto
B_{\text{RF}}\cdot\frac{\left(\omega_{\text{RF}}-\Omega_{L}\right)/\gamma}{\left(\omega_{\text{RF}}-\Omega_{L}\right)^{2}+\gamma^{2}},$
(14) $\displaystyle R=$ $\displaystyle\sqrt{X^{2}+Y^{2}}=|X+iY|\propto
B_{\text{RF}}\cdot\left|\frac{1+i\left(\omega_{\text{RF}}-\Omega_{L}\right)/\gamma}{\left(\omega_{\text{RF}}-\Omega_{L}\right)^{2}+\gamma^{2}}\right|.$
(15)

##  III Caesium

###  III.1 Optical pumping in a paraffin-coated cell

A caesium (Cs) atom steck  has two ground states with hyperfine quantum
numbers $F=3$ and $F=4$ separated by the hyperfine splitting
$\nu_{\text{hf}}=9.192$ GHz. The first excited states have $F^{\prime}=3$ and
$F^{\prime}=4$ with a hyperfine splitting of 1.2 GHz. The optical transition
of interest for this experiment is the Cs D1 $F=4\rightarrow F^{\prime}=3$
transition using $z$-linearly polarised light (see Fig. 1b) with a wavelength
around 895 nm. The optical pumping can be understood by determining the
populations of the 16 magnetic sublevels of the Cs ground state using rate
equations eckel2022 . We will first consider a paraffin-coated cell, where the
decay from the excited state is by spontaneous emission. An example of the
rate of change of the population of the magnetic sublevel $F=3,m=3$,
$dp_{\text{F=3,m=3}}/dt$, i.e., the diagonal element of the density matrix, is

$\begin{split}\frac{dp_{\text{4,3}}}{dt}=&~{}R_{p}(-p_{\text{4,3}}c_{4,3\leftrightarrow
3^{\prime},3^{\prime}}+p_{\text{4,2}}c_{4,2\leftrightarrow
3^{\prime},2^{\prime}}c_{4,3\leftrightarrow 3^{\prime},2^{\prime}}\\\
&+p_{\text{4,3}}c_{4,3\leftrightarrow
3^{\prime},3^{\prime}}c_{4,3\leftrightarrow
3^{\prime},3^{\prime}})-\Gamma_{1}p_{4,3}+\Gamma_{1}/16,\end{split}$ (16)

where $R_{p}$ is the optical pumping rate, $p_{4,3}=p_{4,3}(t)$ is the
population of the magnetic sublevel at time $t$, and $c_{4,2\leftrightarrow
3^{\prime},2^{\prime}}$ is the Clebsch-Gordon coefficient squared steck  for
the $\pi$ transition from $F=4,m=2\leftrightarrow F^{\prime}=3,m^{\prime}=2$,
for example. The longitudinal relaxation rate $\Gamma_{1}$ was not measured
experimentally in this work but is typically much smaller than the transverse
relaxation rate $\gamma$ julsgaard_2003 . The negative terms in Eq. 16
depopulate the magnetic sublevel and the positive terms repopulate the
sublevel. The populations of the 16 magnetic sublevels in the $F=3$ and $F=4$
ground states in the steady state ($dp/dt=0$) are determined numerically and
we plot these in Fig. 2a. We see that the atoms in the $F=4$ sublevels are
symmetrically distributed with most atoms in the $m=\pm 4$ sublevels,
corresponding to a spin-aligned state. Note that many atoms are “lost” to the
other ground state $F=3$. These atoms are not probed as they are 9.192 GHz
detuned from the light.

Figure 2:  Optical pumping from $F=4\rightarrow F^{\prime}=3$ with
$\pi$-polarised light. The populations of the $F=3$ and $F=4$ ground state
magnetic sublevels in the steady state are plotted, with a longitudinal
relaxation rate $\Gamma_{1}=R_{p}/20$ for (a) a paraffin-coated cell where the
dominant de-excitation mechanism from the excited state is spontaneous
emission, and (b) a buffer gas cell where the dominant de-excitation mechanism
is quenching.

###  III.2 Optical pumping in a buffer gas cell

If a buffer gas such as 65 Torr of N${}_{\text{2}}$ is present in a Cs vapour
cell without any paraffin coating, then the Cs atoms will mostly decay via
quenching rather than via spontaneous emission seltzer_romalis_2009 ;
seltzer_thesis , as will now be shown. The many vibrational and rotational
states of the quenching gas molecule, in this case N${}_{\text{2}}$, mean that
when a Cs atom in the excited state collides with a N${}_{\text{2}}$ molecule,
the Cs atom can de-excite without the emission of a photon, instead
transferring its energy to the many vibrational and rotational modes of the
N${}_{\text{2}}$ molecule. The quenching rate $R_{Q}$ is given by

$R_{Q}=n_{Q}\sigma_{Q}v_{\text{Cs,N}_{2}},$ (17)

where $n_{Q}=P/(k_{B}T)=1.91\times 10^{24}$ m-3 is the number density of
N${}_{\text{2}}$ molecules at $T\sim 55\degree$C, $P$ is the pressure, $k_{B}$
is the Boltzmann constant, $\sigma_{Q}=5.5\times 10^{-19}$ m2 seltzer_thesis
is the quenching gas cross-section for Cs and N${}_{\text{2}}$ (at
100$\degree$C) and $v_{\text{Cs,N}_{2}}=\sqrt{8k_{B}T/\pi M}=548$ m/s is the
relative velocity between a Cs atom and N${}_{\text{2}}$ molecule. The mass
$M=3.84\times 10^{-26}$ kg is the effective mass of a Cs atom and
N${}_{\text{2}}$ molecule, given by
$M=m_{\text{Cs}}m_{\text{N}_{2}}/\left(m_{\text{Cs}}+m_{\text{N}_{2}}\right)$.
The quenching factor $Q$ helps determine the dominant decay mechanism, whether
by spontaneous emission ($Q=1$) or by quenching ($Q=0$), and is given by
seltzer_thesis

$Q=\frac{1}{1+R_{Q}\tau_{\text{nat}}}.$ (18)

Calculating $R_{Q}=5.9\times 10^{8}$ s-1 from the parameters stated above for
Cs and 65 Torr N${}_{\text{2}}$ and taking the natural lifetime of the D1
excited state to be $\tau_{\text{nat}}=35$ ns, then $Q=0.05$. This means that,
for the 65 Torr N${}_{\text{2}}$ buffer gas cell used in our experiments, the
dominant de-excitation mechanism from the excited state is quenching. During
quenching, the decay probabilities to the ground states are not governed by
the Clebsch-Gordon coefficients. Instead the atoms decay with equal
probability (1/16) to any of the $F=3$ and $F=4$ ground state magnetic
sublevels. Crucially, though, the $F=4,m=\pm 4$ states will still be dark
states in the presence of N${}_{\text{2}}$, a quenching gas. An example of a
rate equation for the population $dp_{4,3}/dt$ of the $F=4,m=3$ magnetic
sublevel is given by

$\begin{split}\frac{dp_{4,3}}{dt}=&~{}R_{p}(-p_{4,3}c_{4,3\leftrightarrow
3^{\prime},3^{\prime}}+\frac{1}{16}[p_{4,3}c_{4,3\leftrightarrow
3^{\prime},3^{\prime}}\\\ &~{}+p_{4,2}c_{4,2\leftrightarrow
3^{\prime},2^{\prime}}+p_{4,1}c_{4,1\leftrightarrow
3^{\prime},1^{\prime}}+p_{4,0}c_{4,0\leftrightarrow 3^{\prime},0^{\prime}}\\\
&~{}+p_{4,-1}c_{4,-1\leftrightarrow
3^{\prime},-1^{\prime}}+p_{4,-2}c_{4,-2\leftrightarrow
3^{\prime},-2^{\prime}}\\\ &~{}+p_{4,-3}c_{4,-3\leftrightarrow
3^{\prime},-3^{\prime}}])-\Gamma_{1}p_{4,3}+\frac{\Gamma_{1}}{16}.\end{split}$
(19)

The 16 rate equations are solved in the steady state and an illustrative
example of optical pumping with a buffer gas is shown in Fig. 2b. We see that
the distribution of atoms in the ground state sublevels is similar for both
buffer gas and paraffin-coated vapour cells. It is assumed that $Q=0$, which
is a safe assumption to make for the 65 Torr N${}_{\text{2}}$ buffer gas cell
in this paper. In the above we assumed that the excited states $F^{\prime}=3$
and $F^{\prime}=4$ are resolved such that the light is only resonant with the
$F=4\rightarrow F^{\prime}=3$ transition. We note, however, that the
$F^{\prime}=4$ excited state would need to be incorporated into the rate
equations if the buffer gas pressure becomes significantly larger, as
discussed in Sec. V.

###  III.3 Non-linear Zeeman splitting

A Cs atom in the $F=4$ ground state has $2F+1=9$ sublevels $|F,m\rangle$
which, when placed in a small magnetic field $B_{0}$, have the energy
$E(m)=mh\nu_{L}$ due to the linear Zeeman effect. Here $\nu_{L}$ is the Larmor
frequency in Hz. That is to say, the splittings between neighbouring sublevels
are all equal to the Larmor frequency
$\Delta\nu_{m,m-1}\equiv\left(E(m)-E(m-1)\right)/h=\nu_{L}$. In this case, a
single magnetic resonance will be observed when sweeping the RF frequency
$\nu_{\text{RF}}$ (in Hz) across the Larmor frequency $\nu_{L}$ and measuring
the polarization rotation of the transmitted light (see Eq. 13, 14 and 15).
However, at larger magnetic fields, the splittings between sublevels are
slightly different due to the non-linear Zeeman effect. We calculate
julsgaard_2003 ; bao_wickenbrock_rochester_zhang_budker_2018 ; steck

$\Delta\nu_{m,m-1}=\nu_{L}-\delta\left(m-\frac{1}{2}\right),$ (20)

where the non-linear Zeeman splitting (in Hz) is

$\delta=\frac{2\nu_{L}^{2}}{\nu_{\text{hf}}}$ (21)

as illustrated in Fig. 1(b). In particular, the difference in transition
frequencies between $\Delta\nu_{4,3}$ and $\Delta\nu_{-3,-4}$ is

$\left|\Delta\nu_{4,3}-\Delta\nu_{-3,-4}\right|=7\delta.$ (22)

In other words, at larger magnetic fields a total of 8 magnetic resonances
should be observed when sweeping the RF field across the Larmor frequency with
the outermost resonances split by $7\delta$.

##  IV Paraffin-coated cell

A schematic of the experimental setup is shown in Fig. 3. A diode laser system
outputs light resonant with the $F=4\rightarrow F^{\prime}=3$ Cs D1 transition
(895 nm). The light is passed through an optical fiber and is collimated at
its output. The linearly polarised light with an electric field amplitude
$E_{0}\hat{\textbf{z}}$ then passes through a cubic (5 mm)3 hand-blown
paraffin-coated vapour cell (see Fig. 3). The vapour cell is kept at room
temperature ($\sim 18.5\degree$C) and placed inside a magnetic shield
(Twinleaf MS-1). Static $B_{0}\hat{\textbf{z}}$ and oscillating
$B_{\text{RF}}\cos(2\pi\nu_{\text{RF}}t)\hat{\textbf{x}}$ magnetic fields can
be applied using coils inside the magnetic shield. Here $\nu_{\text{RF}}$ is
the RF frequency in Hz, while $\omega_{\text{RF}}=2\pi\nu_{\text{RF}}$ is the
RF frequency in rad/s. Polarimetry is then performed using a half-wave plate,
a polarising beam splitter, and a balanced photodetector (Thorlabs PDB210A/M)
to detect the polarization rotation of the transmitted light. The resultant
photodetector voltage is demodulated at the RF frequency $\nu_{\text{RF}}$
using a lock-in amplifier (SR830) such that in-phase $X$ and out-of-phase $Y$
signals are obtained.

Figure 3: (a) Schematic of an alignment-based magnetometer. The laser light
propagates along the $y$-direction and is $z$-polarised. Components include:
half-wave plates ($\lambda/2$), polarising beam splitters (PBS), a vapour cell
(Cell) and a balanced photodetector (BPD). Static
$\textbf{B}_{0}=B_{0}\hat{\textbf{z}}$ and oscillating magnetic fields
$\textbf{B}_{\text{RF}}(t)=B_{\text{RF}}(t)\hat{\textbf{x}}$ are applied at
the position of the vapour cell. (b) Photo of the paraffin-coated cell. (c)
Photo of the buffer gas cell. (d) Photo of the buffer gas cell surrounded by a
Shapal ceramic cylinder, heating wires and Kapton tape. Figure 4:  Non-linear
Zeeman splitting of the magnetic resonances using a paraffin-coated cell. The
magnitude $R$ is fitted to Eq. 23. The fit is included as a dotted line. The
magnetic resonances for $m=4\rightarrow m=3$ and $m=-3\rightarrow m=-4$, with
different Larmor frequencies, are indicated.

The optical pumping of an aligned state can be experimentally verified by
exploiting the non-linear Zeeman effect. These measurements were done at a
relatively large static magnetic field ($B_{0}=5.84$ G) corresponding to a
Larmor frequency close to 2 MHz. When the RF frequency was swept over the
range 2.037-2.051 MHz, we observe a magnetic resonance spectrum with several
peaks (see Fig. 4). The two largest peaks correspond to the transitions
$m=4\rightarrow m=3$ and $m=-3\rightarrow m=-4$ with transition frequencies
$\Delta\nu_{4,3}$ and $\Delta\nu_{-3,-4}$, respectively. The difference in
transition frequencies $|\Delta\nu_{4,3}-\Delta\nu_{-3,-4}|$ is experimentally
found to be $6.38(0.02)$ kHz, agreeing with the value $7\delta=6.37$ kHz
calculated from Eqs. 21 and 22, confirming that we are observing the non-
linear Zeeman splitting. This difference in transition frequencies was
extracted by fitting the data of $R$ in Fig. 4 to the function julsgaard_2003

$\text{R}=\left|\sum_{m=-3}^{4}\frac{A_{m,m-1}\left[1+i\left(\nu_{\text{RF}}-\nu_{m,m-1}\right)/\tilde{\gamma}\right]}{\left(\nu_{\text{RF}}-\nu_{m,m-1}\right)^{2}+\tilde{\gamma}^{2}}\right|.$
(23)

which is a sum of eight magnetic resonances with resonance frequencies
$\nu_{m,m-1}=\nu_{L}-\delta\left(m-\frac{1}{2}\right)$ and half width at half
maximum (HWHM) $\tilde{\gamma}=1/(2\pi T_{2})$ (in Hz) as seen by comparison
with Eq. 15 and illustrated in Fig. 1. The data was fitted with seven free
parameters: four amplitudes $A_{4,3}$, $A_{3,2}$, $A_{2,1}$, $A_{1,0}$ (as the
magnetic resonance spectrum is symmetric such that $A_{0,-1}=A_{1,0}$,
$A_{-1,-2}=A_{2,1}$, $A_{-2,-3}=A_{3,2}$, $A_{-3,-4}=A_{4,3}$), the Larmor
frequency $\nu_{L}$, the non-linear Zeeman splitting $\delta$, and the width
$\tilde{\gamma}$.

In total, the spectrum has eight peaks, although the middle two are hardly
visible in Fig. 4 due to their smaller height. The height of the individual
peaks corresponding to $A_{m,m-1}/\tilde{\gamma}^{2}$ in Eq. 23 are
proportional to the difference in populations of neighbouring magnetic
sublevels julsgaard_2003 . This is why there are eight peaks in the non-linear
Zeeman splitting, but nine populations in Fig. 2. As the outermost peaks are
largest and have equal height, we conclude that an aligned state is created in
the $F=4$ ground state, with the majority of the atoms pumped into the
$F=4,m=\pm 4$ states. The optical pumping is not perfect as some of the atoms
are pumped into the other magnetic sublevels. This is due to the non-zero
longitudinal relaxation rate $\Gamma_{1}$.

Figure 5: (a) (b) Sensitivity measurement of the paraffin-coated alignment-
based magnetometer ($T\sim 20\degree$C) at a Larmor frequency of
$\nu_{L}=10.25$ kHz. (a) Magnetic resonance with the RF frequency swept
between 9 and 11.5 kHz. (b) A 240 s time trace of the intrinsic OPM noise with
the lock-in amplifier demodulating signals at $\nu_{L}$. (c) (d) Sensitivity
measurement of the 65 Torr N${}_{\text{2}}$ ($T\sim 55\degree$C) at a Larmor
frequency of $\nu_{L}=10.04$ kHz. (c) Magnetic resonance with the RF frequency
swept between 8 and 12.5 kHz. (d) A 240 s time trace of the intrinsic OPM
noise with the lock-in amplifier demodulating signals at $\nu_{L}$.

We now proceed with characterising the magnetic field sensitivity of the
paraffin-coated vapour cell. These measurements were carried out at a smaller
static magnetic field $B_{0}$ corresponding to a Larmor frequency of around 10
kHz. A $10~{}\mu$W light power beam passed through the cell and a 4.22
nT${}_{\text{RMS}}$ (20 mV${}_{\text{RMS}}$) oscillating magnetic field was
applied. A magnetic resonance signal is shown in Fig. 5, where the RF
frequency was swept between 9 kHz and 11.5 kHz and $\nu_{L}=10.25$ kHz. From
this, the peak of the resonance signal is extracted and divided by the applied
oscillating magnetic field to give a conversion between the lock-in amplifier
readout and the corresponding RF field amplitude $B_{\text{RF}}$. Once this
calibration was completed in Fig. 5, the lock-in demodulation frequency was
fixed to the Larmor frequency, the RF amplitude was set to zero
($B_{\text{RF}}=0$), and a 4 minute time trace of the intrinsic noise of the
OPM taken (see Fig. 5). Following on from this the light hitting the balanced
photodetector was completely blocked and another time trace obtained (data not
shown). The sensitivity to small oscillating magnetic fields, i.e., the
intrinsic OPM noise, is 480 fT/$\sqrt{\text{Hz}}$ for $X$ and 460
fT/$\sqrt{\text{Hz}}$ for $Y$. This was calculated using the methods described
in Ref. rushton_2022  by taking the standard deviations (SDs) of 240$\times$1
s averaged segments, which are included in the legend of Fig. 5. The SDs with
the light blocked are only just below at 410 fT/$\sqrt{\text{Hz}}$ for $X$ and
380 fT/$\sqrt{\text{Hz}}$ for $Y$. This noise is mainly due to electronic
noise of the balanced photodetector and also due to a small contribution from
the electronic noise of the data-acquisition system. The signal size and
thereby the sensitivity could be improved by heating the vapour cell
ledbetter_acosta_rochester_budker_pustelny_yashchuk_2007 ; seltzer_thesis  and
using a larger vapour cell. A fundamental limit to the sensitivity is given by
the spin-projection noise
ledbetter_acosta_rochester_budker_pustelny_yashchuk_2007 ;
graf_kimball_rochester_kerner_wong_budker_alexandrov_balabas_yashchuk_2005

$\delta B_{\text{spn}}=\frac{2\hbar}{g_{F}\mu_{B}\sqrt{nVT_{2}}},$ (24)

where $g_{F}=1/4$ for the $F=4$ Cs ground state, $n\sim 2.2\times 10^{16}$ m-3
($T\sim 18.5\degree$C) is the number density of Cs atoms, $T_{2}\sim
1/(\pi(230~{}\text{Hz}))\sim 1.4$ ms is the transverse relaxation time and
$V=(5~{}\text{mm})^{3}$ is the volume of the whole cell, as all the atoms in
the cell are probed. The sensitivity is estimated to be $\delta
B_{\text{spn}}\sim 50$ fT/$\sqrt{\text{Hz}}$ using the numbers above. A
balanced photodetector with reduced electronic noise would help us get closer
to this quantum-limited sensitivity.

##  V Buffer gas cell

We now carry out experiments with a hand-blown cylindrical buffer gas cell (5
mm length, 5 mm diameter) filled with Cs as well as N${}_{\text{2}}$ buffer
gas (see Fig. 3). The buffer gas cell is surrounded by a Shapal ceramic
cylinder, which is chosen for its high thermal conductivity. The ceramic
cylinder is wrapped in a non-magnetic resistive twisted wire and wrapped with
heat insulator aerogel and Kapton tape as shown in Fig. 3. The buffer gas cell
can then be heated and kept at an elevated temperature by running current
through the twisted wire.

The N${}_{\text{2}}$ buffer gas pressure was determined using absorption
spectroscopy as described by Andalkar andalkar_warrington_2002 . The laser
power was kept low for these absorption measurements to avoid any optical
pumping effects. An absorption spectrum of the buffer gas cell is obtained,
plotted on top of an absorption spectrum of a pure Cs cell (75 mm length and
kept at room-temperature) in Fig. 6. The pure cell only contains Cs (and
neither contains paraffin or buffer gas) and is used as a frequency reference.
The absorption spectrum for the pure cell shows four absorption resonances
separated by ground and excited state hyperfine splittings (9.2 GHz, 1.2 GHz)
as expected for Cs D1 spectroscopy. The absorption resonances have a Voigt
lineshape, which is a convolution of a Lorentzian and Gaussian lineshape. For
the pure cell, the Gaussian Doppler width is much larger than the Lorentzian
natural linewidth 4.6 MHz full width at half maximum (FWHM) of the Cs excited
state. For a buffer gas cell, collisions between buffer gas atoms and Cs atoms
lead to Lorentzian pressure broadening as well as frequency shifts of the
absorption resonances, as seen in Fig. 6. The pressure broadening is extracted
by fitting the $F=3\rightarrow F^{\prime}=3$ and $F=3\rightarrow F^{\prime}=4$
absorption resonances to a sum of two Voigt profiles and using their relative
hyperfine strengths (1/4 and 3/4, respectively) and then repeating the
procedure for $F=4\rightarrow F^{\prime}=3$ and $F=4\rightarrow F^{\prime}=4$,
with hyperfine strengths of 7/12 and 5/12, respectively. The Doppler width
$\Gamma_{G}$ is fixed (374 MHz FWHM at 51$\degree$C) and the Lorentzian
$\Gamma_{L}$ (1.26(0.05) GHz) is fitted, corresponding to a pressure of 65(3)
Torr, using the conversion of 19.51 MHz/Torr from andalkar_warrington_2002
for the D1 pressure broadening with N${}_{\text{2}}$. The pressure can also be
extracted from the shift - 0.54(0.01) GHz in resonance frequencies, which
corresponds to a pressure of 65(1) Torr.

Our alignment-based magnetometer uses $\pi$-polarized light resonant with the
$F=4\rightarrow F^{\prime}=3$ transition (see Fig. 1(b)), as in this case, the
$F=4,m=\pm 4$ states are dark states and atoms become optically pumped into
those states with equal probability, creating the spin-alignment, as depicted
in Fig. 2b. Note that for $\pi$-polarized light resonant with the
$F=4\rightarrow F^{\prime}=4$ transition, the $F=4,m=0$ sublevel will be a
dark state instead. With buffer gas pressure broadening, the $F=4\rightarrow
F^{\prime}=3$ and $F=4\rightarrow F^{\prime}=4$ resonances begin to overlap.
From our fit, we deduce that the overlap is only $\sim 10\%$ for our pressure
of 65 Torr N${}_{\text{2}}$ (see Fig. 6 and the thin dotted vertical line). At
higher pressures the two transitions will overlap even more. This is
problematic for an alignment-based magnetometer as the light in this case will
drive both $F=4\rightarrow F^{\prime}=3$ and $F=4\rightarrow F^{\prime}=4$
transitions at the same time. The $F=4,m=\pm=4$ are then not dark states and
significantly less spin-alignment is created.

Figure 6: Absorption spectrum of the D1 line with a 65(3) Torr
N${}_{\text{2}}$ cell alongside a frequency reference which is a pure Cs cell.
The buffer gas cell is heated to 51$\degree$C corresponding to a density of
43.7$\times 10^{16}$ m-3 Cs atoms and a Doppler linewidth $\Gamma_{G}=374$
MHz. The $F=3\rightarrow F^{\prime}=3,4$ and $F=4\rightarrow F^{\prime}=3,4$
transitions are fitted to Voigt profiles and the Lorentzian width $\Gamma_{L}$
and pressure shift are extracted.  Figure 7:  Non-linear Zeeman splitting of
the magnetic resonances using a 65 Torr N${}_{\text{2}}$ buffer gas cell
heated to $\sim 55\degree$C. The magnitude $R$ is fitted to Eq. 23. The
magnetic resonances for $m=4\rightarrow m=3$ and $m=-3\rightarrow m=-4$ are
indicated.

To verify whether optical pumping into the $F=4,m=\pm 4$ states is possible
with the 65 Torr N${}_{\text{2}}$ buffer gas cell where the excited hyperfine
states partially overlap ($\sim 10\%$) and where quenching is the main de-
excitation mechanism as described previously, once again the static field is
adjusted to be large ($B_{0}=8.38$ G) and a magnetic resonance spectrum is
recorded (see Fig. 7). Again we see the magnetic resonances split due to the
non-linear Zeeman effect, and the two outermost resonances have the largest
and equal heights. The frequency difference between the $m=4\rightarrow m=3$
transition and the $m=-3\rightarrow m=-4$ transition is found experimentally
to be $|\Delta\nu_{4,3}-\Delta\nu_{-3,-4}|=13.2(0.1)$ kHz from a fit of the
data in Fig. 7 to Eq. 23, which agrees well with the value $7\delta=13.1$ kHz
calculated from Eqs. 21 and 22.

This experimentally demonstrates that it is possible to generate a spin
aligned stated in the 65 Torr N${}_{\text{2}}$ buffer gas cell by optically
pumping more Cs atoms into the $m=\pm 4$ states than the other magnetic
sublevels in the $F=4$ ground state. It is expected that better optical
pumping into the $m=\pm 4$ states will be achieved if a smaller buffer gas
pressure is used, as there will be less unwanted pumping to the
$F=4\rightarrow F^{\prime}=4$ transition. A higher ratio $R_{p}/\Gamma_{1}$
(see Eq. 19) will also increase pumping into the $m=\pm 4$ states. The
drawback of a lower buffer gas pressure, however, is that the atoms will
diffuse more quickly to the walls, leading to a smaller $T_{2}$ time and hence
a less sensitive OPM. These two processes compete and need to be taken into
consideration when selecting the optimal buffer gas pressure for an alignment-
based magnetometer.

We now characterise the magnetic field sensitivity of the buffer gas cell
using the same procedure which was used for the paraffin-coated cell. The
optimal light power was found to be 30 $\mu$W. A magnetic resonance signal at
10 kHz was obtained with the 65 N${}_{\text{2}}$ Torr cell in Fig. 5. A 240 s
time trace with the RF field turned off is shown in Fig. 5. The sensitivity of
the OPM, defined as the SD of the 240$\times 1$ s data points in Fig. 5, is
310 fT/$\sqrt{\text{Hz}}$ for $X$ and 340 fT/$\sqrt{\text{Hz}}$ for $Y$. The
sensitivity of the buffer gas cell therefore exceeds the paraffin-coated cell
in this paper. The sensitivity is mainly limited by laser shot noise and
electronic noise of the balanced photodetector.

We use Eq. 24 to calculate the predicted quantum-limited spin-projection
noise. The number density $n=60\times 10^{16}$ m-3 at $T=55\degree$C and
$T_{2}=1/(\pi(800~{}\text{Hz}))$. In a buffer gas cell only the atoms inside
the beam are probed, unlike in a paraffin-coated cell where all the atoms in
the cell are probed. We therefore use the volume inside the beam
$V=V_{\text{beam}}=3.9\times 10^{-9}$ m3, where the diameter of the beam is
$\sim 1$ mm and length of the cell is 5 mm. Inserting the numbers above, we
estimate the atomic noise to be $\delta B_{\text{spn}}\sim~{}$100
fT/$\sqrt{\text{Hz}}$. A better sensitivity could be obtained by increasing
the diameter and length of the cell, whilst increasing the size of the beam.
If a 5 mm diameter beam was used, probing the whole cell, the atomic noise is
estimated to be $\delta B_{\text{spn}}\sim 20$ fT/$\sqrt{\text{Hz}}$. Note
that many atoms are lost to the $F=3$ ground state (see Fig. 2), reducing the
number of Cs atoms that are probed. Using a second laser beam (typically
called a repumper) bringing the atoms out of $F=3$ and back into $F=4$ would
also increase the number of probed atoms, improving the sensitivity of the RF
OPM.

##  VI Conclusions

The results presented in this paper demonstrate the first implementation of a
one-beam radio-frequency optically pumped magnetometer (RF OPM), the
alignment-based magnetometer, being used with a buffer gas cell. The
sensitivity of the alignment-based magnetometer with Cs alkali vapour and 65
Torr N${}_{\text{2}}$ buffer gas was $325$ fT/$\sqrt{\text{Hz}}$. This
sensitivity could be further improved upon by using a balanced photodetector
with lower electronic noise. Further studies could investigate the optimal
vapour cell size, operating temperature and buffer gas pressure. Although our
experiments were carried out using hand-blown vapour cells, we expect similar
performance with microfabricated buffer gas cells. Our work opens up the
possibility of the commercialisation of compact, robust and portable RF OPMs
using only one laser beam with buffer gas cells, a much more scalable and
commercially viable option than using paraffin-coated vapour cells.

###### Acknowledgements.

This work was supported by the UK Quantum Technology Hub in Sensing and
Timing, funded by the Engineering and Physical Sciences Research Council
(EPSRC) (Grant No. EP/T001046/1), the QuantERA grant C’MON-QSENS! by EPSRC
(Grant No. EP/T027126/1), the Nottingham Impact Accelerator/EPSRC Impact
Acceleration Account (IAA), and the Novo Nordisk Foundation (Grant No.
NNF20OC0064182). We thank Janek Kolodynski and Marcin Koźbiał for reading and
commenting on the manuscript.

## Data Availability Statement

Further data can be available from the authors upon request.

## References

## References

  * (1) E. Labyt, T. Sander-Thömmes, and R. Wakai.  Flexible high performance magnetic field sensors: On-scalp magnetoencephalography and other applications.  Springer, 2022.
  * (2) M. Auzinsh, D. Budker, and S. M. Rochester.  Optically polarized atoms: understanding light-atom interactions.  Oxford University Press, 2014.
  * (3) D. Budker and M. Romalis.  Optical magnetometry.  Nat. Phys., 3:227, 2007.
  * (4) I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis.  A subfemtotesla multichannel atomic magnetometer.  Nature, 422:596, 2003.
  * (5) W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik.  Quantum noise limited and entanglement-assisted magnetometry.  Phys. Rev. Lett., 104:133601, 2010.
  * (6) W. Chalupczak, R. M. Godun, S. Pustelny, and W. Gawlik.  Room temperature femtotesla radio-frequency atomic magnetometer.  Appl. Phys. Lett., 100:242401, 2012.
  * (7) H. Yao, B. Maddox, and F. Renzoni.  High-sensitivity operation of an unshielded single cell radio-frequency atomic magnetometer.  Opt. Express, 30:42015, 2022.
  * (8) Quspin.  https://quspin.com, December 2022.
  * (9) Fieldline inc.  https://fieldlineinc.com/, December 2022.
  * (10) Twinleaf.  https://twinleaf.com/, December 2022.
  * (11) I. M. Savukov, S. J. Seltzer, M. V. Romalis, and K. L. Sauer.  Tunable atomic magnetometer for detection of radio-frequency magnetic fields.  Phys. Rev. Lett., 95:063004, 2005.
  * (12) S. M. Rochester.  Modeling Nonlinear Magneto-optical Effects in Atomic Vapors.  PhD Thesis. UC Berkeley, 2010.
  * (13) C. Deans, L. Marmugi, and F. Renzoni.  Sub-picotesla widely tunable atomic magnetometer operating at room-temperature in unshielded environments.  Rev. Sci. Instrum., 89:083111, 2018.
  * (14) C. Deans, Y. Cohen, H. Yao, B. Maddox, A. Vigilante, and F. Renzoni.  Electromagnetic induction imaging with a scanning radio frequency atomic magnetometer.  Appl. Phys. Lett., 119:014001, 2021.
  * (15) J. E. Dhombridge, N. R. Claussen, J. Iivanainen, and P. D. D. Schwindt.  High-sensitivity rf detection using an optically pumped comagnetometer based on natural-abundance rubidium with active ambient-field cancellation.  Phys. Rev. Appl., 18:044052, 2022.
  * (16) M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S. Pustelny, and V. V. Yashchuk.  Detection of radio-frequency magnetic fields using nonlinear magneto-optical rotation.  Phys. Rev. A, 75:023405, 2007.
  * (17) T. Zigdon, A. D. Wilson-Gordon, S. Guttikonda, E. J. Bahr, O. Neitzke, S. M. Rochester, and D. Budker.  Nonlinear magneto-optical rotation in the presence of a radio-frequency field.  Opt. Express, 18:25494, 2010.
  * (18) S. M. Rochester.  Nonlinear magneto-optical rotation in a radio-frequency field.  http://rochesterscientific.com/, December 2022.
  * (19) M. V. Balabas, T. Karaulanov, M. P. Ledbetter, and D. Budker.  Polarized alkali-metal vapor with minute-long transverse spin-relaxation time.  Phys. Rev. Lett., 105:070801, 2010.
  * (20) W. Li, M. Balabas, X. Peng, S. Pustelny, A. Wickenbrock, H. Guo, and D. Budker.  Characterization of high-temperature performance of cesium vapor cells with anti-relaxation coating.  J. Appl. Phys., 121:063104, 2017.
  * (21) V. Shah, S. Knappe, P. D. D. Schwindt, and J. Kitching.  Subpicotesla atomic magnetometry with a microfabricated vapour cell.  Nat. Photonics, 1:649, 2007.
  * (22) S. Dyer, P. F. Griffin, A. S. Arnold, F. Mirando, D. P. Burt, E. Riis, and J. P. McGilligan.  Micro-machined deep silicon atomic vapor cells.  J. Appl. Phys., 132:134401, 2022.
  * (23) S. Seltzer.  Developments in alkali-metal atomic magnetometry.  PhD Thesis. Princeton University, 2008.
  * (24) L. M. Rushton, T. Pyragius, A. Meraki, L. Elson, and K. Jensen.  Unshielded portable optically pumped magnetometer for the remote detection of conductive objects using eddy current measurements.  Rev. Sci. Instrum., 93:125103, 2022.
  * (25) C. Deans, L. Marmugi, and F. Renzoni.  Sub-Sm-1 electromagnetic induction imaging with an unshielded atomic magnetometer.  Appl. Phys. Lett., 116:133501, 2020.
  * (26) K. Jensen, M. Zugenmaier, J. Arnbak, H. Stærkind, M. V. Balabas, and E. S. Polzik.  Detection of low-conductivity objects using eddy current measurements with an optical magnetometer.  Phys. Res., 1:033087, 2019.
  * (27) L. Marmugi and F. Renzoni.  Optical magnetic induction tomography of the heart.  Sci. Rep., 6:23962, 2016.
  * (28) C. Deans, L. Marmugi, and F. Renzoni.  Active underwater detection with an array of atomic magnetometers.  Appl. Optics, 57:2346, 2018.
  * (29) P. Bevington, R. Gartman, and W. Chalupczak.  Object detection with an alkali-metal spin maser.  J. Appl. Phys., 130:214501, 2021.
  * (30) P. Bevington, R. Gartman, and W. Chalupczak.  Enhanced material defect imaging with a radio-frequency atomic magnetometer.  J. Appl. Phys., 125:094503, 2019.
  * (31) D. A. Steck.  Cs d line data.  https://steck.us/alkalidata/cesiumnumbers.1.6.pdf, December 2022\.
  * (32) M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk.  Relaxation of atomic polarization in paraffin-coated cesium vapor cells.  Phys. Rev. A, 72:023401, 2005.
  * (33) S. Eckel, D. Barker, E. Norrgard, and J. Scherschligt.  Pylcp: A python package for computing laser cooling physics.  In preparation, 2020.
  * (34) B. Julsgaard.  Entanglement and quantum interactions with macroscopic gas samples.  PhD Thesis. University of Aarhus, 2003.
  * (35) S. J. Seltzer and M. V. Romalis.  High-temperature alkali vapor cells with antirelaxation surface coatings.  J. Appl. Phys., 106:114905, 2009.
  * (36) G. Bao, A. Wickenbrock, S. Rochester, W. Zhang, and D. Budker.  Suppression of the nonlinear zeeman effect and heading error in earth-field-range alkali-vapor magnetometers.  Phys. Rev. Lett., 120:033202, 2018.
  * (37) A. Andalkar and R. B. Warrington.  High-resolution measurement of the Pressure broadening and shift of the Cs D1 and D2 lines by N2 and He buffer gases.  Phys. Rev. A, 65:032708, 2002.