# Deep Learning Based Source Separation Applied To Choir Ensembles

###### Abstract

Choral singing is a widely practiced form of ensemble singing wherein a group
of people sing simultaneously in polyphonic harmony. The most commonly
practiced setting for choir ensembles consists of four parts; Soprano, Alto,
Tenor and Bass (SATB), each with its own range of fundamental frequencies
(F$0$s). The task of source separation for this choral setting entails
separating the SATB mixture into the constituent parts. Source separation for
musical mixtures is well studied and many deep learning based methodologies
have been proposed for the same. However, most of the research has been
focused on a typical case which consists in separating vocal, percussion and
bass sources from a mixture, each of which has a distinct spectral structure.
In contrast, the simultaneous and harmonic nature of ensemble singing leads to
high structural similarity and overlap between the spectral components of the
sources in a choral mixture, making source separation for choirs a harder task
than the typical case. This, along with the lack of an appropriate
consolidated dataset has led to a dearth of research in the field so far. In
this paper we first assess how well some of the recently developed
methodologies for musical source separation perform for the case of SATB
choirs. We then propose a novel domain-specific adaptation for conditioning
the recently proposed U-Net architecture for musical source separation using
the fundamental frequency contour of each of the singing groups and
demonstrate that our proposed approach surpasses results from domain-agnostic
architectures.

##  1 Introduction

Choir music is a well-established and long-standing practice involving a body
of singers performing together. Such ensembles are usually referred to as
choir and may perform with or without instrumental accompaniment. A choir
ensemble is usually structured by grouping the voices into four different
sections, each depicting different frequency ranges for the singers; "Soprano"
($260\text{\,}\mathrm{Hz}$-$880\text{\,}\mathrm{Hz}$), "Alto"
($190\text{\,}\mathrm{Hz}$-$660\text{\,}\mathrm{Hz}$), "Tenor"
($145\text{\,}\mathrm{Hz}$-$440\text{\,}\mathrm{Hz}$), and "Bass"
($90\text{\,}\mathrm{Hz}$-$290\text{\,}\mathrm{Hz}$) [1]. This type of
structural setting is usually referred to as a SATB setting. Although
different variants of this structure exist, the SATB is the most well
documented, with several conservatories across Europe dedicated to the study
and practice of the art form, highlighting its cultural significance. This
will be the main focal point of our study.

The segregation of a mixture signal into its components is a well researched
branch of signal processing, known as source separation. For polyphonic music
recordings, this implies the isolation of the various instruments mixed
together to form the whole. With applications such as music remixing,
rearrangement, audio restoration, and full source extraction, its potential
use in music is of great appeal. While the task remains similar independently
of the type of setting involved, the nature of the sources (e.g.: speech,
musical instrument, singing voice) and their relations may entail various
challenges and, consequently, require different separation methodologies to be
employed.

The most studied case of musical source separation focuses on pop/rock songs,
which typically have three common sources; vocals, drums, bass along with
other instrumental sources which are usually grouped together as others. A
large body of research [2, 3, 4] has been published in this field over the
last few years, beginning with the consolidation of a common dataset for
researchers to train and evaluate their models on. In 2016, DSD100 [5] was
first introduced and made available to the public and was later extended to
MUSDB18 [6], which comprises $150$ full-length music tracks for a total of
approximately $10$ hours of music. To this day, MUSDB18 represents the largest
freely available dataset of its kind.

While source separation for the pop/rock case has come leaps and bounds in the
last few years, it remains largely unexplored for the SATB choir case, despite
its cultural importance. This is partly due to the lack of a consolidated
dataset, similar to the MUSDB18, and partly due to the nature of the task
itself. The sources to be separated in pop/rock have distinct spectral
structure; the voice is a harmonic instrument and has a distinct spectral
shape, defined by a fundamental frequency and its harmonic partials and
formants. The bass element to be separated also has a harmonic structure, but
lacks the formants found in the human voice and has a much lower fundamental
frequency than the human voice. In contrast, the spectrum of a percussive
instrument is generally inharmnoic and energy is usually spread across the
spectrum. In contrast, the sources to be separated in a SATB choir all have a
similar spectral structure with a fundamental frequency, partials and
formants. This makes the task more challenging than its more studied counter
part. However, the distinct ranges of fundamental frequencies in the sources
to be separated can be used to distinguish between them, a key aspect that we
aim to explore in our study.

We build on top of some recently proposed Deep Neural Network (DNN) models to
separate SATB monoaural recordings into each of their respective singing
groups and then propose a specific adaptation to one of the models. The rest
of the paper is organized as follow: Section 2 presents and investigates some
of the recently proposed high performance deep learning based algorithms used
for common musical source separation tasks, such as the U-Net [7] architecture
and its waveform-based adaptation, Wave-U-Net [8]. Section 3 goes over the
dataset curation carried out for this experiment. Section 4 presents our
adaptation of the conditioned U-Net model described in [9], with a control
mechanism conditioned on the input sources’ fundamental frequency (F$0$).
Section 5 defines the evaluation metrics and methodology used in this
experiment. In Section 5.2 we evaluate and compare how existing models and our
proposed adaptation perform on the task of source separation for SATB
recordings. We then present and discuss the results. Section 6 finally
concludes with a discussion around our experiment and provide comments on
future research that we intend to carry out.

##  2 Related Work

While source separation has remained relatively unexplored for the case of
SATB choirs, a number of architectures have been proposed over the last few
years for musical source separation in the pop/rock case. A comprehensive
overview of all proposed models is beyond the scope of this study, but we
provide a summary of some of the most pertinent models that we believe can
easily be adapted to the case in study.

###  2.1 U-Net

The U-Net architecture [7], which was specifically developed to process and
segment biomedical images, inspired many subsequent audio-related adaptations
due to its unprecedented performance.

The original model includes an encoding path, which reduces the initial input
into a latent representation (bottleneck) followed by a decoding path, which
expends the channels’ receptive field back into its original shape while
concatenating the feature maps from the contracting path by the mean of skip
connection layers.

One of the first paper to present a U-Net adaptation towards audio source
separation was proposed by Jansson et al. [10], where they propose an
architecture which specifically targets vocal separation performed on western
commercial music (or pop music). The authors present an architecture directly
derived from the original U-Net one, which takes spectrogram representations
of the sources as input and aims at predicting a soft-mask for the targeted
source (either vocal or instrumental). The predicted mask is then multiplied
element-wise with the original mixture spectrogram in order to obtain the
predicted isolated source. It is worth mentioning that for each of the given
sources, a U-Net instance is trained in order to predict its respective mask.
In the case of SATB mixtures, four U-Net instances are necessary in order to
predict each of the four singing groups.

###  2.2 Conditioned-U-Net

Depending on the nature of the separation task, its underlying process can
easily lead to scaling issues. The conditioned U-Net (C-U-Net) architecture,
described in [9], aims at addressing this limitations by introducing a
mechanism controlled by external data which govern a single U-Net instance.
C-U-Net does not diverge much from the initial U-Net one; as an alternative to
the multiple instances of the model, each of which is specialized in isolating
a specific source, C-U-Net proposes the insertion of feature-wise linear
modulation (FiLM) layers [11], which represents an affine transform defined by
two scalars - $\gamma{}$ and $\beta{}$, across the architecture. This allows
for the application of linear transformations to intermediate feature maps.
These specialized layers conserve the shape of the original intermediate
feature input while modifying the underlying mapping of the filters
themselves.

$\displaystyle FiLM(x)=\gamma{(z)}\dot{x+\beta{(z)}}\ $ (1)

In eq. (1), $x$ is the input of the FiLM layer, $\gamma$ and $\beta$ the
parameters that scale and shift $x$ based on an external information,
$\overline{z}$ [9]. $\gamma{}$ and $\beta{}$ modulates the feature maps
according to an input vector $\overline{z}$, which describes the source to
separate. The condition generator block described in Figure 1 represents a
neural network embedding the one-hot encoding input $\overline{z}$ into the
most optimal values to be used by the FiLM layer.

Figure 1: C-U-Net control mechanism with the vector $\overline{z}$, a one-hot
representation of the source to separate, which dictates the N sets of
$\gamma$ and $\beta$ values to be used by the FiLM layer at each of the block
of the encoding path, $1$ to N.

###  2.3 Wave-U-Net

In [8], the authors present a time-domain adaptation of the U-Net
architecture, which performs the separation operation on the waveform. As the
input is a one-dimensional signal, the feature maps are computed directly from
the waveform samples through $1$D convolution operations. Because Wave-U-Net
takes raw waveforms as input, the initial U-Net model has to be adapted
accordingly in order to accommodate for the input’s nature. Consequently, the
feature maps along both contracting and expanding paths are computed by the
means of single-dimensional convolution layers. Both paths contain twelve
convolutional layers, each, for a total of $24$ layers. In the down-sampling
path, the receptive field is reduced in half after each layer while the input
feature maps are increased by a factor of $24$ every time. On the other hand,
in the up-sampling path the time-context is doubled after every convolutional
layer while the feature maps are reduced, again by $24$, after every layers.
By that mean, the receptive field and the number of channels of the original
input signal will remain preserved at the output stage. Although Wave-U-Net
has proven to deliver satisfying results on common musical source separation
tasks, the fact remains that waveform-based architectures in general require
more data than their spectrogram-based counter parts.

##  3 DATASET

The training data we have curated for this experiment are composed of the
following two datasets:

  * •

Choral Singing Dataset [12] (CSD). Three songs performed by $16$ singers from
the Anton Bruckner Choir
(SATB)111https://zenodo.org/record/1286570#.XyGcHy-z3yU.

  * •

A proprietary dataset with $26$ Spanish SATB songs by $4$ singers, one for
each part.

There are very few publicly available choir music datasets, thus our choice
remains limited. To this day and to the best of our knowledge, there isn’t any
existing dataset which is specifically suited to our task, thus one of the
subsidiary work of this experiment revolves around curating a proper and
complete dataset to train our various models. For our experiment, we take
advantage of the CSD [12]. This dataset was recorded in a professional studio
and contains individual tracks for each of the $16$ singers of a SATB choir,
i.e. $4$ singers per choir section. It comprises three different choral
pieces: Locus Iste, written by Anton Bruckner, Nīno Dios d’Amor Herido,
written by Francisco Guerrero, and El Rossinyol, a Catalan popular song; all
of them were written for $4$-part mixed choir. The dataset is very well suited
for our experiment as the isolated track for each individual singer will allow
us to proceed the same way as in [13], that is to create artificial mixes by
combining various stems from different groups together. Using different
combinations of all $16$ singers, we created $256$ SATB quartets for each
piece, which represent all possible combinations of singers taking into
account the voice type restriction (i.e. exactly one singer per voice is
needed).

The second dataset we use is a proprietary one including $26$ songs for
exactly one singer per part (i.e. $4$ stems per song), which is a well-suited
format for our task as well. All songs offered as part of this dataset are
performed in Spanish and their length revolves around two to three minutes,
for a total of $58$ minutes of audio data.

Our curation work consists in consolidating these two datasets and make sure
all the data that we are using remain consistent and well-formatted across all
the audio stems, which includes length and amplitude normalization, as well as
properties standardization. Most of the files included as part of the initial
datasets were presented as $10$-seconds long snippets as opposed to full-
length songs, which isn’t an ideal format to work with. Hence, some additional
efforts have been devoted to turn these files in a more convenient and
consolidated format.

##  4 APPROACH

Injecting domain knowledge in DNNs has been proved to be an effective way to
learn complex input-output relations with high accuracy when the available
data happen to be scarce and limited [14], such as found in our case.

Each of the singing groups in a SATB recordings performs within its own
respective frequency range, that is, the voices’ F$0$ contour will rarely
overlap across the various groups. This factor makes the sources’ F$0$ a
suitable discriminative feature, which could be injected in the DNN during the
training stage and potentially improve the separation of the various singing
groups in SATB recordings.

In this view, we propose to adapt the original C-U-Net architecture, which
initially embeds the instruments to be separated, $\overline{z}$, in order to
produce the various FiLM parameters ($\gamma,\beta$), and substitute the
external control input data for the F$0$ track of the target source. The new
control input vector $\overline{z}$ will thus hold time as well as frequency
dimensions.

###  4.1 Control Input Representation

As previously mentioned, we use the frame-wise F$0$ as the external control
data of our condition generator network. This entails that a few preliminary
steps are required prior to proceeding to the training stage. We first
automatically extract the sources’ F$0$ track using the DIO algorithm [15].
Once the raw pitch tracks are obtained, we convert each time-step F$0$ into a
one-hot encoded representation as postulated in [16], that is $60$ frequency
bins over $6$ musical octaves, with a base frequency of
$32.7\text{\,}\mathrm{Hz}$ Hertz, for a total of $360$ frequency bins per
time-step. As a result, for $128$ time-steps, our control input will be in the
shape of $[128,360]$.

Figure 2: Control model architecture. The convolution is performed across the
frequency bins for each time-step. The dense layer provides a specific
conditioning for each frequency bin.

###  4.2 Control Model

The control model used in our proposed architecture embeds the one-hot encoded
CQT F$0$ representation for a given time-step into a set of transforms of
identical shape as the spectrogram input. This is achieved by modeling the
condition vectors $\overline{z}$ as $1$-D data with multiple feature channels.
The condition vectors are then fed into a convolutional neural network (CNN)
with a kernel of size $10$ seizing contextual information from the adjacent
time-steps. As a result, all input channels of the initial convolution
contribute to all resulting feature maps in the output of the first
convolutional layer. Finally a dense layer provides a specific conditioning
for each frequency bin at each time-step, taking into account the contextual
information previously captured by the CNN. Figure 2 shows the condition
generator architecture in greater details.

As the temporal relation between the external control input data and the input
spectrogram is crucial, it is important to apply these affine transformations
while the receptive field of the input is still intact. Hence the FiLM layer
is applied prior to the encoding path. Figure 3 shows the overall structure
behind the proposed conditioning architecture.

We propose two variants of the architecture described above, each of which
differs slightly in the way it embeds the control input data; the first
variant applies a unique affine transform for each individual frequency bin at
every input time-step. The resulting scalars in the output of the external CNN
model will thus be in the shape $[512,128]$ for a given input spectrogram of
the same time context. On the other hand, the second variant applies a single
transform for all frequency bins at a given time-step, resulting in a set of
scalars of shape $[1,128]$ for 128 spectrogram frames given in the input. We
refer to the two approaches as "Domain-Specific Global" and "Domain-Specific
Local", respectively and define them with the acronyms C-U-Net D-S G and C-U-
Net D-S L in the rest of this paper. while the three models covered in Section
2 will be referred to as "Domain-Agnostic" models.

Figure 3: C-U-Net Control Mechanism adapted to our task, with the one-hot
vector $\overline{z}$ depicting the various SATB singing groups’ F$0$ contour.

##  5 EVALUATION

To assess our proposed approach and show that injecting domain knowledge as
control input data to the network improves its performance on SATB recordings,
we evaluate the performances of three domain-agnostic state-of-the-art DNN
models; U-Net, its waveform adaptation Wave-U-Net, and the original C-U-Net.
We then compare the results with the two domain-specific models proposed in
Subsection 4.2. We evaluate the performance of a model by computing three
metrics, SDR, SIR, and SAR [5], between the predicted and true audio sources.
The three measurement metrics describe the overall quality of the separation,
the level of interference with other sources as well as the amount of artifact
added by the separation algorithm, respectively. The metrics are computed
using the mir_eval toolbox [17] for each of the SATB singing groups.

###  5.1 Train - Test Split

Given the limited size of our data, we opted to set apart one song from the
proprietary dataset as well as one singer per voice for each song from the CSD
in order to build our first use-case test set. The rest of the data was used
for training. This allowed us evaluate the model on unseen songs and singers.
Our second test case contains unison singing, which was not seen at all during
training. As such, we used the three songs from the CSD with all singers for
evaluation.

###  5.2 Experiment Results

Subsections 5.2.1 and 5.2.2 present the performance results for the two test
cases described earlier; that is, for the test set involving exactly one
singer per part and the other involving exactly four singers per part,
respectively. C-U-Net D-A refers to the domain-agnostic architecture while
C-U-Net D-S L and C-U-Net D-S G refer to our two domain-specific adaptations.
For testing, we use the oracle fundamental frequency of each of the sources,
pre-computed prior to model inference. In a complete source separation
pipeline, we would complement our system with the multi-pitch algorithm
proposed by [18].

(a) Four-Parts SDR Boxplot Results, Use-Case $1$: $4$-Singers Mixture

(b) Four-Parts SDR Boxplot Results, Use-Case $2$: $16$-Singers Mixture

Figure 4: Boxplot SDR results on the five U-Net based models described in
previous sections. Subfigure 4(a) shows the result distribution over the first
use-case test set while 4(b) depicts the results for the second use-case. For
each one of the SATB parts, the model performing with the highest median is
indicated in a dark orange color.

Model | Test Use-Case 1 - SIR (dB) | Test Use-Case 1 - SAR (dB)
---|---|---
| Soprano | Alto | Tenor | Bass | Avg. | Soprano | Alto | Tenor | Bass | Avg.
Wave-U-Net | 5.99$\pm$2.4 | 9.19$\pm$2.9 | 4.62$\pm$2.1 | 8.49$\pm$3.5 | 7.07 | 5.36$\pm$1.7 | 7.11$\pm$2.4 | 4.79$\pm$1.4 | 4.89$\pm$1.4 | 5.54
U-Net | 10.28$\pm$2.4 | 10.77$\pm$4.1 | 6.70$\pm$3.2 | 9.45$\pm$2.0 | 9.30 | 5.35$\pm$1.8 | 7.13$\pm$3.0 | 5.32$\pm$1.8 | 4.94$\pm$1.1 | 5.69
C-U-Net D-A | 10.09$\pm$2.6 | 7.81$\pm$1.6 | 3.32$\pm$3.3 | 7.61$\pm$2.2 | 7.21 | 5.19$\pm$1.6 | 4.41$\pm$2.8 | 2.65$\pm$1.2 | 4.12$\pm$2.0 | 4.09
C-U-Net D-S L | 9.71$\pm$1.7 | 12.37$\pm$1.5 | 9.89$\pm$2.2 | 9.71$\pm$1.7 | 10.42 | 5.44$\pm$1.0 | 8.75$\pm$2.0 | 5.58$\pm$1.3 | 5.51$\pm$1.7 | 6.32
C-U-Net D-S G |  12.72$\pm$1.8 |  14.04$\pm$1.5 |  11.79$\pm$1.5 |  9.78$\pm$2.1 | 12.08 |  7.02$\pm$1.1 |  9.02$\pm$1.6 |  6.86$\pm$1.5 |  5.93$\pm$1.6 | 7.21

Test Use-Case 2 - SIR (dB) Test Use-Case 2 - SAR (dB) Soprano Alto Tenor Bass
Avg. Soprano Alto Tenor Bass Avg. Wave-U-Net 8.13$\pm$2.1 10.02$\pm$0.9
6.80$\pm$2.2 7.45$\pm$2.0 8.10 5.75$\pm$1.0 6.73$\pm$1.1 4.79$\pm$1.5
3.23$\pm$0.9 5.13 U-Net 12.41$\pm$1.8 13.11$\pm$1.2 10.26$\pm$1.1 8.50$\pm$2.3
11.07 6.31$\pm$1.4 7.97$\pm$1.0 6.59$\pm$1.8 5.27$\pm$1.1 6.54 C-U-Net D-A
11.99$\pm$1.8 9.08$\pm$2.8 5.65$\pm$3.1 7.60$\pm$2.0 8.58 5.77$\pm$1.7
4.60$\pm$2.9 3.39$\pm$1.8 4.15$\pm$1.2 4.48 C-U-Net D-S L 10.32$\pm$1.1
13.06$\pm$1.7 10.77$\pm$1.5 8.89$\pm$2.2 10.76 6.02$\pm$0.9 8.59$\pm$1.2
6.45$\pm$1.7 5.59$\pm$1.1 6.66 C-U-Net D-S G 12.08$\pm$1.5 13.50$\pm$2.5
12.05$\pm$1.3 8.91$\pm$2.1 11.63 6.68$\pm$1.2 7.70$\pm$1.3 6.17$\pm$1.6
5.17$\pm$0.8 6.43

Table 1: SIR and SAR mean and standard deviation results on the four SATB
parts as well as their average for the five U-Net based models described in
previous sections. The top table depicts the results obtained from the first
use-case test set while the bottom one the second use-case test set.

####  5.2.1 Use-Case 1:

Table 1 portrays the mean SIR and SAR results on all the SATB parts and
average for all five models mentioned in previous sections. Figure 4(a)
details the SDR score distributions on our first test set. We observe that our
two adaptations, C-U-Net D-S L and C-U-Net D-S G, call attention to a
significant score gap between domain-agnostic and domain-specific models, with
an average increase of about $1$dB SDR and $1.5$dB SIR between the two
different approaches. These improvements underline an overall better quality
of the predicted sources (SDR) as well as a decline in interference between
the various predictions. Our domain-specific architecture hence demonstrates a
better ability to cope with the correlated nature of the various SATB sources
and seem to predict an appropriate spectral mask for each of them. We also
observe that our proposed adaptations return the lowest SDR, SIR and SAR
performances for the Bass part, specifically. This could be due to the fact
that the Bass group, among all SATB groups, shares the highest number of
harmonics with its other source counter parts.

We also note that the mean SDR substantially drops for the Tenor singing group
across nearly all domain-agnostic models, reaching a negative result with the
C-U-Net D-A architecture ($-1.25$dB SDR). We speculate that the reason behind
such decline can be directly related to the close nature of the F$0$ contours
of both Alto and Tenor singing groups, making it harder for domain-agnostic
architecture to distinguish between the two sources. This limitation brings
yet another justification for the conditioning approach we have taken in this
paper.

####  5.2.2 Use-Case 2:

Figure 4(b) as well as the bottom portion of Table 1 presents the SDR, SIR and
SAR scores on the second use-case test set. We observe that the introduction
of more complex mixtures involving a higher number of singers (i.e.: $16$
singers in this case) decreases the performance of our proposed models, with
an average SDR barely surpassing U-Net’s for our C-U-Net D-S G model and
levelling it out for the C-U-Net D-S L model. This can be attributed to the
use of the mean of the various pitches present in a singing group, to
represent the pitch of the unison. Since domain-agnostic models, such as the
plain U-Net, don’t hold this assumption, these architectures are less prone to
errors when exposed to these type of mixture settings 222Audio examples
showcasing how the inference of our proposed models compare against other
state-of-the-art architectures are available online. On the same page is also
included a link to our pre-trained models: https://darius522.github.io/satb-
source-separation-results/.

##  6 CONCLUSIONS AND FUTURE WORK

In this work we have presented the task of musical source separation applied
to SATB choir recordings. We first described the consolidated dataset that
we’ve specifically curated for this experiment and its potential use and
application for future related research. We then assessed how well recent
domain-agnostic deep learning based architectures for musical source
separation performed on this task, given two different use-cases; $4$-singers
mixture and $16$-singers mixture separation. An adaptation of the U-Net
architecture was then proposed, consisting in conditioning some of the network
parameters on the fundamental frequency contour of each of the SATB mixture
sources. The preliminary results showed that taking advantage of domain-
knowledge during the training process improved the performance on both of our
proposed use-cases. For the evaluation presented in this paper, we use the
oracle F$0$ is currently used as external control input data to the network.
In a complete source separation pipeline, we plan on combining the task of
multi-pitch tracking [18] with the system presented in this paper. We also
plan on validating our evaluation with perceptual listening tests and
exploring applications of the SATB separation. including remixing,
transcription and transposition combined with the work presented in [19, 20].

##  7 acknowledgements

The TITANX used for this research was donated by the NVIDIA Corporation. This
work is partially supported by the Towards Richer Online Music Public-domain
Archives (TROMPA H2020 770376) project. Helena Cuesta is supported by the FI
Predoctoral Grant from AGAUR (Generalitat de Catalunya). The authors would
like to thank Rodrigo Schramm and Emmanouil Benetos for sharing their singing
voice datasets for this research.

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# Modulation Transfer Spectroscopy of the $\textrm{D}_{1}$ Transition of
Potassium:
Theory and Experiment.

A. D. Innes1 P. Majumder1 H. R. Noh 2<EMAIL_ADDRESS>S. L. Cornish1
<EMAIL_ADDRESS>1Department of Physics, Durham University, South
Road, Durham, DH1 3LE, United Kingdom2Department of Physics, Chonnam National
University,Gwangju,61186 , Korea.

###### Abstract

We report on a study of modulation transfer spectroscopy of the
$4\textrm{S}_{1/2}\rightarrow 4\textrm{P}_{1/2}$ (D1) transition of naturally
abundant potassium in a room-temperature vapour cell. This transition is
critical for laser cooling and optical pumping of potassium and our study is
therefore motivated by the need for robust laser frequency stabilisation.
Despite the absence of a closed transition, the small ground-state hyperfine
splitting in potassium results in strong crossover features in the D1
modulation transfer spectrum. To emphasise this we compare the D1 and D2
spectra of potassium with those of rubidium. Further, we compare our
experimental results with a detailed theoretical simulation, examining
different pump-probe polarization configurations to identify the optimal
signals for laser frequency stabilisation. We find good agreement between the
experiment and the theory, especially for the
$\textrm{lin}\parallel\textrm{lin}$ polarization configuration.

††preprint: APS/123-QED

##  I Introduction

Experiments employing laser cooling require lasers that are frequency
stabilised to better than the linewidth of the transition used for cooling.
For alkali-metal atoms this corresponds to a frequency stability of $\lesssim$
1 MHz. One method of achieving this is with a spectroscopy technique to obtain
an error signal from an atomic transition. The error signal is then used as
feedback to correct any frequency deviations of the laser. Some of the most
common methods are dithering the current to extract the derivative of the
saturated absorption signal [1, 2, 3], frequency modulation spectroscopy [4,
5], dichroic atomic vapour laser locking (DAVLL) [6, 7, 8, 9], far off
resonance locking with the Faraday effect [10, 11], polarization spectroscopy
[12, 13, 14] and modulation transfer spectroscopy (MTS) [15, 16].

MTS is a pump-probe spectroscopy technique that works by modulating the pump
beam with an electro-optical modulator (EOM) to produce sidebands and carrier
components. When the pump beam is overlapped with the probe beam in an atomic
vapour cell, and near resonance with an atomic transition, a four-wave mixing
process transfers the sidebands from the pump to the probe. The beating
between the sidebands and the probe can then be detected by a fast photodiode.
Demodulation of the photodiode signal leads to the MTS signal. MTS has two key
advantages. Firstly, it generates a dispersive signal on a zero background;
the zero crossing of which occurs when the laser is exactly on resonance with
the associated transition. Secondly, the MTS signal is dominated by cycling
transitions. This can be useful in cases where the hyperfine structure is too
narrow to resolve with other spectroscopic techniques such as the
$\textrm{S}_{1/2}\rightarrow\textrm{P}_{3/2}$ transition in bosonic potassium
[17]. This makes it an effective tool for laser frequency stabilisation.

A considerable body of theoretical and experimental work has been published on
MTS, for example see [15, 16, 18, 19, 20, 21, 22, 23, 24, 17, 25, 26]. There
has already been a comprehensive study on the D2 lines of potassium [17].
However, to our knowledge, very little work has been published on the
$\textrm{D}_{1}$ lines of potassium either in theory or experiment; an MTS
spectrum has been reported but in a wider study of simultaneously locking
multiple lasers to a single cell [27].

The small hyperfine splitting in the $4\textrm{P}_{3/2}$ state of bosonic
potassium makes it difficult to resolve the individual hyperfine components of
the D2 transition spectroscopically and has a detrimental impact on the
efficiency of laser cooling [28]. In contrast, the hyperfine components of the
D1 transition are resolvable due to the simpler structure and slightly larger
splitting. This transition has found important applications for gray molasses
cooling [29, 30] and degenerate Raman sideband cooling [31] and is useful for
efficiently spin-polarizing a sample of potassium atoms.

The bosonic isotopes of potassium also have the interesting feature that the
ground-state hyperfine splitting is smaller than the Doppler width of the D1
and D2 transitions in a room-temperature vapour. This leads to the existence
of ground-state crossover resonances in pump-probe spectroscopy schemes. These
crossovers are something that are notably absent in the D line spectra of
rubidium and caesium [8, 32]. Similar ground-state crossover resonances have
been previously observed in the D2 transition of lithium [23]. The presence of
ground-state crossover features in potassium motivate a comparison between the
MTS spectrum of the D1 and D2 lines, as well as a comparison with a species,
such as rubidium, whose ground-state hyperfine splitting is greater than the
Doppler width.

In this work we present a detailed study of MTS of the D1 transition of
potassium, comparing our results with MTS of the D2 transition. To elucidate
the role of the ground-state crossover, we contrast our results with spectra
obtained for rubidium. We also present spectra for different pump-probe
polarization configurations, showing that the configuration where the beams
have linear and perpendicular polarization offers the strongest signal for
locking. Throughout we compare our experimental results with the predictions
from a theoretical model based upon the solutions to the time dependent
optical Bloch equations without the use of any phenomenological constants. We
find the calculations predict a large dispersive signal for the crossover
features, as observed experimentally.

The layout of the paper is as follows. In Sec. II we present the theoretical
model used to predict the MTS spectrum. In Sec. III we outline the details of
our experiment. In Sec. IV we present our results and compare them against the
predicted theoretical curves. In Sec. V we summarise our work and give an
outlook to possible future extensions of our study.

##  II Theory

Figure 1: The hyperfine structure of potassium relevant to the D1 and D2
transitions, together with the associated line spectra highlighting the
relative transition frequencies. Levels are labelled by the quantum number $F$
for the total angular momentum of the state and hyperfine shifts are given in
MHz using the values from [33]. The horizontal dot-dashed lines show the
positions of crossovers for the S1/2 and P1/2 manifolds. In the line spectra,
lines above the axis correspond to normal transitions and have heights
reflecting the relative oscillator strengths. Lines below the axis correspond
to crossover transitions. Ground-state crossovers are indicated by circles at
the bottom of the line. For clarity, excited-state crossover transitions have
been omitted for the D2 transition. For both spectra, zero detuning
corresponds to the $X(1,2)\rightarrow F^{\prime}=2$ transition in 39K. Note,
in the experiment the spectra will be weighted by the natural abundances of
the isotopes shown in the figure.

In this section, we outline the theory for predicting the MTS signal. Figure 1
shows the hyperfine structure for the 4S1/2, 4P1/2 and 4P3/2 energy levels for
the bosonic isotopes. We illustrate the method of calculation for the D1
transition in 39K, noting that the methodology is directly applicable to 41K
as both isotopes have the same nuclear spin. The final spectra are obtained by
summing the results for 39K and 41K weighted by their natural abundances. We
consider four polarization configurations of the counter-propagating pump and
probe laser beams: (i) $\textrm{lin}\|\textrm{lin}$, (ii)
$\textrm{lin}\bot\textrm{lin}$, (iii) $\sigma^{+}\sigma^{+}$, and (iv)
$\sigma^{+}\sigma^{-}$. Here, for simplicity, the circularly polarized cases
in (iii) and (iv) are labeled by the transitions the beams drive. In the case
of linear polarization configurations, the quantization axis was chosen as the
direction of the electric field of the pump beam, whereas the propagation
direction of the pump beam was chosen as the quantization axis in the case of
circular polarization configurations. In the experiment, we reinforce the
quantization axis with a weak magnetic field parallel to the direction of beam
propagation. The polarization vectors of the pump and probe beams are expanded
as $a_{+}\hat{\epsilon}_{+}+a_{0}\hat{\epsilon}_{0}+a_{-}\hat{\epsilon}_{-}$
and $c_{+}\hat{\epsilon}_{+}+c_{0}\hat{\epsilon}_{0}+c_{-}\hat{\epsilon}_{-}$,
respectively, in the spherical bases. Then, the coefficients for the $\pi$\-
and $\sigma^{+}$ polarized pump beam are $(a_{+},a_{0},a_{-})=(0,1,0)$ for
polarization configurations (i) and (ii) and $(1,0,0)$ for configurations
(iii) and (iv). The corresponding coefficients $(c_{+},c_{0},c_{-})$ for the
probe beam are (i) $(0,1,0)$, (ii) $(-1/\sqrt{2},0,1/\sqrt{2})$, (iii)
$(1,0,0)$, and (iv) $(0,0,1)$.

To find the internal dynamics of the atoms, we solve the following density
matrix equation:

$\displaystyle\dot{\rho}=-(i/\hbar)\left[H_{0}+V,\rho\right]+{\dot{\rho}}_{\rm
relax},$ (1)

where $\rho$ is the density operator, and $H_{0}$ and $V$ are the atomic and
interaction Hamiltonians, respectively. In Eq. (1), ${\dot{\rho}}_{\rm relax}$
represents the term related to relaxations such as spontaneous emission and
transit-time decay [34]. The atomic and interaction Hamiltonians are given by

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle-\sum_{m^{\prime}=-2}^{2}\hbar\delta_{1}\left|F^{\prime}=2,m^{\prime}\right>\left<F^{\prime}=2,m^{\prime}\right|$
(2)
$\displaystyle-\sum_{m^{\prime}=-1}^{1}\hbar\left(\delta_{1}+\Delta_{e}\right)\left|F^{\prime}=1,m^{\prime}\right>\left<F^{\prime}=1,m^{\prime}\right|$
$\displaystyle-\sum_{m=-1}^{1}\hbar\Delta_{g}\left|F=1,m\right>\left<F=1,m\right|,$
$\displaystyle V$ $\displaystyle=$
$\displaystyle\sum_{F=1}^{2}\sum_{F^{\prime}=1}^{2}\sum_{m=-F}^{F}\sum_{q=\pm,0}\frac{\hbar}{2}\left[c_{q}\Omega_{p}e^{-i\delta_{p}t}\right.$
(3) $\displaystyle\left.+a_{q}(\Omega_{c}+\Omega_{s}e^{-i\Omega
t}-\Omega_{s}e^{i\Omega t})\right]$ $\displaystyle\times
C_{F,m}^{F^{\prime},m^{\prime}+q}\left|F^{\prime},m^{\prime}+q\right>\left<F,m\right|+{\rm
h.c.},$

respectively, where h.c. denotes the Hermitian conjugate.

In Eqs. (2) and (3), $\delta_{1}(\equiv\delta+kv)$ is the detuning of the
carrier component of the pump beam. $\delta_{p}$($=-2kv$) is the detuning of
the probe beam relative to the carrier component felt by an atom moving at
velocity $v$ where $\delta$ is the detuning of the laser frequency with
respect to $F=2\rightarrow F^{\prime}=2$ transition of 39K and $k$ is the wave
vector. $F$ ($F^{\prime}$) and $\Delta_{g}$ ($\Delta_{e}$) denote the
hyperfine quantum number and splitting of the ground (excited) state,
respectively. $m$ and $m^{\prime}$ are the projection of the total angular
momentum onto the quantisation axis for the ground and excited states,
respectively. In Eq. (2), the external magnetic field was not taken into
consideration. In Eq. (3), $\Omega_{c}$ and $\Omega_{s}$ represent the Rabi
frequencies of the carrier and sideband components of the pump beam,
respectively; $\Omega_{p}$ is the Rabi frequency of the probe beam. $\Omega$
is the modulation frequency for the sidebands, and
$C_{F,m}^{F^{\prime},m^{\prime}}$ is the normalized transition strength
between the states $\left|F,m\right>$ and $\left|F^{\prime},m^{\prime}\right>$
[18]. In Eq. (3), $c_{q}$ ($a_{q}$) with $q=\pm,0$ are the coefficients of the
electric field vector of the probe (pump) beam in the spherical bases, as
mentioned above.

To solve Eq. (1) using the explicit expression of Hamiltonians given in Eqs.
(2) and (3) in the steady-state regime, the density matrix elements must be
expanded as various Fourier components oscillating in time with various
oscillation frequencies. The detailed description of finding oscillation
frequencies at general polarization configurations were reported in Refs. [18,
35]. Thus, when the three-photon interactions are considered, the optical
coherences have 20 oscillation frequencies and Zeeman coherences and
populations have 11 oscillation frequencies in the cases of
$\textrm{lin}\|\textrm{lin}$ and $\sigma^{+}\sigma^{+}$ configurations [35].
We may use these frequencies in the cases of $\textrm{lin}\bot\textrm{lin}$
and $\sigma^{+}\sigma^{-}$ configurations as well. However, we can select
further non-vanishing components of the density matrix elements. For example,
when the polarization configuration is $\sigma^{+}\sigma^{-}$, the oscillation
frequencies for the optical coherences are given by
$\left\\{-\delta_{p},-\delta_{p}\pm\Omega,-\delta_{p}\pm 2\Omega\right\\}$ and
$\left\\{0,\pm\Omega,\pm 2\Omega,\pm 3\Omega\right\\}$ for $\Delta m(\equiv
m^{\prime}-m)=-1$ and $+1$, respectively. Those frequencies for $\Delta m=-3$
and $+3$ are given by $\left\\{-2\delta_{p},-2\delta_{p}\pm\Omega\right\\}$
and $\left\\{\delta_{p},\delta_{p}\pm\Omega,\delta_{p}\pm 2\Omega\right\\}$,
respectively. In the $\sigma^{+}\sigma^{-}$ configuration, the optical
coherences not satisfying $\Delta m=\pm 1$, and $\pm 3$ vanish. The
oscillation frequencies for the populations are $\left\\{0,\pm\Omega,\pm
2\Omega\right\\}$. We will not present the description of Zeeman coherences
for the $\sigma^{+}\sigma^{-}$ configuration nor the optical and Zeeman
coherences for the $\textrm{lin}\bot\textrm{lin}$ configuration.

Using the expanded density matrix elements, a set of coupled differential
equations for Fourier components of the density matrix elements is obtained
from Eq. (1), which is then solved in a steady-state regime as functions of
$v$ and $\delta$. The MTS signal can be obtained from the relevant optical
coherences with oscillation frequencies of $-\delta_{p}\pm\Omega$, whose real
and imaginary parts are defined as $r_{F,m}^{F^{\prime},m^{\prime},(\pm)}$ and
$s_{F,m}^{F^{\prime},m^{\prime},(\pm)}$, respectively. Then, the in-phase
($I_{n}$) and quadrature ($Q_{n}$) components of the MTS signals are given by

$\displaystyle
I_{n}=\sum_{F=1}^{2}\sum_{F^{\prime}=1}^{2}\sum_{m=-F}^{F}\sum_{q=\pm,0}c_{q}C_{F,m}^{F^{\prime},m+q}$
$\displaystyle\quad\times\int_{-\infty}^{\infty}{\rm d}vf_{\rm
D}(v)\left(s_{F,m}^{F^{\prime},m+q,(-)}+s_{F,m}^{F^{\prime},m+q,(+)}\right),$
(4) $\displaystyle
Q_{n}=\sum_{F=1}^{2}\sum_{F^{\prime}=1}^{2}\sum_{m=-F}^{F}\sum_{q=\pm,0}c_{q}C_{F,m}^{F^{\prime},m+q}$
$\displaystyle\quad\times\int_{-\infty}^{\infty}{\rm d}vf_{\rm
D}(v)\left(-r_{F,m}^{F^{\prime},m+q,(-)}+r_{F,m}^{F^{\prime},m+q,(+)}\right),$
(5)

where $n=39$ and 41 represent 39K and 41K, respectively, and atomic parameters
for specific isotopes are used in the calculation. In Eqs. (II) and (5),
$f_{\rm D}(v)[=1/(\sqrt{\pi}u)e^{-(v/u)^{2}}]$ represents the
Maxwell–Boltzmann velocity distribution function and $u$ is the most probable
speed of the atom in the cell. The final MTS signals including both 39K and
41K are given by

$\displaystyle I(\delta)$ $\displaystyle=$
$\displaystyle\frac{N_{39}}{N_{39}+N_{41}}I_{39}(\delta)+\frac{N_{41}}{N_{39}+N_{41}}I_{41}(\delta-\Delta),$
(6) $\displaystyle Q(\delta)$ $\displaystyle=$
$\displaystyle\frac{N_{39}}{N_{39}+N_{41}}Q_{39}(\delta)+\frac{N_{41}}{N_{39}+N_{41}}Q_{41}(\delta-\Delta),$
(7)

where $N_{39(41)}$ is the atomic density of 39K(41K) in the cell considering
the natural abundances of the isotopes and $\Delta=2\pi\times$ 305 MHz is the
isotope shift of the $F=2\rightarrow F^{\prime}=2$ transition between 39K and
41K.

##  III Experiment

Figure 2: Experimental setup for modulation transfer spectroscopy (MTS). Laser
light is output from a fiber and the red solid line indicates the subsequent
beam path. Two polarizing beam splitters (PBS) and waveplates are used to set
the total power and ratio between the pump and probe beams. The waveplates
labelled $\lambda/n$ are exchanged depending on the polarization
configuration: $n=2$ for linear polarization configurations or $n=4$ for
circular polarization configurations. The transmitted probe beam is detected
on a fast photodiode (FPD). The resulting MTS signal is extracted using a
double-balanced mixer (DBM) and low-pass filter (LPF). nPBS: 50:50 non-
polarizing beam splitter. BD: Beam dump. EOM: Electro-Optical modulator. AFG:
Arbitrary function generator. OSC: Oscilloscope.

The experimental setup is shown in Fig. 2. The laser source is an external
cavity diode laser (Toptica DL Pro). We couple the laser light through a
single mode polarization maintaining fiber such that the subsequent output
beam profile is Gaussian. A pair of polarizing beam splitters (PBS) in
combination with a pair of $\lambda/2$ waveplates are used to control the
total power of laser light delivered to the spectroscopy setup and the ratio
of power between the pump and probe beams. The probe light is passed directly
to the 2 cm long potassium vapour cell. The cell is housed inside a brass
block with a pair of heating elements attached. The cell temperature is raised
to 99(2)°C where we expect a vapour pressure of $3.5\times 10^{-5}$ mbar. The
probe and pump beams are collimated such that their $1/e^{2}$ diameters at the
centre of the cell are $2.06(2)\,\textrm{mm}$ and $1.96(5)\,\textrm{mm}$,
respectively.

The pump light is passed through a homebuilt EOM. The EOM uses a LiTaO3
crystal electrically contacted to a pair of brass capacitor plates. The
addition of an inductor creates a simple LCR circuit that resonantly enhances
the voltage across the crystal. The resonance frequency of the EOM is at
6.054(5) MHz. We always drive the EOM on resonance at its maximum voltage,
producing sidebands each with intensities equal to $15(1)\%$ of the total pump
intensity. More details of the EOM can be found in previous work [8].

We investigate different combinations of laser polarization. To maintain a
well defined quantisation axis we apply a weak magnetic field with a set of
rectangular coils which are concentric with the cell. The coils provide a 1.5
G magnetic field at the centre of the cell in a direction along the beam
propagation axis. By switching around the waveplates indicated in Fig. 2 we
are able to study four different polarization configurations of the laser
light through the cell: circular polarization where the pump and probe drive
opposite transitions ($\sigma^{+}\sigma^{-}$), circular polarization where the
pump and probe drive the same transitions ($\sigma^{+}\sigma^{+}$), linear
polarization where the pump and probe are perpendicular
($\textrm{lin}\bot\textrm{lin}$) and linear polarization where the pump and
probe are parallel ($\textrm{lin}\parallel\textrm{lin}$).

A simple set of electronics apply a phase lock-in technique to demodulate the
MTS signal. The homebuilt fast photodiode detects the beat signal between the
modulated probe carrier and the sidebands. This signal is passed through a
bias-tee [Mini-Circuits model: ZFBT-4R2GW] which filters out the DC component
of the signal corresponding to the standard saturated absorption profile and
sends it to a secondary channel of the oscilloscope. The modulated component
is passed to the double-balanced mixer (DBM) [Mini-Circuits model: ZAD-1H+]
which multiplies the signal with the reference signal provided by the
arbitrary function generator (AFG) [Tekronix model: AFG 3102]. The AFG has two
outputs. One output is used to drive the EOM and the second output is used as
a local oscillator which acts as our reference for the demodulation. The
relative phase of the two outputs can be set using the AFG, allowing us to
fully characterise the phase parameter. The demodulated signal can then be
sent to the oscilloscope for data acquisition or to feedback circuitry to
stabilize the laser frequency.

Figure 3: (a) Absorption signal as filtered by bias-T (top) and MTS signal
(bottom) for potassium in the $\sigma^{+}\sigma^{+}$ configuration. The vapour
cell was 2 cm long and heated to $99~{}^{\circ}\textrm{C}$. Data were taken
with probe and pump beam intensities of $78(2)~{}\textrm{mW cm}^{-2}$ and
$86(5)~{}\textrm{mW cm}^{-2}$ respectively. (b) Signal for rubidium in the
$\sigma^{+}\sigma^{+}$ configuration. The vapour cell was 4 cm long and at
$20~{}^{\circ}\textrm{C}$. Data were taken with probe and pump beam
intensities of $90(2)~{}\textrm{mW cm}^{-2}$ and $53(2)~{}\textrm{mW cm}^{-2}$
respectively. The red lines show the D2 transition and the blue lines show the
D1 transition. For potassium, zero detuning corresponds to the
$X(1,2)\rightarrow F^{\prime}=2$ transition in 39K for the D1 trace and the
$X(2,3)\rightarrow F^{\prime}=3$ transition for the D2 trace. For rubidium,
zero detuning corresponds to the $F=2\rightarrow F^{\prime}=2$ in 87Rb for the
D1 trace and the $F=2\rightarrow F^{\prime}=3$ transition for the D2 trace.
The vertical axes for both (a) and (b) share the same ticks.

##  IV Results

###  IV.1 Comparison with D2 and rubidium MTS

To better illustrate the role of the crossovers and cycling transitions in MTS
we have recorded an MTS trace for the D2 and D1 transitions of both potassium
and rubidium for a brief comparative study. 87Rb is suitable for a comparison
against potassium since they both have the same hyperfine quantum numbers.

Figure 3(a) shows the absorption spectroscopy and MTS signals for the D1 and
D2 transitions in potassium. The vertical axis for the MTS signals are
normalised with respect to the 39K D1 $X(1,2)\rightarrow F=2$ feature. These
data were taken using the $\sigma^{+}\sigma^{+}$ configuration with a probe
beam intensity of $78(2)~{}\textrm{mW cm}^{-2}$ and a pump beam intensity of
$86(5)~{}\textrm{mW cm}^{-2}$. All intensities quoted for the experiment are
the peak intensity where the beam is assumed to be Gaussian. We note from the
absorption profile on Fig. 3(a) that all of the hyperfine transitions lie
within the same Doppler profile. The D2 transition shows the standard MTS
signal, with a single strong feature resulting from the $F=2\rightarrow
F^{\prime}=3$ closed transition on a flat zero background. This feature is
ideal for laser frequency stabilization [17]. For the D1 transition we observe
several strong features in the MTS signal, despite the absence of a closed
transition. The signal is dominated by the ground-state crossover transition,
$X(1,2)\rightarrow F^{\prime}=2$, providing a feature suitable for laser
frequency stabilization. We note that the $X(1,2)$ crossover features involve
both ground-state energy levels such that the pump beam and probe beam can
simultaneously interact with atoms in both states preventing hyperfine optical
pumping, effectively ‘closing’ the transition. Such strong ground-state
crossovers have also been reported in the D2 MTS of lithium [23]. In contrast,
the crossover signals for the D2 transition are weak, as reported previously
[17]. For the $\sigma^{+}\sigma^{+}$ configuration the existence of a cycling
transition may be detrimental; optical pumping on the cycling transition will
populate the $F=2,m_{f}=2$ ground state, but on the crossover at least one of
the pump or probe beams will be interacting with the depopulated $F=1$ state.
This will restrict transfer of the sidebands from the pump to the probe,
weakening the signal. The strong resonances observed for MTS of the D1
transition in the absence of a cycling transition merit further study.

Figure 3(b) shows the absorption spectroscopy and MTS signals for the D1 and
D2 transitions in rubidium. These data were taken taken using the
$\sigma^{+}\sigma^{+}$ configuration with a probe probe beam intensity and
pump beam intensity of $90(2)~{}\textrm{mW cm}^{-2}$ and $53(2)~{}\textrm{mW
cm}^{-2}$, respectively. Again the MTS signal for the D2 transition is
dominated by the cycling transitions. However, in contrast to K, the MTS
signal on the D1 transitions for Rb does not show any measurable features.
This can be attributed to the lack of ground-state crossover transitions; in
Rb the ground-state hyperfine splitting (greater than 3 GHz for both isotopes)
is significantly greater than the Doppler broadening (on the order 500 MHz in
a room temperature cell), as evident in the absorption spectrum.

###  IV.2 MTS signal optimisation and comparison with theory

Figure 4: Gradient (black circles) and amplitude (grey triangles) of the
$X(1,2)\rightarrow F^{\prime}=2$ 39K transition as a function of the beam
intensity in the $\textrm{lin}\parallel\textrm{lin}$ configuration. Results
are shown for (a) the pump beam optimised at a constant probe of
$47.4(8)~{}\textrm{mW cm}^{-2}$ and (b) the probe beam optimised with a
constant pump of $32.8(2)~{}\textrm{mW cm}^{-2}$. Figure 5: (a) MTS signals on
the potassium D1 transition for the $\textrm{lin}\parallel\textrm{lin}$
configuration for different relative phases at the double-balanced mixer. The
data were taken with intensities for the pump carrier, the pump sidebands and
the probe of 25(2)$\textrm{mW cm}^{-2}$, 5.5(4)$\textrm{mW cm}^{-2}$ and
59.5(1.1)$\textrm{mW cm}^{-2}$, respectively. The grey solid trace is the
experimental data and the black dashed trace is the fitted simulation. An
arbitrary vertical offset is added between traces to separate the signals
recorded at different phases. Highlighted regions indicate areas where we see
transitions and have their ground-state shown next to them. Vertical lines
show the transition locations for 39K. Numbers on the left of each phase show
the phase, $\tilde{\phi}$. (b) MTS signal zoomed into the $F=2\rightarrow
F^{\prime}$ (top) and $F=2\rightarrow F^{\prime}$ (bottom) transitions at
$\tilde{\phi}=117.5~{}^{\circ}$. A vertical offset has been added to separate
the two transitions. (c) The amplitude of the $X(1,2)\rightarrow F^{\prime}=2$
feature as a function of the phase for experimental (black circles) and
theoretical predictions (red squares). Figure 6: The MTS signal on the
potassium D1 $X(1,2)\rightarrow F^{\prime}$ transitions for different
polarization configurations. All the traces were recorded using the optimum
parameters for the $\textrm{lin}\parallel\textrm{lin}$ configuration. The gray
solid lines show the experimental data and the black lines indicate the best
fit theory for simulations at 75% intensities. We have also included, for
comparison, the best fit theory data for a simulation at 50% of the quoted
intensities: the red dashed lines. The polarization configurations are (a)
$\textrm{lin}\parallel\textrm{lin}$, (b) $\textrm{lin}\perp\textrm{lin}$, (c)
$\sigma^{+}\sigma^{-}$ and (d) $\sigma^{+}\sigma^{+}.$

We use measurements of the strongest feature resulting from the
$X(1,2)\rightarrow F^{\prime}=2$ transition to optimise the parameters of the
MTS setup for the D1 transition in potassium. Figure 4 shows the gradients and
amplitudes of the signal for the $\textrm{lin}\parallel\textrm{lin}$
configuration as a function of the intensities of (a) the pump beam and (b)
the probe beam, with the intensity of the other beam held constant. We
optimise the intensities to obtain the steepest gradient. The pump intensity
was optimised first with a constant probe intensity of 47.4(8) $\textrm{mW
cm}^{-2}$; then the probe intensity was optimised with a constant pump
intensity of 32.8(2) $\textrm{mW cm}^{-2}$ (chosen arbitrarily). The optimised
intensities were found to be $36(3)~{}\textrm{mW}~{}\textrm{cm}^{-2}$, and
$59.5(1.1)~{}\textrm{mW}~{}\textrm{cm}^{-2}$ for the pump and probe beams,
respectively. Note the saturation intensity is
$1.75~{}\textrm{mW}~{}\textrm{cm}^{-2}$.

In Fig. 5 we investigate the dependence of the MTS signal for the
$\textrm{lin}\parallel\textrm{lin}$ configuration as a function of the
relative phase, $\tilde{\phi}$, at the double-balanced mixer. Traces of
different phase were taken by introducing a phase delay using the AFG and were
taken in $10^{\circ}$ steps. Theoretical traces were calculated as discussed
in Sec. II. The simulation assumed a flat-top intensity distribution rather
than the Gaussian profile used in the experiment. To partially compensate for
this, intensities used in the simulation were set to 75% of the quoted peak
experimental intensities. The width of this top-hat function is equal to the
$1/e^{2}$ diameter of the beam. Two contributions are calculated: an in-phase
component $I(\delta)$ and a quadrature component $Q(\delta)$, where $\delta$
is the detuning. These are superimposed to obtain the signal, $S$, as a
function of the relative phase $\phi$,

$S[\delta,\phi]=A(I(\delta)\cos\tilde{\phi}+Q(\delta)\sin\tilde{\phi}),$ (8)

where $A$ is an arbitrary amplitude factor to match the theory to the
experimental measurements. Experimentally there are two contributions to
$\tilde{\phi}$; the relative phase $\phi$ set by the AFG and an arbitrary
phase offset $\phi_{0}$ arising from cable delays. Thus to compare the theory
and experiment we use $\tilde{\phi}=\phi-\phi_{0}$, and adjust the value of
$\phi_{0}$ to find the best agreement. In practice, traces for all measured
phases of a given polarization are fitted simultaneously to Eq. (8) to extract
the parameters $A$ and $\phi_{0}$.

Figure 5(a) shows example MTS signals for both the experiment (solid lines)
and theoretical predictions (dashed lines) in phase steps of $30^{\circ}$.
There are three distinct regions highlighted corresponding to features arising
from the $F=1$ ground state (right), the $X(1,2)$ crossover (centre) and the
$F=2$ ground state (left). It is also worth noting that the rightmost region
also has a significant contribution from the 41K $X(1,2)$ crossovers as well,
see Fig. 1, but the other features are purely from 39K. Figure 5(b) shows a
zoom in to the $F=2\rightarrow F^{\prime}$ (top) and $F=1\rightarrow
F^{\prime}$ (bottom) ground-state transitions, respectively. The peak-to-peak
amplitude for the $X(1,2)\rightarrow F^{\prime}=2$ feature as a function of
the phase is shown in Fig. 5(c). We note that the optimal peak-to-peak
amplitude is not obtained in either the quadrature or in-phase scenarios, but
rather at a relative phase of $\tilde{\phi}\sim 112^{\circ}$.

We can see from Fig. 5 that there is generally good agreement between theory
and experiment. For phases where the $X(1,2)\rightarrow F^{\prime}$ features
are maximised, the relative peak heights and signal widths agree particularly
well. Further evidence is shown by (c) which compares the experimental
amplitude against the predicted amplitude for the dominant feature over phases
from $0^{\circ}$ to $180^{\circ}$. We certainly see in the region closest to
the maximum that we get better agreement. For the weaker $F=1\rightarrow
F^{\prime}$ and $F=2\rightarrow F^{\prime}$ transitions shown in (b) there is
reasonable agreement, although there are differences in some cases.

###  IV.3 Different polarization configurations

A simple reconfiguration of the waveplates in Fig. 2 allows us to look at the
MTS signal for other polarization configurations. The results are shown in
Fig. 6 for the signals arising from $X(1,2)\rightarrow F^{\prime}$
transitions. The traces were taken with $\tilde{\phi}=117.5~{}^{\circ}$, close
to the optimum phase of $\sim 112~{}^{\circ}$ degrees from Fig. 5(b). The
laser parameters were the same as the optimum values found for the
$\textrm{lin}\parallel\textrm{lin}$ configuration. In the
$\textrm{lin}\perp\textrm{lin}$ case we observe a substantial increase in the
amplitude of the $X(1,2)\rightarrow F^{\prime}=2$ signal and a reduction in
the size of the neighbouring features. There is very good agreement between
the theory and experiment for all cases except for the $\sigma^{+}\sigma^{+}$
configuration. We note that in this case the theory predicts substantially
broader features than are observed experimentally. The Rabi frequencies of
laser beams in the experiment lie between 2–4$\times\Gamma$. In this region of
laser intensities, our assumption of three-photon interactions may be
insufficient, in particular for the specific polarization configuration, and
we may need new calculations of higher order interactions in this case. This
kind of improvement in the calculation is beyond the scope of the current
paper.

Polarization |  $A$ (V) |  $\phi_{0}$ (°)
---|---|---
$\textrm{lin}\parallel\textrm{lin}$ | $1787(7)$ | $110.7(4)$
$\textrm{lin}\perp\textrm{lin}$ | $886(3)$ | $109.4(3)$
$\sigma^{+}\sigma^{-}$ | $1576(5)$ | $108.3(3)$
$\sigma^{+}\sigma^{+}$ | $1232(8)$ | $103.7(7)$
Table 1: Fitted parameters for $\textrm{D}_{1}$ experimental results against
theory. Each of the polarization configurations are listed. $A$ and $\phi_{0}$
are as defined in Eq. (8).

Table 1 shows the fitted values of $A$ and $\phi_{0}$ for all polarization
configurations. The values of $A$ and $\phi_{0}$ mean little on their own
since they are arbitrary and exclusive to our specific setup, however in the
context of theory we expect these values to be constant across different
polarization configurations. We find good agreement between the different
configurations for the phase offset, with all cases falling within a
$7~{}^{\circ}$ range ($1.4^{\circ}$ if we exclude the $\sigma^{+}\sigma^{+}$
data). The amplitude conversion factor, $A$, does not agree between different
configurations. We note that although the $\textrm{lin}\parallel\textrm{lin}$
and the $\sigma^{+}\sigma^{-}$ results are are in reasonable agreement; the
$\textrm{lin}\perp\textrm{lin}$ disagrees by a factor of 2 compared with
$\textrm{lin}\parallel\textrm{lin}$. The theory predicts that the
$X(1,2)\rightarrow F=2$ transition to be at least a factor of two larger in
the linear perpendicular case than in the $\textrm{lin}\parallel\textrm{lin}$
case. However, this is not observed in experiment.

It is of interest to investigate how the theoretical predictions depend upon
the intensities used. As mentioned previously, the simulation assumes a
uniform flat-top intensity distribution but the beam in the experiment is
Gaussian. Figure 6 also includes the simulations for intensities at 50% of the
experimental peak intensities. Across all polarization configurations the
broadening is substantially less. Although the relative heights of the peaks
is somewhat mismatched we note an immediate improvement in the agreement
between the theory and experiment for the $\sigma^{+}\sigma^{+}$. However, the
width of the other features is underestimated by the theory in this case. This
is a crucial point. If simulating the beam at the peak intensity quoted, then
it would be a uniform beam but with a constant intensity at the peak of the
Gaussian beam. This may result in an overestimation of power being delivered
to the atoms. Likewise a uniform beam at 50% of the peak intensity (the
average beam intensity) may underestimate the power at the atoms. It is
evident from the simulations shown in Fig. 6 that there is a strong intensity
dependence from the theory which may assist in explaining some of
discrepancies. We choose, albeit arbitrarily, 75% of the peak beam intensity
obtain theory between these two extremes. Regardless, the simulation still
produces results in reasonable agreement with the experiment for the
$\textrm{lin}\parallel\textrm{lin}$, $\textrm{lin}\bot\textrm{lin}$ and the
$\sigma^{+}\sigma^{-}$ cases. We have not used a Gaussian beam in our
simulations because it is too computationally expensive. In addition, the
difference between experiment and theory might be reduced by using new
calculations of higher order interactions. We are currently elaborating the
calculation to enhance the accuracy.

##  V Conclusions

We have presented an experimental study of the modulation transfer
spectroscopy of the D1 transition in potassium. We have also presented a
theoretical model which can be used to predict the MTS signal which shows
generally good agreement with the experimental measurements. We have shown
that for the D1 transition, the MTS signal shows strong features originating
from ground-state crossover transitions. We have optimized various
experimental parameters, including the beam intensities and polarizations, to
maximise the $X(1,2)\rightarrow F^{\prime}=2$ feature, providing a good
reference for laser frequency stabilisation. We expect these results will be
of interest to researchers employing the D1 transition for laser cooling and
optical pumping of potassium in quantum gas experiments.

## Acknowledgements

We acknowledge insightful discussions with Sarah Bromley, Jonathan Mortlock
and Phil Gregory. This work was supported by the UK Engineering and Physical
Sciences Research Council (EPSRC) Grant EP/P01058X/1 and the National Research
Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.
2020R1A2C1005499).

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