# Discourse Analysis for Evaluating Coherence in Video Paragraph Captions

Arjun R. Akula, Song-Chun Zhu
UCLA Center for Vision, Cognition, Learning, and Autonomy
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Video paragraph captioning is the task of automatically generating a coherent
paragraph description of the actions in a video. Previous linguistic studies
have demonstrated that coherence of a natural language text is reflected by
its discourse structure and relations. However, existing video captioning
methods evaluate the coherence of generated paragraphs by comparing them
merely against human paragraph annotations and fail to reason about the
underlying discourse structure. At UCLA, we are currently exploring a novel
discourse based framework to evaluate the coherence of video paragraphs.
Central to our approach is the discourse representation of videos, which helps
in modeling coherence of paragraphs conditioned on coherence of videos. We
also introduce DisNet, a novel dataset containing the proposed visual
discourse annotations of 3000 videos and their paragraphs. Our experiment
results have shown that the proposed framework evaluates coherence of video
paragraphs significantly better than all the baseline methods. We believe that
many other multi-discipline Artificial Intelligence problems such as Visual
Dialog and Visual Storytelling would also greatly benefit from the proposed
visual discourse framework and the DisNet dataset.

## Introduction

In recent years, video captioning task has drawn a lot of attention from
computer vision community where the task is to explain video semantics using
sentence descriptions (video-paragraphs) (?; ?; ?). With the introduction of
large scale video captioning datasets, great advances have been achieved in
describing the videos using multiple-sentence descriptions. While the success
of these methods is encouraging, they all lack a formal approach to compute
coherence of video-paragraphs (?; ?; ?; ?).

At UCLA, we are working on a novel discourse based framework to model the
coherence in video-captioning task. Text-level discourse structure aids in
understanding a piece of text by linking it with other text units (such as
surrounding clauses, sentences, etc) (?; ?; ?). Linguistic studies
demonstrated that coherence of a text is reflected by its discourse structure
and relations. In other words, coherent text favors certain types of discourse
relation transitions.

Figure 1: (a) Incoherent video-paragraph; (b) Coherent video-paragraph; (c)
Incoherent video-paragraph but becomes coherent when conditioned on the
coherence of input video.

For example, video-paragraph shown in Figure 1(b) is clearly more coherent
than Figure 1(a). This can be attributed to the ordering of the discourse
relations among the sentences. In Figure 1(a), the sentence S1 is the cause
for sentence S3; S2 is the cause for S4. In Figure 1(a), the sentence S1 is
the cause for sentence S2; S3 is the cause for S4. The discourse relation
‘cause’ is more localized in Figure 1(b) and is therefore more coherent
compared to Figure 1(a). Existing video captioning methods fail to perform
such discourse based analysis to assess the coherence of generated video-
paragraphs. Instead they rely on metrics such as BLEU, METEOR, ROUGE, CIDEr,
etc which merely compare the generated video-paragraphs with human paragraph
annotations (?; ?; ?; ?; ?; ?; ?; ?; ?; ?).

Can human-generated annotations be incoherent? We did an experiment to verify
whether if it is possible for humans to write incoherent descriptions. This is
an important question because, if the human paragraphs are always coherent,
then we can infer coherency of generated paragraphs by simply using scores
such as BLEU. However, if human paragraphs are not always coherent, then
relying only on scores such as BLEU is problematic and hence requires explicit
coherence assessment methods like ours. We conducted a human study experiments
where we asked people to describe 100 videos (randomly chosen from ActivityNet
dataset ()) without training them on how to write coherent paragraphs. We then
evaluated the coherency of the descriptions using different set of human
subjects. We found that about 72% of the descriptions are found to be
incoherent (?; ?; ?; ?; ?; ?; ?; ?).

Moreover, video-paragraphs may sometimes appear to be incoherent
linguistically but are actually coherent given the video content. For example,
video-paragraph shown in Figure 1(c) looks incoherent as the topics across the
sentences shift abruptly. However, note that, this is one of the best possible
coherent description one can generate for the video shown in Figure 1(c). This
implies that video-paragraph, in this case, is indeed coherent whereas the
underlying video is incoherent. In this work, we identify such cases by
proposing a discourse representation for videos. We model the coherence of the
video-paragraphs by conditioning on coherence of their corresponding videos.
We also introduce DisNet, a novel dataset containing discourse annotations of
about 3000 videos, to learn a model to automatically infer visual discourse
relations. To the best of our knowledge, this is the first work to introduce
discourse framework for videos (?; ?; ?; ?; ?; ?; ?).

Contributions. The contributions of this work are threefold: (i) novel
discourse representation of videos and video-paragraphs; (ii) a new dataset
containing discourse annotations of videos and video-paragraphs; (iii) a novel
approach to assess the coherence of video-paragraphs based on discourse cues
from both videos as well as video-paragraphs.

The remainder of this paper is organized as follows: First, we define the
discourse representation of videos and video-paragraphs in video-captioning
task. Next, we introduce a new dataset containing the proposed discourse
annotations. We also train models using this dataset and report the baseline
accuracies in automatically inferring the discourse representations of videos
and video-paragraphs. Finally, in the experiments section, we demonstrate that
the proposed discourse framework is effective in evaluating coherence of
paragraphs generated from video-captioning task.

## Visual Discourse Representation

Videos. Defining discourse representation of videos comprises three main
steps, namely: (a) identification of elementary discourse units (EDUs) (?);
(b) identification of discourse acts, which are categories of EDUs that
pertain to their role in the videos and (c) identification of discourse
relations, which aid in understanding relationships among EDUs. In our pilot
experiments, we found that shots are effective EDUs for videos. A shot in a
video refers to a contiguous recording of one or more frames depicting a
continuous action in space and time. Therefore, we first segment the video
into shots and then define the discourse acts and relations among shots. We
used the dense captioning events dataset to sample videos and classified shots
in the videos using a manual iterative process with experts until a stable set
of discourse acts and relations are obtained.

We define the discourse structure of video as directed acyclic graph
$\mathbf{G}$ consisting of a set of nodes $\mathbf{V}$ and a set of labelled
edges $\mathbf{E}$. The interior nodes (non-terminal nodes) denote the
discourse acts and the leaf nodes (terminal nodes) constitute the shots
obtained from video segmentation. $\mathbf{P}$ denotes the root node that
doesn’t have any branches on top of it. Each video contains only one root
node. The edges connecting interior nodes and leaf nodes denote the discourse
relations. Detailed information about each discourse act and the relations
allowed are given below.
Discourse Acts: We found the following three discourse acts to be helpful in
analyzing the coherence of the video:
(a) Primary Context: Shots in the video that contain the main
theme/information are labelled as primary context.
(b) Secondary Context: Shots in the videos that aid in understanding the
primary context of the video are labelled as secondary context.
(c) Auxiliary Context: Shots in the videos that are not essential in
interpreting the primary context of the video are labelled as auxiliary
context.
For instance, consider a typical sports video which usually contain shots
related to players, scores and crowd. Shots showing the players - which is the
main theme in this case - is labelled as primary context; shots showing scores
are labelled as secondary context and shots showing crowd are labelled as
auxiliary context. In our experiments, we found that videos that have only
primary context are usually more coherent than videos that have all the three
(primary, secondary, auxiliary) contexts.

Discourse Relations: In addition to the three discourse acts, we found the
following six discourse relationships to be crucial in analyzing the coherence
of videos:
(a) Interpretation: The edge label between a shot and a discourse act is
labelled as interpretation if the shot summarizes or helps in
interpreting/establishing the context of the discourse act. For a given
discourse act in the video discourse structure, there can be only one
interpretation. Therefore, for most of the videos, the interpretation is
usually (but not necessarily) the shot that appears first in the video among
all the shots in that discourse act.
(b) Sequence: The edge label between a shot and a discourse act is labelled as
sequence if the shot introduces new objects that are not yet introduced in the
previous shots of discourse act.
(c) Sub-Context: The edge label between a shot and a discourse act is labelled
as sub-context if the shot gives detailed view of objects that are already
introduced briefly in the previous shots of discourse act. For instance,
consider a interpretation shot showing a person playing the guitar. If the
next shot zooms in to the guitar to give more details of playing guitar, we
label this new shot using sub-context relationship.
(d) Super-Context: Super-context is a special case of sequence relationship. A
shot in super-context relationship introduces new objects and do not shift the
focus from objects shown in the previous shot. For instance, zooming out gives
the super-context of a shot.
(e) Elaboration: The edge label between a shot and a discourse act is labelled
as elaboration if the shot continues to focus on objects that are already
introduced in the previous shots of discourse act.
(f) Repetition: The edge label between a shot and a discourse act is labelled
as repetition if a shot is repeated from the previously introduced shots of a
discourse act.

Video-Paragraphs. We found that only a subset of discourse relations (cause,
background, sequence, interpretation and elaboration) from a text-level
discourse framework called Rhetorical Structure Theory (RST) are sufficient
enough to analyze coherence of paragraphs in video captioning task. Moreover,
we found the following five additional new discourse relations among sentences
in video-paragraphs, which are specific to video-captioning task and hence are
not part of RST framework:
(a) Expectation: Discourse relation ‘Expectation’ captures the scenarios where
an object/action that is not present in the video but is mentioned in the
video-paragraph. For example, in the below video-paragraph, S2 is related to
S1 using an Expectation discourse relationship.
[It looks like an office.]S1 [There is no chair and desk.]S2
(b) Sentiment: Sometimes sentences in the video-paragraph mention the
sentiment of the video. For instance, in the following video-paragraph, S2 is
related to S1 with a sentiment discourse relationship.
[There are two persons standing next to a bus.]S1 [The scene looks sad.]S2
(c) Continuation: If the sentence continues to describe the same object or
event as in the previous sentence, we use continuation relationship to connect
them.
(d) Parallel: In a video-paragraph, if two sentences describe different
activities happening simultaneously in the video, we connect the sentences
using parallel discourse relationship.
(e) Video-Attribute: It is common in video captioning task to describe the
properties of the video in the video-paragraph. For instance, in the below
video-paragraph, S2 is related to S1 with a video-attribute relationship.
[There are two persons standing next to a bus.]S1 [The scene looks blurry.]S2

## DisNet Dataset

We use the 10,000 videos from the newly-released ActivityNet Captions dataset
and annotate their discourse structures. ActivityNet Captions contains a total
of 20k videos amounting to 800 video hours with 100k total descriptions. These
videos are diverse and comprehensive and their video-paragraphs are collected
by training the human subjects to write coherent descriptions. Therefore their
video-paragraphs are mostly coherent. Therefore they are well-suited for our
discourse representation task. Annotation of discourse acts and relations was
conducted by eight graduate and undergraduate students. Each video and its
video-paragraph was annotated by at least two students. We solved the
disagreements found in the annotations together. Before students began
annotating, they were presented with an instruction manual that explained
definitions of discourse acts and relations. They were also given a few warm-
up videos and video-paragraphs to do as practice before beginning annotation.

As our goal is to better understand the discourse structure, we chose to only
take videos that have at least 3 shots. Also, some videos have hundreds of
shots. As this is too much work for an annotator, we discarded the videos with
more than 30 shots. During annotation, few more videos were discarded due to
bugs that occurred with the shot detector algorithm. Finally, at the end, 3000
videos and the corresponding 3000 video-paragraphs were fully annotated with
their discourse structures. We will now present an overview of the dataset.

### Dataset Statistics

On average, each of the 3000 videos have about 12.6 shots. We found that, as
the number of discourse acts in the video increases, the number of shots also
increases. The average duration of the videos is 120 seconds. The average
number of sentences in the 3000 video-paragraphs is 3.12. We found that 67% of
the shots in the dataset are annotated as primary context, 14% as secondary
context and 19% as auxiliary context. The inter-rater reliability (using
Krippendorf’s Alpha) is 0.81 for primary context, 0.62 for secondary context
and 0.78 for auxiliary context. Furthermore, videos containing all the three
discourse acts, usually but not always, are found to be less coherent than
videos containing only the primary context.

Table 1: Percentage and inter-rater reliability of discourse relations in video-paragraphs Discourse Relation |  Total % |  $\alpha$
---|---|---
Background |  4.3 |  0.66
Continuation |  12.7 |  0.57
Parallel |  16.3 |  0.66
Elaboration |  8.0 |  0.62
Cause |  7.1 |  0.38
Video-Attribute |  4.1 |  0.89
Sequence |  25.4 |  0.81
Interpretation |  15.8 |  0.74
Sentiment |  2.8 |  0.72
Expectation |  3.5 |  0.62

Table 1 shows the proportion of video discourse relations in our dataset. As
can be seen, the relations elaboration, sequence and interpretation make up a
large portion of the dataset. The inter-rater reliability show that the
relation annotations for videos were almost entirely in agreement. The
relations sequence, interpretation, parallel and continuation dominate other
relations in the dataset. In particular, the interpretation relation is found
at least once in most paragraphs. This can be attributed to the fact that most
paragraphs in the dataset begin with a brief summary of video. The inter-rater
reliability of these relations show that some relations had more agreement
between annotators than others. The relation ‘cause’ was the least reliable
category. This was perhaps due to the lack of explicit connectives (such as
because) in the paragraph descriptions. We also noticed that elaboration
relation had some overlap with sequence and background relations.

### Predicting Discourse Acts and Relations in Videos

We use standard machine translation encoder-decoder RNN model (Sutskever et
al., 2014) for learning to predict the sequence of discourse acts from the
input sequence of shots in videos. As RNN suffers from decaying of gradient
and blowing-up of gradient problem, we use LSTM units, which are good at
memorizing long-range dependencies due to forget-style gates (Hochreiter and
Schmidhuber, 1997). The sequence of video shots are passed to the encoder. The
last hidden state of the encoder is then passed to the decoder and then the
decoder generates the sequence of discourse acts. Let the input sequence of
video shots be $\mathbf{x}=\left(x_{1},...,x_{p}\right)$ and the output
sequence of discourse acts as $\mathbf{y}=\left(y_{1},...,y_{n}\right)$. The
distribution of the output sequence w.r.t. the input sequence is:

$p(y_{1},...,y_{n}\mid x_{1},...,x_{p})=\prod_{t=1}^{n}p(y_{t}\mid h_{t}^{d})$
(1)

where $h_{t}^{d}$ is the hidden state at the $t^{th}$ time step of the
decoding LSTM. We further improve our encoder-decoder model using an attention
based sequence-to-sequence model (?). The attention weights act as an
alignment mechanism by re-weighting the encoder hidden states that are more
relevant for decoder time step.

We learn discourse relations in videos using an other encoder-decoder RNN
model similar to the discourse act prediction model. However, in this case, we
first predict the discourse acts and then use them as input features in
addition to the input video scenes.

We use 210 videos for training, 30 videos for validation and the rest 70
videos for testing. We fix our sampling rate to 5fps to bring uniformity in
the temporal representation of actions across all videos. These sampled frames
are then converted into features using VGGNet (?). We initialize the decoder
embedding with Google pre-trained word2vec word embeddings (?). We tune all
hyperparameters using our validation data: learning rate, weight
initializations, hidden states. We use a 1024-dimension RNN hidden state size.
We use Adam optimizer (?). We apply a dropout of 0.5.

Results:Attention based models gave better accuracies than the simple sequence
to sequence models. In particular, the LSTM unit with 512 hidden units and
single encoding layer outperformed other configurations. These results are
encouraging considering the small size of our training dataset.

### Predicting Discourse Relations in Video-Paragraphs

We use RST based disocourse representation to model discourse relations of
video paragraphs as many of our relations belong to RST framework. We use
shift-reduce parser to represent the discourse relations in RST framework.
Shift-reduce discourse parser is an incremental discourse parsing approach,
that maintains a queue containing the EDUs in the paragraph that have not been
processed yet, and a stack of RST subtrees that will eventually be combined to
form a complete tree. All EDUs are placed in the queue and the stack is empty
initially. The parser selects shift or reduce actions in an iterative manner
until a complete RST tree is found or no further actions can be executed. The
problem of selecting the best parsing action given the current parsing state
(i.e., the stack and queue) is modeled as a max margin classification problem.

We use 2200 video paragraphs for training, 200 for validations and the rest
600 for testing. We evaluate the performance on test data using F1 scores for
agreement on unlabeled EDU spans, spans labeled only with nuclearity, and
fully labeled spans (i.e. including relation information). The F1-score is
84.6% for unlabeled spans, 71.6% for labeled spans and 45.9% for fully labeled
spans.

## Conclusion

We are currently exploring a novel discourse based framework to evaluate the
coherence of video paragraphs. Central to our approach is the discourse
representation of videos, which helps in modeling coherence of paragraphs
conditioned on coherence of videos. We also introduce DisNet, a novel dataset
containing the proposed visual discourse annotations of 3000 videos and their
paragraphs.

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# Improving Sensitivity of an Amplitude-Modulated Magneto-Optical Atomic
Magnetometer using Squeezed Light

Jiahui Li  Irina Novikova  Department of Physics, College of William & Mary,
Williamsburg, VA 23187 USA

###### Abstract

We experimentally demonstrate that a squeezed probe optical field can improve
the sensitivity of the magnetic field measurements based on nonlinear magneto-
optical rotation (NMOR) with an amplitude-modulated pump when compared to a
coherent probe field under identical conditions. To realize an all-atomic
magnetometer prototype, we utilize a nonlinear atomic interaction, known as
polarization self-rotation(PSR), to produce a squeezed probe field. An
independent pump field, amplitude-modulated at the Larmor frequency of the
bias magnetic field, allows us to extend the range of most sensitive NMOR
measurements to sub-Gauss magnetic fields. While the overall sensitivity of
the magnetometer is rather low ($>250~{}\mathrm{pT}/\sqrt{\mathrm{Hz}}$, we
clearly observe a $15~{}\%$ sensitivity improvement when the squeezed probe is
used. Our observations confirm the recently reported quantum enhancement in a
modulated atomic magnetometer [1].

Optical atomic magnetometers [2, 3, 4] – devices that enable magnetic field
measurements using changes in optical properties of an atomic medium – are
becoming one of the most promising platforms for precise magnetic field
measurements, and are finding wide applications including geophysics [5, 6],
medical diagnosis [7, 8, 9, 10], fundamental science [11, 12, 13], and many
others [14]. The general operational principle for such devices is based on
the strong relationship between optical absorption and refraction and the spin
orientation or alignment of ground-state electrons in atoms. Once the atomic
spins are initialized by an optical field, their evolution in the magnetic
field is monitored via optical means, giving direct access to the magnetic
field measurements. One of the common diagnostic approaches relies on the
magnetic-optical polarization rotation (NMOR), in which the magnitude of the
magnetic field is calculated from the measured rotation angle of the linear
polarization of an optical probe field [15]. This detection method is utilized
in some of the most sensitive atomic magnetometers [5, 16, 17].

Figure 1: The schematic of the experimental setup. Two independent lasers are
used to produce the pump and probe fields. Polarization squeezing for the
probe optical field is generated in an auxiliary 87Rb (PSR) cell before
entering the main magnetometer cell, and its quadrature is controlled with a
phase retardation plate (PRP). The pump field is amplitude modulated using an
acousto-optical modulator (AOM). The dynamic changes in the probe field
polarization are detected using a balanced photodetector (BPD), and analyzed
either using a spectrum analyzer, or a lock-in amplifier.

Once all technical noises are suppressed, the ultimate sensitivity of the
polarization measurements is limited by the projection noise of an atomic
ensemble [18, 19] and by the shot noise of an optical probe [20]. To improve
the performance beyond these limits, quantum mechanical tools become
necessary. In the regime where the magnetometer performance is shot-noise
limited, the use of squeezed optical probe becomes beneficial. Previous
experiments utilized a linearly polarized optical probe field in which the
coherent vacuum in the orthogonal polarization was replaced by the squeezed
vacuum. They demonstrated the improved magnetometer sensitivity for ac
magnetic field in the spectral range in which squeezing was observable [21,
22, 23, 24]. Technical $1/f$ noise has prevented detection of slowly varying
dc magnetic fields.

Recently, the quantum-enhanced measurements of $43~{}\mu$T magnetic field with
sub-pT precision have been demonstrated [1]. In this work, the problem of the
$1/f$ noises was solved by using an additional pump field modulated at Larmor
frequency. Such modulation shifts the dc-magnetic response into a high-
frequency domain [25, 26], where the technical noises are reduced, and the
magnetometer performance is mostly limited by the fundamental quantum noises.
That experiment demonstrated 20 % improvement in magnetometer sensitivity,
when shot-noise limited, using a crystal-based source of polarization squeezed
optical probe. Additional demonstrated advantages of this approach were the
increased measurement bandwidth and elimination of the back-action noise at
higher atomic densities.

Here we investigate the possibility of realization of an all-atomic quantum
enhanced magnetometer by using Rb atomic vapor cells for both magnetic field
measurements and to produce a non-classical optical probe. For the latter, we
exploit a source of squeezed vacuum based on polarization self-rotation
mechanism (so-called PSR-squeezed vacuum [27, 28]). We show that the use of a
2dB on-resonance squeezed probe increases the signal to noise (SNR) ratio of
the magnetic response at the modulation frequency, thus increasing the
estimated magnetometer sensitivity by 15%. This result confirms the quantum
enhanced modulated magneto-optical magnetometry reported in [1], while
operating in a very different parameter subspace. In addition to a different
source of squeezed light, we also employed an amplitude-modulated, rather than
phase-modulated, pump field, as shown in Fig. 1, inspired by the AMOR
magnetometer configuration [29, 26, 20, 30]. We also aimed to achieve a high
operational dynamic range by using broad optical resonances capable of
operating in terrestrial magnetic fields [30]. For this purpose, we operated
at a rather high probe field power. Also, since PRS squeezing exists only at
certain optical transitions in Rb, we relied on high probe power and optical
saturation to minimize resonant optical losses. We also used a vapor cell with
a small amount of buffer gas (and potentially without buffer gas), as
velocity-changing collisions at high buffer gas pressure cells are known to
reduce the magneto-optical rotation [31]. Unsurprisingly, we found that in
this regime the overall magnetic sensitivity was rather low due to the short
transient spin coherence lifetime. Nevertheless, we observed a clear
improvement with the use of the squeezed probe field. Our results may be of
interest for applications that cannot use buffer gas to restrict Rb motion or
need to operate outside magnetic shielding. Also, the simplicity of the
experimental apparatus offers some practical appeal.

In this experiment, we employ two independent external cavity diode lasers.
The probe beam was tuned near the $5S_{1/2}F=2\rightarrow
5P_{1/2}F^{\prime}=2$ transition to maximize the amount of PSR squeezing. A
detailed description of the PSR squeezing apparatus is provided in Ref.[28,
32], but in short, the propagation of a linearly polarized light through a
87Rb vapor cell produced $\approx 2$ dB of a quadrature-squeezed vacuum in the
polarization orthogonal to the original laser field. For the further
measurements, we treated the total output as a polarization-squeezed optical
field [33]. The phase retardation plate PRP (a wave plate tilted to add a
controllable relative phase between the two polarization components), placed
after the squeezing cell, was used to control the squeezing quadrature and
adjust it to minimize the fluctuations in the polarization rotation angle.
Achieving the maximum squeezing level required operation at relatively high
laser power ($6.5$ mW). We did not take any additional steps to reduce the
probe beam power before the magnetometery cell. However, it is straightforward
to reduce it without compromising the amount of noise suppression using a
simple Mach-Zender interferometer [34], and our measurements suggest that the
magnetometer sensitivity may be improved with additional optimization of the
mean probe power.

A separate external cavity diode laser produced the pump field, and we used an
acousto-optical modulator (AOM) with 50% duty cycle to modulate its power at a
frequency close to the Larmor frequency of the longitudinal bias magnetic
field $\Omega_{L}=\gamma B_{z}$, where $\gamma$ is the gyromagnetic ratio.
Such modulation creates stroboscopic optical pumping of Rb atoms, that in turn
creates a strong magneto-optical polarization response at the modulation
frequency in the (unmodulated) probe optical field [29, 26, 20, 30]. We varied
the pump power from 0.6 mW to 5.6 mW, but since a stronger pump field
amplified the desired magneto-optical response with little impact on the noise
level, we conducted the measurements using the largest available power of 5.6
mW. We also optimized the detuning of the pump beam: while we observed the
modulation-induced polarization rotation resonances for the pump field tuned
to every optical resonance of the $D_{1}$ Rb line, we found that the
magnetometer response is significantly stronger when it was tuned to the
$5S_{1/2}F=1\rightarrow 5^{2}P_{1/2}F^{\prime}=2$ transition.

To suppress possible leakage of the pump field into the probe detection, the
beams are counter-propagating when crossing inside the main magnetometer cell
at an angle of $13$ degrees. This geometry opens an interesting possibility of
measuring a local value of the magnetic field corresponding only to the region
in which the two optical fields intersect, thus enabling a spatial mapping of
the magnetic field.

Figure 2: (a) Direct differential photodetector output as a function of an
applied longitudinal magnetic field. The recorded PBD output is proportional
to the polarization rotation angle. (b) Corresponding lock-in amplifier output
versus the magnetic field. The modulation frequency is set to be 580 kHz
corresponding to 800 mG magnetic field. The lock-in phase is adjusted to
maximize the side resonances.

For the magnetic field measurements, we used a Pyrex vapor cell (length 75 mm,
25 mm diameter) containing isotopically enriched 87Rb and 2.5 Torr of Ne
buffer gas. The cell was mounted inside a magnetic shielding while the
longitudinal magnetic field $B_{z}$ was created by an internal solenoid.
Unless otherwise specified, the cell temperature was maintained at 40.3∘C
corresponding to an atomic density of $5.5\cdot 10^{10}~{}\mathrm{cm}^{-3}$.
The changes in the probe polarization rotation were monitored using a balanced
polarization detector (BPD), consisting of a half-wave plate, orienting the
probe polarization at 45∘ to the axis of a polarization beamsplitter, and the
two identical photodiodes monitoring light power in each of the beamsplitter
output channels. The photodetector photocurrents were subtracted, so that the
output BPD voltage was proportional to the polarization rotation angle. An
example of polarization rotation spectra of the probe field without the
modulated pump is shown in Fig. 2(a). As expected, the polarization rotation
angle as a function of the applied magnetic field presents a dispersion-like
curve, and changes most rapidly around zero magnetic field within a few tens
of mG.

The interaction of atoms with the modulated pump field creates a component of
the atomic coherence oscillating at the pump modulation frequency
$\Omega_{m}$, which consequently creates an optical response at that frequency
in the unmodulated probe field. What is critical for the magnetometer
operation is that the magnitude of this response increases dramatically when
the modulation frequency matches the Larmor precession frequency
$\Omega_{m}\approx\Omega_{L}$, so that the time-dependent atomic coherence
variation induced by the pump field modulation and by the Larmor precession
constructively interfere. In this case, in addition to the stationary
response, we observed a periodic variation of the polarization rotation at
$\Omega_{m}$. This high-frequency response can be easily isolated using a
lock-in amplifier. Fig. 2(b) shows the example of lock-in output for the pump
modulation frequency set to $580$ kHz and slowly varying magnetic field. One
can clearly see the appearance of two sharp resonances at the magnetic field
values about $B_{z}=\pm 800$ mG, corresponding to the $580$ kHz Larmor
frequency. The lock-in phase is adjusted to maximize the slope of the
modulation-induced resonances near $\Omega_{m}\approx\Omega_{L}$. This high
response allows us to detect small changes in the magnetic field on the top of
the large mean value.

Figure 3: Spectrum analyzer signal for the BPD output near the pump modulation
frequency $\Omega_{m}=580~{}\mathrm{kHz}\approx\Omega_{L}$ for the coherent
and squeezed probe field. The spectrum analyzer resolution and video bandwidth
values are 1000 Hz and 1 Hz respectively.

In the parameter space of this experiment, the leading source of noise in the
polarization rotation measurements was the photon shot noise, making the use
of the squeezed light beneficial for reducing the noise level and potentially
improving the magnetometer sensitivity. To confirm this experimentally, we
compared the noise level of the magnetic field measurements performed with
squeezed or coherent probe beams. To switch between these two cases we used a
polarizing beam splitter aligned to reject the polarization of the PSR
squeezed vacuum: when inserted into the probe beam, it effectively “converted”
the squeezed optical field into a coherent one without affecting its other
parameters. One of the features of PSR squeezing is that it occurs only near
the center of an atomic resonance [28, 33, 22]. On one hand, it automatically
ensures the strongest coupling to the atomic transition; on the other hand, it
unavoidably raises the issue of resonant absorption and its negative effect on
squeezing. In this experiment, this absorption was greatly reduced by optical
saturation thanks to a relatively high probe power. We observed about 10%
absorption by the magnetometer Rb cell (including some residual reflection on
cell windows), resulting in squeezing attenuation of 0.3 dB, from -1.9 dB to
-1.6 dB squeezing before and after the cell.

Fig. 3 shows the sample spectrum analyzer traces of the BPD output for
squeezed and coherent probe fields. The central peak at the pump field
modulation frequency $\Omega_{m}=580$ kHz represents the magnitude of the
induced probe polarization rotation. The height of this peak is proportional
to the power of the probe optical field, as well as to the difference between
$\Omega_{L}$ and $\Omega_{m}$, since the atomic response to the modulation is
the strongest near the resonance condition, as discussed above. The noise
floor in our case was determined by the fluctuations of the probe field. For a
coherent probe field, we confirmed that the measurements were shot-noise
limited. If a squeezed probe was introduced, the measured noise was determined
by the noise quadrature measured by the BPD. We controlled this by adjusting
the phase retardation plate, for all the presented measurements the quantum
fluctuations of the detected probe polarization rotation were minimized. Fig.
3 shows the $1.6$ dB drop in the noise level between the coherent and squeezed
probe, as anticipated.

Figure 4: (a) The noise spectra of the lock-in output at the magnetometer
operation point ($\Omega_{m}=\Omega_{L}$) for coherent and squeezed probe
fields (squeezing levels are -1.9 dB/-1.6 dB before/after the magnetometer
cell). The inset shows the zoomed-in section with average noise levels for the
two curves. (b) The estimated sensitivity of the magnetic field measurements
for coherent and squeezed probe fields. The inset shows the measured response
spectrum $\partial S/\partial B(f)$ of the magnetometer as a function of the
detection frequency $f$. The higher-frequency roll-off is due to the limited
minimum $0.3$ ms integration constant of the lock-in amplifier.

The maximum magnetometer sensitivity is achieved for the phase-sensitive
detection with the optimized detection phase, so it was important to verify
that the noise reduction near that modulation frequency translates into
reduced noise of the lock-in readout. In this case, we estimated the magnetic
response $\partial S/\partial B$ by measuring the slope of the lock-in
discrimination curve near $\Omega_{L}$. The noise spectrum of the lock-in data
$\sigma_{S}(f)$ is shown in Fig. 4(a). At the low frequencies ($<100Hz$) the
noise was dominated by the fluctuations of the magnetic field and other
technical noises. The limited minimum integration constant of the lock-in
amplifier ($300~{}\mu$s) caused the slow roll-off at the higher frequencies.
However, in the intermediate region, we observe a clear reduction of the
measured noise with the squeezed probe compare to the coherent field.

Using these measured values, we estimated the sensitivity of such a
magnetometer $\delta B_{min}(f)$ for various detection frequencies $f$ as:

$\delta B_{min}(f)=\frac{\sigma_{S}(f)}{\partial S/\partial B}.$ (1)

To account for the reduction of the signal at higher frequencies we calibrated
its frequency response by adding a small modulation to the applied magnetic
field, and recording its amplitude in the lock-in readout. The resulting
dependence is shown as an inset in Fig. 4(b) where we fit it to the Eq.(3) of
Ref. [1]. As expected, the overall sensitivity is rather low (around
$250~{}\mathrm{pT}/\sqrt{Hz}$ due to the large width of the magneto-optical
resonance. However, one can clearly observe that the squeezed probe provides
better sensitivity across the whole detected spectrum.

Figure 5: (a) Normalized magnetometer response as a function of the
magnetometer cell temperature. The response is normalized to its value at
$T=40.8^{\circ}$C. (b) Spectrum analyzer noise floor at the modulation
frequency $\Omega_{L}=580$ kHz for the coherent and squeezed probe field as a
function of the magnetometer cell temperature. Spectrum analyzer resolution
and video bandwidth values are set to 1000 Hz and 1 Hz respectively. (c)
Normalized quantum noise-limited estimated signal to noise ratio (SNR) as a
function of the magnetometer cell temperature. The SNR is normalized to its
value for the coherent probe at 40.8∘C.

Lastly, we investigated how the magnetometer performance scales with the Rb
vapor temperature. On one hand, the larger number of atoms participating in
the interaction boosts the magneto-optical response. On the other hand, larger
atomic density increases resonant losses, thus reducing the advantages of
squeezing. For the measurements, we changed the temperature of the Rb cell
from 40∘C, the minimum temperature to observe a significant signal, to 70∘C,
where squeezing disappears. As expected, initially the signal grew with the
temperature, reaching its maximum at 55∘C, and then slowly declined, as shown
in Fig. 5(a). Since the noise measurements of the lock-in signal were very
noisy, we instead recorded the noise floor from the spectrum analyzer signal
for both squeezed and coherent probes. As shown in Fig. 5(b), this noise level
increased with the atomic density for both coherent and squeezed probe fields
at higher temperatures, likely due to the excess noise due to the PSR
interaction generated independently in the magnetometer cell [35]. Note,
however, that the squeezed probe provided a lower noise level across the whole
spectrum of temperatures. That observation confirms the finding in [1] that
the modulated magnetometer scheme is not sensitive to the back-action noise.
The increase of the noise level at lower temperatures is quite slow, while the
signal scales much faster below 55∘C. As a result, if we estimate the
temperature dependence of the signal to noise ratio (SNR) using the recorded
spectrum analyzer noise level, we find that it largely follows the signal
temperature dependence, peaking around 55∘C, as shown in Fig. 5(c). Since the
resonant absorption did not noticeably below 55∘C as well, the deterioration
of squeezing was minimal, causing noticeable improvement in estimated SNR for
the squeezed probe field.

In conclusion, in this work we experimentally investigated the prospective of
using a PSR squeezed quantum probe to improve the performance of an amplitude-
modulation-based atomic magnetometer. While the demonstrated sensitivity of
the magnetic field measurements In these experiments we were able to measure
the value of a dc magnetic field of $0.8$ G with the estimated measurement
uncertainty $>250~{}\mathrm{pT}/\sqrt{\mathrm{Hz}}$. Such unimpressive
sensitivity can be explained by strong power broadening and transitional
broadening of the rotation resonance. On the other hand, we observed a clear
noise reduction and the corresponding sensitivity improvement of 15% when the
polarization squeezed probe light was used for the detection. Our work can be
particularly relevant to applications that require large operational bandwidth
[30] or cannot operate at high buffer gas pressure, such as a recently
proposed quantum electron tracker [36].

Acknowledgements We would like to thank Eugeniy Mikhailov for valuable advice
and help, and Savannah Cuozzo for assistance with the squeezing probe
arrangements. We acknowledge the support of the JLab LDRD grant for loaning us
some essential equipment.

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