# SemEval-2022 Task 7: Identifying Plausible Clarifications of
Implicit and Underspecified Phrases in Instructional Texts

Michael Roth Talita Anthonio Anna Sauer
University of Stuttgart
Institute for Natural Language Processing
<EMAIL_ADDRESS>
###### Abstract

We describe SemEval-2022 Task 7, a shared task on rating the plausibility of
clarifications in English-language instructional texts. The dataset for this
task consists of manually clarified how-to guides for which we generated
alternative clarifications and collected human plausibility judgements.111The
task data is available at https://github.com/acidAnn/claire. The task of
participating systems was to automatically determine the plausibility of a
clarification in the respective context. In total, 21 participants took part
in this task, with the best system achieving an accuracy of 68.9%. This report
summarizes the results and findings from 8 teams and their system
descriptions. Finally, we show in an additional evaluation that predictions by
the top participating team make it possible to identify contexts with multiple
plausible clarifications with an accuracy of 75.2%.

##  1 Introduction

Understanding texts in natural language requires that both explicit text
components as well as implicit references and relationships are interpreted
correctly. This applies in particular to instructional texts, which demand a
clear understanding of individual instruction steps in order to reach the
desired goal. Possible uncertainties should therefore already be clarified in
the text. In principle, such clarifications can also be generated
automatically. In that case, however, it will be necessary to investigate the
circumstances under which a clarification is plausible and unambiguous.

As a first step towards such an investigation, this shared task evaluates the
ability of NLP systems to distinguish between plausible and implausible
clarifications of an instruction. Inspired by the success of previous cloze-
based evaluations (see Section 2), we set up our task as a cloze task, in
which clarifications are presented as fillers and systems have to identify
which fillers plausibly fit in a given context (see Table 1). Our focus in
this task is on different types of referring expressions that are either
underspecified or not realized explicitly at all, and we consider possible
clarifications in the form of additional specification or explicitation.

Research in linguistics and psychology has shown that individuals use language
differently Pennebaker and King (1999); Heylighen and Dewaele (2002). In
particular when it comes to implicit and underspecified language, individual
differences can also lead to different interpretations Scholman and Demberg
(2017); Poesio et al. (2019). As a result, worst case scenarios include
medical instructions being followed incorrectly or news being passed on
inaccurately. In view of the fact that language is inherently ambiguous,
however, it is neither sensible nor expedient to produce clarifications for
all occurrences of underspecification. Avoiding worst-case scenarios therefore
goes beyond ranking individual clarifications by plausibility and must take
into account whether multiple (incompatible) clarifications are perceived as
plausible, thus reflecting possible misunderstandings.

Choose a Hair Salon
---
(1) | Check ratings of different salons.
(2) | Visit the salon’s website.
(3) | Call  $\emptyset$ and ask questions.
✓ | the salon | ✗ | a friend
✓ | the number | ✗ | your stylist
✓ | the owner
Table 1: Simplified example from a pilot study: the top shows a sentence (3)
and shortened version of its discourse context (1–2). In the clarified version
of this sentence, the phrase the salon was inserted. Other phrases shown in
the bottom part are automatically generated fillers, annotated as plausible
(✓) or implausible (✗).

We discuss our task and data in more detail in Sections 3 and 4, respectively.
The participating teams are summarized in Section 5 and their results on the
task and additional evaluations in Section 6.

##  2 Related Work

Cloze tasks have become a standard framework for evaluating various discourse-
level phenomena in NLP. Some prominent examples include the narrative cloze
test Chambers and Jurafsky (2008), the story cloze test Mostafazadeh et al.
(2016), and the LAMBADA word prediction task Paperno et al. (2016). In these
tasks, NLP systems are required to make a prediction about the filler of a
cloze that is most likely to continue the discourse. However, it is not always
clear whether exactly one likely filler exists. Evaluations typically
circumvent this issue by requiring systems only to distinguish between a
correct and an incorrect filler, or by evaluating predictions only with a
relative measure. Both of these options ignore the more general challenge that
multiple fillers can be plausible. Our shared task addresses this challenge
explicitly, by requiring systems to classify different clarifying fillers as
either plausible or implausible. This is a natural extension of previous cloze
tasks to discourse contexts in which multiple interpretations are plausible.
This extension makes it possible to evaluate in how far NLP systems can
reflect cases of underspecification and uncertainty as well as possible
sources of misunderstanding.

Our task is based on manual text revisions that can be traced through revision
histories and by which possible needs for clarification can be identified.
Thus, the task follows a number of existing research contributions that deal
with text revisions. A number of previous works examine reasons for and types
of revisions in, for example, Wikipedia Bronner and Monz (2012); Daxenberger
and Gurevych (2012); Yang et al. (2017); Faruqui et al. (2018) and the essay-
based corpus ArgRewrite Zhang and Litman (2015, 2016); Afrin and Litman
(2019); Kashefi et al. (2022). In this work, we use revision histories of
instructional texts because clarifications seem particularly relevant in this
domain.

The starting point of our task is the data set wikiHowToImprove Anthonio et
al. (2020), which comprises revision histories of more than 250,000 how-to
guides from the online platform wikiHow.222www.wikihow.org In our own previous
work, we investigated the extent to which these histories are useful for
examining textual improvements Anthonio and Roth (2020), predicting revision
requirements Bhat et al. (2020), modeling cases of lexical vagueness Debnath
and Roth (2021), and resolving implicit references Anthonio and Roth (2021).
The last two studies in particular have shown that wikiHowToImprove is well
suited as a resource for studying clarifications of semantic phenomena. We
describe one of these studies and how we build upon it in more detail in
Section 3.

##  3 Task and Background

The general idea of the present shared task is to use revisions of English-
language instructional texts as a basis to identify potential clarifications
and to rate them regarding their plausibility. We assume that at least certain
cases of clarifying revisions follow patterns that can be recognized
automatically, by comparing the text before/after revision. An example of such
a pattern is the insertion of a nominal phrase mentioned in context that makes
an implicit reference explicit (see Table 1). We consider additional patterns
as part of this shared task (see Section 4), but only consider cases of
insertion for simplicity.

The focus on insertions allows us to consider clarifications as solutions to a
cloze test, since the revision always fills in a text segment that previously
was not present. Compared to previous cloze tasks, we do not assume that the
revision observed is always unique and plausible. Instead, we also consider
alternative clarification options and obtain plausibility judgments for all
options.

As background, we first summarize findings from a pilot study that we
conducted before setting up the present task (§3.1). Based on this, we then
describe the settings of the shared task (§3.2).

###  3.1 Pilot Study

In our pilot study Anthonio and Roth (2021), we constructed a dataset of
implicit references and potential clarifications in three steps: (1)
heuristically identifying insertions of nominal phrases mentioned in the
previous context, (2) automatically generating alternative clarifications
using generative language modeling (GPT; Radford et al., 2018), and (3)
collecting human plausibility judgements for each clarification option.

The first step of our pilot showed that it is possible to extract about 6,000
relevant clarifications from the revisions in wikiHowToImprove. We further
found that most noisy instances can be filtered by the application of
linguistic constraints and that remaining cases can be identified during the
manual verification in the final step. In the second step, we found GPT to
produce completions for many sentences that seem sensible on the surface
level. Using our best strategy, namely re-ranking based on paragraph-level
perplexity, the best sequence generated by the model was identical to the
human-inserted clarification in over 56% of cases, and the clarification
appeared in the top-10 generated sequences in 78% of cases.

A crucial finding in the third step was that the annotator indicated a
preference for the human-inserted clarification in most cases (68%), but
different, model-generated clarifications were judged as equally good in many
cases (24%). In some cases, the annotator actually preferred a generated
clarification over the human-edited insertion (8%).

The framework for the shared task is strongly motivated by the finding that
alternative clarifications, generated by a computational model, can be as good
or even better than human-produced clarifications. In some cases, we simply
found different verbalizations of the same proposition. In other cases, like
examples (a) and (b) below, we found plausible alternatives that are not fully
compatible semantically.

  * (a)

Call the salon and ask questions.

  * (b)

Call each salon and ask questions.

When multiple incompatible readings exist, there is a risk that instructions
will be misunderstood and not lead to the desired goal. To identify potential
occurrences of such cases, we consider different fillers in the shared task
and rate the plausibility of each filler independently.

Phenomenon | Clarification pattern | Example | Potential filler
---|---|---|---
Implicit Reference | $\emptyset$ $\rightarrow$ [DET] NOUN | Lift your toes up while keeping your leg straight. Hold  $\emptyset$ for a few seconds, then release. Incorporate calf stretches into your yoga routine. | ✓ your pose
✓ the stretch
? a leg
? the chair
| ✗ your head
Fused head | DET/JJ $\emptyset$ $\rightarrow$ DET/JJ NOUN | Traditionally, the groom waits for the bride at the altar, the bride tosses the bouquet and (…). Since this is your wedding, feel free to change these  $\emptyset$ . | ✓ ideas
✓ plans
? symbols
? characters
✗ changes
Noun compound | $\emptyset$ NOUN $\rightarrow$ NOUN NOUN | Heating for cold water tanks isn’t quite of an issue as for tropicals. In fact, you can keep a  $\emptyset$ tank without a heater. | ✓ goldfish
✓ freshwater
✓ water
? fishing
✗ soup
Metonymy |  NP $\emptyset$ $\rightarrow$ NP’s NP $\emptyset$ NP $\rightarrow$ NP of NP | Look at the  $\emptyset$ of the teeth. If you’re unsure of your dog’s age, or want to determine if they are already entering into the senior territory, try the teeth. | ✓ condition
✓ color
✓ thickness
? layout
? points

Table 2: Phenomena, extraction patterns and example clarifications (✓
plausible, ? neutral, ✗ implausible).

###  3.2 Shared Task Settings

The SemEval shared task is set up as follows: Systems are provided with a
cloze sentence, surrounding sentences and a potential clarifying filler as
input, and are required to make a prediction regarding the plausibility of the
filler in the given context. For evaluation, predicted labels are compared
against the manually collected plausibility judgements described in Section 4.
We define two subtasks with different labels and evaluation measures.

#### Task 1: Classification.

In the classification task, systems need to distinguish between three labels
(implausible, neutral and plausible). We use accuracy as the main evaluation
measure, calculated as the proportion of correct predictions among all
predictions of a system.

#### Task 2: Ranking.

In the ranking task, systems need to predict a continuous plausibility score.
We evaluate the predictions based on their correlation with human judgements,
calculated as Spearman’s rank correlation coefficient between all predictions
and all judgements.

We describe the selection of data and collection of human judgements in the
next section. In Section 6, we discuss additional evaluations performed to
assess system performance with regard to the presence of multiple plausible
clarifications.

##  4 Data

We closely follow the three steps of our pilot study, described in Section
3.1, to construct the data for this shared task. Our starting point is the
dataset wikiHowToImprove Anthonio et al. (2020), a resource of sentence-level
revisions and their contexts based on wikiHow. In the first two steps, we
create relevant data from this resource automatically; in the final step, we
collect manual annotations to form a gold standard. In step 1, we apply a
pattern-based approach to identify revisions that involve insertions that
serve specific clarifying functions (§4.1). In step 2, we use transformer-
based language models to produce sets of alternate clarifications that may or
may not be compatible with an observed insertion (§4.2). In step 3, we collect
human plausibility judgements on each clarification independently (§4.3).

###  4.1 Data Extraction

We collect relevant revisions by identifying cases in which a single
contiguous insertion and no other change was made within a sentence. We
compute differences and extract cases automatically based on the Python
library difflib333https://docs.python.org/3/library/difflib.html and the
following preprocessing tools: spaCy444https://github.com/explosion/spaCy for
sentence splitting and tokenization, the Berkeley Neural Parser Kitaev and
Klein (2018) for constituency parsing and Stanza Qi et al. (2020) for POS
tagging, dependency parsing and co-reference resolution. For the shared task,
we focus on four types of phenomena, which are summarized in Table 3.1.

#### Implicit references.

Instances with a non-verbalized reference in the original sentence which was
clarified in the revised sentence through insertion. We select the cases from
Anthonio and Roth (2021) with insertions containing a single noun or a
determiner followed by a noun.

#### Fused heads.

Instances of noun phrases for which the head noun was implicit in the original
sentence and clarified in the revised sentence through insertion. We search
for noun phrases with a determiner or adjective head in the original sentence
and select those instances where a single noun was inserted in the revision.

#### Noun compounds.

Instances of underspecified noun phrases, which were clarified in the revised
sentence through the insertion of a dependent noun to form a more specific
compound. We select instances of single noun insertions in which the inserted
noun is a compound dependent of another noun that has already been present in
the original sentence.

#### Metonymy.

Instances in which a revision adds a noun $y$ to a noun $x$ to make explicit
to which component or aspect of $x$ the text refers. For the genitive pattern
$x$’(s) $y$, we select insertions including an apostrophe and a noun $y$ that
is in a dependency relation nmod:poss with a noun $x$. For the $y$ of $x$
pattern, we select insertions that consist of a noun $y$ and the token of
added right in front of a noun $x$, allowing for intervening determiners and
adjectives.

###  4.2 Constructing Clarifications

We produce a set of possible clarifications for each instance as follows:
First, we generate the top-100 fillers in place of an observed insertion using
language modeling. Second, we select a subset of potentially suitable
clarifications by filtering and clustering the top-100.

#### Filler generation.

For the implicit references, we take the top-100 generated clarifications from
Anthonio and Roth (2021). For the other phenomena, we generate alternative
clarifications automatically using the same approach as Anthonio and Roth
(2021). That is, we feed the original sentence $s$ with the surrounding
sentences from the same paragraph to a language model. We then compute the
top-$100$ completions for the token position(s) where an insertion was added
in the revised sentence. We use BERT Devlin et al. (2019) instead of GPT
Radford et al. (2018) to generate the clarifications, as the required
insertions consist of only one token and BERT makes it possible to also
consider follow-up context directly. The BERT checkpoint bert-base-
uncasedbert-base-uncased555 We also tried bert-base-cased in preliminary
experiments but observed no improvements. in Transformers Wolf et al. (2020)
was used without additional pre-training.

#### Filler selection.

From the top-$100$ clarifications provided by the language model, we select
four fillers with the goal of producing a semantically diverse set of
clarifications. First, we remove unsuitable fillers from the top-100,
including cases that only consist of digits or non-alphanumerical characters
and fillers that do not have the right part of speech based on Stanza
(retaining only NOUN for fused heads and metonymy and NN for noun compounds to
exclude plural nouns).

For all instances with $\geq 4$ candidate fillers, we select the observed
insertion from the revised sentence as one filler. To select semantically
different fillers as alternate candidates, we apply $k$-means clustering with
$k=4$ to the remaining candidates, using the algorithm by Elkan (2003) as
implemented in sklearn Pedregosa et al. (2011). We obtain vector
representations for clustering from BERT (bert-base-uncased) by averaging over
the last hidden state for all tokens in a filler. After clustering, we select
the fillers closest to the four cluster centroids based on cosine similarity.

###  4.3 Plausibility Annotation

#### Task.

After selecting fillers for each sentence, we collect plausibility judgements
on Amazon Mechanical Turk for our train set (19,975 instances, i.e. 3995
sentences with 1 human and 4 generated fillers each6661000 each for noun
compounds and metonymy, 996 for implicit references and 999 for fused heads.),
development and test sets (2,500 instances each, i.e., 125 sentences per
phenomenon with 5 fillers per sentence). Each clarification in the training
set is annotated by 2 crowdworkers. For the development and test set, we
collected annotations from 4 crowdworkers to ensure a consistently high
quality. In each annotation task, we ask participants to indicate on a scale
from 1 to 5 whether the clarification made sense in the given how-to-guide. A
screenshot of the interface for our Human Intelligence Task (HIT) is provided
in Appendix A.

| Train | Dev | Test
---|---|---|---
implausible | 5,474 (27%) | 982 (39%) | 858 (34%)
neutral | 7,162 (36%) | 602 (24%) | 672 (27%)
plausible | 7,339 (37%) | 916 (37%) | 970 (39%)
Total | 19,975 | 2,500 | 2,500
Table 3: Distribution of class labels in our training, development and test sets. Team | Model type | Pre-trained model components | Additional comments
---|---|---|---
X-PuDu | ensemble | DeBERTa, ERNIE, XLM-R | pattern-aware, multi-loss
HW-TSC | ensemble | DeBERTa, RoBERTa, S-BERT | incl. unsupervised model
PALI | ensemble | DeBERTa, RoBERTa, XLM-R | pattern-aware, multi-loss
Nowruz | Transformer | T5 | ordinal regression, multi-loss
JBNU-CCLab | ensemble | DeBERTa | $-$
DuluthNLP | Transformer | ELECTRA | class weighting
Stanford MLab | Transformer | ELECTRA | $-$
niksss | Transformer | BERT | $-$
Table 4: Summary of the best models on the test set according to the submitted
system descriptions.

#### Qualifications.

We use several qualifications to increase the annotation quality. First, we
require participants to be located in the United States or in the United
Kingdom, to increase the chance that the participants are native speakers of
English. Secondly, participants need to have a HIT approval rate $\geq 95\%$
and their number of approved HITS has to be $\geq 1000$. Finally, annotators
are required to pass a qualification test in which they are asked to judge a
list of clearly plausible and implausible cases that were pre-selected
unanimously by the authors.

#### Class labels.

For Task 1 (classification), we average over the real-valued judgements
collected for a clarification and map this plausibility score to one of the
three classes labels. Specifically, we label clarifications with an average
score $\leq 2.5$ as implausible, clarifications with a score $\geq 4.0$ as
plausible, and all clarifications between these thresholds as neutral. The
thresholds have been selected based on manual inspection of the data and
mathematical considerations: in particular, the threshold for plausible
requires scores to be substantially above average (in case of two judgements,
$\geq$3&5 or $\geq$4&4), whereas the implausible threshold allows for a
slightly wider range of judgements. The neutral label covers cases that
received inconclusive individual scores as well as cases of disagreement (e.g.
3&3 as well as 2&5).

#### Statistics.

We show the frequency distribution of the labels in the train, development and
test set in Table 3. It is noteworthy that development and test set
proportionally includes fewer neutral and more implausible clarifications than
the training set. Presumably, this is because we increased the number of
qualification questions from 4 to 6 after collecting the training data to
ensure the quality of the evaluation data.

Since we are particularly interested in cases with multiple plausible
clarifications, we also compute the average number of plausible clarifications
per sentence $s$, which we found to be 1.84, 1.87 and 1.84 in the training,
development and test set, respectively. This means that, on average, each
annotated sentence in the dataset has between 1 and 2 clarifications that the
annotators rated as plausible.

##  5 Participants

A total of 21 users participated in the CodaLab competition set up for the
shared task and 8 teams submitted system description papers. An overview of
the best model by each team is shown in Table 4.777A table with the official
results of the CodaLab competition, including participants who did not submit
system descriptions, is shown in Appendix B. We observe that all systems are
based on Transformer architectures, using one or more of the following pre-
trained models: BERT Devlin et al. (2019), DeBERTa He et al. (2020), ELECTRA
Clark et al. (2019), ERNIE Sun et al. (2019), RoBERTa Liu et al. (2019),
S-BERT Reimers and Gurevych (2019), T5 Raffel et al. (2020), XLM-R Conneau et
al. (2020).

In addition to fine-tuning a single or multiple Transformer models in an
ensemble, some teams have taken additional steps to adapt their system to the
task. We summarize some of these steps below.

#### Consideration of phenomena.

At least two teams took into account that the data set consists of four
phenomena that were identified using different patterns (pattern-aware): PALI
used the phenomenon description that applies to a classification instance as
additional model input; X-PuDu developed an ensemble architecture that
consists of different individual models and hyperparamters for each
phenomenon.

#### Adapted loss functions.

Several teams adapted the loss functions of their models to better account for
various properties of the task. This includes the use of classification and
regression based loss functions in a multi-task learning set-up (multi-loss)
as well as the use of specific loss functions that consider the ordinal nature
of labels (ordinal regression) or differences in label distributions (class
weighting) in the classification task.

#### Unsupervised components.

Given the similarity of our task to general cloze tasks, several teams
experimented with models that were merely self-supervised and not fine-tuned
on task-specific training data. In case of one team, HW-TSC, such an
unsupervised component is also part of the ensemble model that produced the
best results.

##  6 Results and Discussion

The results for Task 1 and 2 are shown in Table 5 and 6, respectively. We
focus our discussion on Task 1: Classification, as the participants of Task 2
form only a subset of the Task 1 participants and the system results rank,
with exception of the last two teams, in the same order. In addition to
showing results by participants, we also provide a human upper bound as well
as results by our own BERT-based baseline model. The upper bound was computed
as the accuracy over all individual annotations when compared against the
(averaged) class label of each test instance.

The human upper bound has an accuracy of 79.4%, indicating that the task is
challenging and potentially involves a number of disagreements. The winning
team of the competition, X-PuDu, achieves an accuracy of 68.9%, only 10.5
percentage points below the human upper bound. The results of all teams lies
substantially above a naive majority class baseline of 39%. All teams but one
also outperform our BERT-based baseline, which is a linear classification
model based on the checkpoint provided by the Transformer library Wolf et al.
(2020) and fine-tuned on our training data.

###  6.1 Findings by Participants

In the following, we briefly summarize a couple of findings by task
participants. More details can be found in the individual task description
papers.

Rank | Team | Accuracy
---|---|---
$-$ | Human (upper bound) | 79.4%
1 | X-PuDu | 68.9%
2 | HW-TSC | 66.1%
3 | PALI | 65.4%
4 | Nowruz | 62.4%
5 | JBNU-CCLab | 61.4%
6 | DuluthNLP | 53.3%
7 | Stanford MLab | 46.6%
8 | niksss | 44.2%
$-$ | BERT (baseline) | 45.7%
Table 5: Results for Task 1 (classification).

#### Different phenomena.

The winning team, X-PuDu, found that different hyperparameters worked best
depending on the phenomenon/extraction pattern. Based on this finding,
different individual models were trained and combined in an ensemble.

#### Label distribution.

Some teams, including DuluthNLP, noticed performance issues related to the
distribution of labels in the development data. As a dedicated solution,
DuluthNLP uses a decreased weight for the neutral label in the loss function.

Rank | Team | Spearman’s $\rho$
---|---|---
1 | X-PuDu | 0.807
2 | PALI | 0.785
3 | HW-TSC | 0.774
4 | Nowruz | 0.707
5 | niksss | 0.252
6 | Stanford MLab | 0.194
Table 6: Results for Task 2 (ranking).

####  neutral label.

Team JBNU-CCLab reported that the neutral label is generally difficult to
distinguish from other labels by different models. An underlying problem could
be that the label represents instances that are seen as somewhat plausible by
multiple annotators as well as instances that are seen as plausible by some
annotators and implausible by others (see Section 4).

#### Noisy data.

Team HW-TSC found that isolated training instances have the label neutral
rather than plausible, even though the respective filler represents a human
insertion (i.e., the filler can be found in the final version of the text in
wikiHow). As the results of our human upper bound in Table 5 show, this is
partly because the right label is sometimes not clear cut even for humans. We
discuss this aspect in more detail in the next section.

###  6.2 Additional Evaluations

We perform two additional evaluations to assess the impact of the neutral
label on system performance and to investigate the possibility of identifying
whether multiple plausible clarifications exist by aggregating the predictions
regarding individual clarifications.

Rank | Team | F1 (all) | F1 (w/o n)
---|---|---|---
1 | X-PuDu | 0.689 | 0.773
2 | HW-TSC | 0.661 | 0.749
3 | PALI | 0.654 | 0.749
4 | Nowruz | 0.624 | 0.714
5 | JBNU-CCLab | 0.551 | 0.627
6 | DuluthNLP | 0.533 | 0.608
7 | Stanford MLab | 0.466 | 0.514
8 | niksss | 0.442 | 0.494
Table 7: Classification results with/without neutral. Rank | Team | Accuracy (#p$\geq$2)
---|---|---
1 | X-PuDu | 75.2%
2 | HW-TSC | 73.2%
3 | PALI | 72.6%
4 | Nowruz | 71.6%
5 | JBNU-CCLab | 63.6%
6 | DuluthNLP | 62.6%
7 | Stanford MLab | 54.8%
8 | niksss | 60.0%
Table 8: Results for identifying contexts with multiple plausible fillers, based on individual model predictions. Correct | #P$\geq$2 | Text | Fillers
---|---|---|---
8 (all) | ✓ |  Galette des rois—or “King Cake” in English—is traditionally made to celebrate the  $\emptyset$ of Epiphany. Especially popular in France during the Christmas season, it is enjoyed elsewhere too. | ✓ holidays ✓ Feast ? hours ? celebration ? proclamation
✗ |  Let the  $\emptyset$ of shoes air dry. You can put them in front of a dehumidifier, a fan, or an open window, but avoid putting them in front of any type of heat source. | ✓ pair ? pile ✗ shoes ✗ color ✗ end
0 (none) | ✓ |  If you want a smoother surface, try a  $\emptyset$ of paper with a higher amount of grains, if you want a faster job but a rougher surface try a paper with a lower amount of grains. | ✓ thickness ✓ piece ? fabric ? product ✗ pile
✗ |  Your cucumber plant will also grow thin, light green shoots that help the plant grasp onto a surface and grow vertically. These  $\emptyset$ grow immediately next to the suckers. | ✓ shoots ? fibers ? tendrils ? foliage ✗ bushes
Table 9: Examples of difficult and easy instances, selected based on how many
systems classified them correctly.

#### Excluding neutral.

For the evaluation without the neutral label, we calculate micro-averaged
precision, recall and F1-scores for the two labels plausible and implausible.
The results in terms of F1-score are shown in Table 7. The results indicate
that all systems perform substantially better in the evaluation setting that
ignores neutral labels. The ranking is identical to the ranking in the
evaluation including all labels. Considering only the plausible and
implausible, Team X-PuDu achieves the highest micro-averaged F1-score of
0.773. In the cases where their system predicts a non-neutral label, it is
correct in 72.7% of cases (precision), and 82.5% of all non-neutral instances
in the data received the correct prediction (recall).

#### Multiple clarifications.

In our final evaluation, we examine whether system predictions can also be
used to determine whether multiple plausible clarifications for a given
context exist. For this, we consider the labels of each individual
clarification and compare system outputs and annotations in terms of whether
two or more clarifications for a cloze and its context received the label
plausible. We show the result of this evaluation in terms of accuracy for each
team in Table 8. Apart from the last two places, the teams rank in the same
order as in the other evaluations. The best performing team, X-PuDu, correctly
predicts whether two or more plausible clarifications exist for 75.2% of all
cases. Table 9 shows examples that were correctly classified by all or none of
the systems.

##  7 Conclusion

In this paper, we presented the task, data, participating systems, and results
of the shared task on clarifying implicit and underspecified phrases in
instructional texts. Our motivation for this task was to explore the
possibility of testing different clarifications for plausibility. In
particular, we were concerned with the question of whether two or more
clarifications can be plausible and whether such cases can be detected
automatically. To create a suitable dataset, we worked with and identified a
set of revisions with manual clarifications, automatically generated possible
alternatives, and then collected human plausibility ratings.

In total, 21 users participated in our shared task. We summarized the systems
and results of 8 teams that submitted descriptions of their systems. The best
systems from each group have in common that they are based on Transformer
architectures or combine them in an ensemble. The best system achieved 68.9%
accuracy, only 10.5 percentage points below a human upper bound. In additional
evaluations, we have shown that an accuracy of up to 75.2% is achieved with
respect to the detection of multiple plausible clarifications.

The results show that the presented task is a difficult one, but that many
cases can already be modeled well by current state-of-the-art methods. There
is further room for improvement with respect to both the data set and models:
with respect to the data, it should be noted that the training set with less
than 20k instances is relatively small and that there are many instances with
a underspecified neutral label (36%). On the model side, we found that the
participating teams make complementary contributions that may allow for
additional improvements in combination.

One shortcoming of the task as presented and performed is that we only
considered four forms of clarifications related to referring expressions. In
addition, clarifications were assessed individually and judgements by
different annotators were aggregated. In the long term, we believe that more
forms of clarifications as well as individual differences regarding their
plausibility need to be considered. Finally, future work will have to
investigate under which circumstances multiple different clarifications are
actually incompatible and can thus reveal potential sources of
misunderstanding.

## Acknowledgements

The research presented in this paper was funded by the DFG Emmy Noether
program (RO 4848/2-1).

## References

  * Afrin and Litman (2019) Tazin Afrin and Diane Litman. 2019.  Identifying editor roles in argumentative writing from student revision histories.  In _International Conference on Artificial Intelligence in Education_ , pages 9–13. Springer.
  * Anthonio et al. (2020) Talita Anthonio, Irshad Bhat, and Michael Roth. 2020.  wikiHowToImprove: A resource and analyses on edits in instructional texts.  In _Proceedings of the 12th Language Resources and Evaluation Conference_ , pages 5721–5729, Marseille, France. European Language Resources Association.
  * Anthonio and Roth (2020) Talita Anthonio and Michael Roth. 2020.  What can we learn from noun substitutions in revision histories? In _Proceedings of the 28th International Conference on Computational Linguistics_ , pages 1359–1370, Barcelona, Spain (Online). International Committee on Computational Linguistics.
  * Anthonio and Roth (2021) Talita Anthonio and Michael Roth. 2021.  Resolving implicit references in instructional texts.  In _Proceedings of the 2nd Workshop on Computational Approaches to Discourse_ , pages 58–71, Punta Cana, Dominican Republic and Online. Association for Computational Linguistics.
  * Bhat et al. (2020) Irshad Bhat, Talita Anthonio, and Michael Roth. 2020.  Towards modeling revision requirements in wikiHow instructions.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 8407–8414, Online. Association for Computational Linguistics.
  * Bronner and Monz (2012) Amit Bronner and Christof Monz. 2012.  User edits classification using document revision histories.  In _Proceedings of the 13th Conference of the European Chapter of the Association for Computational Linguistics_ , pages 356–366, Avignon, France. Association for Computational Linguistics.
  * Chambers and Jurafsky (2008) Nathanael Chambers and Dan Jurafsky. 2008.  Unsupervised learning of narrative event chains.  In _Proceedings of ACL-08: HLT_ , pages 789–797, Columbus, Ohio. Association for Computational Linguistics.
  * Clark et al. (2019) Kevin Clark, Minh-Thang Luong, Quoc V Le, and Christopher D Manning. 2019.  Electra: Pre-training text encoders as discriminators rather than generators.  In _International Conference on Learning Representations_.
  * Conneau et al. (2020) Alexis Conneau, Kartikay Khandelwal, Naman Goyal, Vishrav Chaudhary, Guillaume Wenzek, Francisco Guzmán, Edouard Grave, Myle Ott, Luke Zettlemoyer, and Veselin Stoyanov. 2020.  Unsupervised cross-lingual representation learning at scale.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 8440–8451, Online. Association for Computational Linguistics.
  * Daxenberger and Gurevych (2012) Johannes Daxenberger and Iryna Gurevych. 2012.  A corpus-based study of edit categories in featured and non-featured Wikipedia articles.  In _Proceedings of COLING 2012_ , pages 711–726, Mumbai, India. The COLING 2012 Organizing Committee.
  * Debnath and Roth (2021) Alok Debnath and Michael Roth. 2021.  A computational analysis of vagueness in revisions of instructional texts.  In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Student Research Workshop_ , pages 30–35, Online. Association for Computational Linguistics.
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Elkan (2003) Charles Elkan. 2003.  Using the triangle inequality to accelerate k-means.  In _Proceedings of the 20th international conference on Machine Learning (ICML-03)_ , pages 147–153.
  * Faruqui et al. (2018) Manaal Faruqui, Ellie Pavlick, Ian Tenney, and Dipanjan Das. 2018.  WikiAtomicEdits: A multilingual corpus of Wikipedia edits for modeling language and discourse.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 305–315, Brussels, Belgium. Association for Computational Linguistics.
  * He et al. (2020) Pengcheng He, Xiaodong Liu, Jianfeng Gao, and Weizhu Chen. 2020.  Deberta: Decoding-enhanced bert with disentangled attention.  _arXiv preprint arXiv:2006.03654_.
  * Heylighen and Dewaele (2002) Francis Heylighen and Jean-Marc Dewaele. 2002.  Variation in the contextuality of language: An empirical measure.  _Foundations of science_ , 7(3):293–340.
  * Kashefi et al. (2022) Omid Kashefi, Tazin Afrin, Meghan Dale, Christopher Olshefski, Amanda Godley, Diane Litman, and Rebecca Hwa. 2022.  Argrewrite v. 2: an annotated argumentative revisions corpus.  _Language Resources and Evaluation_ , pages 1–35.
  * Kitaev and Klein (2018) Nikita Kitaev and Dan Klein. 2018.  Constituency parsing with a self-attentive encoder.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 2676–2686, Melbourne, Australia. Association for Computational Linguistics.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.  _arXiv preprint arXiv:1907.11692_.
  * Mostafazadeh et al. (2016) Nasrin Mostafazadeh, Nathanael Chambers, Xiaodong He, Devi Parikh, Dhruv Batra, Lucy Vanderwende, Pushmeet Kohli, and James Allen. 2016.  A corpus and cloze evaluation for deeper understanding of commonsense stories.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 839–849, San Diego, California. Association for Computational Linguistics.
  * Paperno et al. (2016) Denis Paperno, Germán Kruszewski, Angeliki Lazaridou, Ngoc Quan Pham, Raffaella Bernardi, Sandro Pezzelle, Marco Baroni, Gemma Boleda, and Raquel Fernández. 2016.  The LAMBADA dataset: Word prediction requiring a broad discourse context.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1525–1534, Berlin, Germany. Association for Computational Linguistics.
  * Pedregosa et al. (2011) F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. 2011.  Scikit-learn: Machine learning in Python.  _Journal of Machine Learning Research_ , 12:2825–2830.
  * Pennebaker and King (1999) James W. Pennebaker and Laura A. King. 1999.  Linguistic styles: language use as an individual difference.  _Journal of personality and social psychology_ , 77(6):1296–1312.
  * Poesio et al. (2019) Massimo Poesio, Jon Chamberlain, Silviu Paun, Juntao Yu, Alexandra Uma, and Udo Kruschwitz. 2019.  A crowdsourced corpus of multiple judgments and disagreement on anaphoric interpretation.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 1778–1789, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Qi et al. (2020) Peng Qi, Yuhao Zhang, Yuhui Zhang, Jason Bolton, and Christopher D. Manning. 2020\.  Stanza: A python natural language processing toolkit for many human languages.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics: System Demonstrations_ , pages 101–108, Online. Association for Computational Linguistics.
  * Radford et al. (2018) Alec Radford, Karthik Narasimhan, Tim salimans, and Ilya Sutskever. 2018.  Improving language understanding by generative pre-training.  OpenAI.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _Journal of Machine Learning Research_ , 21:1–67.
  * Reimers and Gurevych (2019) Nils Reimers and Iryna Gurevych. 2019.  Sentence-BERT: Sentence embeddings using Siamese BERT-networks.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 3982–3992, Hong Kong, China. Association for Computational Linguistics.
  * Scholman and Demberg (2017) Merel Scholman and Vera Demberg. 2017.  Crowdsourcing discourse interpretations: On the influence of context and the reliability of a connective insertion task.  In _Proceedings of the 11th Linguistic Annotation Workshop_ , pages 24–33, Valencia, Spain. Association for Computational Linguistics.
  * Sun et al. (2019) Yu Sun, Shuohuan Wang, Yukun Li, Shikun Feng, Xuyi Chen, Han Zhang, Xin Tian, Danxiang Zhu, Hao Tian, and Hua Wu. 2019.  Ernie: Enhanced representation through knowledge integration.  _arXiv preprint arXiv:1904.09223_.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45, Online. Association for Computational Linguistics.
  * Yang et al. (2017) Diyi Yang, Aaron Halfaker, Robert Kraut, and Eduard Hovy. 2017.  Identifying semantic edit intentions from revisions in Wikipedia.  In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pages 2000–2010, Copenhagen, Denmark. Association for Computational Linguistics.
  * Zhang and Litman (2015) Fan Zhang and Diane Litman. 2015.  Annotation and classification of argumentative writing revisions.  In _Proceedings of the Tenth Workshop on Innovative Use of NLP for Building Educational Applications_ , pages 133–143, Denver, Colorado. Association for Computational Linguistics.
  * Zhang and Litman (2016) Fan Zhang and Diane Litman. 2016.  Using context to predict the purpose of argumentative writing revisions.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 1424–1430, San Diego, California. Association for Computational Linguistics.

##  Appendix A Annotation Interface

The annotation interface for our crowdsourcing task is depicted in Figure 1.
Annotators see and have to rate a single underlined clarification in its
context.

Figure 1: Interface for collecting annotations.

##  Appendix B CodaLab Leaderboard

In the main part of the paper, we only list results of participants who
provided a description of their system(s) for the shared task. Table 10 shows
a complete set of user names and results of the participants in the CodaLab
competition, including users who did not submit a system description.

Team name | User name | acc. | $\rho$
---|---|---|---
X-PuDu | tt123 | 0.689 | 0.807
HW-TSC | Yinglu_Li | 0.661 | 0.774
$-$ | tiantaijian | 0.661 | 0.763
$-$ | fanxiaoxing | 0.656 | $-$
PALI | stce | 0.654 | 0.785
$-$ | hudou | 0.641 | $-$
$-$ | huangwkk | 0.631 | 0.774
Nowruz | mohammadmahdinoori | 0.624 | 0.707
JBNU-CCLab | OrangeAvocado | 0.614 | $-$
$-$ | CitizenTano | 0.595 | $-$
$-$ | huawei_zhangmin | 0.589 | 0.640
$-$ | parkwonjae | 0.554 | $-$
$-$ | lith | 0.537 | 0.600
$-$ | ywzhang_cr | 0.537 | 0.600
DuluthNLP | Sakrah | 0.533 | $-$
Stanford MLab | patrickliu2011 | 0.466 | 0.194
$-$ | Autism_PAFC | 0.461 | $-$
$-$ | SelinaIW | 0.456 | $-$
niksss | niksss | 0.442 | 0.252
$-$ | andrei.manea | 0.418 | -0.109
$-$ | tanigaki | 0.395 | 0.415
Table 10: Oveview of results, including user submissions without a shared task
system description.


# Quantum effects in rotating reference frames

S. P. Kish  Department of Quantum Science, Research School of Physics &
Engineering,
The Australian National University, Canberra, ACT, Australia.  Centre for
Quantum Computation and Communication Technology,
School of Mathematics and Physics, University of Queensland, Brisbane,
Queensland 4072, Australia  T. C. Ralph  Centre for Quantum Computation and
Communication Technology,
School of Mathematics and Physics, University of Queensland, Brisbane,
Queensland 4072, Australia

###### Abstract

We consider the time delay of interfering single photons oppositely travelling
in the Kerr metric of a rotating massive object. Classically, the time delay
shows up as a phase difference between coherent sources of light. In quantum
mechanics, the loss in visibility due to the indistinguishability of
interfering photons is directly related to the time delay. We can thus observe
the Kerr frame dragging effect using the Hong-Ou-Mandel (HOM) dip, a purely
quantum mechanical effect. By Einstein’s equivalence principle, we can
analogously consider a rotating turntable to simulate the Kerr metric. We look
at the feasibility of such an experiment using optical fibre, and note a
cancellation in the second order dispersion but a direction dependent
difference in group velocity. However, for the chosen experimental parameters,
we can effectively assume light propagating through a vacuum.

††preprint: AIP/123-QED

##  I Introduction

Most experiments performed to date can be explained by a classical theory of
curved space-time or quantum mechanics in flat space-time. These remain to a
large degree mutually exclusive fields of physics. One of the most fundamental
questions of physics today concerns the reconciliation between quantum
mechanics and Einstein’s theory of general relativity. Quantum mechanics in
curved space-time has not been as accessible by experiment. Nonetheless, there
have been many proposals to test quantum phenomena such as superposition and
entanglement in curved space-time kempf ; european . For example, Ref. mzych0
; mzych  considers single photons in superposition at different heights in a
gravitational potential. These photons will experience a classical phase shift
due to time dilation that is proportional to the gravitational acceleration.
The same classical phase can be detected by classical light which does not
exhibit the quantum mechanical effect of superposition. However, if the
photons have a pulse (coherence) time comparable to the time dilation,
significant loss of quantum interference occurs between the photon
wavepackets. A loss in the visibility is predicted at the output.

Similarly, we can consider single photons in a large Sagnac interferometer
around a rotating massive body described by the Kerr metric. Due to frame
dragging, photons co-propagating with the rotation will have a different
arrival time compared with photons that counter-propagate with the rotation.
Thus, if this time difference is comparable to the pulse time of the photon,
loss of quantum interference between photon wavepackets occurs. The loss in
visibility now depends on the Kerr parameter $a$. We can consider an analogous
system on a rotating turntable. By Einstein’s equivalence principle, we might
expect to be able to find situations where the physics is equivalent whether
they are in the space-time of a Kerr metric or accelerating due to the
rotation of the turntable. We can observe the time difference due to light
velocity using the Hong-Ou-Mandel (HOM) effect, a purely quantum mechanical
effect enk . The time delay due to rotation at the single photon level as a
phase shift has been observed previously sag  and also more recently, with a
$N=2$ NOON state fink . The effect of uniform rotation on the
distinguishability of photons was recently observed in a Sagnac-type
interferometer on a rotating platform rest . An extension of the experiment in
Ref rest  as suggested by the authors is to use satellites in Earth’s orbit to
perform tests of quantum mechanics at the interface of general relativity
(i.e. frame-dragging). Moreover, in a recent paper, frame dragging and the HOM
dip has been studied in common-path setup restricted to the surface of the
Earth brady . However, loss in the visibility of the quantum interference in
the Kerr metric has not been investigated in detail in an analogous rotating
frame set-up.

In this paper we consider the time delay in the Kerr metric for the situation
where a stationary observer sends a superposition of a co- and counter-
propagating photon half-way around the massive body. We find that there is a
visibility loss for Gaussian wavepackets due to the time disjoint space-time
of the Kerr metric which agrees with recent work toros . Next, we consider a
turntable experiment to simulate this effect. We use a single-photon Sagnac
interferometer to see the loss in visibility due to the time delay caused by
rotation. We also consider the relativistic effects of the optical fibre
medium. We find that the second order dispersion cancels out in agreement with
previous work steinberg , but small effects due to a difference in group
velocity remain. Nonetheless, experimental parameters accessible with current
technology are proposed for future experiments to observe the visibility loss
due to rotation analogous to what would be observed in a single-photon Sagnac
interferometer in the Kerr metric of a massive object. Ultimately, one may
observe such effects at the interface between quantum mechanics (via single-
photon interference) and non-trivial rotating space-times.

##  II Kerr metric

Although the Hartle-Thorne metric describes the exterior metric of a massive
object with a mass quadrupole moment $q$, the approximate Kerr metric in the
weak field limit disregards $q$ which is of second order in angular momentum.
Therefore, planets or stars can be described by the Kerr metric up to first
order in the angular momentum. For a rotating black hole $q=0$ and the Kerr
metric describes the effect of the dragging of the space-time. The Kerr line
element in Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ is visser

$\begin{split}ds^{2}&=-(1-\frac{r_{s}r}{\Sigma})dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}\\\
&+(r^{2}+a^{2}+\frac{r_{s}ra^{2}}{\Sigma}\sin^{2}{\theta})\sin^{2}{\theta}d\phi^{2}-\frac{2r_{s}ra\sin^{2}{\theta}}{\Sigma}d\phi
dt,\end{split}$ (1)

where $\Delta:=r^{2}-r_{s}r+a^{2}$, $\Sigma:=r^{2}+a^{2}\cos^{2}{\theta}$ and
$a=\frac{J}{Mc}$ where $J$ is the angular momentum of the object of mass $M$.
Note that the Schwarzschild radius is $r_{s}=\frac{2GM}{c^{2}}\equiv 2M$ where
we work in natural units for which $c=1$ and $G=1$. In the equatorial plane of
the metric ($\theta=\frac{\pi}{2}$), a tangential velocity of light can be
obtained in terms of tangential proper distance per Kerr coordinate time,
solved by setting $ds^{2}=0$. The full solution is given by (see also Ref.
kish )

$\begin{split}\frac{dx}{dt}&=\frac{r_{s}a}{r\sqrt{r^{2}+a^{2}(1+r_{s}/r)}}\\\
&\pm\sqrt{\frac{r_{s}^{2}a^{2}}{r^{2}(r^{2}+a^{2}(1+r_{s}/r))}+(1-\frac{r_{s}}{r})}.\end{split}$
(2)

However, if $\frac{a}{r}<<1$ and $\frac{r_{s}}{r}<<1$ we have the weak field
solution valid for massive planets

$\begin{split}\frac{dx}{dt}&\approx\frac{r_{s}a}{r^{2}}\pm\sqrt{1-\frac{r_{s}}{r}}\\\
&\approx\pm(1-\frac{r_{s}}{2r}\pm\frac{r_{s}a}{r^{2}}),\end{split}$ (3)

where we have used the Taylor expansion $\sqrt{1-x}\approx 1-\frac{x}{2}$. We
have two solutions representing counter- and co-rotating light.

###  II.1 Kerr phase difference

According to Eq. (3), the speed of light as seen by a far-away observer in the
weak field limit is $c_{A}=(1-\frac{r_{s}}{2r}+\frac{r_{s}a}{r^{2}})$ and
$c_{B}=-(1-\frac{r_{s}}{2r}-\frac{r_{s}a}{r^{2}})$ for co- and counter-
propagating light, respectively. Hence, the phase difference between co- and
counter- propagating single photon wavepackets is

$\begin{split}\Delta\Phi&=\Phi_{B}-\Phi_{A}=\omega
L(\frac{1}{c_{B}}-\frac{1}{c_{A}})\\\ &\approx\omega
L(\frac{1}{1-\frac{r_{s}}{2r}+\frac{r_{s}a}{r^{2}}}\\\
&-\frac{1}{1-\frac{r_{s}}{2r}-\frac{r_{s}a}{r^{2}}})\\\ &\approx 2\omega
L\frac{r_{s}a}{r^{2}}(1+\frac{r_{s}}{r})\approx
2\omega\pi\frac{r_{s}a}{r}.\end{split}$ (4)

Note that we’ve neglected second order terms in $\frac{a}{r}$ and above, since
$r>>a$ in the weak field limit. For Earth’s parameters $a=3.9$ $m$, $r_{s}=9$
$mm$ on the surface $r=6.37\times 10^{7}$ $m$ with visible light of frequency
$\omega=k=2\times 10^{6}$ $m^{-1}$, the magnitude of the classical Kerr phase
half way around the Earth is $\Delta\Phi_{Kerr}=\omega\Delta t_{Kerr}\approx
7\times 10^{-3}$ where $\Delta t_{Kerr}=2L\frac{r_{s}a}{r^{2}}$. The time-
delay due to the effect of the Kerr metric derived here agrees with Ref. rest
. In principle, we could use a large classical Sagnac interferometer to
measure this phase.

##  III Single photon Sagnac interferometer in the Kerr metric

We consider a thought experiment where a single photon is in a superposition
of two paths around the rotating planet. A stationary single photon source
hovering above the rotating planet releases a photon that passes through a
beamsplitter and forms a superposition of the paths $A$ and $B$ with phases
$\Phi_{A}=\omega t_{A}$ and $\Phi_{B}=\omega t_{B}$. Paths $A$ and $B$ move
semi-circularly around the planet co- and counter- propagating, respectively,
with the rotation direction of the planet. These recombine at a second
beamsplitter that is half-way around the planet and are detected by a photon
number counter.

We begin with the initial state $\ket{1}\ket{0}$ passing through a
beamsplitter

$\ket{1}\ket{0}\rightarrow\frac{1}{\sqrt{2}}(\ket{1}\ket{0}+i\ket{0}\ket{1}).$
(5)

The two arms experience a different phase shift depending on their
trajectories. If the photon is moving with the rotation, it will acquire the
phase $\Phi_{A}$ and against the rotation, it will acquire the phase
$\Phi_{B}$. Thus we have the state after the first beamsplitter and subsequent
propagation to the second beamsplitter

$\frac{1}{\sqrt{2}}(e^{i\Phi_{A}}\ket{1}\ket{0}+ie^{i\Phi_{B}}\ket{0}\ket{1}).$
(6)

After the second beamsplitter, the state becomes

$\frac{1}{2}(e^{i\Phi_{A}}(\ket{1}\ket{0}+i\ket{0}\ket{1}+ie^{i\Phi_{B}}(i\ket{0}\ket{1}+\ket{1}\ket{0})).$
(7)

Thus the number of photons at the output of the second beamsplitter is

$\begin{split}\braket{N}&=\bra{0}\bra{0}\frac{1}{2}(e^{-i\Phi_{A}}-ie^{-i\Phi_{B}})\frac{1}{2}(e^{i\Phi_{A}}+ie^{i\Phi_{B}})\ket{0}\ket{0}\\\
&=\frac{1}{2}(1+\sin{(\Phi_{A}-\Phi_{B})}).\end{split}$ (8)

This particular phase can also be measured using classical coherent probe
states. Due to this reason, it’s usually referred to as a “classical phase".

Consider the single photon mode distribution
$f(\omega)a^{\dagger}_{\omega}\ket{0}$ where

$f(\omega)=(\frac{1}{\pi\sigma^{2}})^{1/4}\exp{(-\frac{1}{2\sigma^{2}}(\omega-\omega_{0})^{2})},$
(9)

is the Gaussian distribution with centre frequency $\omega_{0}$ and $\sigma$
is the pulse width in frequency space. Thus at the output we have

$\begin{split}\braket{N}&=\int{d\omega}|f(\omega)|^{2}\braket{a^{\dagger}_{\omega}a_{\omega}}\\\
&=\frac{1}{2}(1+\int{d\omega(\frac{1}{\pi\sigma^{2}})^{1/2}\exp{(-\frac{1}{\sigma^{2}}(\omega-\omega_{0})^{2})\sin{\omega\Delta
t}}})\\\
&=\frac{1}{2}(1+e^{-(\frac{\Delta\Phi\sigma}{\omega_{0}})^{2}}\sin{\Delta\Phi}),\end{split}$
(10)

where the integral is over all positive frequencies $\omega$ and $\Delta\Phi$
is given by Eq. (4) in the Kerr metric. The visibility is given by

$\mathcal{V}=e^{-(\frac{\Delta\phi\sigma}{\omega})^{2}}=e^{-(\Delta
t\sigma)^{2}}$ (11)

The additional visibility loss is classified as a quantum effect because it is
due to the interference of wavefunctions as opposed to interference of
classical modes of the field had we used coherent probe states.

###  III.1 Two-way velocity of light

We now want to determine the average velocity of light that returns to the
stationary observer. We expect this to be isotropic and equal to $c=1$. We now
consider a double-sided mirror at halfway around the rotating planet that
reflects both of the photons back to the original source and the photons
interfere. The phase of the co-propagating photon $A$ before interfering with
the counter-propagating photon $B$ is

$\begin{split}\Phi_{A}+\Phi_{A}^{\prime}&=\omega
L(\frac{1}{1-\frac{r_{s}}{2r}+\frac{r_{s}a}{r^{2}}})+\omega
L(\frac{1}{1-\frac{r_{s}}{2r}-\frac{r_{s}a}{r^{2}}})\\\ &\approx\omega
L(1+\frac{r_{s}}{2r}),\end{split}$ (12)

where $\Phi_{A}^{\prime}=\Phi_{B}$ since this the counter-propagating phase.
Note we have used the weak field approximation. The mean velocity of the light
as seen by a far-away observer is thus
$c_{mean}=\frac{1}{1+\frac{r_{s}}{r}}\approx 1-\frac{r_{s}}{r}$. Locally, the
velocity of light is $\frac{dx}{d\tau}=c_{mean}\frac{dt}{d\tau}=1$ where
$\frac{dt}{d\tau}=(1-\frac{r_{s}}{r})^{-1}$ from setting
$dr=0,d\phi=0,d\theta=0$ in the Kerr metric. This implies that the local
velocity of light is $c=1$ and isotropic as expected.

Similarly, for the initially counter-propagating photon we have the same
phase. Thus no phase difference is detected and the observer infers that
$c=1$.

###  III.2 Visibility loss in the Kerr metric of Earth

Assuming that the mass quadrupole moment $q$ is negligible, the approximate
metric in the weak field limit is the Kerr metric. Therefore, if we consider
the massive planet to be Earth, the two photon paths will undergo a time delay
due to the frame dragging caused by the Kerr metric. Therefore, we have
$\mathcal{V}=e^{-(\Delta t_{Kerr}\sigma)^{2}}e^{-(\frac{2\pi
r_{s}a\sigma}{r})^{2}}\approx 1-1.5\times 10^{-10}$ where $a=3.9$ $m$,
$r_{s}=0.009$ $m$ and $r=6.37\times 10^{6}$ $m$. The visibility loss in the
quantum interference would be far too small to be detected on Earth’s surface.
The radial position at which the visibility would be significant is $r=2\pi
r_{s}a\sigma\approx 800$ $m$, assuming a neutron star with Earth’s mass.

###  III.3 Extremal black holes

An extremal black hole has a minimum possible mass for a given angular
momentum. We consider the full solution to the speed of light without any weak
field approximations for a near extremal black hole where the black hole mass
is a few orders above the angular momentum. For a black hole, due to symmetry,
the quadrupole mass moment is $q=0$ and the space-time is described by the
Kerr metric. The phase is therefore

$\Delta\Phi=\omega(t_{B}-t_{A})=kL(\frac{1}{c_{B}}-\frac{1}{c_{A}}),$ (13)

where
$c_{A,B}=\frac{r_{s}a}{r_{B}\sqrt{r_{B}^{2}+a^{2}(1+r_{s}/r_{B})}}\pm\sqrt{\frac{r_{s}^{2}a^{2}}{r_{B}^{2}(r_{B}^{2}+a^{2}(1+r_{s}/r_{B}))}+(1-\frac{r_{s}}{r_{B}})}$.
Using units of $r_{s}$, $a\rightarrow a^{\prime}r_{s}$, $r_{A}\rightarrow
r^{\prime}_{A}r_{s}$ and $r_{B}\rightarrow r^{\prime}_{B}r_{s}$. This
simplifies to $c_{A,B}=\frac{1}{r^{\prime}_{B}\sqrt{\frac{r_{B}^{\prime
2}}{a^{\prime 2}}+(1+1/r^{\prime}_{B})}}\pm\sqrt{\frac{1}{r_{B}^{\prime
2}((\frac{r_{B}^{\prime 2}}{a^{\prime
2}}+(1+1/r^{\prime}_{B}))}+(1-\frac{1}{r^{\prime}_{B}})}$. We consider the
values of a near extremal black hole with angular momentum parameter $a$ on
the order of $r_{s}$ with $r_{s}=30$ $km$, $a^{\prime}=\frac{1}{4}$,
interferometer length of $L=2\pi r$ and $\sigma=3.5\times 10^{3}$ $m^{-1}$. In
Fig. 1, we have the phase and the visibility plotted against the radial
distance away from the black hole centre. At a particular radial distance, the
visibility loss is significant and the wavepackets become completely separated
very near the event horizon.

Figure 1: Phase difference (blue) and visibility (red) of single photons
plotted against radial position in units of the Schwarzschild radius $r_{s}$
from centre of a rotating black hole. (Note the extremal black hole parameters
$r_{s}=30$ $km$, $a^{\prime}=\frac{1}{4}$ and $\sigma=3.5\times 10^{3}$
$m^{-1}$.)

##  IV Rotating reference frame

Figure 2: A HOM Sagnac interferometer on a turntable rotating at angular
frequency $\Omega$. A photon source (white) releases two photons that pass
through a beamsplitter and forms a superposition of the paths $A$ and $B$ with
phases $\Phi_{A}$ and $\Phi_{B}$. These recombine at the second beamsplitter
and are detected by a photon counter (gray).

In this section, we will compare the metric of an inertial observer in a
rotating reference frame with an observer in the Kerr metric. Consider an
inertial observer on a rotating turntable at a constant radius $r$ with
angular frequency $\Omega$ as depicted in Fig. 2.

The metric for a rotating observer in the (1+1) space-time is given by
transforming the flat metric using $d\phi\rightarrow d\phi+\Omega dt$

$ds^{2}_{Rotation}=dt^{2}-r^{2}d\phi^{2}\rightarrow(1-v^{2})dt^{2}-2vr_{t}dtd\phi-
r_{t}^{2}d\phi^{2},$ (14)

where the $t$ and $\phi$ coordinates are the inertial coordinates in the
rotating metric and where $v=\Omega r_{t}$ is the tangential velocity. We
compare the rotating metric with the Kerr metric at constant radius in a (1+1)
dimensional space-time

$ds^{2}_{Kerr}=(1-\frac{r_{s}}{r})dt^{2}-\frac{2r_{s}a}{r}dtd\phi-r^{2}d\phi^{2}.$
(15)

We require a coordinate transformation for it to match with equation 14.
Evidently, we require that $rd\phi_{Kerr}=r_{t}d\phi_{Rotation}$ implying the
measured tangential distances are the same. We set the size of the turntable
$r_{t}=r$ which implies that the angular coordinates are equivalent
$d\phi_{Kerr}=d\phi_{Rotation}$. However, we make the following time
coordinate transformation in the Kerr metric

$dt\rightarrow\sqrt{\frac{1-v^{2}}{1-\frac{r_{s}}{r}}}dt.$ (16)

Thus

$\begin{split}ds^{2}_{Kerr}=(1-v^{2})dt^{2}-\frac{2r_{s}a}{r}\sqrt{\frac{1-v^{2}}{1-\frac{r_{s}}{r}}}dtd\phi-r^{2}d\phi^{2}.\end{split}$
(17)

The $g_{t\phi}$ components must match and we have to solve for $v$ in the
equation $vr=\frac{r_{s}a}{r}\sqrt{\frac{1-v^{2}}{1-\frac{r_{s}}{r}}}$. We
find that the tangential velocity of the rotating turntable for simulating a
Kerr reference frame must be

$v=\pm\frac{r_{s}a}{r^{2}\sqrt{1-\frac{r_{s}}{r}+\frac{r_{s}^{2}a^{2}}{r^{4}}}}\approx\pm\frac{r_{s}a}{r^{2}\sqrt{1-\frac{r_{s}}{r}}}.$
(18)

By Einstein’s equivalence principle, the local observer cannot tell whether
they are in the (1+1) local space-time of a Kerr metric or in an inertial
rotating reference frame of a turntable. Therefore, we can simulate tangential
motion in a (1+1) Kerr space-time if we choose an appropriate tangnetial
velocity for the turntable, and use appropriate clocks.

For example, for a turntable of Earth’s radius $r=6.37\times 10^{7}$ $m$, we
can simulate the Earth’s Kerr metric with a turntable tangential velocity of
$v=\frac{r_{s}a}{r^{2}\sqrt{1-\frac{r_{s}}{r}}}\approx 2.6\times 10^{-7}$
$m/s$ or angular frequency $\Omega=4.1\times 10^{-14}$ $rad/s$. Alternatively,
for an Earth mass black hole $r_{s}=0.009\text{ m}$, and turntable radius of
$r=100$ $m$ then the tangential velocity of the turntable with the same radius
would have to be $v=110$ $m/s$ to match the effects of the Kerr metric. On the
other hand, for a $10$ solar mass black hole ($r_{s}=30\text{ km}$) rotating
at $a=r_{s}/100$ with an observer $10r_{s}$ distance from the centre, the
tangential velocity of the turntable at the same radius is $v=30,000\text{
m/s}$. We note that these high tangential velocities are not currently
experimentally feasible.

###  IV.1 Time shift

An alternative scenario is where both observers in either metrics use their
respective coordinate times but we require the time shifts imposed on
tangentially propagating light rays to be equal. We consider the perspective
of an inertial observer in the laboratory frame observing the rotating
platform that sends light for a round trip. The round trip for the light as
seen by the inertial observer is obtained by considering the distance
travelled by the co-moving light $L+vt=t$ where $t$ is the total time of the
round trip. Thus

$\Delta t=\frac{L}{1-v}-L=\frac{2\pi r_{t}}{\sqrt{1-v^{2}}(1-v)}-\frac{2\pi
r_{t}}{\sqrt{1-v^{2}}},$ (19)

where $r_{t}$ is the turntable radius as seen by the inertial observer in the
laboratory frame (note we have accounted for the time dilation). In the Kerr
metric, as seen by a far-away observer the time of the round trip (w.r.t. to
the stationary observer) around a massive planet is (we have used Eq. (3))

$\begin{split}\Delta t_{Kerr}&\approx\frac{2\pi
r}{1-\frac{r_{s}a}{r^{2}}-\frac{r_{s}}{2r}}-\frac{2\pi
r}{1-\frac{r_{s}}{r}}\\\ &\approx 2\pi
r(1+\frac{r_{s}}{2r}+\frac{r_{s}a}{r^{2}})-2\pi
r(1+\frac{r_{s}}{2r})=\frac{2\pi r_{s}a}{r},\end{split}$ (20)

where $r$ is the radius in the Kerr metric. Equating these two far-away times
($\Delta t=\Delta t_{Kerr}$) and without assuming $v$ is small, we have

$\frac{r_{s}a}{rr_{t}}=\frac{1}{\sqrt{1-v^{2}}(1-v)}-\frac{1}{\sqrt{1-v^{2}}}=\frac{v}{\sqrt{1-v^{2}}}.$
(21)

Thus we have the velocity

$v=\frac{r_{s}a}{rr_{t}\sqrt{1+\frac{r_{s}^{2}a^{2}}{r^{4}}}}\approx\pm\frac{r_{s}a}{rr_{t}}$
(22)

For example, for a turntable of radius $r_{t}=0.2$ $m$ and $r=r_{E}$ equal to
the Earth’s radius we have a turntable velocity of $v=\pm 0.8$ $m/s$ or
angular frequency $\Omega=4$ $rad/s$. Compared to the previous example of an
Earth size turntable, the accumulated effect on the smaller turntable is
smaller and thus the velocity must be larger to compensate. Various black-hole
scenarios could be simulated using lab-sized turn-tables using this method. We
note that if the turntable radius is equal to the Kerr metric radius $r_{t}=r$
and we use the metric times in Equations 14 and 16. We obtain the equation

$\frac{r_{s}a}{r^{2}}\sqrt{\frac{1-v^{2}}{1-\frac{r_{s}}{r}}}=v,$ (23)

which, solving for $v$ is equivalent to Eq. (18). Therefore, we have proven
that a transformation of time coordinates isn’t necessary to quantitatively
simulate the Kerr metric with a rotating turntable.

###  IV.2 Phase

For a rotating reference frame, we solve for the light null geodesic to obtain
the tangential velocity of light $\dot{x}=r\frac{d\phi}{dt}$. Setting
$ds^{2}=0$ we have $(1-v^{2})-2v\dot{x}-\dot{x}^{2}=0$. We thus have two
solutions $\dot{x}=1\pm v$. The cross term component $dtd\phi$ once again is
the cause of the anisotropy of light. With the rotation of the turntable, the
phase of the photon is $\Phi_{A}=\omega\frac{L}{c_{A}}=\omega\frac{L}{1+v}$
and against the rotation, with the phase
$\Phi_{B}=\omega\frac{L}{c_{B}}=\omega\frac{L}{1-v}$. Thus the phase
difference for the turntable is

$\Delta\Phi_{Rotation}=\frac{2\omega vL}{1-v^{2}},$ (24)

which is the same phase obtained by a classical Sagnac interferometer.

###  IV.3 Minimum velocity for significant visibility loss

We substitute the phase in Eq. (24) in Eq. (11) to obtain the visibility

$\mathcal{V}=e^{-\sigma^{2}\frac{\Delta\phi^{2}}{\omega^{2}}}=e^{-4\frac{v^{2}L^{2}\sigma^{2}}{(1-v^{2})^{2}}}=e^{-4\frac{v^{2}}{(1-v^{2})}\pi^{2}r^{2}\sigma^{2}}.$
(25)

Note that $L$ is the total contracted length travelled by the light. For the
case where the photons meet half way we have $L=\pi r\sqrt{1-v^{2}}$. For
significant visibility loss the velocity needed for the rotating platform is
given by solving $4\frac{v^{2}}{(1-v^{2})}\pi^{2}R^{2}\sigma^{2}=1$.
Rearranging,

$v_{min}=\frac{1}{\sqrt{4\pi^{2}r^{2}\sigma^{2}+1}}\approx\frac{1}{2\pi
r\sigma}.$ (26)

We consider $\frac{\sigma}{c}=\frac{1}{\Delta t_{p}c}=3.3\times 10^{3}$
$m^{-1}$ corresponding to picosecond pulses and a rotating platform of radius
$r=5$ $m$. Thus $v\approx\frac{1}{2\pi R\sigma}\approx 2900m/s$. If smaller
pulses of $100$ femtoseconds are used then $v\approx 290m/s$. The amount of
g-force for this velocity is $1700$ $g$. However, we can simply increase $L$
by increasing the number of windings around the turntable, thus accumulating
the effect, and making it significant for much lower velocities. Although this
is not feasible without increasing the fibre windings, we discuss and provide
a reasonable optical fibre length using a coherent light source in Section
VI.3.

##  V Two photons input (HOM interference)

In our thought experiment, we considered a superposition of a single photon
travelling two paths. The probability of detection depends on the classical
phase shift $\Delta\Phi$, and also the visibility of the quantum interference.
The Hong-Ou-Mandel (HOM) interferometer uses two photons as input. However, in
this setup, a physical time delay between the paths is explicit in the
probability of detection. The loss of interference is due to loss of
indistinguishability of the photons and has no classical analogue. The HOM
effect can be interpreted as more quantum due to this strong quantum
interference. We consider a source of single photons travelling paths $A$ and
$B$ to a beamsplitter. The state after the beamsplitter becomes

$\begin{split}a^{\dagger}_{1}a^{\dagger}_{2}e^{i\omega\Delta
t}\ket{0}\ket{0}&\rightarrow\frac{1}{2}(a^{\dagger}_{3}+ia^{\dagger}_{4})(a^{\dagger}_{4}+ia^{\dagger}_{3})e^{i\omega\Delta
t}\ket{0}\ket{0}\\\ &=\frac{e^{i\omega\Delta
t}i}{2}(\ket{2}\ket{0}+\ket{0}\ket{2}),\end{split}$ (27)

where $e^{i\omega\Delta t}$ is a global phase difference between the two
modes. We thus have photon pair incidences at the final output of the
beamsplitter. In this case the time delay doesn’t affect the coincidence
probability, since the photons are not distinguishable.

Now we consider a source of photons with arbitrary frequency distributions
$f(\omega_{1})$ and $g(\omega_{2})$ sending the initial state
$\ket{1}\ket{1}=\int{d\omega_{1}f(\omega_{1})a_{1}^{\dagger}(\omega_{1})}\int{d\omega_{2}g(\omega_{2})a_{2}^{\dagger}(\omega_{2})}e^{-i\omega_{2}\Delta
t}\ket{0}\ket{0}$ to a beamsplitter. The transformation is

$\begin{split}a^{\dagger}_{1}a^{\dagger}_{2}\ket{0}\ket{0}&\rightarrow\frac{1}{2}\int{d\omega_{1}f(\omega_{1})}(a^{\dagger}_{3}(\omega_{1})+ia^{\dagger}_{4}(\omega_{1}))\\\
&\times\int{d\omega_{2}g(\omega_{2})}(a^{\dagger}_{4}(\omega_{2})+ia^{\dagger}_{3}(\omega_{3}))e^{-i\omega_{2}\Delta
t}\ket{0}\ket{0}\\\
&=\frac{1}{2}\int{d\omega_{1}f(\omega_{1})}\int{d\omega_{2}g(\omega_{2})e^{-i\omega_{2}\Delta
t}}\\\
&\times(ia^{\dagger}_{3}(\omega_{1})a^{\dagger}_{3}(\omega_{2})+a^{\dagger}_{3}(\omega_{1})a^{\dagger}_{4}(\omega_{2})\\\
&-a^{\dagger}_{4}(\omega_{1})a^{\dagger}_{3}(\omega_{2})+ia^{\dagger}_{4}(\omega_{1})a^{\dagger}_{4}(\omega_{2}))\ket{0}\ket{0}.\end{split}$
(28)

The detection probability of a photon pair in either mode is determined by
modelling the detectors as having a flat frequency response with the projector
$P_{3}=\int{d\omega
a^{\dagger}(\omega)\ket{0}\ket{0}\bra{0}\bra{0}a(\omega)}$. These calculations
have been done in Ref. bran , and for photons of the same frequency
distribution $f(\omega)=g(\omega)$, the probability is

$P=\frac{1}{2}-\frac{1}{2}\int{d\omega_{1}|f(\omega_{1})|^{2}e^{-i\omega_{1}\Delta
t}}\int{d\omega_{2}|f(\omega_{2})|^{2}e^{i\omega_{2}\Delta t}}.$ (29)

For photons with Gaussian frequency distribution of pulse width $\sigma$, we
evaluate Eq. (29) to obtain

$P_{Gauss}=\frac{1}{2}-\frac{1}{2}e^{-\frac{\sigma^{2}\Delta t^{2}}{2}}.$ (30)

For zero time delay $\Delta t=0$, the photons are indistinguishable. At this
point, the visibility $\mathcal{V}=\text{Tr}(\rho_{a}\rho_{b})$ of the two
photon states $\rho_{a}$, $\rho_{b}$ is equal to the purity
$\text{Tr}(\rho^{2})$ of the two photons $\rho=\rho_{b}=\rho_{a}$ bran . Thus
the visibility, defined as $1-P_{0}/P_{max}$, where $P_{0}$ is the coincidence
probability at $\Delta t=0$ and $P_{max}$ is the probability for $\Delta t$
larger than the photon coherence length, is $100\%$ which is known as the
Hong-Ou-Mandel dip. The coincidence count drops to zero when the two input
photons are completely identical. For the case of the rotating turntable, the
photons will become more distinguishable as $\Delta t$ increases. The velocity
at which this becomes significant is the same as in Eq. (26) since the
visibility is the same and the time delay is $\Delta t=\frac{4vL}{1-v^{2}}$.
We can calibrate the dip for $100$ $\%$ visibility using a controlled time
delay in one of the arms and vary the rotational velocity of the turntable.

Compared to using a single photon interferometer, the HOM interferometer is
based on the indistinguishability of the photons interfering with each other.
As in Ref. enk , the HOM effect measures a physical time delay as opposed to a
phase shift.

###  V.1 Two-way velocity of light

Similarly to the Kerr metric, the two-way velocity of light as measured by the
inertial observer on the rotating turntable should be isotropic. We now
consider a double sided mirror that reflects both of the light beams back to
the original source. In this case, our phase difference is
$\phi_{A}^{\prime}=\phi_{A}+\frac{\omega L}{c_{B}}=\omega
L(\frac{1}{1+v}+\frac{1}{1-v})=\frac{2\omega L}{1-v^{2}}$ but $L=\pi
R\sqrt{1-v^{2}}$. Therefore $\phi^{\prime}_{A}=\frac{2\omega
L}{\sqrt{1-v^{2}}}$. Similarly, for the counter-propagating beam of light
$\phi_{B}^{\prime}=\phi_{B}+\frac{\omega L}{c_{A}}=\omega
L(\frac{1}{1-v}+\frac{1}{1+v})=\phi^{\prime}_{A}$. Thus the phase difference
is zero. In other words, the “two-way velocity" of light is isotropic and
$c=1$. This demonstrates that observers riding on the turntable would also
measure $c=1$ between points around the circumference.

##  VI Experimental feasibility in optical fibre

We consider effects specific to the fibre optic set-up and the feasibility of
observing the HOM dip on a rotating platform. In a medium, the dispersion
relation describes the relation of the frequency $\omega$ to its wavenumber
$k$. The dispersion relation is Taylor expanded as
$\omega(k)=\omega(k_{0})+(k-k_{0})v_{g}+\frac{1}{2}(k-k_{0})^{2}\frac{d^{2}\omega}{dk^{2}}$
where $v_{g}=\frac{1}{\alpha}=\frac{d\omega}{dk}$ is the group velocity and
$\frac{1}{\beta}=\frac{d^{2}\omega}{dk^{2}}$ is the inverse group velocity
dispersion. For linear dispersion $\omega=\frac{k}{n}$ implying
$\frac{d^{2}\omega}{dk^{2}}=0$ and the group velocity $v_{g}=\frac{1}{n}$ is
equal to the phase velocity $v_{p}$.

In relativity, the phase velocities of light in a medium in the stationary
reference frame of the laboratory transform according to the Lorentz
transformations.

###  VI.1 Lorentz transformations in a moving medium

We consider using fibre optic cable to guide the photon half-way around the
turntable as seen in Fig. 2. In a medium, the velocity of light is slowed down
by the factor $1/n$ where $n$ is the refractive index. The velocity of light
in the medium depends on direction of the rotation relative to the observer in
the laboratory reference frame.

According to the velocity composition law, for a moving medium, the velocity
of light as seen in the laboratory reference frame of the rotating reference
frame is

$c_{A}=\frac{\frac{c}{n}-v}{1-\frac{v}{cn}},$ (31)

which is equivalently the phase velocity. Therefore,
$L-vt=t\frac{\frac{c}{n}-v}{1-\frac{v}{cn}}$, and $t=\frac{L(n-v)}{1-v^{2}}$
implying that the effective velocity of light as seen by a far-away observer
is $c_{A}^{\prime}=\frac{1-v^{2}}{n-v}$ and similarly,
$c_{B}^{\prime}=\frac{1-v^{2}}{n+v}$.

###  VI.2 Phase velocity

Using the phase velocity in Eq. (31), the new phase is therefore
$\Delta\Phi=\omega_{0}(\frac{L}{c_{A}^{\prime}}-\frac{L}{c_{B}^{\prime}})=\frac{2v}{1-v^{2}}\omega_{0}L$.
Coincidentally, the final phase doesn’t depend on the refractive index if the
fibres are equal length $L$. However, if these are unequal $L_{A}=L$ and
$L_{B}=L+\Delta L$, then $\Delta\Phi^{\prime}=\Delta\Phi+\omega_{0}\Delta
L\frac{n}{1-v^{2}}$. The HOM dip will measure the time delay $\Delta
t^{\prime}=\frac{4vL+2n\Delta L}{1-v^{2}}$. If $\Delta L$ is on the length
scale of the coherence length then we can simply cancel this out with a
controlled time delay.

As in Fig. 3, a time delay initially sets the visibility to $100\%$ with the
turntable at rest. The turntable is slowly rotated and the visibility
decreases to $0\%$. The HOM dip will be initially centered around $v=0$ where
$\Delta t^{\prime}_{0}=2\Delta Ln+\Delta t_{Control}=0$. Thus as the turntable
is slowly rotated we have

$\begin{split}\Delta t^{\prime}&=\frac{4vL}{1-v^{2}}+2\Delta
L\frac{n}{1-v^{2}}-2\Delta Ln\\\ &\approx\frac{4vL}{1-v^{2}}+2\Delta
Lnv^{2}.\end{split}$ (32)

There is as shift in the centre of the HOM dip. $\Delta L$ depends on the
experimental error of the measured fibre lengths and the velocity of the
turntable. For example, for a slowly rotating turntable of $\Omega=2\pi$
$rad/s$ and $R=20$ $cm$ with minimal vibrations, a mismatch in the length of
the optical fibres of $\Delta L=1$ $cm$ would shift the HOM dip by $\Delta
t_{Error}=2\Delta Lnv^{2}\approx 3\times 10^{-11}$ $s$ or relative to the
leading term $\frac{\Delta t_{Error}}{\Delta t}=\frac{\Delta Lnv}{2L}\approx
3\times 10^{-11}$ smaller. Since $v$ is extremely small, the propagated error
in the time difference and thus the zero point of the HOM dip would be
negligible.

Figure 3: Probability of photon detection with angular frequency of turntable
for parameters $L=10$ $km$, $R=20$ $cm$ and $\sigma=4000\pi$.

###  VI.3 Coherence length of photons

We note that we can extend the time of rotation by increasing windings and
therefore $L^{\prime}=(2N+1)\pi R\sqrt{1-v^{2}}$ where $N$ is the number of
windings. Thus the visibility for very slow rotation becomes
$\mathcal{V}=1-P_{Gauss}/P_{Max}=e^{-\frac{\sigma^{2}\Delta
t^{2}}{2}}=e^{-8\frac{v^{2}}{1-v^{2}}(2N+1)^{2}\pi^{2}R^{2}\sigma^{2}}$. Thus
far, we have been assuming timed pulses of light. Alternatively, the source of
down-converted photons could be continuous characterized by a coherence
length. Note that the coherence length is defined as $\Delta
x=\frac{2\pi}{\sigma}$ where $\sigma$ is the width in frequency space. We
consider parameters of $v=\frac{\Omega R}{c}$ where $R=20$ $cm$ and
$\Omega=2\pi$ $Hz$ with optical fibre of length $L^{\prime}=10$ $km$.
Therefore, the coherence length needed for significant visibility loss is
$\Delta x=4\pi L^{\prime}\frac{\Omega R}{c}\approx 500$ $\mu m$ which is
easily coherence length of down-converted photons. In the next section, we
will consider a quasi-continuous source and the effect of dispersion on the
phase.

###  VI.4 Dispersion cancellation

We’ve seen that the phase velocity isn’t affected by the relativity of the co-
and counter- propagating light for fibre of equal length. However, we now
consider the full treatment of the effects of dispersive broadening and the
group velocity in Appendix A.

We find the usual phase but with a correction

$\Delta\phi_{v_{g}}=2\omega^{\prime}L(\frac{1}{v_{g}^{-}}-\frac{1}{v_{g}^{+}})\approx\frac{4\omega_{0}vL}{1-v^{2}}(1+\frac{n^{\prime}(k_{0})v}{1-v^{2}}).$
(33)

Thus, as expected when the index of refraction is a constant $v_{g}=v_{p}$ and
$n^{\prime}(k_{0})=0$ we obtain the usual phase shift with the cancellation of
the refractive index. The correction is negligible for an $L=10$ $km$ long
fibre at extremely slow rotation of $\Omega=2\pi$ $Hz$ and radius $R=20$ $cm$.
For optical fibre $n^{\prime}(k_{0})=-10^{-9}$ $m$, which is much less than
unity and the correction term is negligibly small $5\times 10^{-18}$.

We consider the frequency distribution
$|f(\omega^{\prime})|^{2}=(\frac{1}{\pi\sigma^{2}})^{1/2}\exp(-\frac{\omega^{\prime
2}}{\sigma^{2}})$. The probability is therefore

$\begin{split}P_{c}&=\int{d\omega^{\prime}|f(\omega^{\prime})|^{2}(1-\cos{(2\omega^{\prime}\Delta\alpha
L)}})\\\
&=\frac{1}{2}(1-\exp{(-4\sigma^{2}(\frac{vL}{1-v^{2}}(1+\frac{n^{\prime}(k_{0})v}{1-v^{2}}))^{2})}).\end{split}$
(34)

Thus, we have the visibility with the corrected time delay from the group
velocity. Ultimately, the effect of dispersion due to the material is
cancelled out, and the group velocity effect is far too small
($n^{\prime}(k)\approx-10^{-9}$ in silicon fibre) for the parameters
suggested. For unequal lengths of the optical fibre, the phase term due to the
index of refraction can be cancelled out using a time delay.

##  VII Conclusion

We have shown how to measure the visibility loss of interfering paths of a
photon travelling in the Kerr metric. The effect of the non-trivial Kerr
space-time manifests as a classical phase but the effect of time dilation in
this metric can be measured by the visibility loss of single-photon
interference. We can directly measure the time delay and loss in the
visibility of quantum interference using a single-photon Sagnac interferometer
in optical fibre. We also find that dispersion in the optical fibre cancels
out and the difference in group velocity is negligible for the parameters
considered. Based on our in-depth analysis, we draw parallels between single-
photon interference in a rotating reference frame and the Kerr metric. This
provides a recipe for near term experiments which could reproduce non-trivial
results from the Kerr metric in a rotating system. Ultimately, one would like
to observe such effects at the interface between quantum mechanics and non-
trivial space-times.

Acknowledgements. We thank Matthias Fink and Rupert Ursin from the Institute
for Quantum Optics and Quantum Information of the Austrian Academy of Sciences
for discussions of potential experimental set ups. This work was supported in
part by the Australian Research Council Centre of Excellence for Quantum
Computation and Communication Technology (Project No. CE110001027) and
financial support by an Australian Government Research Training Program
Scholarship. We acknowledge that similar work in Ref. rest  was published at
the same time as thesis submission containing some of the research presented
here kishthesis .

## Data Availability

The data that supports the findings of this study are available within the
article.

####

## Conflict of Interest

The authors have no conflicts to disclose.

## References

  * (1) Quantum imaging using relativistic detectors, Nicholas Bornman, Achim Kempf, and Andrew Forbes, Phys. Rev. D 100, 125004 (2019).
  * (2) Quantum mechanics for non-inertial reference frames, M Moreno and J A del Rio, Eur. J. Phys. 42 045405 (2021).
  * (3) Quantum interferometric visibility as a witness of general relativistic proper time, Magdalena Zych, Fabio Costa, Igor Pikovski & Caslav Brukner, Nature Communications 2: 505 (2011).
  * (4) General relativistic effects in quantum interference of photons, Magdalena Zych, Fabio Costa, Igor Pikovski, Timothy C. Ralph and Caslav Brukner, Class. Quantum Grav. 29 224010 (2012).
  * (5) Phase shifts vs time delays: Sagnac and Hong-Ou-Mandel, S. J. van Enk, arXiv preprint: arXiv:1104.0970(2011).
  * (6) Single-photon Sagnac interferometer, G. Bertocchi, O. Alibart, D.B. Ostrowsky, S. Tanzilli and P. Baldi, J. Phys. B: At. Mol. Opt. Phys. 39, 1011-1016 (2006).
  * (7) Entanglement-enhanced optical gyroscope, Matthias Fink, Fabian Steinlechner, Johannes Handsteiner, Jonathan P. Dowling, Thomas Scheidl, Rupert Ursin, New J. Phys. 21 053010 (2019).
  * (8) Photon Bunching in a Rotating Reference Frame, S. Restuccia, M. Toroš, G. M. Gibson, H. Ulbricht, D. Faccio, and M. J. Padgett, Phys. Rev. Lett. 123, 110401 (2019).
  * (9) Frame dragging and the Hong-Ou-Mandel dip: Gravitational effects in multiphoton interference, A. J. Brady, S. Haldar, Phys. Rev. Research 3, 023024 (2021).
  * (10) Revealing and concealing entanglement with noninertial motion, M. Toros, S. Restuccia, G. M. Gibson, M. Cromb, H. Ulbricht, M. Padgett, D. Faccio, Phys. Rev. A 101, 043837 (2020).
  * (11) Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer, Aephraim M. Steinberg, Paul G Kwiat and Raymond Y. Chiao, Phys Rev A. 1;45(9):6659-6665 (1992).
  * (12) The Kerr spacetime: a brief introduction, D.L. Wiltshire, M. Visser and S.M. Scott eds., Cambridge University Press, Cambridge, U.K. (2009).
  * (13) Quantum metrology in the Kerr metric, S. P. Kish, T. C. Ralph, Phys. Rev. D 99, 124015 (2019).
  * (14) Measurement of subpicosecond time intervals between two photons by interference C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).
  * (15) Hong-Ou-Mandel Interference, A. M. Branczyk, arXiv preprint: arXiv:1711.00080 (2017).
  * (16) Quantum metrology in curved space-time, S. Kish, PhD Thesis, School of Mathematics and Physics, The University of Queensland (2019).

##  Appendix A Dispersion cancellation

In a moving medium, the phase velocity of light for co- and counter-
propagating light is given by

$v_{p\pm}=\frac{\omega}{k}=\frac{1-v^{2}}{n(k)\mp v},$ (35)

and the group velocity is given by

$v_{g\pm}=\frac{1}{\alpha_{\pm}}=v_{p\pm}-\frac{dn(k)}{dk}\frac{1-v^{2}}{(n(k)\mp
v)^{2}}=v_{p\pm}(1-\frac{n^{\prime}(k)}{n(k)\mp v}),$ (36)

where $\alpha_{\pm}=\frac{dk_{\pm}}{d\omega_{\pm}}$ is the inverse group
velocity. The second order effect responsible for broadening is

$\begin{split}\frac{d^{2}\omega_{\pm}}{dk_{\pm}^{2}}&=\frac{1}{\beta_{\pm}}=v_{g\pm}-v_{p\pm}-\frac{d^{2}n(k)}{dk^{2}}\frac{1-v^{2}}{(n(k)\mp
v)^{2}}\\\ &+2(\frac{dn(k)}{dk})^{2}\frac{1-v^{2}}{(n(k)\mp
v)^{3}},\end{split}$ (37)

where the group velocity dispersion (GVD) is defined as
$\beta=\frac{d^{2}k}{d\omega^{2}}$. For example, for fused silica, the index
of refraction is approximately linear with the wavelength. Thus as a function
of $k$, $n(k)=\frac{100000}{k}+1.44$ around the wavenumber $k_{0}=8\times
10^{6}$ $m^{-1}$. Thus $n(k_{0})=1.453$ and the derivative is
$\frac{dn(k_{0})}{dk}=-\frac{10^{5}}{k_{0}^{2}}=-1.6\times 10^{-9}$ $m$. The
second derivative is $\frac{d^{2}n(k_{0})}{dk^{2}}=\frac{2\times
10^{5}}{k_{0}^{3}}=4\times 10^{-16}$ $m^{2}$. Thus
$\frac{d^{2}\omega}{dk^{2}}\approx 1\times 10^{-9}$ $m^{2}/s^{2}$.

We consider the following quasi continuous wave input state of down-converted
light

$\ket{\psi}=\int{d\omega^{\prime}f(\omega^{\prime})\ket{\omega_{0}+\omega^{\prime}}_{A}\ket{\omega_{0}-\omega^{\prime}}_{B}}.$
(38)

After passing through the beamsplitter the modes in the two arms $1$ and $2$
are

$\displaystyle a_{1}(\omega_{1})$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}a_{A}(\omega_{1})e^{ik_{A}(\omega_{1})L}+\frac{1}{\sqrt{2}}a_{B}(\omega_{1})e^{ik_{B}(\omega_{1})L},$
(39) $\displaystyle a_{2}(\omega_{2})$ $\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2}}a_{B}(\omega_{2})e^{ik_{B}(\omega_{2})L}+\frac{1}{\sqrt{2}}a_{A}(\omega_{2})e^{ik_{A}(\omega_{2})L},$
(40)

where
$k_{A}=k_{0}+\alpha_{+}(\omega-\omega_{0})+\beta_{+}(\omega-\omega_{0})^{2}$
is the wavenumber of the co-propagating light in the laboratory reference
frame. Similarly,
$k_{B}=k_{0}+\alpha_{-}(\omega-\omega_{0})+\beta_{-}(\omega-\omega_{0})^{2}$
is the counter- propagating light.

As in Ref. steinberg , we assume a gate window time that is much larger than
the dispersive broadening. This implies that cross terms of annihilation
operators at different frequencies disappear for sufficiently long detector
time scales. Thus the probability $P_{c}$ is

$P_{c}\propto\int{d\omega_{1}}\int{d\omega_{2}}\braket{\psi}{a^{\dagger}_{1}(\omega_{1})a_{2}^{\dagger}(\omega_{2})a_{1}(\omega_{1})a_{2}(\omega_{2})}{\psi}.$
(41)

As it turns out, the phase term $\beta(\omega-\omega_{0})^{2}$ acquired is the
same in both interferometer arms due to the condition that
$\omega_{p}=\omega_{A}+\omega_{B}$. The kernel is

$\begin{split}&\braket{\psi}{a^{\dagger}_{1}(\omega_{1})a_{2}^{\dagger}(\omega_{2})a_{1}(\omega_{1})a_{2}(\omega_{2})}{\psi}\\\
=&|\frac{1}{2}\delta(\omega_{p}-\omega_{1}-\omega_{2})f(\omega^{\prime})[e^{ik_{B}(\omega_{1})L+ik_{A}(\omega_{2})L}\\\
&-e^{-ik_{B}(\omega_{2})L-ik_{A}(\omega_{1})L}]|^{2}\\\
&=|\frac{1}{2}\delta(\omega_{p}-\omega_{1}-\omega_{2})f(\omega^{\prime})[e^{i(\Delta\alpha\omega^{\prime}+\beta^{\prime}\omega^{\prime
2})L}\\\ &-e^{i(-\Delta\alpha\omega^{\prime}+\beta^{\prime}\omega^{\prime
2})L}]|^{2},\end{split}$ (42)

where $\omega^{\prime}=\omega_{1}-\omega_{0}=\omega_{0}-\omega_{2}$,
$\Delta\alpha=\alpha_{+}-\alpha_{-}$ and $\beta^{\prime}=\beta_{+}+\beta_{-}$.
Evaluating the absolute squares

$P_{c}=\int{d\omega^{\prime}|f(\omega^{\prime})|^{2}(1-\cos{(2\omega^{\prime}\Delta\alpha
L))}}.$ (43)