# Multi-Stage Face-Voice Association Learning with Keynote Speaker Diarization

Ruijie Tao  National University of Singapore<EMAIL_ADDRESS>,  Zhan Shi
The Chinese University of Hong Kong<EMAIL_ADDRESS>,  Yidi Jiang
National University of Singapore<EMAIL_ADDRESS>,  Duc-Tuan Truong
Nanyang Technological University<EMAIL_ADDRESS>,  Eng-Siong Chng
Nanyang Technological University<EMAIL_ADDRESS>,  Massimo Alioto
National University of Singapore<EMAIL_ADDRESS>and  Haizhou Li
The Chinese University of Hong Kong<EMAIL_ADDRESS>
(2024)

###### Abstract.

The human brain has the capability to associate the unknown person’s voice and
face by leveraging their general relationship, referred to as “cross-modal
speaker verification”. This task poses significant challenges due to the
complex relationship between the modalities. In this paper, we propose a
“Multi-stage Face-voice Association Learning with Keynote Speaker Diarization”
(MFV-KSD) framework. MFV-KSD contains a keynote speaker diarization front-end
to effectively address the noisy speech inputs issue. To balance and enhance
the intra-modal feature learning and inter-modal correlation understanding,
MFV-KSD utilizes a novel three-stage training strategy. Our experimental
results demonstrated robust performance, achieving the first rank in the 2024
Face-voice Association in Multilingual Environments (FAME) challenge with an
overall Equal Error Rate (EER) of 19.9%. Details can be found in
https://github.com/TaoRuijie/MFV-KSD.

Cross-modal Speaker Verification, Face-voice Association

††copyright: acmlicensed††journalyear: 2024††doi: XXXXXXX.XXXXXXX††conference:
; October 28 - November 1, 2024; Melbourne, Australia.††isbn:
978-1-4503-XXXX-X/18/06††ccs: Information systems Multimedia information
systems

##  1\. Introduction

Figure 1. Cross-modal speaker verification system requires to focus on the
active speech segment of “keynote speaker” (who talks most), and keep a
balance between single-modal understanding and cross-modal correlation
learning.

Speaker recognition aims to verify the identity of a speaker based on their
unique voice characteristics. Deep learning-based speaker representation
systems, with the assistance of large-scale datasets, have achieved robust
performance (Truong et al., 2024b, 2022; Liu et al., 2024; Truong et al.,
2024a). Recent studies have demonstrated that this verification task can also
be extended to the cross-modal scenarios (Tao et al., 2020). Specifically,
even for the unknown identity, the human brain can recognize whether a static
facial image and a heard voice belong to the same individual, relying on
general cross-modal relationships such as gender, nationality, and age (Oh et
al., 2019; Chung et al., 2020b; Nagrani et al., 2018b). Various related
studies have been explored for audio-visual speech processing. For instance,
FaceFilter (Chung et al., 2020a) studied audio-visual speech separation
conditioned on a still face image of a target speaker. Face-based target
speech diarization (Jiang et al., 2024b) employed a static face image as the
prompt to specify the desired speaker.

This year, the FAME Challenge (Saeed et al., 2024) focuses on enhancing the
integration of face and voice embeddings for robust speaker recognition in
multilingual contexts. Instead of relying solely on cross-validation within a
single language, FAME evaluates the effects of multiple languages on the
verification process.

Although the existing cross-modal solutions have achieved remarkable
improvement by leveraging the deep learning methods, there are still two main
problems in this task: As shown in Figure 1, from the data aspect, in real-
world scenarios, raw audios often contain conversational or non-speech
segments. However, speaker recognition models exhibit robust capabilities in
capturing features from single-modal inputs (Liu et al., 2023; Tao et al.,
2024; Chen et al., 2022). The noisy inputs will significantly degrade the
ability of speaker models to extract accurate embeddings. Meanwhile, it is
challenging to find a suitable solution: traditional speaker diarization
methods can hardly distinguish the target speaker’s speech segments in the
audio with imbalanced speaker distribution, where the target speaker’s voice
predominates.

Moreover, from the learning aspect, effectively modeling the correlation
between facial appearance and voice characteristics is significant. It is
crucial to not only generate accurate and robust single-modal (face or speech)
embeddings, but also to establish cross-modal alignment. However, simple
joint-modeling approach for these components can hardly avoid over-fitting,
complicating the understanding of both single modality information and cross-
modal correlations. This requires a smoother training strategy to ensure
effectiveness and generalizability.

To address the challenges outlined above, we propose the “Multi-stage Face-
voice Association Learning with Keynote Speaker Diarization” (MFV-KSD)
framework. MFV-KSD integrates a keynote speaker diarization frontend and a
three-stage training strategy. The keynote speaker diarization frontend is
employed to filter out interfering speech inputs. It can solve the data
problem by enabling more robust and accurate target speaker’s embedding. The
three-stage training strategy consists of intra-modal recognition training,
inter-modal correlation training, and FAME adaptation. By leveraging the
large-scale dataset, this strategy can balance and enhance the intra-modal
learning ability and the inter-modal alignment ability. Our system achieves
19.9% EER in FAME 2024 challenge.

##  2\. Related work

###  2.1. Biometric recognition

As human faces and voices carry unique individual characteristics, various
biometric authentication models have been developed for speech modality, face
modality, or both of them (Dargan and Kumar, 2020).

####  2.1.1. Face recognition

Each human face reflects a unique combination of features such as eyes, nose,
and mouth, which enable the ability to recognize each person based on their
facial characteristics (Baron, 1981). Deep neural networks can extract facial
traits into their representations. Among all solutions, convolutional neural
networks (CNNs) are particularly popular in face recognition systems due to
their robust capability to extract detailed local features, such as the ResNet
model (Deng et al., 2019b).

####  2.1.2. Speaker recognition

The human voice is shaped by unique individual physiological factors,
including the shape of the vocal tract, accent, and rhythm (Kabir et al.,
2021). Consequently, speaker recognition systems can be employed to identify
individuals. For example, the x-vector model (Snyder et al., 2018) is based on
Time Delay Neural Network (TDNN) architecture. The 2D-CNN ResNet architecture
was adapted for speaker recognition tasks, achieving better performance.
Building on these advancements, the ECAPA-TDNN model (Desplanques et al.,
2020) was proposed, which improves upon the x-vector by replacing the original
TDNN layers with multi-level 1D residual Res2Net modules to capture speaker
embeddings effectively.

####  2.1.3. Audio-visual identity recognition

Audio-visual identity recognition systems enhance the effectiveness of
biometric applications across various scenarios. Fusion techniques for audio
and visual modalities in identity recognition are primarily performed at the
feature level (Chen et al., 2020; Das et al., 2020; Sari et al., 2021; Qian et
al., 2022; Rajasekhar and Alam, 2023; Jiang et al., 2023) and the score level
(Sadjadi et al., 2020; Alam et al., 2020). Feature-level fusion projects the
combination of audio and visual representations into a shared embedding space,
whereas score-level fusion aggregates the scores obtained from each modality
into a final prediction score.

###  2.2. Cross-modal speaker verification

Cross-modal speaker verification aims to determine whether a static image and
an audio signal represent the same individual. This task is motivated by the
fact that a person’s face and voice can be associated based on correlations
between physical factors such as age, gender, and ethnicity (Nagrani et al.,
2018c). This audio-visual cross-modal speaker verification system can help
retrieve face-less voice segments or silent video clips of the target
identity.

A common approach in this verification task is to map inputs from different
modalities into a shared space to achieve cross-modal retrieval (Sari et al.,
2021). For instance, (Nagrani et al., 2018d) trained a cross-modal model with
a supervised identity-matching task using a fusion embedding between facial
and voice encoders. Similarly, (Tao et al., 2020) proposed optimizing the
cross-modal architecture to increase the cosine similarity between facial and
voice embeddings within the same identity system. Without requiring identity
labels, (Nagrani et al., 2018a) introduced a cross-modal self-supervised
learning scheme using unlabeled talking face videos. Another crucial aspect of
cross-modal identity verification is linguistic dependence, as demonstrated by
(Saeed et al., 2020), which showed that changing the language in audio input
results in a performance drop.

Figure 2. Multi-stage Face-voice Association Learning with Keynote Speaker
Diarization (MFV-KSD) framework. It comprises a keynote speaker diarization
frontend, face encoder, speaker encoder, and a fusion module. The training
stage consists of three stages: intra-modal recognition training, inter-modal
correlation training, and FAME adaption.

##  3\. Proposed Method: MFV-KSD

In this section, we propose the Multi-stage Face-voice Association Learning
with Keynote Speaker Diarization (MFV-KSD) framework (Figure 2) to solve the
cross-modal speaker verification task.

###  3.1. Overall

MFV-KSD consists of a keynote speaker frontend, face encoder, speaker encoder
and a fusion module. The input is one face image and one speech segment, the
output is the single score to represent if they come from the same person or
not. As shown in the left side of Figure 2, for the given audio and visual
inputs, the keynote speaker diarization frontend is employed to detect the
speech segments of the target speaker. Here, target speaker is the individual
who speaks the most, i.e., keynote speaker, within the speech segment. Then
the face image and speech segments of the target speaker are fed into the face
encoder and speaker encoder, respectively, to obtain the face embedding and
speaker embedding. This speaker encoder can extract the speaker embedding to
capture the voice characteristics of keynote speaker. Similarly, the face
encoder can obtain a face embedding to integrate the facial appearance.

Afterwards, we utilize a cross-modal fusion module to learn the correlation
between audio and visual embeddings. The speaker and face embeddings are
mapped into a shared space by several Fully Connected (FC) layers, with
dropout layers incorporated to prevent over-fitting. We concatenate the multi-
modal embeddings directly due to the efficiency consideration.

###  3.2. Keynote speaker frontend

Ideally, each audio utterance should contain speech from the target speaker
only. However, in the real-world application, such as FAME 2024, the provided
raw audios are conversations, which include speech segments from multiple
speakers. That could affect the robustness of both training and prediction
processes. Therefore, it is crucial to cleanse the samples and achieve high-
quality inputs.

Traditional speaker diarization methods can hardly handle this condition due
to the imbalanced distribution of speakers in each audio utterance. Motivated
by our previous work in target speech diarization (Jiang et al., 2024a), we
propose a keynote speaker diarization system by integrating a keynote speaker
query vector, which is a learnable vector. We adopt the pre-trained keynote
speaker diarization framework as frontend and use it to search the active
segments of the target speaker. This frontend is pre-trained on the simulated
conversation datasets and it is frozen in the MFV-KSD system .

As shown in the right-upper side of Figure 2, this keynote speaker frontend
takes speech segment and trained keynote speaker query vector as inputs and
outputs a frame-level probability sequence, which represents the speaking
activities of the keynote speaker in each frame. The transformer encoder
inputs the speech feature extracted from speech encoder. The transformer
decoder takes prompt vector and transformer encoder speech representation.
Finally, we perform a dot product operation between the decoder output and the
encoder output and apply a sigmoid operation to get the prediction sequence.

This keynote speaker frontend can filter the dataset effectively and minimize
interference signals from the voices of non-relevant speakers, archiving more
accurate model training and evaluation.

###  3.3. Three-stage training

We propose a three-stage training pipeline for MFV-KSD to boost the system
with the cross-modal learning ability. The initial intra-modal recognition
training step aims to accurately capture biometric features from the single
modality, i.e., facial or voice characteristics. Subsequently, inter-modal
correlation training establishes the alignment between the facial appearance
and voice characteristics from the large-scale dataset. Finally, FAME
adaptation is applied to meet the in-domain evaluation. Note that our
experiments have strictly met the unseen-language requirements.

####  3.3.1. Intra-modal recognition training

In the first training stage, our objective is to learn face representation and
speaker representation for recognition purposes on the large-scale single-
modal dataset.

For the speech modality input, we employ the ECAPA-TDNN (Emphasized Channel
Attention, Propagation and Aggregation in Time Delay Neural Network) model
(Desplanques et al., 2020) as the speaker encoder due to its reliable
performance in identifying speakers. For the variable lengths of input
utterances, the speaker encoder can learn the speaker embedding with the fixed
dimension. In this training stage, the speaker embedding is fed into the
classification layer to perform speaker recognition task with the AAM-softmax
loss (Deng et al., 2019a). The learnt embedding can represent the voice
characteristic of the input speech.

For the face modality, we utilize the ResNet-18 model (Deng et al., 2019a) as
the face encoder to effectively extract detailed facial features and recognize
the target speaker. The face encoder is also trained with a classification
task in this stage. It can output face embeddings in MFV-KSD to capture robust
facial appearance features.

####  3.3.2. Inter-modal correlation training

Building on the pre-trained speaker and face encoders from the intra-modal
recognition stage, we perform the inter-modal correlation training to
establish a robust and general alignment between face and voice signals. This
step links the facial features and voice embeddings by learning the large-
scale samples in the wild. A fusion module is introduced to correlate the
multi-modal biometric inputs.

In this stage, each training data sample contains a facial image and
corresponding reference speech from the same individual. They can offer
biometric information from diverse perspectives. During training, the face
component for half of the samples is replaced from the face of the different
person to generate the negative pairs. Based on that, the pre-trained face
encoder, speaker encoder, and fusion module can be trained with the cross-
entropy loss for binary classification. This stage involves as many pairs as
possible from the large audio-visual in-the-wild dataset that can guide the
MFV-KSD to learn the general association.

####  3.3.3. FAME adaptation

Considering the domain mismatch between the pre-trained dataset and the FAME
dataset, we further fine-tune the entire system on specific variants of the
FAME dataset. This adaptation stage integrates our previously pre-trained
framework into the evaluation domain, significantly enhancing the cross-modal
speaker verification performance on the FAME dataset. A very small learning
rate is used in this process.

##  4\. Experimental setup

###  4.1. FAME

The 2024 Face-Voice Association in Multilingual Environments (FAME) challenge
focuses on learning face-voice associations (Saeed et al., 2024). The goal of
FAME is to verify whether a given sample, containing both a face and a voice,
belongs to the same identity. This challenge specifically examines the impact
of multiple languages. The trained model is evaluated on both seen and unseen
languages to assess its performance in cross-language environments.

Table 1. Summary of datasets used for implementation Stage |  Dataset |  Modality
---|---|---
Intra-modal recognition |  VoxCeleb2 |  Speech
|  Glint360K |  Face
Inter-modal correlation |  VoxCeleb1 |  Speech and face
|  MAV-Celeb |
FAME adaption |  MAV-Celeb |  Speech and face

###  4.2. Dataset

As summarized in Table 1, we use large-scale VoxCeleb2 (Chung et al., 2018)
and Glint360K (Deng et al., 2019b) datasets for intra-modal recognition
training. Inter-modal correlation training applies the VoxCeleb1 (Nagrani et
al., 2017) and MAV-Celeb datasets, both of which are audio-visual. MAV-Celeb
is the dataset that used in FAME, it features 154 celebrities speaking three
languages: English, Hindi, and Urdu. In the FAME adaptation stage, only MAV-
Celeb dataset is used.

To meet the unseen language requirements, we cleanse VoxCeleb1 and VoxCeleb2
by VoxLingua107 (Valk and Alumäe, 2021) model, which provide language labels.
For instance, for English-unseen, all three training stages are performed on
VoxCeleb1, VoxCeleb2 or MAV-Celeb datasets without the English language to
meet the requirements. To monitor the performance, we split 6 speakers from
the training set to generate the validation dataset.

##  5\. Results

###  5.1. Performance in the FAME challenge

In FAME 2024, our best results on the test set can be found in Table 2. MFV-
KSD has outperformed the given baseline in FAME (Saeed et al., 2024) and
achieved 19.9% overall EER. Meanwhile, the heard results perform better than
the unheard condition. That proves the importance of language in this task.
Also, we found that the results for Urdu were much better than those for
English.

Table 2. Evaluation results (EER (%)) on MAV-Celeb-V1 test Method | E-seen | E-unseen | U-seen | U-unseen | Overall
---|---|---|---|---|---
Baseline (Saeed et al., 2022) | 29.3 | 37.9 | 25.8 | 40.4 | 33.4
MFV-KSD | 21.8 | 27.3 | 14.7 | 15.8 | 19.9
Table 3. Ablation study on the MAV-Celeb-V1 set for keynote speaker frontend (without inter-modal correlation training) Key-train | Key-test | English | Urdu
---|---|---|---
$\checkmark$ | $\checkmark$ | 28.0 | 15.1
$\times$ | $\times$ | 29.9 | 17.0
$\checkmark$ | $\times$ | 28.8 | 14.7
$\times$ | $\checkmark$ | 30.3 | 17.4
Table 4. Ablation study on the MAV-Celeb-V1 set for three-stage training (with keynote speaker frontend) Intra-modal | Inter-modal | FAME | English | Urdu
---|---|---|---|---
recognition | correlation | adaption
$\checkmark$ | $\checkmark$ | $\checkmark$ | 21.8 | 14.7
$\checkmark$ | $\times$ | $\checkmark$ | 28.0 | 15.1
$\checkmark$ | $\checkmark$ | $\times$ | 30.6 | 16.1

###  5.2. Ablation study

For post-evaluation, we conduct the ablation study on the MAV-Celeb-V1 test
set (seen condition only).

####  5.2.1. Keynote speaker frontend

In Table 3. ‘Key-train’ and ‘Key-test’ denote if we use this Keynote speaker
frontend during training and testing, respectively. The results verify that
this frontend can enhance the dataset and lead to clear improvement when it
performs on the training set and test set together.

####  5.2.2. Multi-stage training

Then we study the effectiveness of our proposed three-stage training strategy
in Table 4. Intra-modal correlation stage involves the information from the
large-scale datasets to boost the generality of the system. FAME adaptation
stage leads the model to understand the in-domain data further. Both of them
are important for the entire framework.

Table 5. Ablation study on the MAV-Celeb-V1 set for model architecture (without inter-modal correlation training) Speaker encoder | Face encoder | English | Urdu
---|---|---|---
ECAPA-TDNN | ResNet18 | 28.0 | 15.1
ECAPA-TDNN | ResNet34 | 29.1 | 17.0
ECAPA-TDNN | ResNet50 | 27.8 | 16.4
ECAPA-TDNN | ResNet18 | 28.0 | 15.1
CAM++ | ResNet18 | 35.5 | 19.6
ResNet34 | ResNet18 | 34.0 | 19.2
Table 6. Ablation study on the learning rate issue (without inter-modal correlation training) Learning-rate | 1e-5 | 1e-4 | 1e-3
---|---|---|---
English | 28.0 | 29.8 | 33.8
Urdu | 15.1 | 19.5 | 24.0

####  5.2.3. Model architecture

We further study the different encoder architectures in Table 5. For the face
modality, a larger face encoder cannot boost the system performance in our
experiments. For the speech modality, ECAPA-TDNN performs better than CAM++
(Wang et al., 2023) and ResNet34 (He et al., 2016).

####  5.2.4. Learning rate

At last, we explore the importance of a small learning rate in Table 6. Since
multi-stage training makes the system learn valuable knowledge, a very small
learning rate can significantly boost the system.

###  5.3. Quality study

We further perform the quality study on our generated validation set to
understand how MFV-KSD works in this task.

####  5.3.1. Confusion matrix

First, we make the confusion matrix on the validation set, which contains 6
people in Figure 3. We find that MFV-KSD performs well for the cross-gender
condition. However, the same gender people with similar age and nationality
are very hard to distinguish (id63 vs id65).

Figure 3. Confusion matrix in the generated validation set. Table 7. Quality study of MFV-KSD for the gender effect Validation set | Normal | Easy | Hard
---|---|---|---
English | 23.9 | 5.4 | 42.5
Urdu | 24.3 | 3.3 | 38.2

####  5.3.2. Gender effect

In Table 7, we set three validation sets: ‘Normal’ has no restriction, ‘Easy’
and ‘Hard’ means all negative pairs are cross-gender or same-gender,
respectively. System can do very well for the easy conditions but hard to work
for these hard conditions. Figure 4 conducts the T-SNE visualization for the
‘Normal’ validation set based on the extracted multi-modal feature, pairs with
the same person (blue dots) have the clear boundary with pairs with different
persons, different gender (green dots). However, pairs with different persons,
the same gender (red dots) are mixed with these blue dots. It is still
challenging to handle the same-gender condition.

Figure 4. T-SNE visualization to study the gender effect.

##  6\. Conclusion

This paper proposes a robust cross-modal speaker verification system to handle
real-world challenging conditions. In future work, a standard benchmark is
required to build, which contains the large-scale, fixed training set,
evaluation set with enough samples, different tracks based on gender, age and
nationality, and diverse tasks (verification, matching and retrieval). That
can assist the community in comparing the various methods fairly.

##  7\. Acknowledgement

The research is supported by: National Natural Science Foundation of China,
Grant No. 62271432; Shenzhen Science and Technology Program
ZDSYS20230626091302006; Shenzhen Science and Technology Research Fund,
Fundamental Research Key Project Grant No. JCYJ20220818103001002; FD-fAbrICS,
Joint Lab for FD-SOI Always-on Intelligent & Connected Systems, Award
I2001E0053. Any opinions expressed in this material do not reflect the views
of the institutions supporting the joint lab.

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# Quantum Clocks and Gravitational Time Dilation and the Quantum Interference

Takeshi Chiba  Department of Physics, College of Humanities and Sciences,
Nihon University,
Tokyo 156-8550, Japan  Shunichiro Kinoshita  Department of Physics, College of
Humanities and Sciences, Nihon University,
Tokyo 156-8550, Japan  Department of Physics, Chuo University, Kasuga, Bunkyo-
ku, Tokyo 112-8551, Japan

###### Abstract

Recently, a proper time observable for a quantum clock is introduced and it is
found that the proper time read by one clock conditioned on another clock
reading a different proper time obeys classical time dilation in accordance
with special relativistic kinematical time dilation. Here, we extend this
proposal to a weak gravitational field in order to investigate whether the
weak equivalence principle holds for quantum matter. We find that for general
quantum states the quantum time dilation in a weak gravitational field obeys a
similar gravitational time dilation found in classical relativity. However,
unlike the special relativistic case, the time dilation involves the external
time (a background coordinate time at the observer on the earth) as well as
the proper times of two clocks. We also investigate a quantum time dilation
effect induced by a clock in a superposition of wave packets localized in
momentum space or in position space and propose the setup to observe the
gravitational effect in the quantum interference effect in the time dilation.

##  I Introduction

The equivalence principle played the fundamental role in constructing general
relativity (GR) by Einstein. The weak equivalence principle (WEP) states that
the motion of a (uncharged) test body is independent of its internal structure
and composition Will (2018). With the WEP together with the local Lorentz
invariance (independence of the results of nongravitational experiments from
the velocity of the local Lorentz frame) and the local position invariance
(the independence of the experimental results from the spacetime position),
the metric couples universally to matter. However, this “universal coupling to
matter” is the root of the problem of the cosmological constant, the validity
of the WEP should be examined further, especially in the quantum regime. Tests
of weak equivalence principle have been performed by measuring relative
acceleration between different atoms using atom interferometers Peters  _et
al._ (1999); Fray  _et al._ (2004); Bonnin  _et al._ (2013); Schlippert  _et
al._ (2014); Albers  _et al._ (2020); Asenbaum  _et al._ (2020); Barrett  _et
al._ (2022).

There are several attempts to formulate the equivalence principle in quantum
regime Zych and Brukner (2018); Anastopoulos and Hu (2018); Hardy (2019). In
addition to the position of a particle which is measured by interferometric
experiments, an important observable quantity regarding the equivalence
principle is the proper time of a clock because the gravitational time
dilation is the direct consequence of the equivalence principle Will (2018).
Measurements of the gravitational time dilation by comparisons of atomic
clocks located at different heights have been performed Chou  _et al._ (2010);
Takamoto  _et al._ (2020); Bothwell  _et al._ (2022); Zheng  _et al._ (2022).

Recently, a proper time observable for a quantum clock is proposed in Smith
and Ahmadi (2020) and the probability that one clock reads a given proper time
conditioned on another clock reading a different proper time is derived.
Moreover, it is shown that when the external degrees of freedom of these clock
particles are described by Gaussian wave packets localized in momentum space,
the clocks observe classical time dilation in accordance with special
relativistic kinematical time dilation.

In this paper, we extend this proposal to a weak gravitational field and study
the time dilation of quantum particles under the Newtonian gravity in order to
investigate whether quantum particles move in the same manner as the classical
particles. Moreover, we are able to derive the quantum time dilation between
two clocks for general quantum states not limited to Gaussian wave packets.
Thanks to the existence of the conserved quantities associated with the
Galilean invariance and the translation invariance of the system, it can be
shown that the gravitational time dilation depends only on the expectation
values of the positions and the velocities of clocks at the initial time. This
may be regarded as the weak equivalence principle for quantum system
Anastopoulos and Hu (2018).

The paper is organized as follows. In Sec. II, we derive the quantum time
dilation formula in a weak gravitational field. In Sec. III, we calculate the
gravitational time dilation for a linear gravitational potential. We also
consider a quantum time dilation effect induced by a clock in a superposition
of wave packets. Sec. IV is devoted to summary. In Appendix A, the time
dilation for Gaussina wavepackets is given, and in Appendix B, the initial
expectation values for a superposition states are given.

##  II Quantum Clock Particles in Spacetime

###  II.1 Classical Particles

We consider a system of $N$ free particles. Each particle whose mass is
$m_{n}~{}(n=1,\dots,N)$ has a set of internal degrees of freedom, labeled by
the configuration variables $q_{n}$ and their conjugate momenta
$P_{q_{n}}$Smith and Ahmadi (2020). These internal degrees of freedom is
supposed to represent the quantum clock.

The action of such a system in a curved spacetime with the metric $g_{\mu\nu}$
is given by

$\displaystyle S=\sum_{n}\int
d\tau_{n}\left(-m_{n}c^{2}+P_{q_{n}}\frac{dq_{n}}{d\tau_{n}}-H_{n}^{\rm
clock}\right),$ (1)

where $\tau_{n}$ is the proper time of $n$th particle and $H_{n}^{\rm
clock}=H_{n}^{\rm clock}(q_{n},P_{q_{n}})$ is the Hamiltonian for its internal
degrees of freedom.

Let $x_{n}^{\mu}$ denote the spacetime position of the $n$th particle. The
trajectory of the $n$th particle $x_{n}^{\mu}(t)$ is parameterized by an
arbitrary external time parameter $t$. Noting that
$cd\tau_{n}=\sqrt{-g_{\mu\nu}\dot{x}_{n}^{\mu}\dot{x}_{n}^{\nu}}dt\equiv\sqrt{-\dot{x}^{2}_{n}}dt$,
where a dot denotes differentiation with respect to $t$, the action is
rewritten as

$\displaystyle S=\int
dt\sum_{n}\frac{1}{c}\sqrt{-\dot{x}_{n}^{2}}\left(-m_{n}c^{2}+P_{q_{n}}c\frac{\dot{q}_{n}}{\sqrt{-\dot{x}_{n}^{2}}}-H_{n}^{\rm
clock}\right)=:\int dt~{}L.$ (2)

The momentum conjugate to $x_{n}^{\mu}$ is given by

$\displaystyle P_{n\mu}=\frac{\partial
L}{\partial\dot{x}_{n}^{\mu}}=\frac{g_{\mu\nu}\dot{x}_{n}^{\nu}}{c\sqrt{-\dot{x}_{n}^{2}}}\left(m_{n}c^{2}+H_{n}^{\rm
clock}\right).$ (3)

Then the Hamiltonian associated with the Lagrangian $L$ is constrained to
vanish:

$\displaystyle
H=\sum_{n}\left(P_{n\mu}\dot{x}_{n}^{\mu}+P_{q_{n}}\dot{q}_{n}\right)-L\approx
0.$ (4)

In terms of the momentum, the constraints can be expressed in the form

$\displaystyle
C_{H_{n}}:=g^{\mu\nu}P_{n\mu}P_{n\nu}c^{2}+\left(m_{n}c^{2}+H_{n}^{\rm
clock}\right)^{2}\approx 0.$ (5)

Using the $(3+1)$ decomposition of the metric in terms of the lapse function
$\alpha$, the shift vector $\beta^{i}$ and the three-metric $\gamma_{ij}$ such
that Misner (2017)

$\displaystyle
ds^{2}=-\alpha^{2}c^{2}dt^{2}+\gamma_{ij}(dx^{i}+\beta^{i}cdt)(dx^{j}+\beta^{j}cdt),$
(6)

the constraint is factorized in the form

$\displaystyle
C_{H_{n}}=-\alpha^{-2}\left(P_{n0}-\beta^{i}P_{ni}\right)^{2}c^{2}+\gamma^{ij}P_{ni}P_{nj}c^{2}+\left(m_{n}c^{2}+H_{n}^{\rm
clock}\right)^{2}=-\alpha^{-2}C_{n}^{+}C_{n}^{-}\approx 0,$ (7)

where $C_{n}^{\pm}$ is defined by

$\displaystyle C_{n}^{\pm}$ $\displaystyle:=$
$\displaystyle\left(P_{n0}-\beta^{i}P_{ni}\right)c\pm h_{n},$ (8)
$\displaystyle h_{n}$ $\displaystyle:=$
$\displaystyle\alpha\sqrt{\gamma^{ij}P_{ni}P_{nj}c^{2}+(m_{n}c^{2}+H_{n}^{\rm
clock})^{2}}.$ (9)

Note that we have set $x^{0}=ct$. Hereafter we assume that the spacetime is
static and the shift vector is vanishing, $\beta^{i}=0$. The coordinates
$x^{\mu}_{n}$ and their conjugate momenta $P_{n\mu}$ satisfy the fundamental
Poisson brackets: $\\{x_{m}^{\mu},P_{n\nu}\\}=\delta_{mn}\delta^{\mu}_{\nu}$.
The canonical momentum $P_{n\mu}$ generates translations in the spacetime
coordinate $x_{n}^{\mu}$. Therefore, if $C_{n}^{\pm}\approx 0$ and the shift
vector vanishes, then $\pm h_{n}$ is the generator of translation in the $n$th
particle’s time coordinate and is the Hamiltonian for both the external and
internal degrees of freedom of the $n$th particle.

###  II.2 Quantization

We canonically quantize the system of $N$ particles by promoting the phase
space variables to operators acting on appropriate Hilbert spaces: $x^{0}_{n}$
and $P_{n0}$ become canonically conjugate self-adjoint operators acting on the
Hilbert space ${\cal H}^{0}_{n}\simeq L^{2}(\mathbb{R})$ associated with the
$n$th particle’s temporal degree of freedom; operators $x_{n}^{i}$ and
$P_{ni}$ acting on the Hilbert space ${\cal H}_{n}^{\rm ext}\simeq
L^{2}(\mathbb{R}^{3})$ associated with the particle’s external degrees of
freedom; operators $q_{n}$ and $P_{q_{n}}$ acting on the Hilbert space ${\cal
H}_{n}^{\rm clock}$ associated with the particle’s internal degrees of
freedom. Then the Hilbert space describing the $n$th particle is ${\cal
H}_{n}\simeq{\cal H}^{0}_{n}\otimes{\cal H}_{n}^{\rm ext}\otimes{\cal
H}_{n}^{\rm clock}$.

The constraint equations (7) now become operator equations restricting the
physical state of the theory,

$\displaystyle C_{n}^{+}C_{n}^{-}|\Psi\rangle\rangle=0,~{}~{}~{}~{}\forall n,$
(10)

where $|\Psi\rangle\rangle\in{\cal H}_{\rm phys}$ is a physical state of a
clock $C$ and a system $S$ and lives in the physical Hilbert space ${\cal
H}_{\rm phys}$.

To specify ${\cal H}_{\rm phys}$, following Page and Wooters Page and Wootters
(1983); Wootters (1984) (see also Giovannetti  _et al._ (2015); Castro-Ruiz
_et al._ (2020); Maccone and Sacha (2020)), the normalization of the physical
state in ${\cal H}_{\rm phys}$ is performed by projecting a physical state
$|\Psi\rangle\rangle$ onto a subspace in which the temporal degree of freedom
of each particle (clock $C$) is in an eigenstate $|t_{n}\rangle$ of the
operator $x_{n}^{0}$ associated with the eigenvalue $t\in\mathbb{R}$ in the
spectrum of $x_{n}^{0}$: $x_{n}^{0}|t_{n}\rangle=ct|t_{n}\rangle$. The state
of $S$ by conditioning $|\Psi\rangle\rangle$ on $C$ reading the time $t$ is
then given by

$\displaystyle|\psi_{S}(t)\rangle=\langle t|\otimes I_{S}|\Psi\rangle\rangle,$
(11)

where $|t\rangle=\otimes_{n}|t_{n}\rangle$ and $I_{S}$ is the identity on
${\cal H}\simeq\bigotimes_{n}{\cal H}_{n}^{\rm ext}\otimes{\cal H}_{n}^{\rm
clock}$. We demand that the state $|\psi_{S}(t)\rangle$ is normalized as
$\langle\psi_{S}(t)|\psi_{S}(t)\rangle=1$ for $\forall t\in\mathbb{R}$ on a
spacelike hypersurface defined by all $N$ particles’ temporal degree of
freedom being in the state $|t_{n}\rangle$. The physical state
$|\Psi\rangle\rangle$ is thus normalized with respect to the inner product
Smith and Ahmadi (2020):

$\displaystyle\langle\langle\Psi|\Psi\rangle\rangle_{PW}:=\langle\langle\Psi||t\rangle\langle
t|\otimes I_{S}|\Psi\rangle\rangle=\langle\psi_{S}(t)|\psi_{S}(t)\rangle=1,$
(12)

and the physical state $|\Psi\rangle\rangle$ can be written as

$\displaystyle|\Psi\rangle\rangle=\int dt|t\rangle\langle t|\otimes
I_{S}|\Psi\rangle\rangle=\int dt|t\rangle|\psi_{S}(t)\rangle.$ (13)

Hereafter, we consider physical states that satisfy

$\displaystyle
C_{n}^{+}|\Psi\rangle\rangle=(P_{n0}c+h_{n})|\Psi\rangle\rangle=0,~{}~{}~{}~{}~{}\forall
n,$ (14)

where $h_{n}$ is the operator corresponding to Eq. (9). This implies that the
conditioned state $|\psi_{S}(t)\rangle$ has positive energy as measured by
$h_{n}$. It can be shown that the conditioned state $|\psi_{S}(t)\rangle$
satisfies the Schrödinger equation with $t$ as a time parameter Smith and
Ahmadi (2020):

$\displaystyle
i\hbar\frac{d}{dt}|\psi_{S}(t)\rangle=H_{S}|\psi_{S}(t)\rangle,$ (15)

where $H_{S}=\sum_{n}h_{n}\otimes I_{S-n}$ with $I_{S-n}$ being the identity
on $\bigotimes_{m\neq n}{\cal H}^{\rm ext}_{m}\otimes{\cal H}^{\rm
clock}_{m}$. Therefore, $|\psi_{S}(t)\rangle$ can be regarded as the time-
dependent state of the $N$-particles with the Hamiltonian $H_{S}$ evolved with
the external time $t$.

###  II.3 Proper Time Observables

In Smith and Ahmadi (2020), a clock is defined to be the quadrupole $\\{{\cal
H}^{\rm clock},\rho_{C},H^{\rm clock},T_{C}\\}$, where $\rho_{C}$ is a
fiducial state and $T_{C}$ is proper time observable. The proper time
observable is defined as a POVM (positive operator valued measure)

$\displaystyle T_{C}:=\left\\{E_{C}(\tau)~{}\forall\tau\in G~{}{\rm
s.t.}\int_{G}d\tau E_{C}(\tau)=I_{C}\right\\},$ (16)

where $E_{C}(\tau)=|\tau\rangle\langle\tau|$ is a positive operator on ${\cal
H}^{\rm clock}$, $G$ is the group generated by $H^{\rm clock}$, and
$|\tau\rangle$ is a clock state associated with a measurement of the clock
yielding the time $\tau$. The clock state $|\tau\rangle$ evolves according to
$U_{C}(\tau)=e^{-iH^{\rm clock}\tau/\hbar}$ as

$\displaystyle|\tau+\tau^{\prime}\rangle=U_{C}(\tau^{\prime})|\tau\rangle.$
(17)

###  II.4 Probabilistic Time Dilation

Consider two clock particles $A$ and $B$ with internal degrees of freedom,
$\\{{\cal H}^{\rm clock}_{A},\rho_{A},H^{\rm clock}_{A},T_{A}\\}$ and
$\\{{\cal H}^{\rm clock}_{B},\rho_{B},H^{\rm clock}_{B},T_{B}\\}$. To probe
time dilation effects between two clocks, we consider the probability that
clock $A$ reads the proper time $\tau_{A}$ conditioned on clock $B$ reading
the proper time $\tau_{B}$ Page and Wootters (1983); Wootters (1984). This
conditional probability is given in terms of the physical state as

$\displaystyle{\rm
Prob}[T_{A}=\tau_{A}|T_{B}=\tau_{B}]=\frac{\langle\langle\Psi|E_{A}(\tau_{A})E_{B}(\tau_{B})|\Psi\rangle\rangle}{\langle\langle\Psi|E_{B}(\tau_{B})|\Psi\rangle\rangle}\,.$
(18)

Consider the case where two clock particles $A$ and $B$ are moving in a curved
spacetime. Suppose that initial conditioned state is unentangled,
$|\psi_{S}(0)\rangle=|\psi_{S_{A}}\rangle|\psi_{S_{B}}\rangle$, and that the
external and internal degrees of freedom of both particles are unentangled,
$|\psi_{S_{n}}\rangle=|\psi_{n}^{\rm ext}\rangle|\psi_{n}^{\rm clock}\rangle$.
Then, from Eq. (13), the physical state takes the form

$\displaystyle|\Psi\rangle\rangle=\int dt|t\rangle|\psi_{S}(t)\rangle=\int
dt\bigotimes_{n\in\\{A,B\\}}e^{-ih_{n}t/\hbar}|\psi_{n}^{\rm
ext}\rangle|\psi_{n}^{\rm clock}\rangle\,.$ (19)

Further suppose that ${\cal H}^{\rm clock}_{n}\simeq L^{2}(\mathbb{R})$ so
that we may consider an ideal clock such that $P_{n}=H^{\rm clock}_{n}/c$ and
$cT_{n}$ are the momentum and position operators on ${\cal H}^{\rm
clock}_{n}$. The canonical commutation relation yields
$[cT_{n},P_{n}]=[T_{n},H^{\rm clock}_{n}]=i\hbar$. Then, the clock states are
orthogonal $\langle\tau|\tau^{\prime}\rangle=\delta(\tau-\tau^{\prime})$ and
satisfy the covariance relation
$|\tau+\tau^{\prime}\rangle=U_{C}(\tau^{\prime})|\tau\rangle$. The conditional
probability Eq. (18) becomes

$\displaystyle{\rm Prob}[T_{A}=\tau_{A}|T_{B}=\tau_{B}]=\frac{\int dt~{}{\rm
tr}[E_{A}(\tau_{A})\rho_{A}(t)]{\rm tr}[E_{B}(\tau_{B})\rho_{B}(t)]}{\int
dt~{}{\rm tr}[E_{B}(\tau_{B})\rho_{B}(t)]},$ (20)

where $\rho_{n}(t)$ is the reduced state of the internal clock degrees of
freedom defined as Smith and Ahmadi (2020)

$\displaystyle\rho_{n}(t)={\rm tr}_{{\cal H}_{S}\backslash{\cal H}_{n}^{\rm
clock}}\left(e^{-iH_{S}t/\hbar}|\psi_{S_{n}}\rangle\langle\psi_{S_{n}}|e^{iH_{S}t/\hbar}\right)$
(21)

with the trace over the complement of the clock Hilbert space.

We assume that the fiducial states of the internal clock degrees of freedom
are the Gaussian wave packets centered at $\tau=0$ with width $\sigma$:

$\displaystyle|\psi_{n}^{\rm clock}\rangle=\frac{1}{\pi^{1/4}\sigma^{1/2}}\int
d\tau~{}e^{-\frac{\tau^{2}}{2\sigma^{2}}}|\tau\rangle\,.$ (22)

###  II.5 Gravitational Time Dilation

In order to investigate the effect of gravity on the quantum time dilation,
consider a Newtonian approximation of spacetime and adopt the metric,
$g_{00}=-\alpha^{2}=-(1+2\Phi({\bf x})/c^{2}),\gamma_{ij}=\delta_{ij}$, where
$\Phi({\bf x})$ is the Newtonian gravitational potential. Then the Hamiltonian
(9) is expanded according to the number of inverse power of $c^{2}$ as

$\displaystyle h_{n}$ $\displaystyle=$ $\displaystyle\alpha
m_{n}c^{2}\sqrt{\frac{\gamma^{ij}P_{ni}P_{nj}}{m_{n}^{2}c^{2}}+\left(1+\frac{H_{n}^{\rm
clock}}{m_{n}c^{2}}\right)^{2}}$ (23) $\displaystyle\simeq$ $\displaystyle
m_{n}c^{2}+H_{n}^{\rm clock}+H_{n}^{\rm ext}+H_{n}^{\rm int}+O(c^{-4}),$

where the rest-mass energy term $m_{n}c^{2}$ is a constant and can be
disregarded in $h_{n}$. The external Hamiltonian $H_{n}^{\rm ext}$ and the
interaction Hamiltonian $H_{n}^{\rm int}$ are given by

$\displaystyle H_{n}^{\rm ext}$ $\displaystyle:=$
$\displaystyle\frac{\delta^{ij}P_{ni}P_{nj}}{2m_{n}}+m_{n}\Phi_{n}\equiv\frac{{\bf
P}_{n}^{2}}{2m_{n}}+m_{n}\Phi_{n},$ (24) $\displaystyle H_{n}^{\rm int}$
$\displaystyle:=$ $\displaystyle-\frac{H_{n}^{\rm ext}H_{n}^{\rm
clock}}{m_{n}c^{2}}+2\frac{m_{n}\Phi_{n}(H_{n}^{\rm clock}+H_{n}^{\rm
ext})}{m_{n}c^{2}}-\frac{(H_{n}^{\rm
ext})^{2}}{2m_{n}c^{2}}-2\frac{(m_{n}\Phi_{n})^{2}}{m_{n}c^{2}}\,,$ (25)

where $\Phi_{n}:=\Phi({\bf x}_{n})$.

The reduced state of the internal clock is then given by

$\displaystyle\rho_{n}(t)$ $\displaystyle=$ $\displaystyle{\rm tr}_{{\cal
H}_{S}\backslash{\cal H}_{n}^{\rm
clock}}\left[e^{-iH_{S}t/\hbar}|\psi_{S_{n}}\rangle\langle\psi_{S_{n}}|e^{iH_{S}t/\hbar}\right]$
(26) $\displaystyle=$ $\displaystyle\bar{\rho}_{n}(t)-it~{}{\rm tr}_{\rm
ext}\left([H_{n}^{\rm int},\bar{\rho}_{n}^{\rm
ext}(t)\otimes\bar{\rho}_{n}(t)]+O((H_{n}^{\rm int}t)^{2})\right)$
$\displaystyle=$ $\displaystyle\bar{\rho}_{n}(t)+it\left(\frac{\langle
H_{n}^{\rm
ext}\rangle}{m_{n}c^{2}}-2\frac{\langle\Phi_{n}\rangle}{c^{2}}\right)[H_{n}^{\rm
clock},\bar{\rho}_{n}(t)]+O(c^{-4}),$

where $\bar{\rho}_{n}(t)=e^{-iH_{n}^{\rm clock}t/\hbar}\rho_{n}e^{iH_{n}^{\rm
clock}t/\hbar}$ and $\bar{\rho}_{n}^{\rm ext}(t)=e^{-iH_{n}^{\rm
ext}t/\hbar}\rho_{n}^{\rm ext}e^{iH_{n}^{\rm ext}t/\hbar}$. The conditional
probability Eq. (18) is evaluated to leading relativistic order as

$\displaystyle{\rm
Prob}[T_{A}=\tau_{A}|T_{B}=\tau_{B}]=\frac{e^{-\frac{(\tau_{A}-\tau_{B})^{2}}{2\sigma^{2}}}}{\sqrt{2\pi}\sigma}\left[1+\left(\frac{\langle
H_{A}^{\rm ext}\rangle}{2m_{A}c^{2}}-\frac{\langle H_{B}^{\rm
ext}\rangle}{2m_{B}c^{2}}-\frac{\langle\Phi_{A}\rangle}{c^{2}}+\frac{\langle\Phi_{B}\rangle}{c^{2}}\right)\left(1-\frac{\tau_{A}^{2}-\tau_{B}^{2}}{\sigma^{2}}\right)\right],$
(27)

where $\langle H_{n}^{\rm ext}\rangle=\langle\psi^{\rm ext}_{n}|H_{n}^{\rm
ext}|\psi_{n}^{\rm ext}\rangle$. Then the average proper time read by clock
$A$ conditioned on clock $B$ indicating the time $\tau_{B}$ is

$\displaystyle\langle T_{A}\rangle$ $\displaystyle=$ $\displaystyle\int
d\tau~{}{\rm Prob}[T_{A}=\tau|T_{B}=\tau_{B}]\tau$ (28) $\displaystyle=$
$\displaystyle\tau_{B}\left(1-\left(\frac{\langle H_{A}^{\rm
ext}\rangle}{m_{A}c^{2}}-2\frac{\langle\Phi_{A}\rangle}{c^{2}}\right)+\left(\frac{\langle
H_{B}^{\rm
ext}\rangle}{m_{B}c^{2}}-2\frac{\langle\Phi_{B}\rangle}{c^{2}}\right)\right)\,.$

Since the external Hamiltonian is defined by Eq. (24), $\langle T_{A}\rangle$
is written in a more suggestive form as

$\displaystyle\langle T_{A}\rangle=\tau_{B}\left(1-\left(\frac{\langle{\bf
P}_{A}^{2}\rangle}{2m_{A}^{2}c^{2}}-\frac{\langle\Phi_{A}\rangle}{c^{2}}\right)+\left(\frac{\langle{\bf
P}_{B}^{2}\rangle}{2m_{B}^{2}c^{2}}-\frac{\langle\Phi_{B}\rangle}{c^{2}}\right)\right)\,.$
(29)

This is one of the main results of this paper and represents the quantum
analog of time dilation formula in the Newtonian gravity.

##  III Gravitational Time Dilation of Quantum Clocks

To evaluate the time dilation, for simplicity, we consider a one-dimensional
problem in the vertical direction and assume that the gravitational potential
is approximated by

$\displaystyle\Phi=gx\,,$ (30)

where $g$ is the gravitational acceleration. $x$ is the vertical coordinate
from the surface of the earth so that the external time coordinate $t$
corresponds to the proper time of an observer at rest on the earth. Here it is
to be noted that unlike the case with the external Hamiltonian being ${\bf
P}_{n}^{2}/2m_{n}$ Smith and Ahmadi (2020), the average of $P_{n}^{2}$ and
$x_{n}$ do evolve in time since $P_{n}$ does not commute with $H^{\rm
ext}_{n}$, Eq. (24).

The external Hamiltonian Eq. (24), $H_{n}^{\rm
ext}=P_{n}^{2}/2m_{n}+m_{n}gx_{n}$, can be derived from the Lagrangian $L_{n}$

$\displaystyle L_{n}=\frac{1}{2}m_{n}\dot{x}_{n}^{2}-m_{n}gx_{n}.$ (31)

Since $L_{n}$ does not depend on the external time $t$ explicitly, $H_{n}^{\rm
ext}$ is a constant. Moreover, there are two additional conserved quantities
(Noether charges) associated with the invariance of $L_{n}$ (up to total
derivative). (i) Noether charge $Q_{G}$ associated with the Galilean
transformation, s.t. $x_{n}\rightarrow x_{n}+vt$,

$\displaystyle Q_{G}=tP_{n}-m_{n}x_{n}+\frac{1}{2}m_{n}gt^{2},$ (32)

and (ii) Noether charge $Q_{x}$ associated with the spatial translation, s.t.
$x_{n}\rightarrow x_{n}+a$

$\displaystyle Q_{x}=P_{n}+m_{n}gt.$ (33)

These three conserved quantities facilitate expressing the time dilation at
the external time $t$ for a general quantum state $|\psi_{n}^{\rm
ext}\rangle$. For example, since the expectation value of $H_{n}^{\rm ext}$ is
conserved, the expectation value evaluated at $t$, $\langle H^{\rm
ext}_{n}\rangle_{t}$, is the same as its initial value:

$\displaystyle\langle H^{\rm ext}_{n}\rangle_{t}=\frac{\langle
P_{n}^{2}\rangle_{t}}{2m_{n}}+m_{n}g\langle x_{n}\rangle_{t}=\langle H^{\rm
ext}_{n}\rangle_{0}=\frac{\langle P_{n}^{2}\rangle_{0}}{2m_{n}}+m_{n}g\langle
x_{n}\rangle_{0}.$ (34)

Similarly, we have

$\displaystyle\langle Q_{G}\rangle_{t}$ $\displaystyle=$ $\displaystyle
t\langle P_{n}\rangle_{t}-m_{n}\langle
x_{n}\rangle_{t}+\frac{1}{2}m_{n}gt^{2}=-m_{n}\langle x_{n}\rangle_{0},$ (35)
$\displaystyle\langle Q_{x}\rangle_{t}$ $\displaystyle=$ $\displaystyle\langle
P_{n}\rangle_{t}+m_{n}gt=\langle P_{n}\rangle_{0}.$ (36)

From these, we obtain

$\displaystyle\langle x_{n}\rangle_{t}$ $\displaystyle=$ $\displaystyle\langle
x_{n}\rangle_{0}+\frac{\langle P_{n}\rangle_{0}}{m_{n}}t-\frac{1}{2}gt^{2},$
(37) $\displaystyle\langle P_{n}\rangle_{t}$ $\displaystyle=$
$\displaystyle\langle P_{n}\rangle_{0}-m_{n}gt.$ (38)

Namely, the evolution of the average of the position is identical to the
evolution of a classical particle with the initial position $\langle
x_{n}\rangle_{0}$ and the initial velocity $\langle P_{n}\rangle_{0}/m_{n}$.
In this sense, the trajectory is independent of the mass of the clock
particle. This may be regarded as the weak equivalence principle for quantum
particles. Anastopoulos and Hu (2018) observed that the probability
distribution of position for a free-falling clock is the same as that of a
free clock with the shift of its mean which is independent of its mass.

Plugging Eq. (34) and Eq. (37) into Eq. (29), the observed average time
dilation between two clocks is given by

$\displaystyle\langle T_{A}\rangle$ $\displaystyle=$
$\displaystyle\tau_{B}\left(1-\left(\frac{\langle{\bf
P}_{A}^{2}\rangle}{2m_{A}^{2}c^{2}}-\frac{\langle\Phi_{A}\rangle}{c^{2}}\right)+\left(\frac{\langle{\bf
P}_{B}^{2}\rangle}{2m_{B}^{2}c^{2}}-\frac{\langle\Phi_{B}\rangle}{c^{2}}\right)\right)$
(39) $\displaystyle=$ $\displaystyle\tau_{B}\left(1-\left(\frac{\langle{\bf
P}_{A}^{2}\rangle_{0}}{2m_{A}^{2}c^{2}}-\frac{\langle{\bf
P}_{B}^{2}\rangle_{0}}{2m_{B}^{2}c^{2}}\right)+\frac{g}{c^{2}}\left(\langle
x_{A}\rangle_{0}+2\frac{\langle P_{A}\rangle_{0}}{m_{A}}t-\langle
x_{B}\rangle_{0}-2\frac{\langle P_{B}\rangle_{0}}{m_{B}}t\right)\right).$

Note that a $\frac{1}{2}gt^{2}$ term disappears because it is independent of
mass (the universality of free fall). It turns out that the differences of the
initial average position and velocity between the two clocks will contribute
to the gravitational effect on the time dilation. This is another main result
of this paper. This time dilation formula holds for a general quantum state,
while an evaluation of the time dilation for Gaussian wave packets is given in
Appendix A. The time dilation in general depends on the external time $t$ as
well as on the initial average of the position and the momentum squared of
each clock. Only if the average of the initial velocity for each clock is
equal, i.e., $\langle P_{A}\rangle_{0}/m_{A}=\langle P_{B}\rangle_{0}/m_{B}$,
the time dilation between the two clocks depends only on the initial average
position and is independent of the external time $t$, that is, the duration of
the experiment.

###  III.1 Time Dilation for a Clock in a Superposition

However, for a state in a superposition, the time dilation would involve a
term different from the classical time dilation. In order to investigate the
effect of the superposition of states, we consider two clocks $A$ and $B$ and
suppose that initially clock $A$ is prepared as a superposition of two
Gaussian wave packets with average momenta $\overline{p}_{A}$ and
$\overline{p}_{A}^{\prime}$ with spread $\Delta_{A}$ and with mean positions
$\overline{x}_{A}$ and $\overline{x}_{A}^{\prime}$ generalizing the situation
considered in Smith and Ahmadi (2020); Khandelwal  _et al._ (2020) :

$\displaystyle\langle P_{A}|\psi^{\rm
ext}_{A}\rangle=\frac{1}{N^{1/2}(\pi\Delta_{A}^{2})^{1/4}}\left(\cos\theta
e^{-i\frac{\overline{x}_{A}({P_{A}}-\overline{p}_{A})}{\hbar}}e^{-\frac{({P_{A}}-\overline{p}_{A})^{2}}{2\Delta_{A}^{2}}}+\sin\theta
e^{i\phi}e^{-i\frac{\overline{x}_{A}^{\prime}({P_{A}}-\overline{p}_{A}^{\prime})}{\hbar}}e^{-\frac{({P_{A}}-\overline{p}_{A}^{\prime})^{2}}{2\Delta_{A}^{2}}}\right)\,,$
(40)

where $\theta\in[0,\pi/2)$ and $\phi\in[0,\pi]$ is the relative phase, and

$\displaystyle N=1+\sin
2\theta\exp\left(-\frac{(\overline{p}_{A}-\overline{p}_{A}^{\prime})^{2}}{4\Delta_{A}^{2}}-\frac{(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{4\sigma_{xA}^{2}}\right)\cos\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)$
(41)

is a normalization factor and $\sigma_{xA}=\hbar/\Delta_{A}$. Further, clock
$B$ is prepared in a Gaussian wave packet with average momentum
$\overline{p}_{B}$ and spread $\Delta_{B}$ and with mean position
$\overline{x}_{B}$.

Then, the average time read by $A$ conditioned on $B$ is given by

$\displaystyle\langle T_{A}\rangle$ $\displaystyle=$
$\displaystyle\tau_{B}\Bigg{\\{}1-\frac{1}{Nc^{2}}\left[\left(\frac{(\overline{p}_{A}-m_{A}gt)^{2}}{2m_{A}^{2}}-g\left(\overline{x}_{A}+\frac{\overline{p}_{A}t}{m_{A}}\right)\right)\cos^{2}\theta+\left(\frac{(\overline{p}_{A}^{\prime}-m_{A}gt)^{2}}{2m_{A}^{2}}-g\left(\overline{x}_{A}^{\prime}+\frac{\overline{p}_{A}^{\prime}t}{m_{A}}\right)\right)\sin^{2}\theta\right.$
(42)
$\displaystyle\left.+\frac{\Delta_{A}^{2}}{4m_{A}^{2}}+\frac{1}{2}g^{2}t^{2}\right]-\frac{\sin
2\theta}{Nc^{2}}e^{-\frac{(\overline{p}_{A}-\overline{p}_{A}^{\prime})^{2}}{4\Delta_{A}^{2}}-\frac{(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{4\sigma_{xA}^{2}}}\Biggl{[}\Biggl{(}\frac{\left(\frac{\overline{p}_{A}+\overline{p}_{A}^{\prime}}{2}-m_{A}gt\right)^{2}}{2m_{A}^{2}}-\frac{\Delta_{A}^{2}(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{8m_{A}^{2}\sigma_{xA}^{2}}+\frac{\Delta^{2}_{A}}{4m_{A}^{2}}$
$\displaystyle-\frac{1}{2}g(\overline{x}_{A}+\overline{x}_{A}^{\prime})-\frac{gt}{2m_{A}}(\overline{p}_{A}+\overline{p}_{A}^{\prime})+\frac{1}{2}g^{2}t^{2}\Biggr{)}\cos\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)$
$\displaystyle-\left(\frac{\hbar(\overline{x}_{A}-\overline{x}_{A}^{\prime})}{2m_{A}^{2}\sigma_{xA}^{2}}\left(\frac{\overline{p}_{A}+\overline{p}_{A}^{\prime}}{2}-m_{A}gt\right)-g\frac{\hbar
t(\overline{x}_{A}-\overline{x}_{A}^{\prime})}{2m_{A}\sigma_{xA}^{2}}+g\frac{\hbar(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\Delta_{A}^{2}}\right)\sin\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)\Biggr{]}$
$\displaystyle+\frac{(\overline{p}_{B}-m_{B}gt)^{2}}{2m_{B}^{2}c^{2}}+\frac{\Delta_{B}^{2}}{4m_{B}^{2}c^{2}}-\frac{g}{c^{2}}\left(-\frac{1}{2}gt^{2}+\overline{x}_{B}+\frac{\overline{p}_{B}t}{m_{B}}\right)\Bigg{\\}}\,.$

Note that for the state given by (40) each initial expectation values,
$\langle P_{A}\rangle_{0}$, $\langle P_{A}^{2}\rangle_{0}$, and $\langle
x_{A}\rangle_{0}$, are evaluated in Appendix B. It coincides with the
expression given in Smith and Ahmadi (2020) when putting $g=0$ and
$\overline{x}_{A}=\overline{x}_{A}^{\prime}=\overline{x}_{B}=0$. It also
reproduces the result by Khandelwal  _et al._ (2020) when putting $\phi=0$ and
$\overline{p}_{A}=\overline{p}_{A}^{\prime}$. The terms proportional to $\sin
2\theta$ represent genuinely quantum time dilation effect.

To make the effect of quantum time dilation manifest, we divide the time
dilation formula Eq. (42) into the classical dilation part $K_{C}$ and the
quantum dilation part $K_{Q}$ as $\langle
T_{A}\rangle=\tau_{B}(1-K_{C}-K_{Q})$. $K_{C}$ is given by the contribution of
statistical mixture of the wave packets with momentum $\overline{p}_{A}$ and
the position $\overline{x}_{A}$ with weight $\cos^{2}\theta$ and
$\overline{p}_{A}^{\prime}$ and $\overline{x}_{A}^{\prime}$ with weight
$\sin^{2}\theta$:

$\displaystyle K_{C}$ $\displaystyle=$
$\displaystyle\frac{(\overline{p}_{A}-m_{A}gt)^{2}}{2m_{A}^{2}c^{2}}\cos^{2}\theta+\frac{(\overline{p}_{A}^{\prime}-m_{A}gt)^{2}}{2m_{A}^{2}c^{2}}\sin^{2}\theta-\frac{(\overline{p}_{B}-m_{B}gt)^{2}}{2m_{B}^{2}c^{2}}+\frac{\Delta_{A}^{2}}{4m_{A}^{2}c^{2}}-\frac{\Delta_{B}^{2}}{4m_{B}^{2}c^{2}}$
(43)
$\displaystyle-\frac{g}{c^{2}}\left(\left(\overline{x}_{A}+\frac{\overline{p}_{A}t}{m_{A}}\right)\cos^{2}\theta+\left(\overline{x}_{A}^{\prime}+\frac{\overline{p}_{A}^{\prime}t}{m_{A}}\right)\sin^{2}\theta-\overline{x}_{B}-\frac{\overline{p}_{B}t}{m_{B}c^{2}}\right)$

and $K_{Q}$ is the rest of the time dilation and is given by

$\displaystyle K_{Q}=\frac{\sin
2\theta}{8m_{A}^{2}c^{2}N}e^{-\frac{(\overline{p}_{A}-\overline{p}_{A}^{\prime})^{2}}{4\Delta_{A}^{2}}-\frac{(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{4\sigma_{xA}^{2}}}\Biggl{\\{}\Biggl{[}2\left(\overline{p}_{A}^{\prime
2}-\overline{p}_{A}^{2}+4m_{A}gt(\overline{p}_{A}-\overline{p}_{A}^{\prime})+2gm_{A}^{2}(\overline{x}_{A}-\overline{x}_{A}^{\prime})\right)\cos
2\theta$
$\displaystyle-(\overline{p}_{A}-\overline{p}_{A}^{\prime})^{2}-\frac{\Delta_{A}^{2}}{\sigma_{xA}^{2}}(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}\Biggr{]}\cos\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)$
$\displaystyle-\left[\frac{2\hbar}{\sigma_{xA}^{2}}(\overline{p}_{A}+\overline{p}_{A}^{\prime}-4m_{A}gt)(\overline{x}_{A}-\overline{x}_{A}^{\prime})+\frac{4\hbar
m_{A}^{2}g}{\Delta_{A}^{2}}(\overline{p}_{A}-\overline{p}_{A}^{\prime})\right]\sin\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)\Biggr{\\}}.$
(44)

Positive $K_{Q}$ implies the enhanced time dilation. We evaluate $K_{Q}$ for
two cases: (1) superposition in momentum space:
$\overline{x}_{A}=\overline{x}_{A}^{\prime}=0$ and
$\overline{p}_{A}\neq\overline{p}_{A}^{\prime}$, and (2) superposition in
position space : $\overline{x}_{A}\neq\overline{x}_{A}^{\prime}$ and
$\overline{p}_{A}=\overline{p}_{A}^{\prime}$.

####  III.1.1  Momentum Superposition

For the superposition in momentum space, $K_{Q}$ is given by

$\begin{split}K_{Q}=\frac{\sin
2\theta}{8m_{A}^{2}c^{2}N}e^{-\frac{(\overline{p}_{A}-\overline{p}_{A}^{\prime})^{2}}{4\Delta_{A}^{2}}}\biggl{\\{}\left[2(\overline{p}_{A}^{\prime
2}-\overline{p}_{A}^{2}+4m_{A}gt(\overline{p}_{A}-\overline{p}_{A}^{\prime}))\cos
2\theta-(\overline{p}_{A}-\overline{p}_{A}^{\prime})^{2}\right]\cos\phi\\\
-\frac{4\hbar
m_{A}^{2}g}{\Delta_{A}^{2}}(\overline{p}_{A}-\overline{p}_{A}^{\prime})\sin\phi\biggr{\\}}.\end{split}$
(45)

The coefficients of $\cos\phi$ and $\sin\phi$ involve the effect of $g$. The
coefficient of $\cos\phi$ depends linearly on $t$, while the coefficient of
$\sin\phi$ depends only on $g$ and is independent of the external time $t$.

However, the coefficient of $\sin\phi$ is negligibly small for an atomic
particle. For an atomic particle, $\sigma_{xA}\sim 10^{-10}{\rm m}$ and
$m_{A}\sim 10^{-25}{\rm kg}$, and
$\Delta_{A}/m_{A}c=\hbar/(m_{A}c\sigma_{xA})\sim 10^{-8}$. Therefore, $\hbar
g(\overline{p}_{A}-\overline{p}_{A}^{\prime})/\Delta_{A}^{2}c^{2}\sim(\hbar/\Delta_{A})(g/c^{2})\sim
g\sigma_{xA}/c^{2}\sim 10^{-26}$. On the other hand,
$((\overline{p}_{A}-\overline{p}_{A}^{\prime})/m_{A}c)^{2}\sim(\Delta_{A}/m_{A}c)^{2}\sim
10^{-16}$.

To compute $K_{Q}$ for the momentum superposition, as in Smith and Ahmadi
(2020), consider two clock particles to be $~{}^{87}{\rm Rb}$ atoms with a
mass of $m=1.4\times 10^{-25}{\rm kg}$ and atomic radius of
$\sigma_{xA}=\hbar/\Delta_{A}=2.5\times 10^{-10}{\rm m}$. Suppose these clock
particles are moving at average velocities of
$\overline{v}_{A}^{\prime}=\overline{p}_{A}^{\prime}/m_{A}=15~{}{\rm m/s}$ and
$\overline{v}_{A}=\overline{p}_{A}/m_{A}=5~{}{\rm m/s}$. We further assume
$t=1~{}{\rm s}$. The strength of the quantum time dilation effect $K_{Q}$ for
several $\phi$ as a function of $\theta$ is shown in Fig. 1.

$K_{Q}$ with gravity has generically opposite sign to $K_{Q}$ without gravity.
The magnitude of $K_{Q}$ with gravity for $\theta<\pi/4$ is larger than
$K_{Q}$ without gravity. The time dilation effect can be measured with present
technology, although the effect for $\phi=\pi/2$ is unmeasurably small. For
$\theta=\pi/8$, $K_{Q}$ ($K_{Q}$ without $g$) is $-2.2\times 10^{-17}$
($1.1\times 10^{-17}$), $-1.6\times 10^{-17}$ ($7.7\times 10^{-18}$),
$2.0\times 10^{-27}$ ($0$), $1.7\times 10^{-17}$ ($-8.1\times 10^{-18}$),
$2.4\times 10^{-17}$ ($-1.2\times 10^{-17}$) for
$\phi=0,\pi/4,\pi/2,3\pi/4,\pi$, respectively.

Figure 1:  $K_{Q}$ for momentum superposition in a gravitational background
(solid curves) and in a non-gravitational background (dashed curves). Curves
are for $\phi=0$ (blue), $\phi=\pi/4$ (orange), $\phi=\pi/2$ (black),
$\phi=3\pi/4$ (green) and $\phi=\pi$ (red). For $\phi=\pi/2$, $K_{Q}$ is of
order $O(10^{-27})$.

####  III.1.2 Spatial Superposition

For the superposition in position space, $K_{Q}$ is given by

$\begin{split}K_{Q}=\frac{\sin
2\theta}{8m_{A}^{2}c^{2}N}e^{-\frac{(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{4\sigma_{xA}^{2}}}\left\\{\left[4m_{A}^{2}g(\overline{x}_{A}-\overline{x}_{A}^{\prime})\cos
2\theta-\frac{\Delta_{A}^{2}}{\sigma_{xA}^{2}}(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}\right]\cos\phi\right.\\\
\left.\quad-\frac{4\hbar(\overline{p}_{A}-2m_{A}gt)}{\sigma_{xA}^{2}}(\overline{x}_{A}-\overline{x}_{A}^{\prime})\sin\phi\right\\}.\end{split}$
(46)

Due to the exponential suppression factor
$\exp(-(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}/4\sigma_{xA}^{2})$ in
$K_{Q}$, the quantum interference effect is possible only when two wave
packets overlap. The coefficients of $\cos\phi$ and $\sin\phi$ involve the
effect of $g$, and this time the coefficient of $\cos\phi$ is independent of
$t$, while the coefficient of $\sin\phi$ depends linearly on $t$. However, the
effect of $g$ in the coefficient of $\cos\phi$ is negligibly small for an
atomic particle. The first term in $K_{Q}$ is of order
$g(\overline{x}_{A}-\overline{x}_{A}^{\prime})/c^{2}\sim
g\sigma_{xA}/c^{2}\sim 10^{-26}$, while
$(\Delta_{A}/m_{A}c)^{2}((\overline{x}_{A}-\overline{x}_{A}^{\prime})/\sigma_{xA})^{2}\sim(\Delta_{A}/m_{A}c)^{2}\sim
10^{-16}$.

$K_{Q}$ for the position superposition is shown in Fig. 2. Here we assumed
$\overline{v}_{A}=\overline{p}_{A}/m_{A}=10~{}{\rm
m/s},\overline{x}_{A}-\overline{x}_{A}^{\prime}=2\sigma_{xA}$ and $t=1{\rm
s}$. For $\phi=0,\pi$, solid curves ($g\neq 0$) coincide with dotted curves
($g=0$) because the effect of $g\neq 0$ is negligible. For $\theta=\pi/4$,
$K_{Q}$ ($K_{Q}$ without $g$) is $-1.3\times 10^{-17}$ ($-1.3\times
10^{-17}$), $5.6\times 10^{-17}$ ($-7.9\times 10^{-17}$), $1.2\times 10^{-16}$
($-1.2\times 10^{-16}$), $1.3\times 10^{-16}$ ($-1.0\times 10^{-16}$),
$2.9\times 10^{-17}$ ($2.9\times 10^{-17}$) for
$\phi=0,\pi/4,\pi/2,3\pi/4,\pi$, respectively.

It is seen that the sign of $K_{Q}$ with gravity is generically positive and
is opposite to $K_{Q}$ without gravity for $0<\phi<\pi$, which implies that
the quantum interference effect with gravity facilitates the time dilation of
clocks.

Figure 2:  $K_{Q}$ for spatial superposition in a gravitational background
(solid curves) and in a non-gravitational background (dashed curves). Curves
are for $\phi=0$ (blue), $\phi=\pi/4$ (orange), $\phi=\pi/2$ (black),
$\phi=3\pi/4$ (green) and $\phi=\pi$ (red). For $\phi=0,\pi$, the effect of
$g\neq 0$ is negligible and two curves overlap.

####  III.1.3 The Effect of Gravity in Quantum Interference

We have seen that the effect of nonzero $g$ in the quantum time dilation can
be potentially observed for clocks prepared in a superposition either in
momentum space or in position space. In both cases, the measurable
contributions originate from the initial expectation value of momentum,
$\langle P_{A}\rangle_{0}$, in Eq. (39). Moreover, we can maximize the effect
of $g$ on $K_{Q}$ by minimizing the effect of $g$ on $K_{C}$. This is achieved
by arranging $\overline{p}_{A}$ and $\overline{p}_{A}^{\prime}$ so that the
center of mass velocity of clock A coincides with the velocity of clock B:

$\displaystyle\frac{\overline{p}_{A}\cos^{2}\theta+\overline{p}_{A}^{\prime}\sin^{2}\theta}{m_{A}}=\frac{\overline{p}_{B}}{m_{B}}.$
(47)

Then the time-dependent terms proportional to $gt$ in Eq. (43) disappear and
$K_{C}$ becomes independent of time:

$\displaystyle
K_{C}=\frac{\overline{p}_{A}^{2}}{2m_{A}^{2}c^{2}}\cos^{2}\theta+\frac{\overline{p}_{A}^{\prime
2}}{2m_{A}^{2}c^{2}}\sin^{2}\theta-\frac{\overline{p}_{B}^{2}}{2m_{B}^{2}c^{2}}+\frac{\Delta_{A}^{2}}{4m_{A}^{2}c^{2}}-\frac{\Delta_{B}^{2}}{4m_{B}^{2}c^{2}}-\frac{g}{c^{2}}\left(\overline{x}_{A}\cos^{2}\theta+\overline{x}_{A}^{\prime}\sin^{2}\theta-\overline{x}_{B}\right),$
(48)

while $K_{Q}$ involves a term proportional to $gt$. Therefore, the effect of
$g$ in $K_{Q}$ would manifest as the time dependence of $K_{Q}$. In principle,
the longer the duration of the experiment is, the more the gravitational
effect is enhanced.

##  IV Summary

In this paper, extending the proper time observable proposed by Smith and
Ahmadi (2020) to a weak gravitational field, we derived a formula of the
average proper time read by one clock conditioned on another clock reading a
different proper time [Eq. (29)]. The time dilation measured by these quantum
clocks has the same form as that in classical relativity.

By considering a linear gravitational potential, we found for a general
quantum state that the evolution of the average of the position of a quantum
clock is identical to the evolution of a classical particle falling under the
gravitational force with the initial position $\langle x_{n}\rangle_{0}$ and
the initial velocity $\langle P_{n}\rangle_{0}/m_{n}$, which may be regarded
as the quantum analog of the weak equivalence principle. We also derived the
time dilation between clocks at the external time $t$ Eq. (39), which holds
not only for Gaussian wave packets but also for general quantum states.

We then considered the case in which the state of one clock is in a
superposition of Gaussian wave packets. We found that the effect arising from
quantum interference appears in the time dilation. The effect of gravity on
such quantum time dilation can be potentially observed for clocks prepared in
a superposition either in momentum space or in position space. We also
identified the condition which maximizes the effect of gravity in the quantum
time dilation.

## Acknowledgments

This work is supported by JSPS Grant-in-Aid for Scientific Research Number
22K03640 (TC) and in part by Nihon University.

##  Appendix A Time Dilation for Wave Packets

Here we give the time dilation for Gaussian wave packets.

First, we consider the situation where the external degrees of freedom of two
clocks are initially prepared in a Gaussian state localized around an average
momentum $\overline{p}_{n}$ with spread $\Delta_{n}$ and around an average
position $\overline{x}_{n}$ with spread $\sigma_{xn}=\hbar/\Delta_{n}$
generalizing the situation considered in Smith and Ahmadi (2020); Khandelwal
_et al._ (2020). Smith and Ahmadi (2020) considered wave packets localized in
momentum space and Khandelwal  _et al._ (2020) considered wave packets
localized in position space.

The external state in the momentum representation is then given by

$\displaystyle\langle P_{n}|\psi^{\rm
ext}_{n}\rangle=\frac{1}{(\pi\Delta_{n}^{2})^{1/4}}e^{-i\frac{\overline{x}_{n}({P_{n}}-\overline{p}_{n})}{\hbar}}e^{-\frac{({P_{n}}-\overline{p}_{n})^{2}}{2\Delta_{n}^{2}}}\,,$
(49)

and the initial average of $P_{n}^{2}$ is then given by

$\displaystyle\langle P_{n}^{2}\rangle_{0}=\langle\psi^{\rm
ext}_{n}|P_{n}^{2}|\psi^{\rm ext}_{n}\rangle=\int dp~{}p^{2}|\langle
p|\psi^{\rm
ext}_{n}\rangle|^{2}=\overline{p}_{n}^{2}+\frac{\Delta_{n}^{2}}{2}\,.$ (50)

The external state in the position representation is given by

$\displaystyle\langle x_{n}|\psi^{\rm ext}_{n}\rangle$ $\displaystyle=$
$\displaystyle\int dp\langle x_{n}|p\rangle\langle p|\psi^{\rm ext}\rangle$
(51) $\displaystyle=$
$\displaystyle\frac{1}{(\pi\sigma_{xn}^{2})^{1/4}}e^{\frac{i}{\hbar}x_{n}\overline{p}_{n}}e^{-\frac{(x_{n}-\overline{x}_{n})^{2}}{2\sigma_{xn}^{2}}}\,,$

where $\sigma_{xn}=\hbar/\Delta_{n}$, and the initial average of the position
is $\overline{x}_{n}$, $\langle x_{n}\rangle_{0}=\int dx~{}x|\langle
x|\psi^{\rm ext}_{n}\rangle|^{2}=\overline{x}_{n}$.

Plugging $\langle{\bf P}_{n}^{2}\rangle_{0}$ into Eq. (39), the time dilation
between two clocks is given by

$\displaystyle\langle T_{A}\rangle$ $\displaystyle=$
$\displaystyle\tau_{B}\left(1-\left(\frac{\langle{\bf
P}_{A}^{2}\rangle_{0}}{2m_{A}^{2}c^{2}}-\frac{\langle{\bf
P}_{B}^{2}\rangle_{0}}{2m_{B}^{2}c^{2}}\right)+\frac{g}{c^{2}}\left(\langle
x_{A}\rangle_{0}+2\frac{\langle P_{A}\rangle_{0}}{m_{A}}t-\langle
x_{B}\rangle_{0}-2\frac{\langle P_{B}\rangle_{0}}{m_{B}}t\right)\right)$ (52)
$\displaystyle=$
$\displaystyle\tau_{B}\left[1-\left(\frac{(\overline{p}_{A}-m_{A}gt)^{2}}{2m_{A}^{2}c^{2}}-\frac{(\overline{p}_{B}-m_{B}gt)^{2}}{2m_{B}^{2}c^{2}}+\frac{\Delta_{A}^{2}}{4m_{A}^{2}c^{2}}-\frac{\Delta_{B}^{2}}{4m_{B}^{2}c^{2}}\right)+\frac{g}{c^{2}}\left(\overline{x}_{A}+\frac{\overline{p}_{A}t}{m_{A}}-\overline{x}_{B}-\frac{\overline{p}_{B}t}{m_{B}}\right)\right]\,.$

Therefore, supposing $\Delta_{A}/m_{A}=\Delta_{B}/m_{B}$, the time dilation
between two quantum clocks agree with the classical relativistic time
dilation.

##  Appendix B Initial expectation values for a superposition of states

We summarize the expressions of expectation values of momentum, momentum
squared, and position operators for a superposition of Gaussian wave packets
given by Eq. (40):

$\displaystyle\langle P_{A}\rangle$
$\displaystyle=\overline{p}_{A}\cos^{2}\theta+\overline{p}_{A}^{\prime}\sin^{2}\theta-\frac{\sin
2\theta}{N}\,e^{-\frac{(\overline{p}_{A}^{2}-\overline{p}_{A}^{\prime})^{2}}{4\Delta_{A}^{2}}-\frac{(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{4\sigma_{xA}^{2}}}$
$\displaystyle\qquad\times\left[\frac{(\overline{p}_{A}-\overline{p}_{A}^{\prime})\cos
2\theta}{2}\cos\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)+\hbar\frac{\overline{x}_{A}-\overline{x}_{A}^{\prime}}{2\sigma_{xA}^{2}}\sin\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)\right],$
(53) $\displaystyle\langle P_{A}^{2}\rangle$
$\displaystyle=\overline{p}_{A}^{2}\cos^{2}\theta+\overline{p}_{A}^{\prime
2}\sin^{2}\theta+\frac{\Delta_{A}^{2}}{2}-\frac{\sin
2\theta}{N}\,e^{-\frac{(\overline{p}_{A}^{2}-\overline{p}_{A}^{\prime})^{2}}{4\Delta_{A}^{2}}-\frac{(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{4\sigma_{xA}^{2}}}$
$\displaystyle\quad\times\Biggl{\\{}\left[\frac{\overline{p}_{A}^{2}-\overline{p}_{A}^{\prime
2}}{2}\cos
2\theta+\left(\frac{\overline{p}_{A}-\overline{p}_{A}^{\prime}}{2}\right)^{2}+\Delta_{A}^{2}\left(\frac{\overline{x}_{A}-\overline{x}_{A}^{\prime}}{2\sigma_{xA}}\right)^{2}\right]\cos\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)$
$\displaystyle\qquad+\hbar\frac{(\overline{p}_{A}+\overline{p}_{A}^{\prime})(\overline{x}_{A}-\overline{x}_{A}^{\prime})}{2\sigma_{xA}^{2}}\sin\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)\Biggr{\\}},$
(54) $\displaystyle\langle x_{A}\rangle$
$\displaystyle=\overline{x}_{A}\cos^{2}\theta+\overline{x}_{A}^{\prime}\sin^{2}\theta-\frac{\sin
2\theta}{N}e^{-\frac{(\overline{p}_{A}^{2}-\overline{p}_{A}^{\prime})^{2}}{4\Delta_{A}^{2}}-\frac{(\overline{x}_{A}-\overline{x}_{A}^{\prime})^{2}}{4\sigma_{xA}^{2}}}$
$\displaystyle\quad\times\left[\frac{\overline{x}_{A}-\overline{x}_{A}^{\prime}}{2}\cos
2\theta\cos\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)-\hbar\frac{\overline{p}_{A}-\overline{p}_{A}^{\prime}}{2\Delta_{A}^{2}}\sin\left(\phi-\frac{(\overline{x}_{A}+\overline{x}_{A}^{\prime})(\overline{p}_{A}-\overline{p}_{A}^{\prime})}{2\hbar}\right)\right].$
(55)

Note that the terms proportional to $\frac{\sin 2\theta}{N}$ in the above
expectation values represent contributions of quantum interference.

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