# Mixing and CPV in beauty and charm at LHCb

Laís Soares Lavra on behalf of the LHCb collaboration  Université Clermont
Auvergne, Laboratoire de Physique de Clermont, CNRS/IN2P3,
Clermont-Ferrand, France

###### Abstract

This document presents eight recent and new analyses from the LHCb experiment
covering results on CKM angle $\gamma$ and high precision measurements on
charm mixing parameters. In addition, the latest CP violation measurements in
charm and beauty decays are reported.

##  1 CKM angle $\gamma$

The angle $\gamma\equiv arg(-V_{ud}V^{*}_{ub}/V_{cd}V^{*}_{cb})$ of the CKM
unitarity triangle can be measured directly using tree-level b-hadron decays.
These measurements are obtained by exploiting the interference between
$b\rightarrow u$ and $b\rightarrow c$ transitions in decays of
$B^{\pm}\rightarrow Dh^{\pm}$, where $h=K,\pi$ 111Hereinafter $h$ refer to
either a charged kaon or a charged pion.. Since $\gamma$ is shared by all such
decays, the best precision is obtained through the combination of measurements
from several decay modes.

###  1.1 Update of the LHCb combination of the CKM angle $\gamma$

Using LHCb data collected during the first two runs of the LHC, this analysis
[1] presents a combination of $\gamma$ measurements using several
$B\rightarrow Dh$ decay modes and LHCb results from different $D$-decay modes
for the first time. Alongside $\gamma$, charm mixing and CP violation
parameters are simultaneously determined in the combination. The results are
obtained using frequentist treatment, and auxiliary inputs from other
experiments. The combination result in $\gamma=(65.4^{+3.8}_{-4.2})^{\circ}$
provides the highest precision measurement from a single experiment to date.

###  1.2 Constraints on the CKM angle $\gamma$ from $B^{\pm}\rightarrow
Dh^{\pm}$ decays

In this analysis [2], $B^{\pm}\rightarrow Dh^{\pm}$ decays are studied using
$D\rightarrow{\pi}{K}\pi^{0}$, $D\rightarrow KK\pi^{0}$ and
$D\rightarrow\pi\pi\pi^{0}$ final states. Measurements of $\gamma$ and related
parameters $r_{B}$ (the magnitude of the ratio of amplitudes of the
interfering decays) and $\delta_{B}$ (the strong phase between them) are
obtained using 9 fb-1 of data collected with the LHCb detector. The CP
observables are determined through a simultaneous mass fit to all decay modes.
The most suppressed decay mode
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{0}]_{D}K^{\pm}$ is observed for the
first time with a significance of 7.8$\sigma$ and signal yields of 155 $\pm$
19\. The fit results and external inputs are used to interpret the results in
terms of $\gamma$, $r_{B}$, $\delta_{B}$ and confidence intervals evaluated
using the profile likelihood method. The results are
$\gamma=(56.4^{+24}_{-19})^{\circ}$, $\delta_{B}=(122^{+19}_{-23})^{\circ}$
and $r_{B}=(9.3^{+1.0}_{-0.9})\times 10^{-2}$. These measurements shall be
used in future combinations to constrain the angle $\gamma$.

##  2 CPV and mixing in charm

Charm meson oscillations are characterised by the mixing parameters,
$x=\Delta{M}/{\Gamma}$ and $y=\Delta{\Gamma}/{\Gamma}$, where $\Delta{M}$ and
$\Delta{\Gamma}$ are the mass and decay width difference of the two mass
eigenstates $\left\lvert D_{1,2}\right\rangle=p\left\lvert
D^{0}\right\rangle+q\left\lvert\bar{D}^{0}\right\rangle$, and $\Gamma$ is the
average decay width. Related to the mixing, CP violation can be described by
two parameters $|q/p|$ and $\phi\equiv{arg(q/p)}$. These parameters are
heavily suppressed in the SM and expected to be very small, and their precise
determination constitute an important test to SM predictions.

###  2.1 First observation of the mass difference between neutral charm-meson
eigenstates

Precise measurements of mixing and CPV parameters in $D^{0}\rightarrow
K_{S}^{0}\pi^{+}\pi^{-}$ decay are performed using 5.4 fb-1 data sample
collected by LHCb from 2016-2018 [3]. This analysis is Dalitz plot and time-
dependent and the mixing parameters are determined using the bin-flip method
[4]. The results obtained for $x$, $y$, $|q/p|$ and $\phi$ are

$\displaystyle x=(3.98^{+0.56}_{-0.54})\times 10^{-3},$ $\displaystyle
y=(4.6^{+1.5}_{-1.4})\times 10^{-3},$ $\displaystyle|q/p|=0.996\pm 0.052,$
$\displaystyle\phi=0.056^{+0.047}_{-0.051}.$

The mixing parameter $x$ is found to be a non-zero value with a significance
greater than 7$\sigma$ being the first observation of the mass difference in a
$D^{0}$ system. All results are consistent with CP symmetry and improve the
knowledge of mixing-induced CPV in the charm sector.

###  2.2 High precise measurement of ${y}_{CP}$-$y_{CP}^{K\pi}$ parameter
using two-body $D^{0}$ meson decays

Using 6.0 fb-1 data sample collected by LHCb from 2015-2018, the charm mixing
parameter ${y}_{CP}$-$y_{CP}^{K\pi}$ 222known as $y_{CP}$ ($y_{CP}\approx y$
in the limit of CP conservation). is precisely determined through the
measurement of the lifetime difference between the $D^{0}\rightarrow f$
($f=K^{-}K^{+}$ or $\pi^{-}\pi^{+}$) and $D^{0}\rightarrow K^{-}\pi^{+}$
decays [5]. To determine the $y_{CP}-y^{K\pi}_{CP}$ parameter, a weighted
average of statistically independent measurements of
$y^{KK}_{CP}-y^{K\pi}_{CP}$ and $y^{\pi\pi}_{CP}-y^{K\pi}_{CP}$ is performed.
These parameters are accessed with an exponential fit to the signal yield
ratios of $D^{0}\rightarrow f$ over $D^{0}\rightarrow K^{-}\pi^{+}$ as a
function of the decay time $t$,

$R^{f}(t)=\frac{N(D^{0}\rightarrow f,t)}{N(D^{0}\rightarrow
K^{-}\pi^{+},t)}\propto{e}^{-({y^{f}_{CP}-y^{K\pi}_{CP}})t/\tau_{D^{0}}}\times\frac{\varepsilon(f,t)}{\varepsilon(K^{-}\pi^{+},t)}$
(1)

where $\tau_{D^{0}}$ is the measured $D^{0}$ lifetime and
$\varepsilon(h^{-}h^{+},t)$ is the time-dependent efficiency. Since the
efficiencies differ for each decay mode, a kinematic matching and a weighting
procedure are employed to equalise the efficiencies so that they cancel in the
ratio. The analysis procedure is validated with three different methods and
the results of the fit to the corresponding $R^{f}(t)$ are

$\displaystyle y^{\pi\pi}_{CP}-y^{K\pi}_{CP}=(6.57\pm 0.53\pm 0.16)\times
10^{-3},$ $\displaystyle y^{KK}_{CP}-y^{K\pi}_{CP}=(7.08\pm 0.30\pm
0.14)\times 10^{-3},$

where the first uncertainty is statistical and the second systematic.
Combining these two measurements yields $y_{CP}-y^{K\pi}_{CP}=(6.96\pm 0.26\pm
0.13)\times 10^{-3}$, which is compatible with the world average and more
precise by a factor of four.

###  2.3 Measurement of $\mathit{CP}$ asymmetries in
$D^{+}_{(s)}\rightarrow\eta\pi^{+}$ and
$D^{+}_{(s)}\rightarrow\eta^{{}^{\prime}}\pi^{+}$ decays

Searches for direct CP asymmetries in $D^{+}_{(s)}\rightarrow\eta\pi^{+}$ and
$D^{+}_{(s)}\rightarrow\eta^{\prime}\pi^{+}$ decays are performed using 6 fb-1
LHCb dataset from 2015-2018, with $\eta^{(^{\prime})}$ mesons reconstructed in
the final state $\gamma\pi^{+}\pi^{-}$ [6]. The CP asymmetry
$\mathcal{A}^{CP}$ of each decay mode is obtained trough the raw asymmetry
subtracted by the production and detection asymmetries, determined using the
control channels $D^{+}_{(s)}\rightarrow\phi\pi^{+}$, where the asymmetries
are assumed to be zero ($D^{+}_{s}$) and well constrained from an external
input ($D^{+}$). A simultaneous likelihood fit to the data is performed for
$D^{\pm}_{(s)}$ candidates in intervals of $\eta^{(^{\prime})}$ mass to
determine the raw asymmetries. The measured CP asymmetries are

$\displaystyle\mathcal{A}^{CP}_{D^{+}\rightarrow\eta\pi^{+}}=(0.34\pm 0.66\pm
0.16\pm 0.05)\%,$
$\displaystyle\mathcal{A}^{CP}_{D_{s}^{+}\rightarrow\eta\pi^{+}}=(0.32\pm
0.51\pm 0.12)\%,$
$\displaystyle\mathcal{A}^{CP}_{D^{+}\rightarrow\eta{{}^{\prime}}\pi^{+}}=(0.49\pm
0.18\pm 0.06\pm 0.05)\%,$
$\displaystyle\mathcal{A}^{CP}_{D_{s}^{+}\rightarrow\eta^{\prime}\pi^{+}}=(0.01\pm
0.12\pm 0.08)\%,$

where the first uncertainty is statistical, then systematic and the last,
relevant for the $D^{+}$ modes, is from the uncertainty on the CP asymmetry of
the $D^{+}\rightarrow\phi\pi^{+}$ control channel. All the results are
consistent with CP symmetry, and with previous measurements.

##  3 CPV in charmless b-meson decays

Charmless b-meson decays provide an interesting environment to observe CP
violation effects since these decays have contributions from both penguin and
tree-level processes with amplitudes of similar size. In addition, in multi-
body decays the different intermediate states can interfere with each other,
producing large strong-phase differences, which can lead to enhanced CP
violation in certain regions of phase space.

###  3.1 Direct $\mathit{CP}$ violation in charmless three-body decays of
$B^{\pm}$ mesons

Measurements of direct $\mathit{CP}$ violation in $B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}$, $B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}$,
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ and
$B^{\pm}\rightarrow\pi^{\pm}K^{+}K^{-}$ decays are performed using 5.9 fb-1
data sample collected by LHCb from 2015-2018 [7]. The CP asymmetry
$\mathcal{A}^{CP}$ of each decay mode are accessed directly from the raw
asymmetry, corrected for nuisance asymmetries. The raw asymmetries are
obtained from the phase-space efficiency corrected yields of $B^{+}$ and
$B^{-}$ fitted simultaneously. Then the CP asymmetries are obtained by
subtracting the production asymmetry, calculated using the control mode
$B^{\pm}\rightarrow J/\psi K^{\pm}$. The results for the phase-space
integrated CP asymmetries are

$\displaystyle\mathcal{A}^{CP}_{B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}=(+1.1\pm 0.2\pm 0.3\pm 0.3)\%,$
$\displaystyle\mathcal{A}^{CP}_{B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}=(-3.7\pm
0.2\pm 0.2\pm 0.3)\%,$
$\displaystyle\mathcal{A}^{CP}_{B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}}=(+8.0\pm
0.4\pm 0.3\pm 0.3)\%,$
$\displaystyle\mathcal{A}^{CP}_{B^{\pm}\rightarrow\pi^{\pm}K^{+}K^{-}}=(-11.4\pm
0.7\pm 0.3\pm 0.3)\%,$

where the first uncertainty is statistical, second systematic, and third
uncertainty due to the CP asymmetry of the $B^{\pm}\rightarrow J/\psi K^{\pm}$
control channel. For $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ and
$B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}$ decays, CP violation is observed for
the first time with significance of 14.1$\sigma$ and 8.5$\sigma$,
respectively. For $B^{\pm}\rightarrow\pi^{\pm}K^{+}K^{-}$ the observed CP
violation is compatible and more precise than the previous measurement while
the result for the $B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}$ decay is
consistent with CP conservation. The CP asymmetry distributions across the
Dalitz Plot of these four modes are investigated using the Mirandizing
approach [8]. Nine different regions of the Dalitz plot are studied among the
four $B^{\pm}\rightarrow h^{\pm}h^{+}h^{-}$ decays. The local CP asymmetries
are measured from fits to the invariant mass of candidates in the regions
studied, following the same procedure as for integrated measurements. In Fig.
1 shows the results for a selected region in
$B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ decay. The projection onto
$m^{2}(\pi^{+}\pi^{-})_{\text{high}}$ reveals an indication of the
$\chi_{c0}$(1P) resonance for the first time. The local CP asymmetry measured
in this region is $\mathcal{A}^{CP}=({+74.5}\pm 2.7\pm 1.8\pm 0.3)\%$, which
is the highest CP asymmetry ever measured to date. In addition, significant
localised CP asymmetries in the $\pi\pi\rightarrow KK$ rescattering region are
observed in the four analysed channels.

Figure 1: The $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$: (left)
$m^{2}(\pi^{+}\pi^{-})_{\text{high}}$ projection for the region
$4<m^{2}(\pi^{+}\pi^{-})_{\text{high}}<16$ GeV${}^{2}/$c4 and
$4<m^{2}(\pi^{+}\pi^{-})_{\text{low}}<15$ GeV${}^{2}/$c4 with (right) mass
fits for this region ($B^{-}$ on the left, $B^{+}$ on the right).

###  3.2 Search for direct $\mathit{CP}$ violation in charged charmless
$B\rightarrow PV$ decays

Measurements of CP asymmetries are reported in charmless $B$ decays to a
pseudo-scalar and a vector resonance ($B\rightarrow PV$) using 5.9 fb-1 data
sample collected by LHCb from 2015-2018 [9]. A new model-independent method
based on the angular distributions is employed to measure CP asymmetries in
low mass and narrow vector resonances [10]. Five different $B\rightarrow PV$
decays from $B^{\pm}\rightarrow h^{\pm}h^{+}h^{-}$ final states are analysed.
For the $B^{\pm}\to\rho(770)^{0}K^{\pm}$ decay a CP asymmetry of
$\mathcal{A}^{CP}=(+15.0\pm 1.9\pm 1.1)\%$ is observed for the first time with
a significance of 6.8$\sigma$. For the other modes, results are consistent
with CP symmetry.

###  3.3 Search for $CP$ violation using $\hat{T}$-odd correlations in
$B^{0}\rightarrow p\bar{p}K^{+}\pi^{-}$ decays

Using 8.4 fb-1 LHCb dataset from 2011-2012 and 2016-2018, a search for CP and
P violation in charmless four-body $B^{0}\rightarrow p\bar{p}K^{+}\pi^{-}$
decays is performed with observables constructed using triple-product
asymmetries techniques [11, 12]. These observables are measured both globally
and in regions of the phase-space, where their deviation from zero is probed.
The signal yields and the observables are obtained from a simultaneous fit to
$p\bar{p}K^{+}\pi^{-}$ invariant mass distributions. No evidence of CP and P
asymmetries are found in integrated measurements. Results in regions of the
phase are consistent with CP symmetry and local P-violations with significance
greater than 5.8$\sigma$ are observed.

## References

## References

  * [1] LHCb collaboration, Aaij, R. et al, JHEP 12, 141 (2021).
  * [2] LHCb collaboration, Aaij, R. et al, LHCb-PAPER-2021-036, arXiv:2112.10617 [hep-ex].
  * [3] LHCb collaboration, Aaij, R. et al, Phys. Rev. Lett. 127, 111801 (2021).
  * [4] A. Di Canto et al., Phys. Rev. D 99, 012007 (2019).
  * [5] LHCb collaboration, Aaij, R. et al, LHCb-PAPER-2021-041, arXiv:2202.09106 [hep-ex].
  * [6] LHCb collaboration, Aaij, R. et al, LHCb-PAPER-2021-051, arXiv:2204.12228 [hep-ex].
  * [7] LHCb collaboration, Aaij, R. et al, LHCb-PAPER-2021-049, in preparation.
  * [8] Bediaga I. et al, Phys. Rev. D 80, 096006 (2009).
  * [9] LHCb collaboration, Aaij, R. et al, LHCb-PAPER-2021-050, in preparation.
  * [10] Alvarenga Nogueira J. H. et al, Phys. Rev. D 94, 054028 (2016).
  * [11] LHCb collaboration, Aaij, R. et al, LHCb-PAPER-2022-003, arXiv:2205.08973 [hep-ex].
  * [12] G. Duriex and Y. Grossman, Phys. Rev. D 92, 076013 (2015).


# Exploring Speaker-Related Information in Spoken Language Understanding for
Better Speaker Diarization

Luyao Cheng, Siqi Zheng, Zhang Qinglin, Hui Wang, Yafeng Chen, Qian Chen
Speech Lab of DAMO Academy, Alibaba Group
<EMAIL_ADDRESS>
###### Abstract

Speaker diarization(SD) is a classic task in speech processing and is crucial
in multi-party scenarios such as meetings and conversations. Current
mainstream speaker diarization approaches consider acoustic information only,
which result in performance degradation when encountering adverse acoustic
conditions. In this paper, we propose methods to extract speaker-related
information from semantic content in multi-party meetings, which, as we will
show, can further benefit speaker diarization. We introduce two sub-tasks,
Dialogue Detection and Speaker-Turn Detection, in which we effectively extract
speaker information from conversational semantics. We also propose a simple
yet effective algorithm to jointly model acoustic and semantic information and
obtain speaker-identified texts. Experiments on both AISHELL-4 and AliMeeting
datasets show that our method achieves consistent improvements over acoustic-
only speaker diarization systems.

##  1 Introduction

Speaker diarization(SD) is the task of answering the question “who speaks
when" by partitioning audio into segments with speaker identities. In most
application settings, the results of speaker diarization are perceived by
readers through the assignment of speaker labels to the corresponding words or
sentences transcribed from an Automatic Speech Recognition(ASR) system.

Despite the rich profusion of transcribed texts, mainstream speaker
diarization systems consider only acoustic information Park et al. (2021);
Horiguchi et al. (2020); Park et al. (2020); Fujita et al. (2019); Zheng et
al. (2021); Du et al. (2022a). Traditional SD systems usually consist of the
following components: (1) Voice activity detection (VAD) to filter out non-
speech frames. (2) Extraction of speaker embeddings from the short audio
segments, using popular models such as i-vectorDehak et al. (2011),
d-vectorZhang and Koishida (2017) and x-vectorSnyder et al. (2018). (3)
Clustering embeddings into several classes using algorithms such as
agglomerative hierarchical clustering (AHC) Day and Edelsbrunner (1984),
spectral clustering(SC) Wang et al. (2017), and HDBSCAN Zheng et al. (2022).
Various speaker embedding model and clustering methods have been explored and
proposed in Yu et al. (2021); Dawalatabad et al. (2021); He et al. (2021);
Zheng and Suo (2022); Du et al. (2022b).

Utilizing only acoustic information has significant limitations. For example,
the performance of SD system suffers from obvious degradation in adverse
acoustic conditions such as noise, reverberation, and far-field recordings. In
addition, we often encounter speakers with similar voice characteristics,
which pose serious challenge to clustering them into expected classes. Given
the abundance of transcribed texts present in meetings and conversations, it
is of sufficient interest to explore the possibilities of utilizing semantic
information to go beyond the limits of acoustic-only speaker diarization.

Some previous works tried to use semantic information to classify roles in
two-speaker conversations, such as doctor-patient conversation and pilot-air
traffic controller dialogueZuluaga-Gomez et al. (2021); Flemotomos and
Narayanan (2022). However, these methods are only suitable for specific two-
speaker scenarios where the roles are clearly-defined, such as medical
diagnosis, job interviews, and air traffic communications. In this work we
focus on open multi-party meeting scenarios where the number of speakers is
unknown and the relations between speakers are unspecified.

Figure 1: We propose a multi-modal speaker diarization system that utilizes
the SLP module to extract speaker-related information from transcribed text.
The multi-modal fusion and semantic backend modules combine both acoustic and
semantic information to improve the accuracy of speaker diarization. The
system’s output includes text segments with corresponding speaker
identification.

Speaker identity information has been proven to be beneficial to many
downstream NLP tasksChi et al. (2017a); Zhang et al. (2018); Chi et al.
(2017b). However, these works only consider speaker identities as given ground
truth Carletta et al. (2006); Janin et al. (2003); Zhong et al. (2021), which
is impractical in real world settings. Therefore, it is crucial to make valid
inference of speaker identities on the transcribed conversations using a well-
performed speaker diarization system.

The main contributions of this paper include:

(1) We propose two semantic tasks to extract speaker-related information from
automatically transcribed texts, namely Dialogue Detection and Speaker-Turn
Detection.

(2) We design a simple yet effective integration method to effectively combine
semantic and acoustic information for more robust speaker diarization.

##  2 Proposed Methods

###  2.1 A Novel Multi-modal Framework

Figure 1 illustrates the proposed semantic-acoustic speaker diarization
system, along with its relation with upstream ASR components and downstream
NLP applications. We introduce a Spoken Language Processin(SLP) module
involving two sub-tasks to extract speaker-related information from
transcribed texts. The acoustic-based speaker diarization system is used to
process original audio, perform segmentation and estimate speaker embeddings
for each segments. To associate speaker embeddings with corresponding text
phrases, a forced alignment component was introduced to our system. Finally,
we propose an integration method to collectively process outputs from SLP
module, acoustic SD module, and forced alignment module.

###  2.2 Learning Speaker Information From Texts

To extract semantic speaker-related information, we define two sub-tasks:
dialogue detection and speaker-turn detection.

Dialogue detection takes a sequence of sentences as input and determines
whether this is transcribed from a multi-speaker dialogue or a single-speaker
speech. Dialogue-detection can be defined as a binary classification problem.

Speaker turn detection tries to predict, for each given sentence in the
sequence, the probability of the occurrence of speaker change. Speaker turn
detection can be defined as a sequence labeling problem, where the goal is to
determine whether the given position represents a point of change in speaker
role from a semantic perspective.

Both dialogue detection and speaker turn detection models are fine-tuned from
a pre-trained BERT language model. Design of training samples and details of
experiments are discussed in next section.

###  2.3 Integrating Semantic-Acoustic Information

In this section we describe how speaker-related information extracted from
semantic content can assist us in improving upon acoustic-only SD system. A
traditional SD system typically involves an audio segmentation module and an
embedding clustering module. Poor segmentation and incorrect clustering are
the most common problems in speaker diarization. Semantic information from
dialogue detection helps improve clustering accuracy and speaker turn
detection helps to find more precise place in text where a change of speaker
occurs.

Note that dialogue detection and speaker turn detection tasks can be solved
either by acoustic-only approach or semantic-only approach. Semantic-only
approach is described above. Acoustic-only results can be derived directly
from acoustic-based speaker clustering. The speaker clustering algorithm
assigns a cluster label to each speaker-segment. Acoustic results for dialogue
detection can be obtained simply by checking whether the number of different
speaker labels is larger than 1. Results for speaker turn detection can be
obtained by analyzing the transition patterns of speaker-segment labels or
predicting change points using an acoustic-based neural networks such as
Target-Speaker VADHe et al. (2021).

Semantic-Acoustic Dialogue Detection. Let $z^{(s)}$ denote the result of
binary classification output of semantic dialogue detection and $z^{(a)}$ be
the counterpart of acoustic dialogue detection. We also define $D_{p}$ to be
the distance of the largest speaker cluster present in the dialogue to its
furthest cluster, and $D_{q}$ to be the standard deviation of the cosine
distances among all speaker embeddings present in the selected speech. $D_{p}$
measures how spread out different clusters are and $D_{q}$ measures how tight
embeddings in one cluster are grouped together. Then the fusion score
$\hat{s}$ is estimated by:

$\hat{s}=z^{(a)}z^{(s)}+z^{(a)}(p_{s}+\alpha_{1}D_{p})+z^{(s)}(p_{s}+\alpha_{2}D_{q}),$
(1)

where $\alpha_{1}$ and $\alpha_{2}$ are learnable and $p_{s}$ is logit output
from semantic dialogue detection.

For some threshold $\theta$, the binary output of semantic-acoustic dialogue
detection is represented by the indicator function:

$\displaystyle\hat{z}^{\text{fusion}}_{dd}=\mathbf{1}_{\hat{s}>\theta}$ (2)

Once semantic-acoustic dialogue detection obtain results for all sentence
sequences that cover the entire transcribed meeting, we re-adjust the
acoustic-based clustering results. By doing this we are able to incorporate
semantic information to improve speaker clustering. More details can be found
in Appendix B.

Semantic-Acoustic Speaker Turn Detection. Semantic-only speaker turn detection
outputs a sequence of probability of the occurrence of speaker change. Let
$p_{n}$ be the probability at position $n$, and $q_{n}$ represents the speaker
change probability from an acoustic-only model near position $n$. $q_{n}$ is
obtained by taking the maximum probability of the closest 200 frames estimated
by the Target-Speaker VAD model. Then the integrated speaker-change
probability is given by

$\tilde{p}_{n}=\beta_{1}p_{n}+\beta_{2}q_{n}$ (3)

for some learnable hyperparameters $\beta_{1}$ and $\beta_{2}$.

Boundary and Outlier Correction. We use semantic information to correct
boundary errors caused by errors and mismatches from the forced-alignment and
ASR models. We also use semantic information to correct outliers in embedding
extraction. To improve system robustness, we exclude audio segments that are
too short from clustering. Outliers and left-out embeddings are assigned to
the closest cluster.

##  3 Experiments and Results

###  3.1 Datasets

We conduct experiments on AISHELL-4Fu et al. (2021) and AliMeetingYu et al.
(2022) datasets. Both focus on multi-party meeting scenario, where all speech
content are manually annotated. Table 1 listed detailed information about the
datasets. We perform experiments using both ground truth (GT) text and text
transcribed from ASR system.

|  | AISHELL-4
---
Train/Eval
| Alimeeting
---
Train/Eval
Session | 191/20 | 209/20
#Avg. Duration(s) | 1939.03/2245.94 | 1915.52/1924.7
#Avg. Speakers | 4.8/5.8 | 3.27/3.0
#Avg. Speaker-Turn | 343.95/220.8 | 649.34/552.75
Avg. Text Len. | 8904.8/9990.9 | 13249.55/12067.05

Table 1: Details of AISHELL-4 and AliMeeting data. TaskName | Text | Methods | AISHELL-4 | Alimeeting
---|---|---|---|---
Precision | Recall | F1 | Precision | Recall | F1
Dialogue Detection | GT | Acoustic-Only | 74.402 | 84.995 | 79.346 | 93.012 | 92.259 | 92.634
Semantic-Only | 74.649 | 96.976 | 84.360 | 94.669 | 98.009 | 96.310
Multi-Modal | 86.308 | 93.402 | 89.715 | 96.450 | 97.600 | 97.020
ASR | Acoustic-Only | 80.405 | 96.936 | 87.900 | 96.482 | 98.428 | 97.445
Semantic-Only | 55.731 | 84.414 | 67.138 | 93.649 | 88.688 | 91.101
Multi-Modal | 82.461 | 95.826 | 88.642 | 96.641 | 98.320 | 97.476
Speaker-Turn Detection | GT | Acoustic-Only | 53.962 | 51.272 | 52.583 | 54.329 | 52.997 | 53.655
Semantic-Only | 69.569 | 89.514 | 78.291 | 76.696 | 93.141 | 84.123
Multi-Modal | 81.652 | 77.240 | 79.385 | 76.861 | 92.849 | 84.102
ASR | Acoustic-Only | 61.657 | 79.162 | 69.322 | 66.105 | 67.696 | 66.891
Semantic-Only | 42.299 | 63.386 | 50.738 | 61.046 | 50.100 | 55.034
Multi-Modal | 68.132 | 73.878 | 70.889 | 67.593 | 66.960 | 67.276
Table 2: The results of two sub-tasks on AISHELL-4 and Alimeeting test set. Text Type | Methods | AISHELL-4 | Alimeeting
---|---|---|---
cp-wer | cp-wer-all | speaker-wer | cp-wer | cp-wer-all | speaker-wer
GT Text | Baseline - only acoustic info. | 17.309 | 19.099 | 5.974 | 41.669 | 52.617 | 18.888
Semantic-acoustic info. - A | 15.540 | 18.798 | 6.558 | 36.360 | 45.772 | 14.700
Semantic-acoustic info. - B | 15.225 | 18.364 | 6.281 | 36.145 | 45.462 | 14.500
ASR Text | Baseline - only acoustic info. | 33.905 | 35.590 | 3.647 | 45.678 | 49.778 | 8.404
Semantic-acoustic info. - A | 33.355 | 34.318 | 2.650 | 38.467 | 40.182 | 2.413
Semantic-acoustic info. - B | 33.290 | 34.210 | 2.575 | 38.459 | 40.154 | 2.389
Table 3: The speaker diarization results of different systems on the AISHELL-4
and Alimeeting test set

###  3.2 Experimental Setups

In our experiments, the acoustic modules, including ASR, ASR Post-Processing,
Embedding Extractor, and Forced Alignment models, are fixed and used
consistently throughout all our experiments. In details, the ASR system we
introduced was based on UniASRGao et al. (2020). The ASR Post-Processing
contained Punctuation-PredictionChen et al. (2020) and Text-Smoothing which
are common used in meeting scenrio. The Forced Alignment module we introduced
was based on McAuliffe et al. (2017). For acoustic speaker diarization system,
we employed a speaker embedding extractor based on ECAPA-TDNNDesplanques et
al. (2020), while for speaker clustering, we utilized Spectral Clustering
algorithm with p-percentileWang et al. (2017).

We fine-tune the semantic models for dialogue detection and speaker turn
detection tasks based on the pre-trained BERT language model111Based on bert-
base-chinese from HuggingFace using the text from AISHELL-4 and AliMeeting
training sets. Training samples are sequences of sentences generated by a
sliding-window method with a window length of 64 and a shift of 16 and the
label for these two semantic subtasks can be generated by the speaker label
from the speech content manually annotated in the datasets.

For dialogue detection task, we fine-tune for 3 epochs on train dataset with a
learning rate of 5e-6 and a batch size of 64. For speaker-turn detection task,
we also fine-tune for 3 epochs on train dataset with a learning rate of 1e-6
and a batch size of 64.

###  3.3 Results and Discussions

We compare our proposed methods with the classic speaker diarization system
mentioned in Section 1.

Table 2 shows the results of dialogue detection and speaker-turn detection
tasks from acoustic-only, semantic-only, and multi-modal models. We not only
compare results using ASR-transcribed texts, but also conduct experiments
using ground truth texts as inputs, in order to see the optimal improvements
introduced by semantic information. The multi-modal model surpasses single-
modal results on both GT and ASR text. The experiments demonstrate that
semantic model can effectively supplement acoustic-only model, resulting in
more precise speaker representation. It is expected that the introduction of
semantic information on ASR text does not result in a significant improvement
due to the a lot of errors present in the text. However, our multi-modal
approach shows consistent improvement in both GT and ASR-based results,
indicating the robustness of our method.

We use the cp-WER metric to measure the speaker diarization task. We introduce
a new metric speaker-WER that aims to measure the word error rate caused
solely by speaker errors. More details about the metrics can be found in
Appendix A. Table 3 shows the final results of our speaker diarization system.
Compared to the baseline, which only uses acoustic information, our system
incorporating semantic information shows a significant improvement. The
results for “Semantic-acoustic info. - A" indicate that only semantic
information is used for sub-tasks Dialogue Detection and Speaker Turn
Detection, while “Semantic-acoustic info. - B" indicates that both semantic
and acoustic information are used in the two sub-tasks.

##  4 Conclusions

We propose a novel multi-modal speaker diarization system that utilizes two
spoken language processing tasks, dialogue detection and speaker-turn
detection, to extract speaker-related information from text. These information
are then combined with acoustic information to improve the overall performance
of speaker diarization. Our experiments demonstrate that incorporating
semantic information can effectively address the limitations of single-modal
speech.

##  5 Limitations

The performance of SLP tasks rely heavily on the accuracy of ASR system.
Poorly-transcribed texts can lead to degradation of our multi-modal method.
Since we cannot easily obtain accurate speaker-turn information from the ASR
text, the training set for SLP tasks based on ASR text is also not easy to
obtain. In future work, we will try more methods, like Data Arguments, to get
better results on ASR text.

Overlapping speech is another challenge for the task, as a monaural ASR system
can no longer capture all spoken words from all speakers. In future work, we
plan to explore methods such as speech separation or multi-party ASR to handle
overlapping speech.

## References

  * Carletta et al. (2006) Jean Carletta, Simone Ashby, Sebastien Bourban, Mike Flynn, Mael Guillemot, Thomas Hain, Jaroslav Kadlec, Vasilis Karaiskos, Wessel Kraaij, Melissa Kronenthal, et al. 2006.  The ami meeting corpus: A pre-announcement.  In _International workshop on machine learning for multimodal interaction_ , pages 28–39. Springer.
  * Chen et al. (2020) Qian Chen, Mengzhe Chen, Bo Li, and Wen Wang. 2020.  Controllable time-delay transformer for real-time punctuation prediction and disfluency detection.  _ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 8069–8073.
  * Chi et al. (2017a) Ta-Chung Chi, Po-Chun Chen, Shang-Yu Su, and Yun-Nung Chen. 2017a.  Speaker role contextual modeling for language understanding and dialogue policy learning.  _IJCNLP 2017_ , page 163.
  * Chi et al. (2017b) Ta-Chung Chi, Po-Chun Chen, Shang-Yu Su, and Yun-Nung (Vivian) Chen. 2017b.  Speaker role contextual modeling for language understanding and dialogue policy learning.  In _International Joint Conference on Natural Language Processing_.
  * Dawalatabad et al. (2021) Nauman Dawalatabad, Mirco Ravanelli, Franccois Grondin, Jenthe Thienpondt, Brecht Desplanques, and Hwidong Na. 2021.  Ecapa-tdnn embeddings for speaker diarization.  In _Interspeech_.
  * Day and Edelsbrunner (1984) William H. E. Day and Herbert Edelsbrunner. 1984.  Efficient algorithms for agglomerative hierarchical clustering methods.  _Journal of Classification_ , 1:7–24.
  * Dehak et al. (2011) Najim Dehak, Patrick Kenny, Réda Dehak, Pierre Dumouchel, and Pierre Ouellet. 2011.  Front-end factor analysis for speaker verification.  _IEEE Transactions on Audio, Speech, and Language Processing_ , 19:788–798.
  * Desplanques et al. (2020) Brecht Desplanques, Jenthe Thienpondt, and Kris Demuynck. 2020.  Ecapa-tdnn: Emphasized channel attention, propagation and aggregation in tdnn based speaker verification.  In _Interspeech_.
  * Du et al. (2022a) Zhihao Du, Shiliang Zhang, Siqi Zheng, and Zhijie Yan. 2022a.  Speaker embedding-aware neural diarization: an efficient framework for overlapping speech diarization in meeting scenarios.  _CoRR_ , abs/2203.09767.
  * Du et al. (2022b) Zhihao Du, Shiliang Zhang, Siqi Zheng, and Zhijie Yan. 2022b.  Speaker overlap-aware neural diarization for multi-party meeting analysis.  _CoRR_ , abs/2211.10243.
  * Flemotomos and Narayanan (2022) Nikolaos Flemotomos and Shrikanth S. Narayanan. 2022.  Multimodal clustering with role induced constraints for speaker diarization.  In _INTERSPEECH_.
  * Fu et al. (2021) Yihui Fu, Luyao Cheng, Shubo Lv, Yukai Jv, Yuxiang Kong, Zhuo Chen, Yanxin Hu, Lei Xie, Jian Wu, Hui Bu, et al. 2021.  Aishell-4: An open source dataset for speech enhancement, separation, recognition and speaker diarization in conference scenario.  _arXiv preprint arXiv:2104.03603_.
  * Fujita et al. (2019) Yusuke Fujita, Naoyuki Kanda, Shota Horiguchi, Yawen Xue, Kenji Nagamatsu, and Shinji Watanabe. 2019.  End-to-end neural speaker diarization with self-attention.  _2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU)_ , pages 296–303.
  * Gao et al. (2020) Zhifu Gao, Shiliang Zhang, Ming Lei, and Ian Mcloughlin. 2020.  Universal asr: Unifying streaming and non-streaming asr using a single encoder-decoder model.  _ArXiv_ , abs/2010.14099.
  * He et al. (2021) Maokui He, Desh Raj, Zili Huang, Jun Du, Zhuo Chen, and Shinji Watanabe. 2021.  Target-speaker voice activity detection with improved i-vector estimation for unknown number of speaker.  In _Interspeech_.
  * Horiguchi et al. (2020) Shota Horiguchi, Yusuke Fujita, Shinji Watanabe, Yawen Xue, and Kenji Nagamatsu. 2020.  End-to-end speaker diarization for an unknown number of speakers with encoder-decoder based attractors.  _ArXiv_ , abs/2005.09921.
  * Janin et al. (2003) Adam L. Janin, Don Baron, Jane Edwards, Daniel P. W. Ellis, David Gelbart, Nelson Morgan, Barbara Peskin, Thilo Pfau, Elizabeth Shriberg, Andreas Stolcke, and Chuck Wooters. 2003.  The icsi meeting corpus.  _2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP ’03)._ , 1:I–I.
  * McAuliffe et al. (2017) Michael McAuliffe, Michaela Socolof, Sarah Mihuc, Michael Wagner, and Morgan Sonderegger. 2017.  Montreal forced aligner: Trainable text-speech alignment using kaldi.  In _Interspeech_.
  * Park et al. (2020) Tae Jin Park, Kyu J. Han, Manoj Kumar, and Shrikanth S. Narayanan. 2020.  Auto-tuning spectral clustering for speaker diarization using normalized maximum eigengap.  _IEEE Signal Processing Letters_ , 27:381–385.
  * Park et al. (2021) Tae Jin Park, Naoyuki Kanda, Dimitrios Dimitriadis, Kyu J. Han, Shinji Watanabe, and Shrikanth S. Narayanan. 2021.  A review of speaker diarization: Recent advances with deep learning.  _ArXiv_ , abs/2101.09624.
  * Snyder et al. (2018) David Snyder, Daniel Garcia-Romero, Gregory Sell, Daniel Povey, and Sanjeev Khudanpur. 2018.  X-vectors: Robust dnn embeddings for speaker recognition.  _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 5329–5333.
  * Wang et al. (2017) Quan Wang, Carlton Downey, Li Wan, P. A. Mansfield, and Ignacio Lopez-Moreno. 2017\.  Speaker diarization with lstm.  _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 5239–5243.
  * Yu et al. (2022) Fan Yu, Shiliang Zhang, Yihui Fu, Lei Xie, Siqi Zheng, Zhihao Du, Weilong Huang, Pengcheng Guo, Zhijie Yan, Bin Ma, Xin Xu, and Hui Bu. 2022.  M2met: The icassp 2022 multi-channel multi-party meeting transcription challenge.  In _ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pages 6167–6171.
  * Yu et al. (2021) Ya-Qi Yu, Siqi Zheng, Hongbin Suo, Yun Lei, and Wu-Jun Li. 2021.  Cam: Context-aware masking for robust speaker verification.  In _IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2021, Toronto, ON, Canada, June 6-11, 2021_ , pages 6703–6707. IEEE.
  * Zhang and Koishida (2017) Chunlei Zhang and Kazuhito Koishida. 2017.  End-to-end text-independent speaker verification with triplet loss on short utterances.  In _Interspeech_.
  * Zhang et al. (2018) Rui Zhang, Honglak Lee, Lazaros Polymenakos, and Dragomir Radev. 2018.  Addressee and response selection in multi-party conversations with speaker interaction rnns.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 32.
  * Zheng et al. (2021) Siqi Zheng, Weilong Huang, Xianliang Wang, Hongbin Suo, Jinwei Feng, and Zhijie Yan. 2021.  A real-time speaker diarization system based on spatial spectrum.  In _IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2021, Toronto, ON, Canada, June 6-11, 2021_ , pages 7208–7212. IEEE.
  * Zheng and Suo (2022) Siqi Zheng and Hongbin Suo. 2022.  Reformulating speaker diarization as community detection with emphasis on topological structure.  In _IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2022, Virtual and Singapore, 23-27 May 2022_ , pages 8097–8101. IEEE.
  * Zheng et al. (2022) Siqi Zheng, Hongbin Suo, and Qian Chen. 2022.  PRISM: pre-trained indeterminate speaker representation model for speaker diarization and speaker verification.  In _Interspeech 2022, 23rd Annual Conference of the International Speech Communication Association, Incheon, Korea, 18-22 September 2022_ , pages 1431–1435. ISCA.
  * Zhong et al. (2021) Ming Zhong, Da Yin, Tao Yu, Ahmad Zaidi, Mutethia Mutuma, Rahul Jha, Ahmed Hassan Awadallah, Asli Celikyilmaz, Yang Liu, Xipeng Qiu, and Dragomir R. Radev. 2021.  Qmsum: A new benchmark for query-based multi-domain meeting summarization.  In _North American Chapter of the Association for Computational Linguistics_.
  * Zuluaga-Gomez et al. (2021) Juan Zuluaga-Gomez, Seyyed Saeed Sarfjoo, Amrutha Prasad, Iuliia Nigmatulina, Petr Motlícek, Karel Ondrej, Oliver Ohneiser, and Hartmut Helmke. 2021.  Bertraffic: Bert-based joint speaker role and speaker change detection for air traffic control communications.

##  Appendix A Metrics

The performance metrics for speaker diarization is concatenated minimum-
permutation Word error rate(cpWER). The cpWER is computed as follows:

  1. 1.

Concatenate each speaker’s utterances both from reference and hypothesis
results.

  2. 2.

Compute the WER between the reference and all possible speaker permutation of
hypothesis.

  3. 3.

Choose the lowest WER among the results from all the speaker permutations as
the final cpWER.

Specifically, the Word Error Rate(WER) is calculated by:

$E_{wer}=\frac{N_{Ins}+N_{Subs}+N_{Del}}{N_{Total}}\times 100\%$ (4)

The cpWER is affected both by the speaker diarization system and speech
recognition system.

Note that the number of speakers in the system result and the reference result
are not equal. If we ignore the cpWER measured by this part of the text, we
will record it as $E_{cp-matched}$. If we think that this part of the text
should be considered all wrong, we will record the cpWER this time for $E_{cp-
all}$.

We know that WER calculates the minimum edit distance from the system result
to the reference result. In the calculation process, the cost of changing one
sequence into another sequence by using three operations of insertion,
deletion and replacement is counted, while the calculation of cpWER
additionally introduces the operation of modifying the speaker ID of a word to
make two speaker-labeled texts become consistent.

Since the speaker diarization system cannot modify the text results of speech
recognition, we calculate the speaker-WER by removing the errors caused by ASR
results from cpWER. Compared to the three string operations in WER, we
additionally define an operation to convert one text result with speaker ID to
another by modifying the speaker ID of a word. Similar to the WER algorithm,
we use dynamic programming to count the number of operations for changing the
speaker ID. The speaker-WER results is calculated by:

$E_{speakerWER}=\frac{N_{Spk-Cost}}{N_{Total}}$ (5)

##  Appendix B Pseudocodes to Update Speaker Diarization from Semantic results

The following psuedocodes show how results from Dialogue Detection and Speaker
Turn Detection are utilized to re-adjust and update speaker diarization
results.

We create a sequence of speaker change occurrences,
$P_{stp}=\\{p_{1},p_{2},...,p_{N}\\}$, where $N$ is the number of speaker
changes, by combining the results of the Dialogue Detection and Speaker-Turn
Detection tasks.

We propose a split process, as shown in Algorithm 1, to adjust the speaker IDs
of certain segments to align the cluster results with the speaker change
results as closely as possible by increasing the number of speakers
appropriately.

Algorithm 1 The Semantic Split Process

$P_{stp}=\\{p_{1},p_{2},...,p_{N}\\}$ The set of the speaker turn point,
$p_{i}\in\\{0,1\\}$

$D=\\{d_{1},d_{2},...,d_{N}\\}$ Each text segments divided by speaker change,

$E=\\{e_{1},e_{2},...,e_{N}\\}$ Set of mean embedding belong to $d_{k}$,

$\tau_{split}$ The split threshold

$\hat{S}\leftarrow[1]$, $B\leftarrow[e_{1}]$, $N_{spk}\leftarrow 1$,
$i\leftarrow 2$

while $i\leq N$ do

$\text{dist}=f(e_{i},B)\in R^{\mathop{Size}(B)}$

if $p_{i}=1$ then

if $\text{dist}<\tau_{split}$ then

$s_{i}=\mathop{argmin}_{s_{i}\in S}\text{dist}$

$\hat{S}\leftarrow\hat{S}\text{.append}(s_{i})$

else

$N_{spk}\leftarrow N_{spk}+1$

$s_{i}\leftarrow N_{spk}$

$\hat{S}\leftarrow\hat{S}\text{.append}(s_{i})$

end if

else

$s_{i}\leftarrow s_{i-1}$

$\hat{S}\leftarrow\hat{S}\text{.append}(s_{i})$

end if

$B\leftarrow B\cup e_{i}$

end while

After the split process, a merge process is implemented to eliminate redundant
speaker IDs. We consider both acoustic information, such as the similarity
distance between speaker embeddings, and semantic information, such as the
score differences between the merge of speakers $i$ and $j$, computed from the
Dialogue-Detection and Speaker-Turn Detection results of the utterances with
the speaker IDs in two adjacent text segments. The pseudocode for the merge
process is outlined in Algorithm 2.

Algorithm 2 The Semantic Merge Process

$E$ Set of mean embedding of all the speakers

$S$ Set of speakers after split process

$\tau_{\text{merge}}$ the merge threshold

while Can Not Merge do

$\text{cost}_{sim}\leftarrow\mathop{cosine\\_distance}(E)$

$\text{cost}_{dd}\leftarrow\text{score}_{s}^{\text{after}}-\text{score}_{s}^{\text{before}}$

$i_{merge},j_{merge}=\mathop{argmin}_{i,j\in
S}(\text{cost}_{sim}+\text{cost}_{s})$

if $\text{cost}_{\text{all}}[i_{merge},j_{merge}]<\tau_{merge}$ then

update $S\text{ by merge speaker }i_{merge},j_{merge}$

update $E\text{ by merge speaker }i_{merge},j_{merge}$

end if

end while


#  INR-TH-2023-005 Resonant generation of electromagnetic modes
in nonlinear electrodynamics:
Quantum perturbative approach

Ilia Kopchinskii1,2 , Petr Satunin2,1
1 Moscow State University,
Leninskiye Gory, 119991 Moscow, Russia
2 Institute for Nuclear Research of the Russian Academy of Sciences,
60th October Anniversary Prospect, 7a, 117312 Moscow, Russia e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The paper studies resonant generation of higher-order harmonics in a closed
cavity in Euler-Heisenberg electrodynamics from the point of view of pure
quantum field theory. We consider quantum states of the electromagnetic field
in a rectangular cavity with conducting boundary conditions, and calculate the
cross-section for the merging of three quanta of cavity modes into a single
one ($3\to 1$ process) as well as the scattering of two cavity mode quanta
($2\to 2$ process). We show that the amplitude of the merging process vanishes
for a cavity with an arbitrary aspect ratio, and provide an explanation based
on plane wave decomposition for cavity modes. Contrary, the scattering
amplitude is nonzero for specific cavity aspect ratio. This $2\to 2$
scattering is a crucial elementary process for the generation of a quantum of
a high-order harmonics with frequency $2\omega_{1}-\omega_{2}$ in an
interaction of two coherent states of cavity modes with frequencies
$\omega_{1}$ and $\omega_{2}$. For this process we calculate the mean number
of quanta in the final state in a model with dissipation, which supports the
previous result of resonant higher-order harmonics generation in an effective
field theory approach [1].

##  1 Introduction

Nonlinear interaction of electromagnetic field in vacuum, induced by the
interaction with virtual electrons, is one of the phenomena theoretically
predicted at the dawn of quantum electrodynamics (QED) [2, 3] but have not
been experimentally detected yet due to the weakness of the nonlinear
interaction. Specifically, nonlinear dynamics of a strong electromagnetic
field is governed by the effective Euler-Heisenberg Lagrangian [3]. In
addition to the QED contribution the theory may obtain correction from
hypothetical photophilic scalar and pseudoscalar particles [4, 5, 6].

The variety of phenomena predicted by nonlinear electrodynamics include both
perturbative and nonperturbative effects (see e. g. [7, 8, 9, 10] for a
review). A significant part of perturbative effects resembles similar effects
in nonlinear optical crystals, which include vacuum birefringence of a photon
in an external field [11, 12], and light-by-light scattering [2, 3, 13]. The
latter effect can be tested at high-intensity laser facilities (ELI [14],
CoReLS [15] etc) by an experiment of three laser beam interaction [16, 17,
18], see also [19, 20, 21] for related processes.

An alternative approach to test the generation of high-order harmonics is to
study this effect for standing electromagnetic modes in high-quality microwave
cavities instead of the traveling waves of lasers [22, 23]. Although the
electromagnetic fields in cavities reach weaker (but still strong) values than
in lasers, the signal field may be additionally enhanced due to the resonant
character of the interaction which does not occur in the region of lasers
cross-section. Assuming specific set of two pump modes of frequencies
$\omega_{1}$ and $\omega_{2}$ in the cavity of specific resonant aspect ratio,
one obtains resonant enhancement for a mode of frequency
$2\omega_{1}-\omega_{2}$ [23, 24]. Based on this effect, an experimental
proposal aimed on search for nonlinear electrodynamics has been suggested [24,
25] which is under design [26] in Superconducting Quantum Materials and
Systems center (SQMS) [27] at Fermi National Accelerator Laboratory.

Another interesting issue is to study this nonlinear process for an arbitrary
set of cavity modes, which can be done analytically for the rectangular shape
of the cavity. Thus, when studying nonlinear effects for a single pump mode in
one-dimensional cavity it turns out that there is no resonant generation of
the third harmonics [28]. The generalization of the method to the case of two
pump modes (frequencies $\omega_{1}$ and $\omega_{2}$) and three-dimensional
rectangular cavity is given in [1]. It turns out that the third harmonic as
well as the “plus” signal mode are not resonantly amplified in the cavity, the
only resonant case is the “minus” signal mode of frequency
$2\omega_{1}-\omega_{2}$. Aforementioned calculations were performed in the
effective field theory approach, in which the classical nonlinear wave
equations were solved. It implies that the electromagnetic states are
classical states. Remarkably, the final state of higher-order harmonics
generation can include very small number of quanta; so one may expect the
breakdown of the classical effective field theory approach. Alternatively, one
may apply pure quantum approach based on the calculation of the corresponding
matrix elements, which is the purpose of current work. We expect that the
quantum calculation will shed more light on the results obtained in the
classical calculations.

The difference with the standard quantum perturbative calculation is that the
quantum states are eigenstates of the cavity, not the plane waves. The
quantization of electromagnetic modes in cavities was previously considered in
the context of the Casimir effect [29, 30], see also [31] for the symplectic
quantization. Although the classical electromagnetic modes relate to the
coherent quantum states, we first calculate simple one-particle amplitudes for
$3\to 1$ merging and $2\to 2$ scattering processes. We show that $3\to 1$
merging amplitude vanishes so the third harmonics generation does not occur;
$2\to 2$ scattering, in turn, rules the generation of the “minus” signal mode.

The paper is organized as follows: in section 2 we outline the results of
quantization of the free electromagnetic field in rectangular one-dimensional
and three-dimensional perfectly conducting cavities. In section 3 we include
Euler-Heisenberg self-interaction of the electromagnetic field into the
Lagrangian and treat it perturbatively with modified $\sf S$-matrix formalism.
Several $\sf S$-matrix elements are calculated, which correspond to the key
processes discussed on the classical level in [1]. Finally, the interpretation
of results is given in section 4.

##  2 Quantum excitations of electromagnetic field in rectangular cavity

In this section we consider the quantization procedure for free
electromagnetic field in a closed cavity with perfectly conducting boundary
conditions, and obtain the spectrum of quantum excitations in one-dimensional
and three-dimensional rectangular cavities (cf. [29, 30, 31]). Generally, the
free Lagrangian of the electromagnetic field and perfectly conducting boundary
conditions read111Natural system of units $\hbar=c=1$ is assumed.,

$\displaystyle\mathcal{L}_{0}=-\frac{1}{4}\quantity\big(F_{\mu\nu})^{2},$ (1)
$\displaystyle\evaluated{\mathbf{E}\dotproduct{\bf\it\tau}}_{C}$
$\displaystyle=0,\qquad\evaluated{\mathbf{B}\dotproduct\mathbf{n}}_{C}=0,$ (2)

where $F_{\mu\nu}$ is a 4-tensor of electromagnetic field, whose components
are taken from the strength of electric field $\mathbf{E}$ and the force of
magnetic field $\mathbf{B}$. The perfectly conducting boundary conditions (2)
are posed on the projections of field vectors onto the tangent vector
${\bf\it\tau}$ and normal vector $\mathbf{n}$ evaluated at the surface $C$ of
the cavity.

The absence of free charges in the cavity allows us to fix both the Coulomb
gauge $\divergence\mathbf{A}(t,\mathbf{r})=0$, and additionally $A_{0}=0$. Let
us suppose that $\quantity{{\bf\it\mathcal{A}}_{n}(\mathbf{r})}$ is a complete
orthonormal system of eigenfunctions, satisfying the perfectly conducting
boundary conditions. Here $n$ is an integer number in case of a single finite
dimension, or a set of integers in case of larger number of finite dimensions.
The vector-potential is decomposed into discrete Fourier series over spatial
variable $\mathbf{r}$,

$\mathbf{A}(t,\mathbf{r})=\mathbf{A}^{+}(t,\mathbf{r})+\mathbf{A}^{-}(t,\mathbf{r}),\qquad\mathbf{A}^{\pm}(t,\mathbf{r})=\frac{1}{\sqrt{V}}\sum\limits_{n}a^{\pm}_{n}{\bf\it\mathcal{A}}_{n}(\mathbf{r})\frac{e^{\pm
i\omega_{n}t}}{\sqrt{2\omega_{n}}},$ (3)

where $V$ is the cavity’s volume. The dimensionless amplitudes $a^{\pm}_{n}$
in the quantization procedure become the creation and annihilation operators
$\hat{a}^{\pm}_{n}$ satisfying commutation relations
$\commutator{\hat{a}^{-}_{n}}{\hat{a}^{+}_{k}}\equiv\delta_{nk}$. The
operators $\hat{a}^{\pm}_{n}$ act on the Fock space containing the vacuum
state $\ket{0}$, which is vanished by acting the annihilation operator
$\hat{a}^{-}_{n}\ket{0}=0$ for arbitrary $n$; $m$-particle states
$\ket{m_{n}}$ are built with $a^{+}_{n}$ acting on the vacuum $\ket{0}$,
$\ket{m_{n}}=\frac{\quantity(a_{n}^{+})^{m}}{\sqrt{m!}}\ket{0}$. Classical
electromagnetic waves are related to the coherent states $\ket{\xi_{n}}$
defined as $\ket{\xi_{n}}=e^{-\frac{\absolutevalue{\xi}^{2}}{2}}e^{\xi
a^{+}_{n}}\ket{0}$.

###  2.1 One-dimensional cavity

Let us consider free electromagnetic field confined in an effective one-
dimensional cavity $D=(0,L_{x})$ between two large perfectly conducting planes
of area $S=L_{y}L_{z}$ such that $\sqrt{S}\gg L_{x}$, and volume $V=L_{x}S$.
The vector-potential depends only on $(t,x)$ coordinates, so that condition
$\divergence{\mathbf{A}}=0$ implies $\mathbf{A}\perp\mathbf{e}_{x}$. Electric
boundary conditions
$\evaluated{\mathbf{E}\dotproduct\mathbf{e}_{x}}_{x=0,L_{x}}=0$ translate
literally for the vector-potential, which is therefore decomposed into sine
eigenfunctions,

$A_{i}^{\pm}(t,x)=\sqrt{\frac{2}{V}}\sum\limits_{n=1}^{\infty}a^{\pm}_{i,n}\sin(k_{n}x)\frac{e^{\pm
i\omega_{n}t}}{\sqrt{2\omega_{n}}},\qquad k_{n}=\omega_{n}=\frac{\pi
n}{L_{x}},\qquad i\in\\{y,z\\}.$ (4)

Two independent ($y$ and $z$ directed) components are related to two linear
polarizations. Any one-particle state for fixed $n$ in one-dimensional cavity
is expressed as a linear combination of two orthogonal states with $y$ and $z$
linear polarizations, $\ket{1_{n,y}}=a^{+}_{n,y}\ket{0}$ and
$\ket{1_{n,z}}=a^{+}_{n,z}\ket{0}$.

Note that the cavity modes are not the eigenstates of the spatial momentum
operator. However, they can be easily decomposed to two plane waves traveling
in opposite directions which are the spatial momentum eigenstates,
$\ket{1_{n}}=\frac{1}{2i}\quantity(\ket{k_{n}}-\ket{-k_{n}})$ (polarization
index is omitted). Here $\ket{k_{n}}$ denote the state with definite momentum
$(k_{n},0,0)$. In turn, the plane waves $\ket{k_{n}}$ do not satisfy the
conducting boundary conditions so cannot be considered as the cavity
eigenstates.

###  2.2 Rectangular cavity

The following case is a three-dimensional rectangular cavity of dimensions
$L_{x},L_{y},L_{z}$ with perfectly conducting walls. The volume of the cavity
is $V=L_{x}L_{y}L_{z}$. The electromagnetic potential $\mathbf{A}$ decomposes
into cavity eigenmodes ${\bf\it\mathcal{A}}^{\lambda}_{npq}(\mathbf{r})$ which
are conventionally grouped into TE and TM modes,

$\mathbf{A}^{\pm}(t,\mathbf{r})=\sqrt{\frac{1}{V}}\sum\limits_{\lambda,npq}^{\infty}a^{\lambda\pm}_{npq}{\bf\it\mathcal{A}}^{\lambda}_{npq}(\mathbf{r})\frac{e^{\pm
i\omega_{npq}t}}{\sqrt{2\omega_{npq}}},\qquad\omega_{npq}=\sqrt{\frac{\pi^{2}n^{2}}{L_{x}^{2}}+\frac{\pi^{2}p^{2}}{L_{y}^{2}}+\frac{\pi^{2}q^{2}}{L_{z}^{2}}},\qquad\lambda\in\\{\text{TM},\,\text{TE}\\}.$
(5)

Here $n,p,q$ are integer numbers related to the harmonics number over each
spatial dimension. The spectrum $\omega_{npq}$ is degenerate if all $n,p,q$
are nonzero: $\mbox{TE}_{npq}$ and $\mbox{TM}_{npq}$ are two independent
solutions related to the same eigenfrequency, so the TE/TM property may be
treated as a polarization basis. The orthonormal eigenfunctions of the
rectangular cavity ${\bf\it\mathcal{A}}^{\lambda}_{npq}(\mathbf{r})$ are
proportional to the dimensionless eigenfunctions of the electric field, see
[32],

$\scriptsize{\bf\it\mathcal{A}}^{\text{TM}}_{npq}(\mathbf{r})=\frac{\sqrt{4(2-\delta_{q0})}}{\omega_{npq}}\begin{pmatrix}\dfrac{k_{x}k_{z}}{\sqrt{k_{x}^{2}+k_{y}^{2}}}\cos(k_{x}x)\sin(k_{y}y)\sin(k_{z}z)\\\
\dfrac{k_{y}k_{z}}{\sqrt{k_{x}^{2}+k_{y}^{2}}}\sin(k_{x}x)\cos(k_{y}y)\sin(k_{z}z)\\\
-\sqrt{k_{x}^{2}+k_{y}^{2}}\sin(k_{x}x)\sin(k_{y}y)\cos(k_{z}z)\end{pmatrix}\\!\\!,\
{\bf\it\mathcal{A}}^{\text{TE}}_{npq}(\mathbf{r})=\\!\sqrt{4(2-\delta_{n0}-\delta_{p0})}\begin{pmatrix}+\dfrac{k_{y}}{\sqrt{k_{x}^{2}+k_{y}^{2}}}\cos(k_{x}x)\sin(k_{y}y)\sin(k_{z}z)\\\
-\dfrac{k_{x}}{\sqrt{k_{x}^{2}+k_{y}^{2}}}\sin(k_{x}x)\cos(k_{y}y)\sin(k_{z}z)\\\
0\end{pmatrix},$

where $k_{x}(n)=\frac{\pi n}{L_{x}}$, $k_{y}(p)=\frac{\pi p}{L_{y}}$ and
$k_{z}(q)=\frac{\pi q}{L_{z}}$. Here the normalization factors
$\sqrt{4(2-\delta_{q0})}$ and $\sqrt{4(2-\delta_{n0}-\delta_{p0})}$ are
introduced to satisfy the normalization condition
$\norm{{\bf\it\mathcal{A}}^{\lambda}_{npq}}^{2}=V$ for zero and nonzero
wavenumbers simultaneously. One-particle states are produced by the action of
the creation operator, $\ket{1^{\lambda}_{npq}}=a^{\lambda+}_{npq}\ket{0}$.
Similar to the one-dimensional cavity case, the rectangular cavity eigenstates
decompose to finite (now $8$) number of plane waves.

##  3 Transition amplitudes in the presence of the electromagnetic field self-
interaction

In this section we discuss the electromagnetic field self-interaction, and
calculate certain transition amplitudes between cavity quantum states. The
self-interaction of the electromagnetic field is described by a quartic
nonlinear Lagrangian,

$\mathcal{L}_{\text{int}}=\kappa\left((F_{\mu\nu}F^{\mu\nu})^{2}+\beta(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}\right)=4\kappa\quantity(\mathbf{E}^{4}-2\mathbf{B}^{2}\mathbf{E}^{2}+\mathbf{B}^{4}+4\beta(\mathbf{B}\mathbf{E})^{2}).$
(6)

The values $\kappa=\frac{\alpha_{e}^{2}}{90m_{e}^{4}}$ and $\beta=\frac{7}{4}$
related to the standard Euler-Heisenberg Lagrangian [3] are obtained by
integrating out electrons in pure QED. Nevertheless, we keep arbitrary values
of $\kappa$ and $\beta$ in order to take into account possible new physics
contribution. The calculation of certain transition amplitudes
$\innerproduct{f(t_{f})}{i(t_{i})}$ between cavity eigenstates $\ket{i}$ and
$\ket{f}$ for a time interval $(t_{f}-t_{i})$ can be performed in the
perturbative approach assuming $\mathcal{L}_{\text{int}}$ to be small. The
transition probability $w_{fi}$ is a squared modulus of the amplitude,
$w_{fi}(t_{f}-t_{i})=|\innerproduct{f(t_{f})}{i(t_{i})}|^{2}$.

An essential difference from the standard perturbation theory in relativistic
quantum field theory (QFT) is that the initial and final states are no longer
plane waves but cavity modes captured in a closed volume. Thus, one cannot
straightly define asymptotic states at spatial infinity which are necessary
for the standard S-matrix formalism. In order to solve it one can either
construct the perturbation theory for cavity amplitudes from the first quantum
mechanical principles or try to reduce the task to the standard one. The
latter approach is simpler, so we apply it assuming $t_{f(i)}\to+(-)\infty$.
As long as the nonlinear Euler-Heisenberg term (6) remains small in comparison
to the classical linear term (1), i.e. $\kappa\mathbf{B}^{4}\ll\mathbf{B}^{2}$
or $\absolutevalue{\mathbf{B}}\ll m_{e}^{2}/\alpha_{e}\sim 10^{9}$ T,

we can decompose the cavity modes as linear combinations of plane waves
$\ket{i}=\Sigma_{n}c^{i}_{n}\ket{k_{n}}$222Here $n$ is a single integer in
case of a single closed dimension, and a set of integers in case of more
closed dimensions.. Then the transition amplitude
$\innerproduct{f(t_{f})}{i(t_{i})}$ is expanded into the linear combination of
transition amplitudes between plane wave states which are expressed as $\sf
S$-matrix elements,

$\innerproduct{f(t_{f})}{i(t_{i})}=\Sigma_{nm}(c^{f}_{m})^{*}c^{i}_{n}\matrixelement{k_{m}}{\sf
S}{k_{n}}.$ (7)

The following calculations may proceed in two ways. First, one can calculate
plane wave amplitudes and take a sum (7) over them with the appropriate
coefficients. The complexity of the summation quadratically grows with the
number of plane waves in the decomposition. Alternatively, due to the
linearity of (7) with respect to the states, the sums can be taken forming the
initial and final states back, $\matrixelement{f}{\sf S}{i}$; the $\sf
S$-matrix and the corresponding matrix element read the standard form,

${\sf
S}=\mathrm{Texp}\,\quantity(i\int{:}\mathcal{L}_{\text{int}}(x){:}\differential[4]{x})\quad\Rightarrow\quad\matrixelement{f}{\sf
S}{i}=i\int\matrixelement{f}{{:}\mathcal{L}_{\text{int}}(x){:}}{i}\differential[4]{x}+\order{\kappa^{2}},$
(8)

where we expand the T-exponent up to the first non-vanishing term
$\propto\kappa$. In order to perform the calculation, one needs to determine
the Wick’s contractions of ladder operators and electromagnetic fields, for
one-dimensional cavity,

$\wick[below]{\c{a}^{-}_{i,n}\,\c{E}_{j}(t,x)}=\delta_{ij}\,i\sqrt{\frac{\omega_{n}}{V}}\sin(k_{n}x)\,e^{i\omega_{n}t},\qquad\wick[below]{\c{a}^{-}_{z,n}\,\c{B}_{y}(t,x)}=-\sqrt{\frac{\omega_{n}}{V}}\cos(k_{n}x)\,e^{i\omega_{n}t}=-\wick[below]{\c{a}^{-}_{y,n}\,\c{B}_{z}(t,x)},$
(9)

and for rectangular cavity,

$\wick[below]{\c{a}^{\lambda-}_{npq}\,\c{\vb
E}(t,\mathbf{r})}=i\sqrt{\frac{\omega_{npq}}{2V}}{\bf\it\mathcal{A}}^{\lambda}_{npq}(\mathbf{r})\,e^{i\omega_{npq}t},\qquad\wick[below]{\c{a}^{\lambda-}_{npq}\,\c{\vb
B}(t,\mathbf{r})}=\frac{1}{\sqrt{2\omega_{npq}V}}\curl{\bf\it\mathcal{A}}^{\lambda}_{npq}(\mathbf{r})\,e^{i\omega_{npq}t}.$
(10)

These contractions can be straightforwardly verified by substituting electric
and magnetic fields’ expansions (based on (4), (5)) into the l.h.s of (9),
(10).

Computing the $\sf S$-matrix element (8) with the Wick contractions given in
(9), (10), one integrates over the four-dimensional volume. The spatial
integration $\int d^{3}x$ is carried out over the cavity modes in the volume
of the cavity; the time integration is carried out over the time oscillating
exponents for initial and final states,
$\int_{-\infty}^{+\infty}dt\;e^{i(\Sigma\omega_{i}-\Sigma\omega_{f})t}=2\pi\delta(\Sigma\omega_{i}-\Sigma\omega_{f})$,
giving the delta-function which represents the energy conservation. In the
standard scattering problem (see a textbook [33]) the factor $2\pi\delta(0)$
is interpreted as the time of the interaction $T$ between wavepackets which
goes to infinity in case of plane waves; the scattering probability grows
linearly with $T$.

Two changes appear in our task. First, the interaction time is finite and
should be bounded by a coherence time,
$\int_{t_{coh}}dt\;e^{i(\Sigma\omega_{i}-\Sigma\omega_{f})t}=t_{coh}$ if
$\Sigma\omega_{i}=\Sigma\omega_{f}$. Second, one should also take into account
dissipation which is introduced as a small imaginary additive to frequency,
$\omega_{f}-i\frac{\omega_{f}}{Q}$. Thus, the time integral gives
$\max\quantity(t_{coh},\frac{Q}{\omega_{f}})$.

This treatment of dissipation is valid in the limit of high quality factor of
the cavity $Q\gg 1$. It ceases to be applicable when the field cannot be
quantized as a set of harmonic oscillators (3), since interaction with the
walls (being source of dissipation) makes the oscillations sufficiently an-
harmonic. In general case the dissipation can be treated rigorously on quantum
level, e.g. within the non-equilibrium diagram technique, although we restrict
our treatment to the case $Q\gg 1$.

###  3.1 The merging process $3\to 1$ in one-dimensional cavity

Let us first consider the linear cavity, and calculate the amplitude for the
merging of three quanta of pump modes into a single quantum of signal mode.
The cases of three quanta of the same pump mode, as well as two quanta of one
pump mode $\omega_{1}$ and a quantum of a mode $\omega_{2}$ can be considered
simultaneously. Additionally, any quantum state may be taken of different
polarization. Generally, the initial and final quantum field states read,

$\ket{i}=\ket{1^{i}_{n}}\otimes\ket{1^{j}_{n}}\otimes\ket{1^{l}_{p}}=a^{+}_{i,n}a^{+}_{j,n}a^{+}_{l,p}\ket{0},\qquad\qquad\ket{f}=\ket{1^{s}_{2n+p}}=a^{+}_{s,2n+p}\ket{0}.$
(11)

The indices $n$ and $p$ relate to wavenumbers of two pump modes ($n=p$ means
the case of single pump mode); the indices $i,j,l,s$ relate to polarization
for every modes. The calculation of the $\sf S$-matrix element proceeds as
follows:

$\matrixelement{f}{\sf
S}{i}=i\kappa\int\limits_{-\infty}^{+\infty}\differential{t}\iint\differential{S}\int\limits_{0}^{L_{x}}\differential{x}\matrixelement{f}{{:}(\mathbf{E}\mathbf{E})^{2}-2\mathbf{B}^{2}\mathbf{E}^{2}+(\mathbf{B}\mathbf{B})^{2}+4\beta(\mathbf{B}\mathbf{E})^{2}{:}}{i},$
(12)

where $\iint\differential{S}=S$ denotes the area of transverse cavity
dimensions. Each term in (12) reads (see Appendix A for details),

$\expectationvalue{(\mathbf{E}\mathbf{E})^{2}}=\expectationvalue{(\mathbf{B}\mathbf{B})^{2}}=-\frac{1}{2}\expectationvalue{\mathbf{B}^{2}\mathbf{E}^{2}}=\frac{\sqrt{(2n+p)n^{2}p}\pi^{2}}{L_{x}^{3}}\quantity[\delta_{ij}\delta_{ls}(1+2\delta_{is})+(1-\delta_{ls})(1-\delta_{ij})],$
(13)

and $\langle(\mathbf{E}\mathbf{B})^{2}\rangle=0$. Here no summation over
polarization indices $i,j,l,s$ is assumed. The brackets $\langle...\rangle$
denote the integral over $dx$ for the matrix element of the corresponding
operator. Taking all the contributions together we obtain that the amplitude
vanishes, $\matrixelement{f}{\sf S}{i}=0$, for any $\beta$: the amplitude
components related to two electromagnetic invariants vanish independently,
$\expectationvalue{(F_{\mu\nu})^{4}}=\expectationvalue{(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}}=0$.

Hence, neither a monochromatic excitation of linear cavity $\omega$ cannot
generate the third harmonic $3\omega$ nor two pump modes $\omega_{1}$ and
$\omega_{2}$ cannot merge into a mode $2\omega_{1}+\omega_{2}$ for any
polarization of incoming photons. More insight for the reason of this
prohibition can be obtained with the plane wave decomposition.

#### Plane wave decomposition.

Let us provide a more clear interpretation for the reason of non-generation of
the third harmonics or “plus” combined signal mode. The amplitude
$\matrixelement{f}{\sf S}{i}=\matrixelement{1^{s}_{2n+p}}{\sf
S}{1^{i}_{n},1^{j}_{n},1^{l}_{p}}$ decomposes (see Sec. 2.1) into 16 plane
wave amplitudes,

$\matrixelement{f}{\sf
S}{i}=\sum_{(\pm)^{4}}\frac{(-1)^{\pm}}{(2i)^{4}}\matrixelement{\pm
k^{s}_{2n+p}}{\sf S}{\pm k_{n}^{i},\pm k_{n}^{j},\pm k_{p}^{l}},$ (14)

where $(-1)^{\pm}=1$ in case of even number of signs “$+$” in the notations
for all initial and final plane wave states, and $-1$ otherwise. The momentum
conservation requires (see eq. 4) that all signs in the states notation should
coincide, “$+$” or “$-$” simultaneously. Hence, the only remaining nonzero
terms in (14) are

$\matrixelement{f}{\sf
S}{i}=\frac{1}{(2i)^{4}}\matrixelement{k^{s}_{2n+p}}{\sf
S}{k_{n}^{i},k_{n}^{j},k_{p}^{l}}+\frac{1}{(2i)^{4}}\matrixelement{-k^{s}_{2n+p}}{\sf
S}{-k_{n}^{i},-k_{n}^{j},-k_{p}^{l}}=\frac{2}{(2i)^{4}}\matrixelement{k^{s}_{2n+p}}{\sf
S}{k_{n}^{i},k_{n}^{j},k_{p}^{l}}.$ (15)

Thus, the 4-point amplitude describing $3\to 1$ merging process for cavity
eigenstates reduces to the corresponding amplitude for plane waves; all plane
wave momenta must be parallel. The plane wave matrix element, in turn, satisfy
Lorentz Invariance. Hence, the amplitude should contain only Lorentz scalars
which can be only the scalar products of momenta and polarization vectors. For
massless transverse photons all these scalar products vanish at the kinematic
configuration of parallel momenta.

#### Plane wave decomposition for three-dimensional cavity.

Aforementioned consideration can be in a direct way generalized to the case of
$3\to 1$ merging process
$\matrixelement{1^{s}_{(2n+n^{\prime}),(2p+p^{\prime}),(2q+q^{\prime})}}{\sf
S}{2^{\lambda}_{npq},1^{\lambda^{\prime}}_{n^{\prime}p^{\prime}q^{\prime}}}$333Both
the third harmonics generation ($n^{\prime}=n,p^{\prime}=p,q^{\prime}=q$) and
the generation for a mode of combined frequency $2\omega_{1}+\omega_{2}$ are
considered. in three-dimensional rectangular cavity. In fact, although the
plane wave decomposition give $8^{4}$ plane wave amplitudes, due to the
momentum conservation the only nonzero terms contain only the states of
parallel momenta, which in turn vanishes due to the reasons given in the
previous paragraph.

###  3.2 The scattering process $2\to 2$ in three-dimensional cavity

The classical EFT analysis [1], supported with the quantum calculation of Sec.
3.1, shows that in three-dimensional cavity with pump modes of frequencies
$\omega_{1}$ and $\omega_{2}$ there is no resonant generation of signal modes
$3\omega_{1}$ ($3\omega_{2}$) and $2\omega_{1}+\omega_{2}$
($2\omega_{2}+\omega_{1}$). However, in case of special resonant cavity
geometry there is a classical solution describing resonant generation of the
mode $2\omega_{1}-\omega_{2}$. The energy conservation shows that the energy
of a final quantum in the merging process of three quanta into a single one is
a sum of incoming quanta energies $2\omega_{1}+\omega_{2}$. This apparent
contradiction can be explained as follows.

The precise quantum correspondent of a classical wave is a coherent state
which is an eigenstate of the electromagnetic field operator but not an
eigenstate of operator of a particle number. Hence, the full number of quanta
in a process involving the coherent states in initial and final states is not
fixed. Since, one may suppose that the $2\omega_{1}-\omega_{2}$ signal mode
generation at a quantum one-particle level is a result of $2\to 2$ “scattering
process” instead of $3\to 1$ merging. The only option for this elementary
$2\to 2$ process, satisfying the energy conservation, is a scattering of two
quanta of the pump mode $\omega_{1}$ to a single quantum of mode $\omega_{2}$
and a quantum of the signal mode $2\omega_{1}-\omega_{2}$. The energy
conservation holds $2\omega_{1}\to\omega_{2}+(2\omega_{1}-\omega_{2})$.

Let us perform the calculation of $2\to 2$ for a particular configuration with
a simple set of modes previously considered in classical effective theory
approach in [1], precisely scattering of $2$ quanta TE011 to a quantum of
TM110 and a quantum of TM130. The energy conservation reads,

$2\omega_{011}=\omega_{110}+\omega_{130}.$ (16)

Considering cavity of dimensions $L_{x}:L_{y}:L_{z}=1:1:r$ and taking into
account the expressions (5) for given mode frequencies

$\omega_{011}=\frac{\pi}{L_{z}}\sqrt{r^{2}+1},\qquad\omega_{110}=\frac{\pi}{L_{z}}\sqrt{2}\,r,\qquad\omega_{130}=\frac{\pi}{L_{z}}\sqrt{10}\,r,$
(17)

one obtains $r=\sqrt{\sqrt{5}-2}$, which is identical to the resonant
condition in the effective theory approach [1]. It is worth mentioning that
the resonant condition can be satisfied in other rectangular configurations
where $L_{x}\neq L_{y}$, the choice (17) is made only for simplicity. The
general analysis of resonant conditions for a rectangular cavity, performed in
[1], suggests that resonance is not prohibited for many other combinations of
pump and signal modes, the selected combination of TE011, TM110 and TM130
being just a particular example.

The initial and final states for the $2\to 2$ process read,

$\ket{i}=\ket{2^{\text{TE}}_{011}}=\frac{1}{\sqrt{2}}\left(a^{\text{TE}+}_{011}\right)^{2}\ket{0},\qquad\ket{f}=\ket{1^{\text{TM}}_{110}}\otimes\ket{1^{\text{TM}}_{130}}=a^{\text{TM}+}_{110}a^{\text{TM}+}_{130}\ket{0}.$
(18)

The matrix element of the $2\to 2$ process $\matrixelement{f}{\sf S}{i}\equiv
2\pi i\delta\left(\Sigma\omega_{i}-\Sigma\omega_{f}\right){\sf M}_{2\to 2}$
reads,

${\sf M}_{2\to
2}=4\kappa\int\limits_{V}\differential^{3}x\matrixelement{f}{\,{:}\mathbf{E}^{4}-2\mathbf{B}^{2}\mathbf{E}^{2}+\mathbf{B}^{4}+4\beta(\mathbf{B}\mathbf{E})^{2}{:}\,}{i}=\expectationvalue{\mathbf{E}^{4}}-2\expectationvalue{\mathbf{B}^{2}\mathbf{E}^{2}}+\expectationvalue{\mathbf{B}^{4}}+4\beta\expectationvalue{(\mathbf{B}\mathbf{E})^{2}},$
(19)

where the brackets around an operator imply the corresponding part of the
matrix element ${\sf M}_{2\to 2}$. Performing the Wick contractions (10) and
integrating over the cavity volume one obtains (see Appendix B for details),

$\displaystyle\expectationvalue{\mathbf{E}^{4}}=-\frac{4\kappa}{\sqrt{2}V}\sqrt{\omega_{011}^{2}\omega_{110}\omega_{130}},\qquad\qquad\expectationvalue{\mathbf{B}^{4}}=\frac{4\kappa}{\sqrt{2}V}\frac{\pi^{4}}{L_{z}^{4}}2r^{2}\frac{2r^{2}-3}{\sqrt{\omega_{011}^{2}\omega_{110}\omega_{130}}},$
(20)
$\displaystyle\expectationvalue{\mathbf{B}^{2}\mathbf{E}^{2}}=\frac{4\kappa}{2\sqrt{2}V}\frac{\pi^{2}}{L_{z}^{2}}\left(4r^{2}\sqrt{\frac{\omega_{011}^{2}}{\omega_{110}\omega_{130}}}-(r^{2}-1)\sqrt{\frac{\omega_{110}\omega_{130}}{\omega_{011}^{2}}}\right),$
(21)
$\displaystyle\expectationvalue{(\mathbf{B}\mathbf{E})^{2}}=\frac{4\kappa}{2\sqrt{2}V}\frac{\pi^{2}}{L_{z}^{2}}r^{2}\left(3\sqrt{\frac{\omega_{011}^{2}}{\omega_{110}\omega_{130}}}+3\sqrt{\frac{\omega_{110}}{\omega_{130}}}-\sqrt{\frac{\omega_{130}}{\omega_{110}}}-\sqrt{\frac{\omega_{110}\omega_{130}}{\omega_{011}^{2}}}\right).$
(22)

Collecting all contributions (20)–(22) together and taking into account the
expressions (16) and (17), one obtains,

${\sf M}_{2\to
2}=\frac{-\kappa}{L_{z}^{5}}\frac{(2\pi)^{2}r^{3}}{\sqrt[4]{5}\sqrt{1+r^{2}}}\quantity[5+2\sqrt{5}-\beta\quantity(\sqrt{1+r^{2}}+\sqrt{2}r)^{2}].$
(23)

The probability for the $2\to 2$ process is a squared modulo of the matrix
element, including the restored energy delta-function part. Following the
dissipation arguments from Sec. 3, the delta-function part should be
interpreted as the mode minimal quality factor, which decreases with the mode
frequency for SRF cavity [34]. Precisely, we replace
$2\pi\delta\left(\Sigma\omega_{i}-\Sigma\omega_{f}\right)$ with
$Q/\omega_{130}$. Finally,

$P_{2\to 2}=\absolutevalue\big{\matrixelement{f}{\sf
S}{i}}^{2}=G_{1}^{2}\frac{\kappa^{2}Q^{2}}{L_{z}^{8}},\quad
G_{1}^{2}=\frac{2^{3}\pi^{2}r^{4}}{5^{3/2}(1+r^{2})}\quantity[5+2\sqrt{5}-\beta\quantity(\sqrt{1+r^{2}}+\sqrt{2}r)^{2}]^{2}.$
(24)

This indicates that a single cavity pump mode can generate two signal modes if
both of them are resonant. This process may be potentially probed
experimentally. However, the effect of Bose enhancement shows that if one of
the signal modes is already excited at the $n$-th level ($n$ quanta), the
corresponding amplitude will be enhanced in $\sqrt{n}$ times. So, we come to
the case of two pump modes implying the “minus” signal mode generation.

###  3.3 “Minus” signal mode generation by the coherent states

In this subsection we apply the $2\to 2$ matrix element to the generation of
one-particle state of the signal mode $\ket{1^{\text{TM}}_{130}}$ in the field
of two coherent states of pump modes. Here we will focus on a small mean
number of signal mode quanta $\expectationvalue{N_{s}}\lesssim 1$ in the
steady regime, so we neglect the processes with two and more quanta of signal
mode in the final state for simplicity444In general, one should take into
account many-particle states of the signal mode as well.. Precisely, the
initial and final states read,

$\displaystyle\ket{i}$
$\displaystyle=\ket{\xi^{\text{TE}}_{011}}\otimes\ket{\eta^{\text{TM}}_{110}}=e^{-\frac{|\xi|^{2}+|\eta|^{2}}{2}}\sum\limits_{i,j=0}^{\infty}\frac{\xi^{i}\eta^{j}}{i!j!}\quantity(a^{\rm
TE+}_{011})^{i}\quantity(a^{\rm TM+}_{110})^{j}\ket{0},$
$\displaystyle\ket{f}$
$\displaystyle=\ket{\xi^{\text{TE}}_{011}}\otimes\ket{\eta^{\text{TM}}_{110}}\otimes\ket{1^{\text{TM}}_{130}}=\ket{i}\otimes\ket{1^{\text{TM}}_{130}}=a^{\text{TM}+}_{130}\ket{i}.$

The parameters $\xi,\eta$ are associated with the mean number of quanta in the
pump modes,

$\expectationvalue{N_{\text{TE}_{011}}}=|\xi|^{2},\qquad\expectationvalue{N_{\text{TM}_{110}}}=|\eta|^{2}.$
(25)

The matrix element related to the coherent generation connected to the $2\to
2$ matrix element as,

${\sf
M}_{\text{coh}}=e^{-|\xi|^{2}-|\eta|^{2}}\int_{V}d^{3}x\sum\limits_{i,j,k,l=0}^{\infty}\frac{\xi^{i}\eta^{j}}{i!j!}\times\frac{\quantity(\xi^{\ast})^{k}\quantity(\eta^{\ast})^{l}}{k!l!}\expectationvalue{a^{\text{TM}-}_{130}\quantity(a^{\text{TE}-}_{011})^{k}\quantity(a^{\text{TM}-}_{110})^{l}\,{:}{\cal
L}_{\rm
int}{:}\,\quantity(a^{\text{TE}+}_{011})^{i}\quantity(a^{\text{TM}+}_{110})^{j}}{0},$
(26)

where ${\cal L}_{\rm int}$ determined in (6). Performing the Wick contractions
similarly to that done in Sec.3.2., we obtain the matrix element (23) with
additional combinatorial factor $li(i-1)$ and the factor
$(l-1)!(i-2)!\delta_{l-1,j}\delta_{k,i-2}$ from remaining after the Wick
contraction field operators in the initial and final states, yielding

${\sf
M}_{\text{coh}}=e^{-|\xi|^{2}-|\eta|^{2}}\sum\limits_{i,j,k,l=0}^{\infty}\frac{\xi^{i}\eta^{j}}{j!}\times\frac{\quantity(\xi^{\ast})^{k}\quantity(\eta^{\ast})^{l}}{k!}\delta_{l-1,j}\delta_{k,i-2}\times\sqrt{2}\,M_{2\to
2}=\xi^{2}\eta^{*}\times\sqrt{2}\,{\sf M}_{2\to 2}.$ (27)

Therefore, the probability for the coherent process reads (recall (25)),

$P_{\text{coh}}=2\expectationvalue{N_{\text{TE}_{011}}}^{2}\cdot\expectationvalue{N_{\text{TM}_{110}}}\cdot
P_{2\to 2}.$ (28)

We want to emphasize the natural appearance of a Bose enhancement factor
$\expectationvalue{N_{\text{TM}_{110}}}$ that relates to the process initiated
by a single pump mode TE011.

The transition probability (28) can be interpreted as the mean steady number
of signal quanta in the final state. Precisely, averaging the operator of the
number of signal mode quanta $\hat{N}_{s}\equiv a^{\dagger}_{s}a_{s}$ (for our
choice of modes $a_{s}=a_{\rm TM_{130}}$) over the initial state $\ket{i}$
includes the time evolution from $t_{i}$ to $t_{f}$ and back,

$\expectationvalue{N_{s}}=\matrixelement{i}{\hat{N}_{s}}{i}=\int
df^{\prime}df^{\prime\prime}\matrixelement{i}{S^{\dagger}}{f^{\prime}}\matrixelement{f^{\prime}}{\hat{N}_{s}}{f^{\prime\prime}}\matrixelement{f^{\prime\prime}}{S}{i}.$
(29)

Here $\ket{f^{\prime}}$ and $\ket{f^{\prime}}$ represent the full number of
states in theory at the moment $t_{f}$. We suppose that the pump mode coherent
states remain unchanged; the only difference are the excited states of the
signal mode. Finally, only $\ket{f^{\prime}}=\ket{f^{\prime\prime}}=\ket{f}$
give nonzero outcome, so we have $\expectationvalue{N_{s}}=P_{\text{coh}}$.

Remind the connection between the mean photon number and the classical
amplitude

$\expectationvalue{N_{p}}=\int_{V}\frac{\mathbf{E}^{2}_{p}+\mathbf{B}^{2}_{p}}{2\omega_{p}}\differential^{3}x=\frac{F_{0}^{2}V}{2\omega_{p}},$
(30)

where $F_{0}$ is the amplitude for electric and magnetic field for each pump
mode. Finally, one obtains

$\expectationvalue{N_{s}}=\tilde{G}_{1}^{2}\times\kappa^{2}Q^{2}F_{0}^{6}L_{z}^{4},\qquad\tilde{G}_{1}^{2}=\frac{4}{(10)^{3/2}\pi
r^{3}(1+r^{2})^{2}}\quantity[5+2\sqrt{5}-\beta\quantity(\sqrt{1+r^{2}}+\sqrt{2}r)^{2}]^{2}.$
(31)

Numerically, one obtains $\tilde{G}_{1}^{2}=3.3$ for $r=\sqrt{\sqrt{5}-2}$ and
$\beta=7/4$. The result (31) coincides in terms of the mean number of signal
quanta (30) with one obtained by solving classical equation in the effective
field theory approach [1].

###  3.4 Towards experimental probes

Alghough the article is mostly devoted to the details of the matrix element
calculation responcible to the signal mode generation, it is worth to review
the perspectives of experimental detection of the nonlinear effect. We follow
[23, 24] and experimental proposals mentioned in [26, 25].

The main issue of experimental detection is to discriminate the signal events
from the thermal noise. For this reason the Dicke radiometer equation is used
in the literature [23, 24],

$SNR=\frac{P_{s}}{T}\sqrt{\frac{t}{B}}=\frac{\langle
N_{s}\rangle}{T}\frac{\omega_{s}}{L_{z}Q}\sqrt{\frac{t}{B}},$ (32)

Here $P_{s}$ is the signal power, $T$ — the cavity temperature, $t$ — is the
total measurement time, and $B$ is the bandwidth which can be taken as
$B=1/t$. Fixing SNR as a characteristic of a given detector, one inverses the
equation (31), and obtain the necessary measuring time necessary to detect
nonlinearity with the coupling constant $\kappa$,

$t=\frac{SNR\,\cdot
T}{\tilde{G}_{1}^{2}QF_{0}^{6}L_{z}^{3}\omega_{s}\kappa_{EH}^{2}}\left(\frac{\kappa_{EH}}{\kappa}\right)^{2},$
(33)

where $\kappa_{EH}=\frac{\alpha_{em}^{2}}{90m_{e}^{4}}$ related to the Euler-
Heisenberg theory. Let us estimate the meausuring time (33) numerically.
Assuming $L_{z}=20$ cm (hence $\omega_{s}=2.4\cdot 10^{-6}$ eV), $F_{0}=0.1$
T, $Q=10^{10}$ and $T=1$ K, $\mathrm{SNR}\simeq 5$, one can estimate that the
Euler-Heisenberg interaction can be detected in $22$ seconds.

Additional issues include the filtering the signal mode from the pump modes.
For this reason the authors of Ref. [24] proposed the special filtering cavity
bottle-shape geometry in which only the signal penetrates into a bottleneck.
The large values for the pump mode field strength can be supported by the
cavity walls since the magnetic field strength of pump modes is supposed to be
less than $0.2$ T which is the critical magnetic field for cavity walls made
from pure Niobium.

##  4 Discussion

We applied the technique of quantum perturbative calculations for
electromagnetic field with nonlinear 4-photon interaction in a closed cavity.
We calculated the amplitudes and probabilities of $3\to 1$ merging and $2\to
2$ scattering elementary processes.

We clarify the following points which remain unclear in the classical
effective theory calculation [1]. First, the matrix elements for the third
harmonics generation and “plus” combination modes vanish since the matrix
elements in plane wave decomposition contain only scalar products of momenta
which vanish in presence of Lorentz invariance.

Secondly, we provide a clear quantum interpretation for the signal mode
“minus” generation by an interaction of two pump modes. Thus, it turns out
that the crucial elementary process is $2\to 2$ process in terms of quanta.
The overall amplitude is nonzero since the coherent states which describe
classical waves contain a fluctuating number of quanta so the whole number
does not have to conserve. We have shown that the same elementary process is
responsible for two signal modes generation by a single pump mode; the process
driven by two pump modes is the same including Bose enhancement.

The developed formalism of perturbative calculations of transition
probabilities between cavity states may be generalized to other choices of
initial and final states, which may include, for example, squeezed states etc.
One more way of generalization is to consider average values of various
operators, correlators (see Schwinger-Keldysh formalism for an interacting
scalar field in one-dimensional cavity [35]). An application of the formalism
to cavity processes including $4$-photon interaction in a nonlinear media may
be also interesting.

#### Acknowledgments

The Authors thank Maxim Fitkevich, Dmitry Kirpichnikov, Dmitry Levkov,
Valery Rubakov, Alexey Rubtsov and Dmitry Salnikov for helpful discussions.
The work is supported by RSF grant 21-72-10151.

##  Appendix A Elements of calculation of the matrix element for the $3\to 1$
merging process

In this Appendix we provide more details for the merging matrix element
calculation (12), whose expansion into the components of electromagnetic field
strength operators has the following form,

$\displaystyle\matrixelement{f}{\sf S}{i}$ $\displaystyle=4i\kappa
S\int\limits_{-\infty}^{+\infty}\differential
t\int\limits_{0}^{L_{x}}\differential
x\expectationvalue{a^{-}_{l,3n}{:}\mathbf{E}^{4}-2\mathbf{B}^{2}\mathbf{E}^{2}+\mathbf{B}^{4}+4\beta(\mathbf{B}\mathbf{E})^{2}{:}\quantity(a^{+}_{i,n})^{2}a^{+}_{j,n}}{0}=$
$\displaystyle=\expectationvalue{\mathbf{E}^{4}}-2\expectationvalue{\mathbf{B}^{2}\mathbf{E}^{2}}+\expectationvalue{\mathbf{B}^{4}}+4\beta\expectationvalue{(\mathbf{B}\mathbf{E})^{2}}=$
(34)
$\displaystyle=\quantity\Big{\expectationvalue{E_{y}^{4}}+2\expectationvalue{E_{y}^{2}E_{z}^{2}}+\expectationvalue{E_{z}^{4}}}\leavevmode\nobreak\
-\leavevmode\nobreak\
2\quantity\Big{\expectationvalue{B_{y}^{2}E_{y}^{2}}+\expectationvalue{B_{y}^{2}E_{z}^{2}}+\expectationvalue{B_{z}^{2}E_{y}^{2}}+\expectationvalue{B_{z}^{2}E_{z}^{2}}}$
$\displaystyle+\leavevmode\nobreak\
\quantity\Big{\expectationvalue{B_{y}^{4}}+2\expectationvalue{B_{y}^{2}B_{z}^{2}}+\expectationvalue{B_{z}^{4}}}\leavevmode\nobreak\
+\leavevmode\nobreak\
4\beta\quantity\Big{\expectationvalue{B_{y}^{2}E_{y}^{2}}+2\expectationvalue{B_{y}E_{y}B_{z}E_{z}}+\expectationvalue{B_{z}^{2}E_{z}^{2}}}.$

Here the brackets for the field strength operators are defined in the first
line of (34). In total, $13$ terms appear in eq. (34). Let us present an
evaluation for one of them:

$\displaystyle\expectationvalue{E_{y}^{4}}$ $\displaystyle=4i\kappa
S\int\limits_{-\infty}^{+\infty}\differential
t\int\limits_{0}^{L_{x}}\differential
x\wick[below]{\expectationvalue{\c{1}a^{-}_{l,3n}\,{:}\c{1}E_{y}\c{3}E_{y}\c{2}E_{y}\c{1}E_{y}{:}\,\c{1}a^{+}_{i,n}\c{2}a^{+}_{i,n}\c{3}a^{+}_{j,n}}{0}}=$
$\displaystyle=4i\kappa S\int\limits_{-\infty}^{+\infty}\differential
t\int\limits_{0}^{L_{x}}\differential
x\,4!\,\quantity(+i\delta_{ly}\sqrt{\frac{\omega_{3n}}{V}}\sin(k_{3n}x)\,e^{+i\omega_{3n}t})\leavevmode\nobreak\
\times$ $\displaystyle\qquad\qquad\times\leavevmode\nobreak\
\quantity(-i\delta_{iy}\sqrt{\frac{\omega_{n}}{V}}\sin(k_{n}x)\,e^{-i\omega_{n}t})^{2}\quantity(-i\delta_{jy}\sqrt{\frac{\omega_{n}}{V}}\sin(k_{n}x)\,e^{-i\omega_{n}t})=$
$\displaystyle=4i\kappa
S\,4!\,\delta_{iy}\delta_{jy}\delta_{ly}\,i(-i)^{3}\,\sqrt{\frac{\omega_{3n}\omega_{n}^{3}}{V^{4}}}\int\limits_{-\infty}^{+\infty}e^{i(\omega_{3n}-3\omega_{n})t}\differential
t\int\limits_{0}^{L_{x}}\sin(k_{3n}x)\sin^{3}(k_{n}x)\differential x=$
$\displaystyle=2\pi\delta(0)\,\frac{12\sqrt{3}i\pi^{2}n^{2}\kappa}{L_{x}^{3}S}\,\delta_{iy}\delta_{jy}\delta_{ly}.$

Here $4!$ is a combinatorial factor arising due to the identity of four
$E_{y}$ operators. At the last step we use $\omega_{n}=k_{n}=\frac{\pi
n}{L_{x}}$ and the similar relation for the $3n$ subscript.

##  Appendix B Details of the matrix element $2\to 2$ calculation

Let us present the calculation for the terms of the matrix element (19) in
more detail. The first term $\expectationvalue{(\mathbf{E}\mathbf{E})^{2}}$
reads,

$\displaystyle\expectationvalue{(\mathbf{E}\mathbf{E})^{2}}:\qquad\qquad$
$\displaystyle\int\differential^{3}{\rm
x}\matrixelement{0}{\tfrac{1}{\sqrt{2}}a^{\text{TE}}_{110}a^{\text{TE}}_{110}\,{:}\mathbf{(EE)(EE)}{:}\,a^{\text{TM}+}_{110}a^{\text{TM}+}_{130}}{0}=(+i)^{2}(-i)^{2}\frac{\sqrt{\omega_{130}\omega_{011}^{2}\omega_{110}}}{\sqrt{2}\sqrt{2^{4}V^{4}}}\times$
$\displaystyle\times$ $\displaystyle\int\differential^{3}{\rm
x}\,8\Bigl{[}\underbrace{({\bf\it\mathcal{A}}^{\text{TM}}_{130}{\bf\it\mathcal{A}}^{\text{TM}}_{110})({\bf\it\mathcal{A}}^{\text{TE}}_{011}{\bf\it\mathcal{A}}^{\text{TE}}_{011})}_{\int=-V/2}\leavevmode\nobreak\
+\leavevmode\nobreak\
2\underbrace{({\bf\it\mathcal{A}}^{\text{TM}}_{130}{\bf\it\mathcal{A}}^{\text{TE}}_{011})({\bf\it\mathcal{A}}^{\text{TM}}_{110}{\bf\it\mathcal{A}}^{\text{TE}}_{011})}_{\int=0}\Bigr{]}=\frac{-\sqrt{\omega_{130}\omega_{011}^{2}\omega_{110}}}{\sqrt{2}V}.$
(35)

Here 8 and 2 are combinatorial coefficients, for shortness we indicate the
result of integration with underbraces. Thus, the second term in (35)
vanishes.

Analogically, the following term
$\expectationvalue{(\mathbf{B}\mathbf{B})^{2}}$ reads,

$\displaystyle\expectationvalue{(\mathbf{B}\mathbf{B})^{2}}:\qquad\int\differential^{3}{\rm
x}\matrixelement{0}{\tfrac{1}{\sqrt{2}}a^{\text{TE}}_{110}a^{\text{TE}}_{110}\,{:}\mathbf{(BB)(BB)}{:}\,a^{\text{TM}+}_{110}a^{\text{TM}+}_{130}}{0}=(+i)^{2}(-i)^{2}\frac{1}{\sqrt{2}\sqrt{2^{4}V^{4}}\sqrt{\omega_{130}\omega_{011}^{2}\omega_{110}}}\times$
$\displaystyle\times\int\differential^{3}{\rm x}\,8\Bigl{[}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{130}\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{110})(\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011}\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011})}_{\int=V/2\cdot\pi^{4}(L_{x}^{-2}+3L_{y}^{-2})(L_{y}^{-2}-L_{z}^{-2})}\leavevmode\nobreak\
+\leavevmode\nobreak\ 2\underbrace{(\rm
rot{\bf\it\mathcal{A}}^{\text{TM}}_{130}\rm
rot{\bf\it\mathcal{A}}^{\text{TE}}_{011})(\rm
rot{\bf\it\mathcal{A}}^{\text{TM}}_{110}\rm
rot{\bf\it\mathcal{A}}^{\text{TE}}_{011})}_{\int=-V/2\cdot\pi^{4}\,L_{x}^{-2}L_{z}^{-2}}\Bigr{]}=$
(36)
$\displaystyle=\frac{\pi^{4}}{\sqrt{2}V}\frac{(L_{x}^{-2}+3L_{y}^{-2})(L_{y}^{-2}-L_{z}^{-2})-2L_{x}^{-2}L_{z}^{-2}}{\sqrt{\omega_{130}\omega_{011}^{2}\omega_{110}}}=\frac{1}{\sqrt{2}V}\frac{\pi^{4}}{L_{z}^{4}}2r^{2}\frac{2r^{2}-3}{\sqrt{\omega_{011}^{2}\omega_{110}\omega_{130}}}.$

The combinatorial coefficients $8$ and $2$ coincide with the previous case.
However, the result of the integration differs. The first of the mixed terms
$\expectationvalue{(\mathbf{B}\mathbf{B})(\mathbf{E}\mathbf{E})}$ reads,

$\displaystyle\expectationvalue{(\mathbf{B}\mathbf{B})(\mathbf{E}\mathbf{E})}:\qquad\qquad\qquad\int\differential^{3}{\rm
x}\matrixelement{0}{\tfrac{1}{\sqrt{2}}a^{\text{TE}}_{110}a^{\text{TE}}_{110}\,{:}\mathbf{(BB)(EE)}{:}\,a^{\text{TM}+}_{110}a^{\text{TM}+}_{130}}{0}=\frac{1}{\sqrt{2^{4}V^{4}}}\,\int\differential^{3}{\rm
x}\,4\,\times$
$\displaystyle\times\Biggl{[}(-i)^{2}\sqrt{\frac{\omega_{011}^{2}}{\omega_{130}\omega_{110}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{130}\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{110})(\mathbf{\mathcal{A}}^{\text{TE}}_{011}\mathbf{\mathcal{A}}^{\text{TE}}_{011})}_{\int=-V/2\cdot\pi^{2}(L_{x}^{-2}+3L_{y}^{-2})}\leavevmode\nobreak\
+\leavevmode\nobreak\
i^{2}\sqrt{\frac{\omega_{130}\omega_{110}}{\omega_{011}^{2}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011}\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011})(\mathbf{\mathcal{A}}^{\text{TM}}_{130}\mathbf{\mathcal{A}}^{\text{TM}}_{110})}_{\int=V/2\cdot\pi^{2}(L_{y}^{-2}-L_{z}^{-2})}\leavevmode\nobreak\
+$ (37) $\displaystyle+\leavevmode\nobreak\
2i(-i)\sqrt{\frac{\omega_{110}}{\omega_{130}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{130}\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011})(\mathbf{\mathcal{A}}^{\text{TM}}_{110}\mathbf{\mathcal{A}}^{\text{TE}}_{011})}_{\int=0}\leavevmode\nobreak\
+\leavevmode\nobreak\
2i(-i)\sqrt{\frac{\omega_{130}}{\omega_{110}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{110}\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011})(\mathbf{\mathcal{A}}^{\text{TM}}_{130}\mathbf{\mathcal{A}}^{\text{TE}}_{011})}_{\int=0}\Biggr{]}=$
$\displaystyle=\frac{1}{2\sqrt{2}V}\frac{\pi^{2}}{L_{z}^{2}}\left(4r^{2}\sqrt{\frac{\omega_{011}^{2}}{\omega_{110}\omega_{130}}}-(r^{2}-1)\sqrt{\frac{\omega_{110}\omega_{130}}{\omega_{011}^{2}}}\right).$

The last term,

$\displaystyle\expectationvalue{(\mathbf{B}\mathbf{E})^{2}}:\qquad\qquad\qquad\int\differential^{3}{\rm
x}\matrixelement{0}{\tfrac{1}{\sqrt{2}}a^{\text{TE}}_{110}a^{\text{TE}}_{110}\,{:}\mathbf{(BE)(BE)}{:}\,a^{\text{TM}+}_{110}a^{\text{TM}+}_{130}}{0}=\frac{1}{\sqrt{2^{4}V^{4}}}\,\int\differential^{3}{\rm
x}\,4\,\times$
$\displaystyle\times\Biggl{[}(-i)^{2}\sqrt{\frac{\omega_{011}^{2}}{\omega_{130}\omega_{110}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{130}\mathbf{\mathcal{A}}^{\text{TE}}_{011})(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{110}\mathbf{\mathcal{A}}^{\text{TE}}_{011})}_{\int=-3V/2\cdot\pi^{2}L_{y}^{-2}}\leavevmode\nobreak\
+\leavevmode\nobreak\
i^{2}\sqrt{\frac{\omega_{130}\omega_{110}}{\omega_{011}^{2}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011}\mathbf{\mathcal{A}}^{\text{TM}}_{130})(\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011}\mathbf{\mathcal{A}}^{\text{TM}}_{110})}_{\int=V/2\cdot\pi^{2}L_{y}^{-2}}\leavevmode\nobreak\
+$ (38) $\displaystyle+\leavevmode\nobreak\
i(-i)\sqrt{\frac{\omega_{110}}{\omega_{130}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{130}\mathbf{\mathcal{A}}^{\text{TE}}_{011})(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{110}\mathbf{\mathcal{A}}^{\text{TE}}_{011})}_{\int=0}\leavevmode\nobreak\
+\leavevmode\nobreak\
i(-i)\sqrt{\frac{\omega_{130}}{\omega_{110}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{110}\mathbf{\mathcal{A}}^{\text{TM}}_{130})(\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011}\mathbf{\mathcal{A}}^{\text{TE}}_{011})}_{\int=0}+$
$\displaystyle+\leavevmode\nobreak\
i(-i)\sqrt{\frac{\omega_{110}}{\omega_{130}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{130}\mathbf{\mathcal{A}}^{\text{TE}}_{011})(\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011}\mathbf{\mathcal{A}}^{\text{TM}}_{110})}_{\int=3V/2\cdot\pi^{2}L_{y}^{-2}}\leavevmode\nobreak\
+\leavevmode\nobreak\
i(-i)\sqrt{\frac{\omega_{130}}{\omega_{110}}}\underbrace{(\rm
rot\mathbf{\mathcal{A}}^{\text{TM}}_{110}\mathbf{\mathcal{A}}^{\text{TE}}_{011})(\rm
rot\mathbf{\mathcal{A}}^{\text{TE}}_{011}\mathbf{\mathcal{A}}^{\text{TM}}_{130})}_{\int=0}\Biggr{]}=$
$\displaystyle=\frac{1}{2\sqrt{2}V}\frac{\pi^{2}}{L_{z}^{2}}r^{2}\left(3\sqrt{\frac{\omega_{011}^{2}}{\omega_{110}\omega_{130}}}+3\sqrt{\frac{\omega_{110}}{\omega_{130}}}-\sqrt{\frac{\omega_{130}}{\omega_{110}}}-\sqrt{\frac{\omega_{110}\omega_{130}}{\omega_{011}^{2}}}\right).$

Finally, restoring the coefficient $4\kappa$, the result (20)–(22) is
obtained.

## References

  * [1] I. Kopchinskii and P. Satunin.  Resonant generation of electromagnetic modes in nonlinear electrodynamics: Classical approach.  Phys. Rev. A, 105:013508, 01 2022.
  * [2] H. Euler and B. Kockel.  The scattering of light by light in Dirac’s theory.  Naturwiss., 23(15):246–247, 1935.
  * [3] W. Heisenberg and H. Euler.  Consequences of Dirac’s theory of positrons.  Z. Phys., 98(11-12):714–732, 1936.
  * [4] A. A. Anselm.  Axion $\leftrightarrow$ Photon Oscillations in a Steady Magnetic Field. (In Russian).  Yad. Fiz., 42:1480–1483, 1985.
  * [5] L. Maiani, R. Petronzio, and E. Zavattini.  Effects of Nearly Massless, Spin Zero Particles on Light Propagation in a Magnetic Field.  Phys. Lett. B, 175:359–363, 1986.
  * [6] Stefan Evans and Johann Rafelski.  Virtual axion-like particle complement to Euler-Heisenberg-Schwinger action.  Phys. Lett. B, 791:331–334, 2019.
  * [7] Gerald V. Dunne.  The Heisenberg-Euler Effective Action: 75 years on.  Int. J. Mod. Phys. A, 27:1260004, 2012.
  * [8] W. Dittrich and H. Gies.  Probing the quantum vacuum. Perturbative effective action approach in quantum electrodynamics and its application, volume 166 of Springer Tracts in Modern Physics.  Springer Berlin, Heidelberg, 2000.
  * [9] Alexander Kuznetsov and Nickolay Mikheev, editors.  Electroweak Processes in External Active Media, volume 252 of Springer Tracts in Modern Physics.  Springer Berlin, Heidelberg, 2013.
  * [10] A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson.  Advances in QED with intense background fields.  Phys. Rept., 1010:1–138, 2023.
  * [11] R. Baier and P. Breitenlohner.  Photon Propagation in External Fields.  Acta Phys. Austriaca, 25:212–223, 1967.
  * [12] Z. Bialynicka-Birula and I. Bialynicki-Birula.  Nonlinear effects in Quantum Electrodynamics. Photon propagation and photon splitting in an external field.  Phys. Rev. D, 2:2341–2345, 1970.
  * [13] N.N. Rozanov.  Four-wave interactions of intense radiation in vacuum.  JETP, 76:991, Jun 1993.
  * [14] ELI.  https://eli-laser.eu/.
  * [15] CoReLS.  https://corels.ibs.re.kr/.
  * [16] E. Lundström, G. Brodin, J. Lundin, M. Marklund, R. Bingham, J. Collier, J. T. Mendonça, and P. Norreys.  Using high-power lasers for detection of elastic photon-photon scattering.  Phys. Rev. Lett., 96:083602, Mar 2006.
  * [17] Holger Gies, Felix Karbstein, Christian Kohlfürst, and Nico Seegert.  Photon-photon scattering at the high-intensity frontier.  Phys. Rev. D, 97:076002, Apr 2018.
  * [18] B. King, H. Hu, and B. Shen.  Three-pulse photon-photon scattering.  Phys. Rev. A, 98:023817, Aug 2018.
  * [19] A. M. Fedotov and N. B. Narozhny.  Generation of harmonics by a focused laser beam in vacuum.  Phys. Lett. A, 362:1–5, 2007.
  * [20] P. V. Sasorov, Francesco Pegoraro, Timur Zh. Esirkepov, and Sergei V. Bulanov.  Generation of high order harmonics in Heisenberg–Euler electrodynamics.  New J. Phys., 23(10):105003, 2021.
  * [21] Chantal Sundqvist and Felix Karbstein.  Two-beam laser photon merging.  Phys. Rev. D, 108(5):056028, 2023.
  * [22] Gert Brodin, Mattias Marklund, and Lennart Stenflo.  Proposal for Detection of QED Vacuum Nonlinearities in Maxwell’s Equations by the Use of Waveguides.  Phys. Rev. Lett., 87:171801, 2001.
  * [23] Daniel Eriksson, Gert Brodin, Mattias Marklund, and Lennart Stenflo.  A Possibility to measure elastic photon-photon scattering in vacuum.  Phys. Rev. A, 70:013808, 2004.
  * [24] Zachary Bogorad, Anson Hook, Yonatan Kahn, and Yotam Soreq.  Probing Axionlike Particles and the Axiverse with Superconducting Radio-Frequency Cavities.  Phys. Rev. Lett., 123(2):021801, 2019.
  * [25] Yonatan Kahn, Bianca Giaccone, Andrei Lunin, Alexandr Netepenko, Roman Pilipenko, and Michael Wentzel.  Searching for axions and light-by-light scattering with superconducting RF cavities.  Proc. SPIE Int. Soc. Opt. Eng., 12016:29, 2022.
  * [26] B. Giaccone et al.  Design of axion and axion dark matter searches based on ultra high Q SRF cavities.  arXiv preprint, 2207.11346, 7 2022.
  * [27] SQMS.  https://sqmscenter.fnal.gov/.
  * [28] Kazunori Shibata.  Intrinsic resonant enhancement of light by nonlinear vacuum.  Eur. Phys. J. D, 74(10):215, 2020.
  * [29] S. Hacian, R. Jauregui, F. Soto, and C. Villarreal.  Spectrum of electromagnetic fluctuations in the Casimir effect.  J. Phys. A, 23:2401–2412, 1990.
  * [30] S. Hacyan, R. Jauregui, and C. Villarreal.  Spectrum of quantum electromagnetic fluctuations in rectangular cavities.  Phys. Rev. A, 47:4204–4211, 1993.
  * [31] S. Chenarani and A. Shirzad.  Quantization In Finite Volumes Using Symplectic Quantization Programm.  arXiv preprint, 1311.0361, 11 2013.
  * [32] David Hill.  Electromagnetic fields in cavities: Deterministic and statistical theories.  Antennas and Propagation Magazine, IEEE, 56:306–306, 02 2014.
  * [33] Matthew D. Schwartz.  Quantum Field Theory and the Standard Model.  Cambridge University Press, 3 2014.
  * [34] H. Padamsee.  Calculations for breakdown induced by ”large defects” in superconducting niobium cavities.  IEEE Transactions on Magnetics, 19(3):1322–1325, 1983.
  * [35] Dmitrii A. Trunin.  Enhancement of particle creation in nonlinear resonant cavities.  Phys. Rev. D, 107(6):065004, 2023.