# Virtual transitions in an atom-mirror system in the presence of two scalar
photons

Ashmita Dasa∗, Soham Sena† and Sunandan Gangopadhyaya‡ a Department of
Astrophysics and High Energy Physics,
S.N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata 700106, India

###### Abstract

We examine the virtual transition of an atom-mirror system with the
simultaneous emission of two scalar photons, where the atom and the mirror
admit a relative acceleration between them. For the single photon emission,
literature (Phys. Rev. Lett. 121 (2018) 071301) dictates that the transition
probabilities of two individual systems, such as an atom accelerating with
respect to the mirror and its reverse, turn out to be equivalent under the
exchange of the frequencies of atom and the field. Addressing the
observational merit of such excitation process, a detectable probability
($P\sim 10^{-2}$) is also reported in the above literature. In the present
manuscript our finding dictates that the simultaneous emission of dual photon
instead of one, destroys the equivalence between the transition probabilities
as reported in the above literature.

## 1 Introduction

††
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
For the past several years we have seen a rich amalgamation of several
individual disciplines of physics, which led us to many interesting phenomena.
The classic example is none other than the combination of general
relativity[1, 2] and thermodynamics in the context of black holes and
cosmological spacetime [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Furthermore in
recent times the blending of quantum optics and the acceleration radiation in
flat/curved spacetime became somewhat popular and considered to be an
alternative mechanism in explaining the Unruh-Fulling (UF) effect [14, 15, 16,
17, 18, 19, 20, 21, 22]. In the context of black hole spacetime this approach
predicts the presence of an entropy which has different origin than that of
the Bekenstein-Hawking entropy, named as horizon brightened acceleration
radiation (HBAR) entropy [15]. More along this line is the use of cavity
quantum electrodynamics in the study of acceleration radiation which shows
that the transition probability of an accelerated atom can be increased to
many orders in magnitude than we usually obtain in the standard Unruh
radiation [14]. Since the proposition of UF effect, it has proven its
prominence in case of the Hawking radiation in black hole spacetime [3, 4],
particle emission by cosmological horizon [23], Schwinger mechanism [24],
quantum entanglement [25], etc. The idea of UF effect dictates that in
Minkowski spacetime a uniformly accelerating observer perceives the vacuum
state of a field as a thermal bath in equilibrium with temperature
$T=\frac{\hbar a}{2\pi k_{B}c}$. Here $k_{B},\,c,\,\hbar$ denote the Boltzmann
constant, speed of light in vacuum and the reduced Planck’s constant
respectively, and $a$ symbolises the acceleration of the observer [26, 27].
The observational aspects of UF effect are widely explored by modelling a
uniformly accelerated two level atomic detector, known as Unruh-DeWitt
detector (UD) [28, 29, 30]. The construction of UD detector states that, in
flat spacetime, a point like uniformly accelerated detector records particle
excitation due to the interaction with a scalar field, located in its
Minkowski vacuum. It captures the main essence of the UF effect in terms of
the transition rate/ power spectrum of the UD detector. For some brief review
on UF effect and modelling of UD detector, we refer our readers to [31, 29].
At this stage we resume the discussion related to the alternative approaches
in the studies of acceleration radiation/UF effect. It is well known that in
quantum theory of fields, the virtual processes due to the vacuum fluctuations
have disclosed many new quantum phenomena such as Lamb shift in the hydrogen
atom which led to the construction of quantum electrodynamics, Raman
scattering in the field of spectroscopy etc. In terms of the atom-field
interaction it dictates that a two level atom makes a transition to its
excited state while a virtual photon is simultaneously emitted. Subsequently
the atom promptly comes back to its ground state by absorbing the photon. It
was proposed in [14] that the UF effect can also be perceived by interrupting
a virtual process, where the emitted virtual photon turns into the real
observable photon. Due to the accelerated motion of the atom, it get away from
its original point of virtual emission which yields a nonzero probability for
the atom not absorbing the emitted virtual photon. This transforms a virtual
photon into a real one in the final state of the system [14, 15]. It is worth
to be mentioned that the atom acquires acceleration by extracting energy from
some external force agency which drives the centre of mass motion of the atom
[14, 15, 16]. This explains the notion of accelerated radiation/UF effect via
this alternative approach. This mechanism has unveiled many interesting
findings which can be found in the following references [15, 16, 21, 22].
Using this alternative mechanism, the authors of [16], have considered a
system which contains an atom and a mirror with a relative acceleration
between them. They have examined two possible setup such as the atom is
uniformly accelerating with respect to a fixed mirror and the reverse.
Subsequently, the transition probabilities for both the systems are obtained
due to the virtual processes. It was shown that the transition probabilities
are different from each other, however, both the probabilities are related
under the exchange of atom and field frequencies. The authors have defined
this connection as the “application of equivalence principle in QED”. Moreover
a spatial oscillatory behaviour of the transition probabilities has been
reported in [16] due to the interference of the “incident” and “reflected”
photons. Note that this oscillatory feature cannot be removed by integrating
out the probabilities over all values of field frequency. In relation to the
above equivalence, we mention that a violation of this equivalence relation
has recently been reported in the background of a generalized uncertainty
principle framework [32]. For experimental verification of such excitations,
the authors of [16] have proposed a setup where an ensemble of $N$ polar
molecules is coupled with a superconducting transmission line microwave cavity
terminated by a SQUID (semiconducting quantum interference device) [33, 34,
35]. This setup mimics an accelerating mirror and two level atomic system
[16]. It was shown that for all $N\sim 10^{4}-10^{6}$ polar molecules and the
acceleration $(a)$ of the mirror larger than the $c$ times the field frequency
($c\nu$), the transition probability lies within the detectable range (i.e
$P_{exc}\sim 10^{-2}$). The observational merit of this model gives us a
strong motivation to extend the model to the simultaneous emission of two
scalar photons instead one.

In recent times the authors of [36] have proposed a toy model of UD detector
simultaneously interacting with multiple scalar fields. This work has revealed
some interesting features in terms of the transition probability of the
detector which cannot be achieved in case of an atom and single field
interaction. Following [36], we consider the interaction Lagrangian as
$\sim\sigma(\tau)\phi_{1}[x(\tau)]\,\phi_{2}[x(\tau)]$ for the simultaneous
emission of two photons. Here $\phi_{1}[x(\tau)],\phi_{2}[x(\tau)]$ depict two
scalar photon fields and $\sigma(\tau)$ represents the operators in detector
sector. $\tau$ denotes the proper time of the atomic detector. At this stage
we briefly mention the origin of such interaction Lagrangian [36]. The
standard interaction term of a UD detector and single scalar field is of the
form, $\mathcal{L}_{int}\sim\sigma({\tau})\phi[x(\tau)]$. When the field makes
a transition from its vacuum to 1-particle state and the detector moves to its
energy eigenstate $\ket{E}_{f}$ from $\ket{E}_{i}$, a nonzero transition
amplitude can be achieved within the first order in perturbation theory. The
transition amplitude for such system becomes as follows [31, 29],

$A(E_{f}|E_{i})\sim\bra{E_{f}}\sigma(\tau)\ket{E_{i}}\bra{1_{p}}\phi[x(\tau)]\ket{0}~{},$
(1)

Following the single field interaction one can intuitively write the
interaction Lagrangian for the dual photons emission as,

$\mathfrak{L}_{int}=\mathbbm{g}\,\sigma(\tau)(\phi_{1}[x(\tau)]+\phi_{2}[x(t)])~{},$
(2)

Note unlike single field case, now we have several choices of final states due
to the emission of two fields. Below we briefly discuss such two possible
cases. For detailed discussion we refer our readers to the section.(VI) of
[36]

Case 1: One of the possible final state can be written as,

$\displaystyle\begin{split}\ket{\small{\mbox{initial}}}&=\ket{E_{i}}\otimes\ket{0_{1}}\otimes\ket{0_{2}}~{};\\\
\ket{\small{\mbox{final}}}&=\ket{E_{f}}\otimes\Bigl{(}\ket{1_{p_{1}}}\otimes\ket{0_{2}}+\ket{0_{1}}\otimes\ket{1_{p_{2}}}\Bigr{)}~{}.\end{split}$
(3)

Here $\ket{0_{1,2}}$ and $\ket{1_{p_{1,2}}}$ depict the vacuum and 1-particle
state for each field. In this scenario the detector is interacting with two
scalar fields where at a particular instant a single field interacts with the
detector and creates 1-particle state, while another is lying in its vacuum.
Therefore in this case the transition amplitude varies as the summation of the
two transitions i.e $\sim(A_{1}+A_{2})$, where $A_{1},\,A_{2}$ denote the
amplitudes as similar to eq.(1). This case has its own merits and may reveal
interesting features [36].

Case 2: In this case both the fields make simultaneous transition to their
respective 1-particle states due to the interaction with the detector at the
same point in spacetime. This leads to the following final state of the system
:

$\ket{\small{\mbox{final}}}=\ket{E_{f}}\otimes\ket{1_{p_{1}}}\otimes\ket{1_{p_{2}}}~{},$
(4)

when the initial state is same as written in Eq.(3). For the interaction
Lagrangian in Eq.(2), initial and final states as in Eqs.(3) and (4)
respectively, a nonzero transition amplitude can be achieved, if one considers
the higher order term in perturbation theory i.e,
$\sim\bra{\small{\mbox{final}}}\mathfrak{L}_{int}^{2}\ket{\small{\mbox{initial}}}$.
The term $\mathfrak{L}_{int}^{2}$, will bring in the following combination of
fields,

$\sim\phi_{1}^{2}+\phi_{2}^{2}+2\phi_{1}\phi_{2}~{}.$ (5)

Note that the interaction like $\sim\phi_{1}^{2}$ and $\sim\phi_{2}^{2}$
contribute to the transition amplitude when the fields make a transition from
the vacuum to their 2-particle state. In the present work we restrict our
analysis within the transition of the fields to 1-particle states as
energetically 1-particle state is most favoured [36]. Therefore in the present
context, only the third term of Eq.(5) contribute to the transition amplitude.
This analysis implies that in working with the Lagrangian
$\sim\phi_{1}[x(\tau)]\phi_{2}[x(\tau)]$ is not an arbitrary choice, rather it
originates from a fundamental interaction such as Eq.(2). The emergence of
various final states and interaction Lagrangian are solely due to the dual
field interaction and cannot be achieved in single field setup. Therefore
these features render the distinctive characters of single and dual field
emission processes.

With respect to the previous literature, it is seen that apart from the field
theoretic interests, multiple fields analyses have earned loads of attentions
in the sector of beyond the standard model of particle physics, cosmology,
dark matter physics, etc. It is well known that in search for the signatures
of extra dimensions in the collider experiments (such that Large Hadron
Collider), the Kaluza-Klein (KK) modes of bulk graviton and bulk standard
model are the potential candidates [37, 38]. These KK modes are originated due
to the compactification of the spatial extra dimensions and interpreted as the
multiple fields in 4-dimensional effective theory [39]. Furthermore, in [40],
a large number of elementary particle species have been considered in
resolving the hierarchy problem of the two fundamental length scales such as
electroweak and Planck scale. Later, it was shown that an appropriate
abundance of dark matter can also be achieved while considering a large number
of copies of the standard model in [41, 42]. In cosmology, the multiple field
inflationary models are considered to explore the inflationary dynamics,
inflationary trajectories and particle production [43, 44, 45]. Two-field
inflationary scenario have been studied in the context of swampland de Sitter
conjecture [46], reheating in cosmology [47], dark energy, [48] etc. These
promising features of multiple field theories instigate us to explore the
implications of multi-field interaction with two level atomic detector in the
present manuscript. Thus we propose a system which contains a two-level atom
and a mirror with an existent relative acceleration between them. The system
becomes excited due to the virtual processes in the vacuum states of the
photon fields. However, due to the relative acceleration of the atom and
mirror the two real scalar photons are simultaneously emitted and detected at
the same time. Concurrently, The atom makes a transition to its higher excited
state. We make an attempt to map our system within the experimental setup as
described in [16] as following. Within a cavity setup an atom makes a
transition to its higher excited state while simultaneously two photons are
emitted with two different single frequency modes due to the interaction of
the atom and dual photon. Thus in contrast to the single field emission
process, one obtains two quanta of detectable photons carrying two different
frequencies within a cavity. We examine the transition probabilities of such
systems under the conditions where the atom(mirror) is accelerating with
respect to fixed mirror (atom). We enlist our findings as below,

  * •

The so called “equivalence principle in QED”[32, 49, 50, 51, 52, 53, 54] is no
more holding in case of the dual field emission process. This solely happens
due to the emergence of a cross term in the transition amplitude for the
emission of dual photon modes.

  * •

We achieve a correction term as a function of $(a,\,\nu,\,\omega)$, over the
single field emission case, where, $\omega$ represents the angular frequency
of the atom. This correction term plays an important role as we observer that
for an accelerating mirror (fixed atom) system the transition probability
becomes a nontrivial function of $(a,\,\nu)$ even under the condition $a\gg
c\nu$. This is in contrast with the outcome in [16]. Undoubtedly, this outcome
promotes more investigations concerning with the atomic detector and multiple
fields interaction and also signifies the observational prominence of the
present setup.

We organise our manuscript as follows. In section(2, 3), we briefly describe
the two systems and obtain the corresponding transition probabilities. In
section(4), we study the relation between the excitation probabilities of
these two systems and subsequently discuss the phenomenological aspects of our
model in section(5) We discuss our result in section(6).

## 2 A uniformly accelerating atom and a fixed mirror

We consider a system which contains a uniformly accelerated two level atom
with respect to a fixed mirror. The atom follows a trajectory as below,

$\displaystyle t(\tau)=$
$\displaystyle\frac{c}{a}\sinh\left(\frac{a\tau}{c}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,z(\tau)=$
$\displaystyle\frac{c^{2}}{a}\cosh\left(\frac{a\tau}{c}\right),$ (6)

where $(a,\,\tau)$ symbolise the uniform acceleration of the atom along the
$z$-direction and the proper time of the atom respectively. The atom is
interacting with two scalar photons at the same time and possesses a
transition angular frequency $\omega$. We restrict the wave vectors of the two
photons ($\mathbf{k}_{1},\mathbf{k}_{2}$) to be parallel to the $z$-axis and
the frequencies of the same are denote by $\nu_{1}$ and $\nu_{2}$. In the
initial state of the system the atom lies in its ground state while the fields
are located in their respective Minkowski vacuum. Thus the initial state can
be written as :
$\ket{\mbox{initial}}=\ket{g}\otimes\ket{0_{1}}\otimes\ket{0_{2}}$, where
$\ket{g}$ is the ground state of the atom and the $\ket{0_{1,2}}$ stand for
the vacuum states of two scalar photons. Following the discussion in [36], it
can be perceived that the simultaneous emission of two photons leads to a
final state of the system as below,

$\ket{\small{\mbox{final}}}=\ket{e}\otimes\ket{1_{\nu_{1}}}\otimes\ket{1_{\nu_{2}}}~{}.$
(7)

Here $\ket{e}$ represents the excited state of the atom. Allowing upto the
first order in perturbation theory, a nonzero transition amplitude can be
obtained only for the choice of interaction Hamiltonian as following,

$\hat{\mathcal{H}}_{I}(\tau)=\hbar\mathcal{G}\left[\hat{b}_{\nu_{1}}\phi_{\nu_{1}}+h.c.\right]\left[\hat{b}_{\nu_{2}}\phi_{\nu_{2}}+h.c.\right]\left[\hat{\sigma}e^{-i\omega\tau}+h.c.\right]$
(8)

Here $\mathcal{G}$ denotes the atom-fields coupling constant,
$\phi_{\nu_{1,2}}(x(\tau))$ are the mode solutions of two scalar photon fields
and ($\hat{\sigma},\,\hat{\sigma}^{\dagger}$) stand for the lowering and
raising operators of the atom. $\hat{b}_{\nu_{1,2}}$ denote the annihilation
operators corresponding to the two scalar photons. We assume that the mirror
is fixed at $z_{0}(<\frac{c^{2}}{a})$ along the $z$-axis which yields the
following boundary condition for the mode solutions,

$\phi_{\nu_{1,2}}\biggr{|}_{z=z_{0}}=0~{}.$ (9)

Eq.(9) dictates that the normal modes of the photon fields ought to be the
superposition of the incident and the reflected waves which in turn produces
standing waves as below,

$\phi_{\nu_{1,2}}=e^{-i\nu_{1,2}t-ik_{1,2}(z-z_{0})}-e^{-i\nu_{1,2}t+ik_{1,2}(z-z_{0})}~{}.$
(10)

We take the duration of atom-fields interaction to be infinite and thus the
transition amplitude for this excitation can be written as,

$\begin{split}\mathcal{A}_{|i\rangle\rightarrow|f\rangle}=&-\frac{i}{\hbar}\int_{-\infty}^{+\infty}d\tau\langle
1_{\nu_{1}},1_{\nu_{2}},e|\hat{\mathcal{H}}_{I}(\tau)|0_{\nu_{1}},0_{\nu_{2}},g\rangle\\\
=&-i\mathcal{G}\int_{-\infty}^{+\infty}d\tau~{}e^{i(\nu_{1}+\nu_{2})t}e^{i\omega\tau}\left(e^{ik_{1}(z-z_{0})}-c.c.\right)\\\
&\times\left(e^{ik_{2}(z-z_{0})}-c.c.\right)\end{split}$ (11)

We replace $k_{1,2}=\frac{\nu_{1,2}}{c}$ in the above equation and consider
that the emitted scalar photons have identical frequencies i,e
$\nu_{1}=\nu_{2}=\nu\implies\,k_{1}=k_{2}=\frac{\nu}{c}$. Therefore the
transition amplitude in eq.(11) takes the following form,

$\mathcal{A}_{|i\rangle\rightarrow|f\rangle}=-i\mathcal{G}\int_{-\infty}^{\infty}d\tau~{}e^{2i\nu
t}e^{i\omega\tau}\left(e^{\frac{2i\nu}{c}(z-z_{0})}+c.c.-2\right)~{}.$ (12)

This yields the transition probability of the system as below,

$\begin{split}\mathcal{P}_{exc}=\mathcal{G}^{2}\biggr{|}\int d\tau~{}e^{2i\nu
t(\tau)}e^{i\omega\tau}\left(e^{\frac{2i\nu}{c}(z(\tau)-z_{0})}+c.c.-2\right)\biggr{|}^{2}~{}.\end{split}$
(13)

At this stage we use the trajectory for the accelerated atom as in Eq.(6) in
eq.(13) and obtain the excitation probability as follows,

$\begin{split}\mathcal{P}_{exc}=&\mathcal{G}^{2}\biggr{|}\int_{-\infty}^{+\infty}d\tau\biggr{(}e^{\frac{2i\nu
c}{a}e^{\frac{a\tau}{c}}-2ikz_{0}}+e^{-\frac{2i\nu
c}{a}e^{-\frac{a\tau}{c}}+2ikz_{0}}\\\ &-2e^{\frac{2i\nu
c}{a}\sinh\left(\frac{a\tau}{c}\right)}\biggr{)}e^{i\omega\tau}\biggr{|}^{2}\end{split}$
(14)

Now we evaluate the three integrals in Eq.(14) separately where the first
integral is given by,

$I_{1}=\int_{-\infty}^{+\infty}d\tau~{}e^{\frac{2i\nu
c}{a}e^{\frac{a\tau}{c}}-2ikz_{0}}e^{i\omega\tau}~{}.$ (15)

Changing the variable such that $x=\frac{2\nu c}{a}e^{\frac{a\tau}{c}}$,
Eq.(15) reads,

$\begin{split}I_{1}=&\frac{c}{a}\left(\frac{a}{2\nu c}\right)^{\frac{i\omega
c}{a}}\int_{0}^{\infty}dx~{}e^{ix}x^{\frac{i\omega c}{a}-1}e^{-2ikz_{0}}\\\
=&\frac{c}{a}\left(\frac{a}{2\nu c}\right)^{\frac{i\omega
c}{a}}e^{-2ikz_{0}}e^{-\frac{\pi c\omega}{2a}}\Gamma\left[\frac{i\omega
c}{a}\right]~{}.\end{split}$ (16)

Following the similar change of variables the second integral in eq.(14)
yields,

$\begin{split}I_{2}=&\frac{c}{a}\left(\frac{a}{2\nu c}\right)^{-\frac{i\omega
c}{a}}\int_{0}^{\infty}dx^{\prime}~{}e^{-ix^{\prime}}{x^{\prime}}^{-\frac{i\omega
c}{a}-1}e^{2ikz_{0}}\\\ =&\frac{c}{a}\left(\frac{a}{2\nu
c}\right)^{-\frac{i\omega c}{a}}e^{2ikz_{0}}e^{-\frac{\pi
c\omega}{2a}}\Gamma\left[-\frac{i\omega c}{a}\right]~{}.\end{split}$ (17)

The third integral, which solely appears due to the dual field interaction in
eq.(14) is given by,

$I_{3}=-2\int_{-\infty}^{+\infty}d\tau~{}e^{\frac{2i\nu
c}{a}\sinh\left(\frac{a\tau}{c}\right)}e^{i\omega\tau}~{}.$ (18)

Under the coordinate transformation $\tilde{x}=e^{\frac{a\tau}{c}}$, $I_{3}$
becomes,

$\begin{split}I_{3}=&-\frac{2c}{a}\int_{0}^{\infty}d\tilde{x}~{}{\tilde{x}}^{\frac{i\omega
c}{a}-1}e^{\frac{i\nu
c}{a}\left(\tilde{x}-\frac{1}{\tilde{x}}\right)}~{}.\end{split}$ (19)

Performing the above integration the final form of $I_{3}$ is obtained as
follows [55, 56],

$I_{3}=-\frac{4c}{a}e^{-\frac{\pi\omega c}{2a}}K_{\frac{i\omega
c}{a}}\left(\frac{2\nu c}{a}\right),$ (20)

where $K_{\frac{i\omega c}{a}}\left(\frac{2\nu c}{a}\right)$ is the modified
Bessel function. Combining the Eq.(16) and Eq.(17), we obtain

$I_{1}+I_{2}=\frac{c}{a}e^{-\frac{\pi\omega c}{2a}}\left(\left(\frac{a}{2\nu
c}\right)^{\frac{i\omega c}{a}}e^{-2ikz_{0}}~{}\Gamma\left[\frac{i\omega
c}{a}\right]+c.c.\right)~{}.$ (21)

For a complex parameter $z$ and its complex conjugate $z^{*}$, one obtains,
$z+z^{*}=2\Re(z)$. One can also write the complex number $z$ as,
$z=|z|\left[\cos(\text{arg(z)})+i\sin(\text{arg(z)})\right]$. Using these
relations, we recast the sum in eq.(21) as,

$\begin{split}I_{1}+I_{2}=&\frac{2c}{a}e^{-\frac{\pi
c\omega}{2a}}\Re\left(\left(\frac{a}{2\nu c}\right)^{\frac{i\omega
c}{a}}e^{-2ikz_{0}}~{}\Gamma\left[\frac{i\omega c}{a}\right]\right)\\\
=&\frac{2c}{a}e^{-\frac{\pi
c\omega}{2a}}\Re\biggr{[}\biggr{(}\cos\left(\frac{2\nu z_{0}}{c}-\frac{\omega
c}{a}\ln\left[\frac{a}{2\nu c}\right]\right)\\\ -&i\sin\left(\frac{2\nu
z_{0}}{c}-\frac{\omega c}{a}\ln\left[\frac{a}{2\nu
c}\right]\right)\biggr{)}\biggr{(}\Re\left[\Gamma\left[\frac{i\omega
c}{a}\right]\right]\\\ +&i\Im\left[\Gamma\left[\frac{i\omega
c}{a}\right]\right]\biggr{)}\biggr{]}\\\ =&\frac{2ce^{-\frac{\pi
c\omega}{2a}}}{a}\biggr{[}\cos\left[\frac{2\nu z_{0}}{c}-\frac{\omega
c}{a}\ln\left[\frac{a}{2\nu c}\right]\right]\Re\left[\Gamma\left[\frac{i\omega
c}{a}\right]\right]\\\ +&\sin\left[\frac{2\nu z_{0}}{c}-\frac{\omega
c}{a}\ln\left[\frac{a}{2\nu c}\right]\right]\Im\left[\Gamma\left[\frac{i\omega
c}{a}\right]\right]\biggr{]}~{}.\end{split}$ (22)

In the above equation we substitute the following relations,

$\displaystyle\Re\left[\Gamma\left(\frac{i\omega
c}{a}\right)\right]=\left|\Gamma\left(\frac{i\omega
c}{a}\right)\right|\cos\left(\text{arg}\left[\Gamma\left(\frac{i\omega
c}{a}\right)\right]\right)$ (23)
$\displaystyle\Im\left[\Gamma\left(\frac{i\omega
c}{a}\right)\right]=\left|\Gamma\left(\frac{i\omega
c}{a}\right)\right|\sin\left(\text{arg}\left[\Gamma\left(\frac{i\omega
c}{a}\right)\right]\right)~{}.$ (24)

and obtain the final form of ($I_{1}+I_{2}$) as,

$\begin{split}I_{1}+I_{2}=&\frac{2c}{a}e^{-\frac{\pi
c\omega}{2a}}\left|\Gamma\left(\frac{i\omega
c}{a}\right)\right|\cos\left(\theta\right)\end{split}$ (25)

where,

$\theta=\frac{2\nu z_{0}}{c}-\frac{\omega c}{a}\ln\left[\frac{a}{2\nu
c}\right]-\text{arg}\left[\Gamma\left(\frac{i\omega c}{a}\right)\right]~{}.$
(26)

We obtain the transition probability while summing up the Eqs.(20, 25) as
below,

$\begin{split}\mathcal{P}_{exc}=&\frac{4\mathcal{G}^{2}c^{2}}{a^{2}}e^{-\frac{\pi\omega
c}{a}}\left|\Gamma\left(\frac{i\omega
c}{a}\right)\right|^{2}\cos^{2}(\theta)\biggr{|}\biggr{(}1\\\
-&2\sec(\theta)\left|\Gamma\left(\frac{i\omega
c}{a}\right)\right|^{-1}K_{\frac{i\omega c}{a}}\left(\frac{2\nu
c}{a}\right)\biggr{)}\biggr{|}^{2}\\\
=&\frac{8\pi\mathcal{G}^{2}c}{a\omega}\frac{\cos^{2}(\theta)}{e^{\frac{2\pi\omega
c}{a}}-1}\biggr{(}1-\frac{4\sec(\theta)}{\left|\Gamma\left(\frac{i\omega
c}{a}\right)\right|}K_{\frac{i\omega c}{a}}\left(\frac{2\nu c}{a}\right)\\\
+&\frac{4\omega c\sec^{2}(\theta)}{a\pi}\biggr{|}K_{\frac{i\omega
c}{a}}\left(\frac{2\nu c}{a}\right)\biggr{|}^{2}\sinh\left(\frac{\pi\omega
c}{a}\right)\biggr{)}\end{split}$ (27)

Here, $K_{\frac{i\omega c}{a}}\left(\frac{2\nu c}{a}\right)$ is a real number
provided $\nu,c,a$ and $\omega$ are real. Note that the excitation probability
in eq.(27) depends on the parameters, $\mathcal{G},a,\nu,\omega$ and $z_{0}$.
We explore the relevant phenomenology of this result in the section (5).

## 3 A uniformly accelerating mirror and a fixed atom

A uniformly accelerating mirror with respect to a fixed atom is considered in
this section. As similar to the section(2), photons are simultaneously
produced from their respective vacua due to the accelerated motion of the
mirror and subsequently detected by the atom at the same time in Minkowski
spacetime. With the simultaneous detection of the two field quanta, the atom
gets excited and jumps to its higher energy state. The atom is fixed at
$z=z_{0}<\frac{c^{2}}{a}$ in Minkowski spacetime. At this stage we take a
widely used coordinate transformation such as,

$\displaystyle t=$
$\displaystyle\frac{c}{a}e^{\frac{a\tilde{z}}{c^{2}}}\sinh\left(\frac{a\tilde{t}}{c}\right),\,\,\,\,\,\,\,\,\,\,\,z=$
$\displaystyle\frac{c^{2}}{a}e^{\frac{a\tilde{z}}{c^{2}}}\cosh\left(\frac{a\tilde{t}}{c}\right)~{}.$
(28)

Here $(\tilde{t},\,\tilde{z})$ denote the spacetime coordinates for the moving
mirror, ($t,\,z$) are the Minkowski coordinates and $a$ is a constant
parameter. When the mirror is following the trajectory such as $\tilde{z}=0$,
it represents a uniformly accelerated mirror in the Minkowski spacetime with
an acceleration $a$. This picture can be mapped to the Rindler space (defined
in terms of coordinates $(\tilde{t},\tilde{z})$), where one can realize this
incident such as the mirror is at rest and atom is accelerating. Consequently,
the vacuum state of the fields become Rindler vacuum which one defines with
respect to the static mirror in Rindler space. On the other hand now the
accelerated atom in Rindler space perceives particle production from the
Rindler vaccum of the fields. Note that in Rindler space the Rindler vacuum
does not perceive any horizon as it is defined with respect to the static
mirror. Whereas the accelerated atom in Rindler space perceives Rindler like
horizon. Using Eq.(28), the Minkowski line element
($ds^{2}=c^{2}dt^{2}-dz^{2}$) can be recasted in the form of the Rindler
metric, which reads,

$ds^{2}=e^{\frac{2a\tilde{z}}{c^{2}}}(c^{2}d\tilde{t}^{2}-d\tilde{z}^{2})~{}.$
(29)

As the Rindler metric in Eq.(29) is conformally flat to the Minkowski line
element, the positive frequency mode solutions of the scalar photons remain
same as that of the standard mode solutions in Minkowski spacetime. Therefore
in Rindler space the normal modes of the scalar photons can be expressed as
standing waves, given by the following relation,

$\phi_{\nu_{1,2}}(\tilde{t},\tilde{z})=e^{-i\nu_{1,2}\tilde{t}+ik_{1,2}\tilde{z}}-e^{-i\nu_{1,2}\tilde{t}-ik_{1,2}\tilde{z}}$
(30)

where $\nu_{1}$ and $\nu_{2}$ are the angular frequencies of photons 1 and 2
in the Rindler space [16]. The normal modes in Eq.(30) depict the positive
frequency mode solutions with respect to the Rindler time ($\tilde{t}$), which
signifies that the vacuum states corresponding to these modes belong to the
Rindler vacuum in Rindler space. From Eq.(28), we write the coordinates
($\tilde{t},\tilde{z}$) in terms of the ($t,z$) as below,

$\displaystyle\tilde{t}=$
$\displaystyle\frac{c}{2a}\ln\left[\frac{z+ct}{z-ct}\right],\,\,\,\,\,\,\,\,\,\,\tilde{z}=$
$\displaystyle\frac{c^{2}}{2a}\ln\left[\frac{a^{2}}{c^{4}}(z^{2}-c^{2}t^{2})\right]~{}.$
(31)

Using Eq.(31) in Eq.(30), we achieve,

$\begin{split}\phi_{\nu_{1,2}}(t,z)=&e^{+\frac{i\nu_{1,2}c}{a}\ln\left[\frac{a}{c^{2}}(z-ct)\right]}\Theta[z-ct]\\\
-&e^{-\frac{i\nu_{1,2}c}{a}\ln\left[\frac{a}{c^{2}}(z+ct)\right]}\Theta[z+ct]\end{split}$
(32)

where $\Theta$ denotes the Heaviside theta function. In line with the
treatment as carried out in [16], we consider that the mode solutions as
written in Eq.(32), in terms of Minkowski coordinates, initially contain no
particles. This implies that these modes are in the Rindler vacuum of the
corresponding fields which are defined with respect to the static mirror in
Rindler space. Such a Rindler vacuum can be prepared outside a massive star
just before it starts to collapse [16]. We carry our subsequent analysis in
Rindler space and obtain the transition probability while using the trajectory
of the atom as written in Eq.(31). However, this analysis could have been done
with a more physically realistic initial state in which the field modes are
assumed to be in the Minkowski vacuum before they reflect off the mirror. A
way to do this would be to follow the formalism in [31, 57] which we shall not
carry out in this paper. The mode solutions in Eq.(32) portray that the right
moving field modes can access only the region such that $z>\,c|t|$, which is
known as right Rindler wedge (RRW). On the other hand left moving modes can
only access the region $-z>\,c|t|$, called the left Rindler wedge (LRW).
Individually the mode solution in RRW and LRW is not complete in whole
Minkowski spacetime. However their combination as appears in Eq.(32) is
complete in all Minkowski spacetime and can be analytically continued to the
future ($t>|z|/c$) and past ($-t>|z|/c$) regions.

Now we write the excitation probability of the system due to the simultaneous
emission of two photons and the transition of atom to its higher excited state
as below,

$\mathcal{P}^{\prime}_{exc}=\mathcal{G}^{2}\biggr{|}\int_{-\infty}^{+\infty}dt\,e^{i\omega
t}\phi_{\nu_{1}}^{*}(t,z)\,\phi_{\nu_{2}}^{*}(t,z)\biggr{|}^{2}~{}.$ (33)

For simplicity we consider the two photons have equal angular frequency $\nu$.
We substitute the position of the atom $z=z_{0}$ in the above equation, which
yields,

$\begin{split}\mathcal{P}^{\prime}_{exc}=&\mathcal{G}^{2}\biggr{|}\int_{-\infty}^{\infty}dt~{}e^{i\omega
t}\biggr{(}e^{-\frac{i\nu
c}{a}\ln\left[\frac{a}{c^{2}}(z_{0}-ct)\right]}\Theta[z_{0}-ct]\\\
&-e^{\frac{i\nu
c}{a}\ln\left[\frac{a}{c^{2}}(z_{0}+ct)\right]}\Theta[z_{0}+ct]\biggr{)}^{2}\,\biggr{|}^{2}~{}.\end{split}$
(34)

Note that $\Theta[z_{0}-ct]^{2}$ has non zero value only in the region
$-\infty<t<\frac{z_{0}}{c}$ while $\Theta[z_{0}+ct]^{2}$ has non zero value in
the region $-\frac{z_{0}}{c}<t<\infty$. For the cross-term,
$\Theta[z_{0}-ct]\Theta[z_{0}+ct]$, one obtains a non zero value in the regime
$-\frac{z_{0}}{c}<t<\frac{z_{0}}{c}$. Using these conditions, we write the
probability in eq.(34) as,

$\begin{split}\mathcal{P}^{\prime}_{exc}=&\mathcal{G}^{2}\biggr{|}\int_{-\infty}^{\frac{z_{0}}{c}}dt~{}e^{i\omega
t}e^{-\frac{2i\nu c}{a}\ln\left[\frac{a}{c^{2}}(z_{0}-ct)\right]}\\\
&+\int^{\infty}_{-\frac{z_{0}}{c}}dt~{}e^{i\omega t}e^{\frac{2i\nu
c}{a}\ln\left[\frac{a}{c^{2}}(z_{0}+ct)\right]}\\\
&-2\int_{-\frac{z_{0}}{c}}^{\frac{z_{0}}{c}}dt~{}e^{i\omega t}e^{\frac{i\nu
c}{a}\ln\left[\frac{z_{0}+ct}{z_{0}-ct}\right]}\biggr{|}^{2}~{}.\end{split}$
(35)

For the first integral in the Eq.(35), we change the variable $t\rightarrow-t$
and obtain,

$\begin{split}\mathcal{P}^{\prime}_{exc}&=\mathcal{G}^{2}\biggr{|}\int_{-\frac{z_{0}}{c}}^{\infty}dt~{}e^{-i\omega
t}e^{-\frac{2i\nu c}{a}\ln\left[\frac{a}{c^{2}}(z_{0}+ct)\right]}\\\
&+\int^{\infty}_{-\frac{z_{0}}{c}}dt~{}e^{i\omega t}e^{\frac{2i\nu
c}{a}\ln\left[\frac{a}{c^{2}}(z_{0}+ct)\right]}\\\
&-2\int_{-\frac{z_{0}}{c}}^{\frac{z_{0}}{c}}dt~{}e^{i\omega t}e^{\frac{i\nu
c}{a}\ln\left[\frac{z_{0}+ct}{z_{0}-ct}\right]}\biggr{|}^{2}~{}.\end{split}$
(36)

Further we take $x=\omega\left(t+\frac{z_{0}}{c}\right)$ in the above
equation, which leads us to the following form of
$\mathcal{P}^{\prime}_{exc}$.

$\begin{split}\mathcal{P}^{\prime}_{exc}&=\mathcal{G}^{2}\biggr{|}\frac{1}{\omega}\int_{0}^{\infty}dx~{}e^{-i\left(x-\frac{\omega
z_{0}}{c}\right)}x^{-\frac{2i\nu c}{a}}e^{-\frac{2i\nu
c}{a}\ln\left[\frac{a}{c\omega}\right]}\\\
&+\frac{1}{\omega}\int_{0}^{\infty}dx~{}e^{i\left(x-\frac{\omega
z_{0}}{c}\right)}x^{\frac{2i\nu c}{a}}e^{\frac{2i\nu
c}{a}\ln\left[\frac{a}{c\omega}\right]}\\\ &-\frac{2e^{-\frac{i\omega
z_{0}}{c}}}{\omega\left(\frac{2\omega z_{0}}{c}\right)^{\frac{i\nu
c}{a}}}\int_{0}^{\frac{2\omega z_{0}}{c}}dx~{}e^{ix}x^{\frac{i\nu
c}{a}}\left(1-\frac{cx}{2\omega z_{0}}\right)^{-\frac{i\nu
c}{a}}\biggr{|}^{2}~{}.\end{split}$ (37)

We denote the first two integrals in the above eq.(37) as $I^{\prime}_{1}$ and
$I^{\prime}_{2}$ and obtain the following relation

$\begin{split}I^{\prime}_{1}+I^{\prime}_{2}&=-\frac{2\nu
c}{a\omega}\left(\frac{a}{\omega c}\right)^{-\frac{2i\nu
c}{c}}e^{\frac{i\omega z_{0}}{c}}e^{-\frac{\pi\nu
c}{a}}\Gamma\left[-\frac{2i\nu c}{a}\right]\\\ &-\frac{2\nu
c}{a\omega}\left(\frac{a}{\omega c}\right)^{\frac{2i\nu
c}{c}}e^{-\frac{i\omega z_{0}}{c}}e^{-\frac{\pi\nu
c}{a}}\Gamma\left[\frac{2i\nu c}{a}\right]\\\ &=-\frac{4\nu
c}{a\omega}e^{-\frac{\pi\nu c}{a}}\left|\Gamma\left[-\frac{2i\nu
c}{a}\right]\right|\cos\left(\theta^{\prime}\right)\end{split}$ (38)

where,

$\begin{split}\theta^{\prime}=&\frac{\omega z_{0}}{c}-\frac{2\nu
c}{a}\ln\left[\frac{a}{\omega
c}\right]+\text{arg}\left[\Gamma\left[-\frac{2i\nu c}{a}\right]\right]\\\
=&\frac{\omega z_{0}}{c}-\frac{2\nu c}{a}\ln\left[\frac{a}{\omega
c}\right]-\text{arg}\left[\Gamma\left[\frac{2i\nu
c}{a}\right]\right].\end{split}$ (39)

In the last line of eq.(39), we use
$\text{arg}\left[\Gamma[i\alpha]\right]=-\text{arg}\left[\Gamma[-i\alpha]\right]$
($\forall~{}\alpha\in\mathbb{R}$). The third integral in eq.(36) is given by,

$I^{\prime}_{3}=-\frac{2e^{-\frac{i\omega
z_{0}}{c}}}{\omega\left(\frac{2\omega z_{0}}{c}\right)^{\frac{i\nu
c}{a}}}\int_{0}^{\frac{2\omega z_{0}}{c}}dx~{}e^{ix}x^{\frac{i\nu
c}{a}}\left(1-\frac{cx}{2\omega z_{0}}\right)^{-\frac{i\nu c}{a}}~{}.$ (40)

The form of the integral in eq.(40) takes the following form

$\begin{split}I_{3}^{\prime}=&-\frac{2e^{-\frac{i\omega
z_{0}}{c}}}{\omega\left(\frac{2\omega z_{0}}{c}\right)^{\frac{i\nu
c}{a}}}\int_{0}^{\frac{2\omega z_{0}}{c}}dxe^{ix}x^{\frac{i\nu
c}{a}}\left(1-\frac{cx}{2\omega z_{0}}\right)^{-\frac{i\nu c}{a}}\\\
=&-\frac{4\pi\nu z_{0}}{a\sinh\left[\frac{\pi\nu
c}{a}\right]}e^{\frac{-i\omega
z_{0}}{c}}\biggr{[}~{}_{2}\\!F_{3}\left[a_{1},a_{2};\frac{1}{2},1,\frac{3}{2};-\frac{\omega^{2}z_{0}^{2}}{c^{2}}\right]\\\
&+\frac{i\omega z_{0}}{c}\left(1+\frac{i\nu
c}{a}\right)~{}_{2}\\!F_{3}\left[a_{2},a_{3};\frac{3}{2},\frac{3}{2},2;-\frac{\omega^{2}z_{0}^{2}}{c^{2}}\right]\biggr{]}~{},\end{split}$
(41)

where $a_{1}=\frac{1}{2}+\frac{i\nu c}{2a}$, $a_{2}=1+\frac{i\nu c}{2a}$,
$a_{3}=\frac{3}{2}+\frac{i\nu c}{2a}$, and $~{}_{2}\\!F_{3}$ denotes the
generalized hypergeometric function. In the above equation we write,

$\begin{split}\mathcal{B}_{\mathcal{f}}=&~{}_{2}\\!F_{3}\left[a_{1},a_{2};\frac{1}{2},1,\frac{3}{2};-\frac{\omega^{2}z_{0}^{2}}{c^{2}}\right]\\\
&+\frac{i\omega z_{0}}{c}\left(1+\frac{i\nu
c}{a}\right)~{}_{2}\\!F_{3}\left[a_{2},a_{3};\frac{3}{2},\frac{3}{2},2;-\frac{\omega^{2}z_{0}^{2}}{c^{2}}\right]~{},\end{split}$
(42)

where $\mathcal{B}_{\mathcal{f}}$ can be expressed as,

$\mathcal{B}_{\mathcal{f}}=\left|\mathcal{B}_{\mathcal{f}}\right|e^{i\zeta}$
(43)

where $\zeta=\text{arg}\left[\mathcal{B}_{\mathcal{f}}\right]$. Adding Eqs.
(41) and (38) we obtain,

$\begin{split}&I_{1}^{\prime}+I_{2}^{\prime}+I_{3}^{\prime}\\\ &=-\frac{4\nu
c}{a\omega}e^{-\frac{\pi\nu c}{a}}\left|\Gamma\left[-\frac{2i\nu
c}{a}\right]\right|\cos(\theta^{\prime})-\frac{4\pi\nu
z_{0}}{a\sinh\left[\frac{\pi\nu c}{a}\right]}e^{-\frac{i\omega
z_{0}}{c}}\mathcal{B}_{\mathcal{f}}\\\ &=-\frac{4\nu
c}{a\omega}\sqrt{\frac{\pi a}{\nu
c}}\frac{\cos(\theta^{\prime})}{\sqrt{e^{\frac{4\pi\nu
c}{a}}-1}}\left[1+\frac{2\,\chi\,\omega
z_{0}}{c}\,\frac{\mathcal{B}_{\mathcal{f}}\sec(\theta^{\prime})e^{-\frac{i\omega
z_{0}}{c}}}{1-e^{-\frac{2\pi\nu c}{a}}}\right]\end{split}$ (44)

where $\chi=\sqrt{\frac{2\pi\nu c}{a}\sinh\left[\frac{2\pi\nu c}{a}\right]}$.
Using eq.(44) in eq.(37), we can rewrite the exact analytical form of the
transition probability as follows,

$\begin{split}\mathcal{P}_{exc}^{\prime}&=\frac{16\pi\,\mathcal{G}^{2}\,\nu
c}{a\,\omega^{2}}\frac{\cos^{2}(\theta^{\prime})}{e^{\frac{4\pi\nu
c}{a}}-1}\,\,\biggr{[}1+\frac{4\,|\mathcal{B}_{\mathcal{f}}|\,\chi\omega\,z_{0}}{c}\frac{\sec(\theta^{\prime})}{1-e^{-\frac{2\pi\nu
c}{a}}}\\\ &\times\cos\left(\zeta-\frac{\omega
z_{0}}{c}\right)+\frac{4\,|\mathcal{B}_{\mathcal{f}}|^{2}\chi^{2}\omega^{2}z_{0}^{2}}{c^{2}}\frac{\sec^{2}(\theta^{\prime})}{(1-e^{-\frac{2\pi\nu
c}{a}})^{2}}\biggr{]}~{}.\end{split}$ (45)

In order to examine the equivalence we further simplify the above equation and
consider the leading order behaviour of the same. However in sec.(5) we show
that the exact (Eq.(45)) and leading order (Eq.(47)) analysis yield same order
of estimation for the transition probability. Following the experimental
proposition as considered in [35, 16], one may take $\omega$ to be $\sim 1$
gHz. The fixed position of the atom $(z_{0})$ can assumed to be a small
parameter such as $\mathcal{O}(0.01)$ m. This leads us to write,
$\frac{2\omega z_{0}}{c}<1$, where $c=3\times 10^{8}$ m/s. In Eq.(40), the
upper limit on $x$ dictates that $x<\frac{2\omega z_{0}}{c}$, which implies
$\frac{cx}{2\omega z_{0}}<1$. This allows us to write the Taylor expansion in
the term $\left(1-\frac{cx}{2\omega z_{0}}\right)^{-\frac{i\nu c}{a}}$ and as
well as the expansion of $e^{ix}$ upto the linear order in $x$. This yields,

$\begin{split}I^{\prime}_{3}\cong&-\frac{2e^{-\frac{i\omega
z_{0}}{c}}}{\omega\left(\frac{2\omega z_{0}}{c}\right)^{\frac{i\nu
c}{a}}}\int_{0}^{\frac{2\omega z_{0}}{c}}dx~{}x^{\frac{i\nu
c}{a}}\left(1+ix+\frac{i\nu c^{2}x}{2a\omega z_{0}}\right)\\\
\cong&-\frac{4z_{0}}{c}e^{-\frac{i\omega z_{0}}{c}}\left(\frac{1}{1+\frac{i\nu
c}{a}}+\frac{2i\omega z_{0}}{c}\frac{1+\frac{\nu c^{2}}{2a\omega
z_{0}}}{2+\frac{i\nu c}{a}}\right)~{}.\end{split}$ (46)

Combining the Eqs.(38,46), we get the excitation probability as follows,

$\begin{split}\mathcal{P}^{\prime}_{exc}&\cong\frac{16\pi\mathcal{G}^{2}\nu
c}{a\omega^{2}}\frac{\cos^{2}(\theta^{\prime})}{e^{\frac{4\pi\nu
c}{a}}-1}\biggr{[}\biggr{(}1+\frac{a\omega z_{0}}{\nu
c^{2}}\frac{e^{\frac{\pi\nu
c}{a}}\sec(\theta^{\prime})}{\left(1+\frac{\nu^{2}c^{2}}{a^{2}}\right)}\\\
&\times\frac{\left(\cos\left(\frac{\omega z_{0}}{c}\right)-\frac{\nu
c}{a}\sin\left(\frac{\omega
z_{0}}{c}\right)\right)}{\left|\Gamma\left[-\frac{2i\nu
c}{a}\right]\right|}\biggr{)}^{2}\\\ &+\biggr{(}\frac{a\omega z_{0}}{\nu
c^{2}}\frac{e^{\frac{\pi\nu
c}{a}}\sec(\theta^{\prime})}{\left(1+\frac{\nu^{2}c^{2}}{a^{2}}\right)}\frac{\left(\frac{\nu
c}{a}\cos\left(\frac{\omega z_{0}}{c}\right)+\sin\left(\frac{\omega
z_{0}}{c}\right)\right)}{\left|\Gamma\left[-\frac{2i\nu
c}{a}\right]\right|}\biggr{)}^{2}\biggr{]}~{}.\end{split}$ (47)

The final forms of the excitation probabilities corresponding to the two cases
is summarised in Eqs.(27,47).

## 4 Non-equivalence in dual photon scenario

In this section we examine the equivalence between the two systems which are
discussed in sections (2, 3), in the context of the dual photon emission. In
our case excitation of the ground state atom (equivalent to $\hbar\omega$) is
accompanied by the simultaneous emission of two photons with frequency $\nu$
(equivalent to a total $2\hbar\nu$ amount of energy). Therefore, following
[16], we interchange the atomic frequency $(\omega)$ with twice the frequency
of the emitted photons $(2\nu)$ for our setup. We interchange
$\nu=\frac{\omega}{2}$ in Eqs.(27, 47) respectively and obtain the two
probabilities as below,

$\begin{split}\mathcal{P}_{exc}=&\frac{8\pi\mathcal{G}^{2}c}{a\omega}\frac{\cos^{2}(\bar{\theta})}{e^{\frac{2\pi\omega
c}{a}}-1}\biggr{(}1-\frac{4\sec(\bar{\theta})}{\left|\Gamma\left(\frac{i\omega
c}{a}\right)\right|}K_{\frac{i\omega c}{a}}\left(\frac{\omega c}{a}\right)\\\
+&\frac{4\omega c\sec^{2}(\bar{\theta})}{a\pi}\biggr{|}K_{\frac{i\omega
c}{a}}\left(\frac{\omega c}{a}\right)\biggr{|}^{2}\sinh\left(\frac{\pi\omega
c}{a}\right)\biggr{)}\end{split}$ (48)

and

$\begin{split}\mathcal{P}^{\prime}_{exc}&\cong\frac{8\pi\mathcal{G}^{2}c}{a\omega}\frac{\cos^{2}(\theta^{\prime\prime})}{e^{\frac{2\pi\omega
c}{a}}-1}\biggr{[}\biggr{(}1+\frac{2az_{0}}{c^{2}}\frac{e^{\frac{\pi\omega
c}{2a}}\sec(\theta^{\prime\prime})}{\left(1+\frac{\omega^{2}c^{2}}{4a^{2}}\right)}\\\
&\times\frac{\left(\cos\left(\frac{\omega z_{0}}{c}\right)-\frac{\omega
c}{2a}\sin\left(\frac{\omega
z_{0}}{c}\right)\right)}{\left|\Gamma\left[-\frac{i\omega
c}{a}\right]\right|}\biggr{)}^{2}\\\
&+\biggr{(}\frac{2az_{0}}{c^{2}}\frac{e^{\frac{\pi\omega
c}{2a}}\sec(\theta^{\prime})}{\left(1+\frac{\omega^{2}c^{2}}{4a^{2}}\right)}\frac{\left(\frac{\omega
c}{2a}\cos\left(\frac{\omega z_{0}}{c}\right)+\sin\left(\frac{\omega
z_{0}}{c}\right)\right)}{\left|\Gamma\left[-\frac{i\omega
c}{a}\right]\right|}\biggr{)}^{2}\biggr{]}~{}.\end{split}$ (49)

Here,

$\bar{\theta}=\frac{\omega z_{0}}{c}-\frac{\omega
c}{a}\ln\left[\frac{a}{\omega
c}\right]-\text{arg}\left[\Gamma\left(\frac{i\omega c}{a}\right)\right]~{}$
(50)

and

$\theta^{\prime\prime}=\frac{\omega z_{0}}{c}-\frac{\omega
c}{a}\ln\left[\frac{a}{\omega
c}\right]-\text{arg}\left[\Gamma\left[\frac{i\omega c}{a}\right]\right]~{}.$
(51)

Comparing Eq.(49) with Eq.(48), we observe that the excitation probabilities,
although has a same Planckian pre-factor, the terms in the parenthesis do not
quite match. This indicates that the probabilities are not related to each
other under such exchanges and therefore may imply a violation of the
equivalence relation. This is in line with the observation made in [16] where
a subtle symmetry between the excitation of a stationary atom due to an
accelerating mirror in Minkowski spacetime with the excitation of an
accelerating atom in Minkowski spacetime relative to a stationary mirror was
shown. This symmetry was regarded as a manifestation of the equivalence
principle. Although, there is a lot of debate about the status of the
equivalence principle applying to non-gravitational processes in a
gravitational field [51, 54], however, the observation made in [16] indeed
shows a nice symmetry between the two set ups in the one photon case which is
found to be broken in our case. Also from the exact result i.e Eq.(45) one can
obtain the same insight leading to the violation of equivalence relation. In
the upcoming section we study the observational prominence of the dual photon
emission scenario.

## 5 Phenomenological aspects of the model

In this section we study the behaviour of the system while the atom is excited
to its higher energy state along with the simultaneous emission of two
photons. Therefore in fig.(1), we plot the transition probability as depicted
in Eq.(27) with respect to the angular frequency of the detector $(\omega)$.
We fix the parameters such as, $\nu\sim 10^{4}$ Hz, $z_{0}=0.01$ m. We also
take the effective coupling strength between the $N$ $(\sim 10^{4}-10^{8}$)
polar molecules and the photons as, $g_{{\rm eff}}=g\sqrt{N}\sim 10^{7}$ Hz.
The acceleration of the atom is fixed to be $\sim 10^{15}$ m/s2, which can be
achieved by using the superconducting circuits [58].

Figure 1: Plot of $\mathcal{P}_{exc}$ vs. $\omega$. Atom is accelerating with
respect to the fixed mirror and two photons are simultaneously emitted.

Fig.(1) depict the standard Planckian distribution and also shows an
oscillatory behaviour as similar to the single photon emission case.

Now we explore the other system which is the mirror accelerating with respect
to a fixed atom. Largely, we fix the values for the parameters as considered
in [16], so that the observational prominence of our model may also be
verified with respect to the similar experimental setup as proposed in [16].
This leads us to fix $\omega\sim 10^{9}$ Hz, $z_{0}=0.01$ m, $g_{{\rm
eff}}=g\sqrt{N}\sim 10^{7}$ Hz and acceleration of the mirror to be $\sim
10^{15}$ m/s2 in eq.(45) and obtain the behaviour of the transition
probability with the variation of the field frequency $(\nu)$ in Fig. (2).

Figure 2: Plot of $\mathcal{P}^{\prime}_{exc}$ vs. $\nu$. Mirror is
accelerating with respect to a fixed atom and two photons are simultaneously
emitted.

This plot suggests that in case of the dual photon emission process, the
transition probability becomes $\mathcal{O}(10^{-4})$ which is similar to the
outcome of single field emission. We comment that in case of the dual photon
emission the excitation probability can be enhanced to the detectable
probability (i.e $P\sim\mathcal{O}(10^{-2})$) [16] by taking into account the
probabilities for $100$ such cavity modes. We can also observe similar
behaviour of the approximate form of the excitation probability in eq.(47) if
we use the same parameters as in Fig.(2). Now we analyse Eq.(49) under the
condition $\frac{c\nu}{a}\ll 1$. We keep fix rest of the parameters
$(\omega,z_{0},g_{{\rm eff}},a)$ same as taken in the Fig.(2). This yields,

$\begin{split}&\mathcal{P}^{\prime}_{exc}|_{\nu c\ll
a}\,\cong\frac{4\mathcal{G}^{2}\,\cos^{2}(\theta^{\prime})}{\omega^{2}}\biggr{[}1+\frac{2a\omega
z_{0}}{\nu c^{2}}\frac{\sec(\theta^{\prime})}{\left|\Gamma\left[-\frac{2i\nu
c}{a}\right]\right|}\\\ &+\biggr{(}\frac{a\omega z_{0}}{\nu
c^{2}}\frac{\sec(\theta^{\prime})}{\left|\Gamma\left[-\frac{2i\nu
c}{a}\right]\right|}\biggl{)}^{2}\biggr{\\{}1+\frac{2\pi\nu
c}{a}\biggl{(}1+\frac{1}{\pi}\biggr{)}\sin\biggr{(}\frac{\omega
z_{0}}{c}\biggl{)}\\\ &+\sin^{2}\biggr{(}\frac{\omega
z_{0}}{c}\biggl{)}\biggl{\\}}\biggl{]}~{}.\end{split}$ (52)

We further plotted Eq.(52) with respect to $\nu$ and obtained the Fig.(3).

Figure 3: Plot of $\mathcal{P}^{\prime}_{exc}$ vs. $\nu$. for $\frac{\nu
c}{a}\ll 1$

## 6 Discussion

Transforming a virtual photon to a real detectable photon has become a
remarkable tool in studying the acceleration radiation and its several
consequences in the flat/curved spacetime. Besides its theoretical aspects,
this mechanism is also promising from the observational perspective. In the
present manuscript we consider an atom-mirror system where the atom and the
mirror possess relative acceleration between them. Two configurations are
possible such as the atom is accelerating with respect to a fixed mirror and
vice versa. We further consider that due to the virtual processes, which take
place in the vacuum states of the two scalar photons, the atom makes a
transition to its higher excited state along with the simultaneous emission of
the two virtual photons. Applying the mechanism of transforming the virtual
photons to real one, we evaluate the transition probabilities corresponding to
the above configurations. Our results turn out to be distinct in nature than
that of the single field emission case. We obtain that the dual field emission
process breaks the “equivalence principle in QED”. An interference term
emerges in the transition probability due to the emission of the two photon
quanta, which in turn breaks the equivalence between the two setup.
Undoubtedly, this demands more such investigations to perceive the notion,
applications and the limitations of such equivalence principle. As similar to
the single filed emission process, our results suggest that both the
transition probabilities as in Eqs.(27, 47) exhibit spatial oscillatory
behaviour. An additional interference term excluding the standard Planckian
factor is solely responsible for this oscillatory behaviour in the excitation
probability. For our case too, a similar experimental proposition as mentioned
in [16] can be constructed as following. One may consider that $N(\sim
10^{4}-10^{8})$ number of polar molecules are collectively forming an ensemble
where each molecule is simultaneously interacting with the two photons within
a cavity setup. We refer our readers to [16, 33, 34, 35] for a thorough
understanding of the experimental setup which demonstrate the motion of
accelerated mirror and capture its several consequences. Note that for
mathematical simplicity we translate the moving mirror and static atom case in
Minkowski spacetime to Rindler spacetime, where the mirror becomes static and
the atom accelerates. This permits us to define a Rindler vacuum state with
respect to the static mirror in Rindler spacetime. In this context, we point
out that it is a very difficult or somewhat impossible task to construct a
Rindler vacuum state in the laboratory. For this reason, the experimental
implementation of this setup (which is briefly mentioned earlier) is
constructed in the Minkowski spacetime where the mirror accelerates and the
atom is static [16, 33, 34, 35].

We observe that for an accelerating mirror system, under the condition $a\gg
c\nu$, the transition probability of the system continues to be the function
of $(a,\,c,\nu)$. This implies that the parameter space in the dual photon
emission process is larger than that of the single field emission case, even
under the above condition. Therefore we gain the freedom to study the
transition probability with respect to the field frequency, acceleration of
the mirror etc. Further, we obtain a detectable transition probability by
conveniently choosing the range of field frequency and the acceleration of the
mirror with respect to the fixed atom. It will be interesting to extended the
present work with the emission of more than two rather larger number of
photons. We are planning to report this elsewhere.

## References

  * [1] A. Einstein, Sitzungsber Preuss Akad Wiss (1915) 844.
  * [2] A. Einstein, Ann. der Physik 49 (1916) 769.
  * [3] S.W. Hawking, Nature 248 (1974) 30.
  * [4] S.W. Hawking, Commun. Math. Phys 43 (1975) 199.
  * [5] S.W. Hawking, Phys. Rev. D 13 (1976) 191.
  * [6] J.D. Bekenstein, Lett. Nuovo Cimento 4 (1972) 737.
  * [7] J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333.
  * [8] D. N. Page, Phys. Rev. D 13 (1976) 198.
  * [9] D. N. Page, Phys. Rev. D 14 (1976) 3260.
  * [10] D. N. Page, Phys. Rev. D 16 (1977) 2402.
  * [11] S. A. Fulling, Phys. Rev. D 7 (1973) 2850.
  * [12] P. Davies, J. Phys. A 8 (1975) 609.
  * [13] B. S. DeWitt, General Relativity: An Einstein Centenary Survey, eds S. W. Hawking, W. Israel (Cambridge Univ Press (1979), Cambridge, UK).
  * [14] M. O. Scully, V. V. Kocharovsky, A. Belyanin, E. Fry and F. Capasso,Phys. Rev. Lett. 91 (2003) 243004.
  * [15] M. O. Scully, S. A. Fulling, D. M. Lee, D. N. Page, W. P. Schleich and A. A. Svidzinsky, Proc. Natl. Acad. Sci. U.S.A. 115 (2018) 8131.
  * [16] A. A. Svidzinsky, S. J. Ben-Benjamin, S. A. Fulling and D. N. Page, Phys. Rev. Lett 121 (2018) 071301.
  * [17] H. E. Camblong, A. Chakraborty and C. R. Ordóñez, Phys. Rev. D 102 (2020) 085010.
  * [18] A. Azizi, H. E. Camblong, A. Chakraborty, C. R. Ordóñez and M. O. Scully, Phys. Rev. D 104 (2021) 065006.
  * [19] A. Azizi, H. E. Camblong, A. Chakraborty, C. R. Ordóñez and M. O. Scully, Phys. Rev. D 104 (2021) 084086.
  * [20] A. Azizi, H. E. Camblong, A. Chakraborty, C. R. Ordóñez and M. O. Scully, Phys. Rev. D 104 (2021) 084085.
  * [21] S. Sen, R. Mandal and S. Gangopadhyay, Phys. Rev. D 105 (2022) 085007.
  * [22] S. Sen, R. Mandal and S. Gangopadhyay, Phys. Rev. D 106 (2022) 025004.
  * [23] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752.
  * [24] J. Schwinger, Phys. Rev. 82 (1951) 664.
  * [25] M. A. Nielsen and I. A. Chuang, “Quantum Computation and Quantum Information: 10th Anniversary Edition” (2011), Cambridge University Press, United States.
  * [26] W. G. Unruh, Phys. Rev. D 14 (1976) 870.
  * [27] W. G. Unruh, Phys. Rev. D 15 (1977) 365.
  * [28] W. G. Unruh and R. M. Wald, Phys. Rev. D 29 (1984) 1047.
  * [29] L. C. B. Crispino, A. Higuchi, G. E. A. Matsas, Rev. Mod. Phys. 80 (2008) 787.
  * [30] Aindriú Conroy,Phys. Rev. D 105 (2022) 123513.
  * [31] N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,” (Cambridge University Press, Cambridge) 1982.
  * [32] R. Chatterjee, S. Gangopadhyay and A. S. Majumdar, Phys. Rev. D 104 (2021) 124001.
  * [33] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, Nature (London) 479, (2011) 376.
  * [34] J. R. Johansson, G. Johansson, C. M. Wilson and F. Nori, Phys. Rev. Lett. 103 (2009) 147003.
  * [35] M. Wallquist, K. Hammerer, P. Rabl, M. Lukin and P. Zoller, Phys. Scr. T137 (2009) 014001.
  * [36] C. Chowdhury, A. Das and B. R. Majhi, Eur. Phys. J. Plus 134 (2019) 65.
  * [37] H. Davoudiasl, J. L. Hewett and T. G. Rizzo, Phys. Rev. Lett. 84, 2080 (2000)
  * [38] S. Chang, J. Hisano, H. Nakano, N. Okada and M. Yamaguchi, Phys. Rev. D 62, 084025 (2000)
  * [39] K. R. Dienes, 2002 TASI Lectures, “New directions for new dimensions: An introduction to Kaluza-Klein theory, large extra dimensions, and the brane world”
  * [40] G. Dvali, Fortsch. Phys. 58, 528 (2010)
  * [41] G. Dvali and M. Redi, Phys. Rev. D 80, 055001 (2009)
  * [42] G. Dvali, I. Sawicki and A. Vikman, JCAP 0908, 009 (2009)
  * [43] D. Wands, Lect. Notes Phys. 738, 275-304 (2008)
  * [44] J. O. Gong, Int. J. Mod. Phys. D 26, no.01, 1740003 (2016)
  * [45] L. Alvarez-Gaume, C. Gomez and R. Jimenez, JCAP 03, 017 (2012)
  * [46] S. Noori Gashti, JHAP 2, no.1, 13-24 (2022).
  * [47] J. Martin and L. Pinol, JCAP 12, no.12, 022 (2021).
  * [48] Y. Akrami, M. Sasaki, A. R. Solomon and V. Vardanyan, Phys. Lett. B 819, 136427 (2021)
  * [49] T. G. Philbin, et al., Science 319 (2008) 1367.
  * [50] F. Rohrlich, Ann. Phys. 22 (1963) 169.
  * [51] M. Pauri and M. Vallisneri, Foun. Phys. 29 (1999) 1499.
  * [52] D. Singleton and S. Wilburn, Phys. Rev. Lett. 107 (2011) 081102.
  * [53] D. Singleton and S. Wilburn, Int. J. Mod. Phys. D 25 (2016) 1644009.
  * [54] S. A. Fulling and J. H. Wilson, Phys. Scr. 94 (2019) 014004.
  * [55] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition (Academic Press, Amsterdam, 2007).
  * [56] F. Hammad, A. Landry and D. Dijamco, Phys. Rev. D 102 (2021) 085010.
  * [57] D. Su, C. T. M. Ho, R. B. Mann, and T. C. Ralph, New J. Phys. 19 (2017) 063017.
  * [58] N. Friis, A. R. Lee, K. Truong, C. Sabin, E. Solano, G. Johansson and I. Fuentes,Phys. Rev. Lett. 110 (2013) 113602.


# Explicit Domain Adaptation through implicit knowledge: Post-hoc Data
Homogenization

Kurt Willis & Luis Oala
Fraunhofer HHI
Berlin, Germany
<EMAIL_ADDRESS>

# Guided Data Homogenization for Post-Hoc Domain Adaptation

Kurt Willis & Luis Oala
Fraunhofer HHI
Berlin, Germany
<EMAIL_ADDRESS>

# Post-Hoc Domain Adaptation via Guided Data Homogenization

Kurt Willis & Luis Oala
Fraunhofer HHI
Berlin, Germany
<EMAIL_ADDRESS>

###### Abstract

Addressing shifts in data distributions is an important prerequisite for the
deployment of deep learning models to real-world settings. A general approach
to this problem involves the adjustment of models to a new domain through
transfer learning. However, in many cases, this is not applicable in a post-
hoc manner to deployed models and further parameter adjustments jeopardize
safety certifications that were established beforehand. In such a context, we
propose to deal with changes in the data distribution via guided data
homogenization which shifts the burden of adaptation from the model to the
data. This approach makes use of information about the training data contained
implicitly in the deep learning model to learn a domain transfer function.
This allows for a targeted deployment of models to unknown scenarios without
changing the model itself. We demonstrate the potential of data homogenization
through experiments on the CIFAR-10 and MNIST data sets.

A routine assumption in machine learning is that data in the training and
testing environment is identically distributed. In reality, data distributions
may differ (Hendrycks et al., 2020), leading to performance degradations for
models trained with standard deep learning algorithms. Transfer learning (Pan
& Yang, 2010) has emerged as a field that aims to transfer knowledge of models
from a learned source task to a target task by exploiting underlying
commonalities. Domain adaptation in particular, addresses a mismatch in either
the input space $\mathcal{X}$ or the data distribution $\mathbb{P}$ of a
source and target task:
$(\mathcal{X}_{s},\mathbb{P}_{s})\neq(\mathcal{X}_{t},\mathbb{P}_{t})$. A
common approach to this challenge is to encourage the model to learn a domain
invariant representation of the data. This is typically done through the use
of the joint distribution, requiring access to the source data set.

This work proposes domain adaptation through homogenization of the data in a
post-hoc-manner. This eliminates the need to anticipate numerous possible
test-time scenarios during training and the original model is kept intact - a
desirable property for security and robustness certification (Balunovic et
al., 2019; Oala et al., 2020). In detail, an explicit domain mapping function
is learned through an optimization objective adopted from work by Yin et al.
(2020). This does not assume access to the source data set; instead, the data
statistics measured in an appropriate feature space are sufficient. These
statistics are readily available in networks that make use of the widely
adapted batch normalization layers (Ioffe & Szegedy, 2015).

An example for the relevance of post-hoc domain adaptation is seen in optical
systems which provide inputs for deep learning applications. Sensor corrosion
or re-calibration of hardware devices has proven to be a significant barrier
to the reliable application of deep learning systems in fields such as
medicine (Heaven, 2020) or autonomous driving (Michaelis et al., 2020).

We show the ability to homogenize non-identically distributed data in
experiments on the CIFAR-10 (Krizhevsky, 2012) and MNIST (LeCun & Cortes,
2010) data sets. The method is further tested in the unsupervised setting,
where no labeled data in the target domain is available.

Related work focuses on learning domain-invariant representations in order to
bridge the discrepancy in distributions. In this effort, Ghifary et al. (2015)
propose training an auto-encoder that is able to encode both domains. Tzeng et
al. (2015) make use of the criterion loss along with a domain confusion loss
which is implemented by adding an adaptation layer on top of the classifier’s
last layer. This auxiliary layer aims to make the learned representations
indistinguishable between domains by maximally confusing a domain classifier.
Long et al. (2015) further extend the use of adaptation layers to multiple
layers throughout the network. Zellinger et al. (2019) and Sun & Saenko (2016)
formulate a distribution loss incorporating higher-order moments. Saito et al.
(2019) propose to classify data of the target domain by comparing feature
representations to prototypes for each class. Zhu et al. (2017) propose
CycleGAN, a generative adversarial network that learns explicit domain
transformations, by formulating an adversarial loss and measuring
reconstruction accuracy.

This work can also be viewed in the context of inverse problems, and more
specifically, related to the task of deblurring or deconvolution. Whereas Xu
et al. (2014) and Kobler et al. (2017) train a deconvolution model for image
reconstruction on handcrafted perturbation functions, this work solely makes
use of the prior knowledge learned by an image classification network and
statistics to correct for general distribution shifts that also entail
deconvolutions.

##  1 Method

Data set
distributionSource$\mathcal{A}$Original$\mathcal{B}_{\text{true}}$Perturbed
target$\mathcal{B}$Reconstructed$\rho(\mathcal{B})$$\delta$$\rho$OptimizeUnknown$\varphi$Feature
mapLossStatistics

Figure 1: Overview of the homogenization setting. A general transformation
$\rho$ is optimized, so that a neural network $\Phi$ regains its performance
on the transformed dataset $\rho(\mathcal{B})$. The feature maps are obtained
from $\Phi$.

Figure 1 summarizes the main idea of guided data homogenization. Suppose we
are given a neural network that was trained on - or performs reasonably well
on - a source data set $\mathcal{A}$ and a target data set $\mathcal{B}$,
typically smaller in size ($n_{\mathcal{B}}<n_{\mathcal{A}}$). Further, the
distribution of $\mathcal{B}$ differs crucially from $\mathcal{A}$, i.e. the
neural network is unable to attain high performance on data coming from
$\mathcal{B}$. The task is to learn a transformation $\rho$ that adjusts
$\mathcal{B}$ so that the network’s performance is regained.

An underlying assumption is that $\mathcal{B}$ originates from the same
distribution as $\mathcal{A}$, but has been corrupted or transformed by an
unknown perturbation $\delta$. The transformation $\rho$ is modeled to correct
for this perturbation, in the sense that
$\rho(\mathcal{B})\approx\mathcal{B}_{\text{true}}$. This is done by
minimizing a dissimilarity-score between $\rho(\mathcal{B})$ and $\mathcal{A}$
in the feature space given by an appropriate feature map $\varphi$. Towards
this goal, simple statistics (mean and variance) are recorded for both data
sets after they have been transformed by $\varphi$. By use of backpropagation,
$\rho$ is optimized, in order to make the statistics of the target
$\mathcal{B}$ approach those of the source $\mathcal{A}$. Whether matching
statistics in feature space actually increases similarity between
$\mathcal{A}$ and $\mathcal{B}$ strongly depends on $\varphi$ and its
representation of the data set.

For a given feature mapping $\varphi:\mathbb{R}^{d}\to\mathbb{R}^{m}$, the
dissimilarity score, or the statistics-loss, is defined as follows.

$\textit{L}_{\varphi}(\mathcal{A},\mathcal{B})=\|\operatorname{\mu}({\varphi(\mathcal{A})})-\operatorname{\mu}({\varphi(\mathcal{B})})\|_{2}+\|\operatorname{\sigma^{2}}({\varphi(\mathcal{A})})-\operatorname{\sigma^{2}}({\varphi(\mathcal{B})})\|_{2}$
(1)

$\operatorname{\mu}$ and $\operatorname{\sigma^{2}}$ denote the empirical
feature mean and variance over the feature channels, as is generally done in
the computations of BatchNorm-layers. A data set mapping is obtained by
applying a feature map $\varphi$ to every input:
$\varphi(\mathcal{A}):=\\{(\varphi(\mathbf{x}),y)\mid(\mathbf{x},y)\in\mathcal{A}\\}\;$.
This notion can be extended to a collection of feature maps:
$\textit{L}_{[\varphi_{1},\dots,\varphi_{n}]}(\mathcal{A},\mathcal{B})=\sum_{i=1}^{n}\textit{L}_{\varphi_{i}}(\mathcal{A},\mathcal{B})$.
Furthermore, a class-dependent loss can be obtained by splitting the data set
into subsets for each class and accumulating the resulting losses:
$\textit{L}^{\textit{CC}}_{\varphi}(\mathcal{A},\mathcal{B})=\sum_{c=1}^{C}\textit{L}_{\varphi}(\mathcal{A}|_{c},\mathcal{B}|_{c})\,.$
Here, $\mathcal{A}|_{c}=\\{(\mathbf{x},y)\in\mathcal{A}\mid y=c\\}$ is the
subset of $\mathcal{A}$ constrained to samples of label $c$. This case can be
compared to related work, where class-prototypes are used.

##  2 Experiments

(a)

(b)

(c)

Figure 2: Excerpt of results on the homogenization experiments on CIFAR10 data
set in a supervised setting after 100 epochs for NN ALL method. $\kappa=0.15$,
$n_{\mathcal{B}}=512$. The full results are depicted in Figure 3 of the
Appendix.

The experiments111The code for guided data homogenization and all experiments
can be accessed at https://github.com/willisk/Thesis. are conducted on the
popular image recognition data sets MNIST and CIFAR-10. The target data set
$\mathcal{B}$ is obtained by applying a randomized, parameter-controlled
perturbation to a small unseen subset of the original data set. The
perturbation model is a composition of additive Gaussian noise and two noise-
controlled 2d-convolutions:
$\delta(\mathbf{x})=\mathbf{K}_{\kappa}^{(2)}(\mathbf{K}_{\kappa}^{(1)}(\mathbf{x}+\kappa\bm{\mu})$.
The kernel matrices of the convolutions are each of size $3\times 3$ and
initialized to the identity kernel with added Gaussian noise. The overall
noise level is controlled by the parameter $\kappa$. The reconstruction model
is a vanilla four-layer convolutional residual network of width 16.

### Feature maps

As explained in Section 1, the statistics-loss depends on the feature mapping
$\varphi$. Various feature mappings are compared in the experiments. The main
approach (NN ALL) involves all hidden states of a neural network
$\Phi=(\Phi_{\text{L}}\circ\dots\Phi_{1})$. These are seen as the outputs of a
collection of feature maps [$\varphi_{0},\dots,\varphi_{\text{L-1}}$], where

$\displaystyle\varphi_{\ell}$
$\displaystyle:\mathbb{R}^{d}\to\mathbb{R}^{d_{\ell}}$
$\displaystyle=(\Phi_{\ell}\circ\dots\Phi_{1})\,,\textnormal{ for
}\ell=1,\dots,L-1$ $\displaystyle\varphi_{\textnormal{0}}$
$\displaystyle:\mathbb{R}^{d}\to\mathbb{R}^{d}$ $\displaystyle=\text{id}\,.$

Another feature mapping serving as a comparison is induced by the penultimate
layer $\Phi_{\text{L}-1}$, the layer before the logits layer. It is reported
as NN.

$\varphi:\mathbb{R}^{d}\to\mathbb{R}^{d_{\textnormal{L-1}}}=(\Phi_{\textnormal{L-1}}\circ\dots\circ\Phi_{1})$

In order to further delineate the benefit of using representations stemming
from the neural network, a feature mapping involving only one linear layer
with a width of 512 is considered. Comparing the performance of the
reconstruction using shallow versus deep features can further warrant the
necessity of complex feature maps. This one linear layer can be seen as a set
of random linear projections; it is reported as RP.

The hidden representations of the same neural network model with randomly
initialized parameters (RANDOM NN) serves as another baseline comparison. This
is done in order to test the utility of an optimized feature representation.

The complete loss formulation for the supervised setting is then given by

$r_{\text{stats}}\textit{L}_{\varphi}(\mathcal{A},\mathcal{B})+r_{\text{crit}}\textit{L}_{\text{crit}}(\mathcal{B},y_{\mathcal{B}})\,,$

where $y_{\mathcal{B}}$ are the target labels and $\textit{L}_{\text{crit}}$
is the criterion-loss, which, in this case, is the cross-entropy loss. The
unsupervised setting only makes use of $\textit{L}_{\varphi}$ (i.e.
$r_{\text{crit}}=0$).

##  3 Results

For measuring the overall distortion of the reconstruction, the l2-error
$\varepsilon_{2}$ of the unit vectors is calculated.
$\varepsilon_{2}^{2}=\frac{1}{d}\sum_{i=0}^{d}\|\rho(\delta(\mathbf{e}_{i}))-\mathbf{e}_{i}\|_{2}^{2}\,,$
where $\mathbf{e}_{i}$ is the i-th unit vector of the standard basis. Further,
a metric commonly encountered in image quality assessment, the peak signal-to-
noise ratio (PSNR) is reported. One last metric, that doesn’t rely on the
mean-squared error, the structural similarity index measure (SSIM) (Horé &
Ziou, 2010) is considered. It is re-scaled to fit the interval $[0,1]$ and is
given as a percentage (%).

In order to measure the generalization ability of the learned transformation,
a second, independently trained neural network’s ($\Phi_{\text{ver}}$)
accuracy is reported as the verification accuracy. Further, the same
transformations are applied to an unseen validation set $\mathcal{C}$. The
accuracy attained by $\Phi_{\text{ver}}$ on this new set constitutes the
validation accuracy.

The baseline evaluation metrics, along with the final results of the
reconstruction task on the CIFAR-10 data set are given in Table 1 for each
method. Less than $1\%$ of the original training set sample size is used in
these experiments. After the perturbation has been applied to $\mathcal{B}$
the scores drop significantly. Relying solely on the criterion results in
overfitting the reconstruction function, as is seen by the disparity in
generalization scores. Random projections (RP) and the randomly initialized
convolutional neural network fail to capture vital information about the data
set distribution; although they are still able to improve various scores. The
best-performing method is NN ALL.

While shallow features are able to perform the reconstruction task on simple
data sets like MNIST (see Table 4 of the Appendix), their benefit on a more
complex data set like CIFAR-10 is limited. Further results for the class-
conditional formulation $\textit{L}^{\textit{CC}}$ are given in Appendix A.
Using this formulation, a significant performance boost is noted when
$r_{\text{crit}}=0$, however, no notable benefit is obtained when the
criterion loss is included.

The evaluation metrics for the unsupervised setting, where no label
information is provided are given in Table 3 of the Appendix. The scores are
close to the supervised case and show promising results even for small sample
sizes.

Table 1: Metrics of homogenization results in the supervised setting after 100
optimization epochs for CIFAR-10. The mean and the standard deviation over 5
runs are given. The number of samples $n_{\mathcal{B}}=512$.

Data set |  accuracy [%] |  validation accuracy [%] |  verification accuracy [%] |  $\downarrow$ l2-error |  $\uparrow$ PSNR |  $\uparrow$ SSIM [%]
---|---|---|---|---|---|---
Data set $\mathcal{A}$ | 99.7 | – | 99.8 | 0 | $\infty$ | 100
Data set $\mathcal{B}_{\text{true}}$ | 99.8 | 93.2 | 99.8 | 0 | $\infty$ | 100
Data set $\mathcal{B}$ | 26.2 | 26.6 | 26.0 | 24.3 | 16.5 | 84.0
Homogenized $\mathcal{B}$ |  |  |  |  |  |
CRITERION | 99.6 $\pm$ 0.4 | 77.9 $\pm$ 2.1 | 75.9 $\pm$ 2.7 | 14.0 $\pm$ 2.7 | 18.0 $\pm$ 0.4 | 86.7 $\pm$ 2.2
NN | 95.5 $\pm$ 0.6 | 83.1 $\pm$ 0.6 | 80.8 $\pm$ 0.8 | 11.6 $\pm$ 1.6 | 19.5 $\pm$ 0.3 | 90.1 $\pm$ 0.5
NN ALL | 99.3 $\pm$ 0.4 | 88.2 $\pm$ 0.6 | 84.9 $\pm$ 1.0 | 10.0 $\pm$ 2.6 | 23.3 $\pm$ 0.3 | 95.4 $\pm$ 0.2
RANDOM NN | 51.9 $\pm$ 5.0 | 48.4 $\pm$ 5.7 | 42.7 $\pm$ 4.2 | 29.0 $\pm$ 10.6 | 11.3 $\pm$ 0.7 | 59.6 $\pm$ 4.7
RP | 51.1 $\pm$ 2.1 | 51.0 $\pm$ 1.4 | 47.3 $\pm$ 1.1 | 76.4 $\pm$ 5.2 | 18.1 $\pm$ 0.2 | 82.6 $\pm$ 0.6

##  4 Conclusions

Neural networks, and convolutional neural networks in particular for image
data sets, are powerful feature extractors. The learned feature
representations carry implicit knowledge about the training data, and by
tracking simple statistics of the latent states, more information about the
data distribution is gained. Yin et al. (2020) have shown that it is possible
to recover high-fidelity, plausible images from only having access to the
BatchNorm-statistics. This work demonstrates that it is also possible to use
this information in the context of domain adaptation for data homogenization
and, in particular, shows its applicability to deconvolution or re-calibration
tasks. Guided data homogenization proves to be viable even when few data in
the target domain is available or even when no label information in the target
domain is given. Further exploration of the connection between implicit model
knowledge and post-hoc model adaptation offers an intriguing avenue for future
research.

## References

  * Balunovic et al. (2019) Mislav Balunovic, Maximilian Baader, Gagandeep Singh, Timon Gehr, and Martin Vechev.  Certifying geometric robustness of neural networks.  In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett (eds.), _Advances in Neural Information Processing Systems_ , volume 32, pp. 15313–15323. Curran Associates, Inc., 2019.  URL https://proceedings.neurips.cc/paper/2019/file/f7fa6aca028e7ff4ef62d75ed025fe76-Paper.pdf.
  * Ghifary et al. (2015) Muhammad Ghifary, W Bastiaan Kleijn, Mengjie Zhang, and David Balduzzi.  Domain generalization for object recognition with multi-task autoencoders.  In _Proceedings of the IEEE international conference on computer vision_ , pp. 2551–2559, 2015.
  * Heaven (2020) Will Douglas Heaven.  Google’s medical ai was super accurate in a lab. real life was a different story. — mit technology review, April 2020.  URL https://technologyreview.com.
  * Hendrycks et al. (2020) Dan Hendrycks, Steven Basart, Norman Mu, Saurav Kadavath, Frank Wang, Evan Dorundo, Rahul Desai, Tyler Zhu, Samyak Parajuli, Mike Guo, Dawn Song, Jacob Steinhardt, and Justin Gilmer.  The many faces of robustness: A critical analysis of out-of-distribution generalization, 2020.
  * Horé & Ziou (2010) Alain Horé and Djemel Ziou.  Image quality metrics: Psnr vs. ssim.  pp. 2366–2369, 08 2010.  doi: 10.1109/ICPR.2010.579.
  * Ioffe & Szegedy (2015) Sergey Ioffe and Christian Szegedy.  Batch normalization: Accelerating deep network training by reducing internal covariate shift.  In _International conference on machine learning_ , pp. 448–456. PMLR, 2015.
  * Kobler et al. (2017) Erich Kobler, Teresa Klatzer, Kerstin Hammernik, and Thomas Pock.  Variational networks: Connecting variational methods and deep learning.  In Volker Roth and Thomas Vetter (eds.), _Pattern Recognition_ , pp. 281–293, Cham, 2017. Springer International Publishing.  ISBN 978-3-319-66709-6.
  * Krizhevsky (2012) Alex Krizhevsky.  Learning multiple layers of features from tiny images.  _University of Toronto_ , 05 2012.
  * LeCun & Cortes (2010) Yann LeCun and Corinna Cortes.  Mnist handwritten digit database.  2010\.  URL http://yann.lecun.com/exdb/mnist/.
  * Long et al. (2015) Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan.  Learning transferable features with deep adaptation networks.  In _International conference on machine learning_ , pp. 97–105. PMLR, 2015.
  * Michaelis et al. (2020) Claudio Michaelis, Benjamin Mitzkus, Robert Geirhos, Evgenia Rusak, Oliver Bringmann, Alexander S. Ecker, Matthias Bethge, and Wieland Brendel.  Benchmarking robustness in object detection: Autonomous driving when winter is coming, 2020.
  * Oala et al. (2020) Luis Oala, Jana Fehr, Luca Gilli, Pradeep Balachandran, Alixandro Werneck Leite, Saul Calderon-Ramirez, Danny Xie Li, Gabriel Nobis, Erick Alejandro Muñoz Alvarado, Giovanna Jaramillo-Gutierrez, Christian Matek, Arun Shroff, Ferath Kherif, Bruno Sanguinetti, and Thomas Wiegand.  Ml4h auditing: From paper to practice.  In _Proceedings of the Machine Learning for Health NeurIPS Workshop_ , volume 136 of _Proceedings of Machine Learning Research_ , pp. 280–317. PMLR, 11 Dec 2020.  URL http://proceedings.mlr.press/v136/oala20a.html.
  * Pan & Yang (2010) S. J. Pan and Q. Yang.  A survey on transfer learning.  _IEEE Transactions on Knowledge and Data Engineering_ , 22(10):1345–1359, 2010.  doi: 10.1109/TKDE.2009.191.
  * Saito et al. (2019) Kuniaki Saito, Donghyun Kim, Stan Sclaroff, Trevor Darrell, and Kate Saenko.  Semi-supervised domain adaptation via minimax entropy.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pp. 8050–8058, 2019.
  * Sun & Saenko (2016) Baochen Sun and Kate Saenko.  Deep coral: Correlation alignment for deep domain adaptation.  In _European conference on computer vision_ , pp. 443–450. Springer, 2016.
  * Tzeng et al. (2015) Eric Tzeng, Judy Hoffman, Trevor Darrell, and Kate Saenko.  Simultaneous deep transfer across domains and tasks.  In _Proceedings of the IEEE international conference on computer vision_ , pp. 4068–4076, 2015.
  * Xu et al. (2014) Li Xu, Jimmy SJ Ren, Ce Liu, and Jiaya Jia.  Deep convolutional neural network for image deconvolution.  In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K. Q. Weinberger (eds.), _Advances in Neural Information Processing Systems_ , volume 27, pp. 1790–1798. Curran Associates, Inc., 2014.  URL https://proceedings.neurips.cc/paper/2014/file/1c1d4df596d01da60385f0bb17a4a9e0-Paper.pdf.
  * Yin et al. (2020) Hongxu Yin, Pavlo Molchanov, Jose M Alvarez, Zhizhong Li, Arun Mallya, Derek Hoiem, Niraj K Jha, and Jan Kautz.  Dreaming to distill: Data-free knowledge transfer via deepinversion.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 8715–8724, 2020.
  * Zellinger et al. (2019) Werner Zellinger, Thomas Grubinger, Edwin Lughofer, Thomas Natschläger, and Susanne Saminger-Platz.  Central moment discrepancy (cmd) for domain-invariant representation learning, 2019.
  * Zhu et al. (2017) Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros.  Unpaired image-to-image translation using cycle-consistent adversarial networks.  In _Proceedings of the IEEE international conference on computer vision_ , pp. 2223–2232, 2017.

##  Appendix A Full results

###  A.1 CIFAR-10

Table 2: Metrics on homogenization results in the supervised setting after 100
optimization epochs for CIFAR-10, including the class-conditional formulation
(CC). The mean and the standard deviation over 5 runs are given.
$n_{\mathcal{B}}=512$.

Homogenized $\mathcal{B}$ |  accuracy [%] |  validation accuracy [%] |  verification accuracy [%] |  $\downarrow$ l2-error |  $\uparrow$ PSNR |  $\uparrow$ SSIM [%]
---|---|---|---|---|---|---
CRITERION | 99.6 $\pm$ 0.4 | 77.9 $\pm$ 2.1 | 75.9 $\pm$ 2.7 | 14.0 $\pm$ 2.7 | 18.0 $\pm$ 0.4 | 86.7 $\pm$ 2.2
NN | 95.5 $\pm$ 0.6 | 83.1 $\pm$ 0.6 | 80.8 $\pm$ 0.8 | 11.6 $\pm$ 1.6 | 19.5 $\pm$ 0.3 | 90.1 $\pm$ 0.5
NN CC | 99.9 $\pm$ 0.1 | 85.8 $\pm$ 0.7 | 83.0 $\pm$ 0.6 | 9.9 $\pm$ 2.2 | 20.9 $\pm$ 0.6 | 92.0 $\pm$ 0.9
NN ALL | 99.3 $\pm$ 0.4 | 88.2 $\pm$ 0.6 | 84.9 $\pm$ 1.0 | 10.0 $\pm$ 2.6 | 23.3 $\pm$ 0.3 | 95.4 $\pm$ 0.2
NN ALL CC | 99.8 $\pm$ 0.0 | 87.8 $\pm$ 0.9 | 84.7 $\pm$ 1.0 | 12.8 $\pm$ 0.9 | 23.7 $\pm$ 0.5 | 95.4 $\pm$ 0.4
RANDOM NN | 51.9 $\pm$ 5.0 | 48.4 $\pm$ 5.7 | 42.7 $\pm$ 4.2 | 29.0 $\pm$ 10.6 | 11.3 $\pm$ 0.7 | 59.6 $\pm$ 4.7
RANDOM NN CC | 55.6 $\pm$ 1.6 | 53.0 $\pm$ 2.7 | 48.1 $\pm$ 1.9 | 38.2 $\pm$ 10.8 | 13.5 $\pm$ 0.6 | 67.4 $\pm$ 4.3
RP | 51.1 $\pm$ 2.1 | 51.0 $\pm$ 1.4 | 47.3 $\pm$ 1.1 | 76.4 $\pm$ 5.2 | 18.1 $\pm$ 0.2 | 82.6 $\pm$ 0.6
RP CC | 42.6 $\pm$ 0.9 | 42.5 $\pm$ 0.9 | 41.4 $\pm$ 1.2 | 42.2 $\pm$ 1.7 | 17.0 $\pm$ 0.2 | 81.5 $\pm$ 0.4

Table 3: Metrics on homogenization results in the unsupervised setting after
100 optimization epochs for CIFAR-10. The mean and the standard deviation over
5 runs are given. $n_{\mathcal{B}}=512$.

Homogenized $\mathcal{B}$ |  accuracy [%] |  validation accuracy [%] |  verification accuracy [%] |  $\downarrow$ l2-error |  $\uparrow$ PSNR |  $\uparrow$ SSIM [%]
---|---|---|---|---|---|---
NN | 85.6 $\pm$ 1.3 | 79.8 $\pm$ 1.7 | 76.4 $\pm$ 0.3 | 18.1 $\pm$ 2.3 | 17.4 $\pm$ 0.7 | 85.3 $\pm$ 1.0
NN ALL | 91.8 $\pm$ 0.6 | 84.5 $\pm$ 0.7 | 83.6 $\pm$ 1.0 | 14.1 $\pm$ 2.3 | 22.1 $\pm$ 0.6 | 94.5 $\pm$ 0.3
RANDOM NN | 57.1 $\pm$ 6.4 | 53.2 $\pm$ 4.9 | 49.3 $\pm$ 5.4 | 34.3 $\pm$ 17.8 | 12.1 $\pm$ 0.7 | 67.2 $\pm$ 3.4
RP | 47.7 $\pm$ 1.1 | 45.8 $\pm$ 1.0 | 42.8 $\pm$ 0.9 | 36.1 $\pm$ 5.1 | 15.5 $\pm$ 0.3 | 79.0 $\pm$ 0.7

Ground Truth

Distorted

CRITERION

NN

NN CC

NN ALL

NN ALL CC

RANDOM NN

RANDOM NN CC

RP

RP CC

Figure 3: Results of the homogenization on CIFAR-10 data set after 100 epochs

Ground Truth

Distorted

NN

NN ALL

RANDOM NN

RP

Figure 4: Results of the homogenization on CIFAR-10 data set in an
unsupervised setting after 100 epochs.

###  A.2 MNIST

Table 4: Metrics on homogenization results after 100 optimization epochs for
MNIST. The mean and the standard deviation over 5 runs are given.
$n_{\mathcal{B}}=512$.

Homogenized $\mathcal{B}$ |  accuracy [%] |  validation accuracy [%] |  verification accuracy [%] |  $\downarrow$ l2-error |  $\uparrow$ PSNR |  $\uparrow$ SSIM [%]
---|---|---|---|---|---|---
CRITERION | 99.8 $\pm$ 0.0 | 97.7 $\pm$ 0.0 | 90.4 $\pm$ 0.2 | 9.8 $\pm$ 0.1 | 15.7 $\pm$ 0.0 | 80.6 $\pm$ 0.0
NN | 97.5 $\pm$ 0.2 | 98.0 $\pm$ 0.1 | 89.7 $\pm$ 1.9 | 16.9 $\pm$ 0.5 | 18.0 $\pm$ 0.1 | 88.5 $\pm$ 0.4
NN CC | 97.9 $\pm$ 0.2 | 98.2 $\pm$ 0.1 | 97.8 $\pm$ 0.3 | 14.3 $\pm$ 1.1 | 20.1 $\pm$ 0.4 | 91.9 $\pm$ 1.6
NN ALL | 98.3 $\pm$ 0.3 | 98.1 $\pm$ 0.3 | 98.0 $\pm$ 0.9 | 9.5 $\pm$ 1.7 | 21.2 $\pm$ 0.1 | 90.0 $\pm$ 1.1
NN ALL CC | 97.7 $\pm$ 0.3 | 98.3 $\pm$ 0.2 | 96.7 $\pm$ 0.4 | 13.2 $\pm$ 1.1 | 22.2 $\pm$ 0.1 | 93.3 $\pm$ 1.0
RANDOM NN | 99.8 $\pm$ 0.0 | 97.7 $\pm$ 0.1 | 90.3 $\pm$ 0.7 | 9.9 $\pm$ 0.3 | 15.8 $\pm$ 0.0 | 80.7 $\pm$ 0.0
RANDOM NN CC | 99.8 $\pm$ 0.0 | 97.8 $\pm$ 0.1 | 90.5 $\pm$ 0.4 | 10.0 $\pm$ 0.2 | 15.8 $\pm$ 0.0 | 80.7 $\pm$ 0.0
RP | 99.8 $\pm$ 0.0 | 98.0 $\pm$ 0.1 | 97.3 $\pm$ 0.1 | 7.2 $\pm$ 0.1 | 15.3 $\pm$ 0.0 | 81.0 $\pm$ 0.0
RP CC | 99.8 $\pm$ 0.0 | 97.9 $\pm$ 0.0 | 96.6 $\pm$ 0.1 | 7.6 $\pm$ 0.2 | 15.4 $\pm$ 0.0 | 81.0 $\pm$ 0.0

Ground Truth

Distorted

CRITERION

NN

NN CC

NN ALL

NN ALL CC

RANDOM NN

RANDOM NN CC

RP

RP CC

Figure 5: Results of the reconstruction task on MNIST data set after 100
epochs

##  Appendix B Parameter-Settings

Table 5: Parameters and specifications of the experiments. $n_{\mathcal{A}}$,
$n_{\mathcal{B}}$, $n_{\mathcal{C}}$ denote the sample size of the source,
target and validation data set respectively. $C$ is the number of classes and
$d$ the number of input dimensions of the data set. $\kappa$ is the distortion
parameter of the perturbation model. $s_{\text{width}}$ and $s_{\text{width}}$
are the width and the depth of the residual network used for reconstruction.
$s_{\text{RP}}$ is the number of random projections, the width used for RP.
$r_{\text{crit}}$ and $r_{\text{stats}}$ are the loss weighting factors of the
objective function.

MNIST Data Set
---
$n_{\mathcal{A}}$ | = 60,000
$n_{\mathcal{B}}$ | = 512
$n_{\mathcal{C}}$ | = 1024
Properties |
$C$ | = 10
input shape | = (1, 28, 28)
$d$ | = 784
Distortion |
$\kappa$ | = 0.3
Neural Network |
Architecture |
main network: | ResNet20
verifier network: | ResNet9
Reconstruction Model
$s_{\text{width}}$ | = 8
$s_{\text{depth}}$ | = 8
$s_{\text{RP}}$ | = 512
Optimization |
learning rate | = 0.1
batch size | = 128
$r_{\text{crit}}$ | = 1
$r_{\text{stats}}$ | = 0.001

CIFAR10 Data Set
---
$n_{\mathcal{A}}$ | = 50,000
$n_{\mathcal{B}}$ | = 512
$n_{\mathcal{C}}$ | = 1024
Properties |
$C$ | = 10
input shape | = (3, 32, 32)
$d$ | = 3072
Distortion |
$\kappa$ | = 0.15
Neural Network |
Architecture |
main network: | ResNet34
verifier network: | ResNet18
Reconstruction Model
$s_{\text{width}}$ | = 8
$s_{\text{depth}}$ | = 8
$s_{\text{RP}}$ | = 512
Optimization |
learning rate | = 0.1
batch size | = 128
$r_{\text{crit}}$ | = 1
$r_{\text{stats}}$ | = 10