# Long-term Reproducibility for Neural Architecture Search

David Towers, Matthew Forshaw, A. Stephen McGough
Newcastle University, UK
{d.towers2, matthew.forshaw, stephen.mcgough}
@newcastle.ac.uk Amir Atapour-Abarghouei
Durham University, UK
amir.atapour-abarghouei
@durham.ac.uk

###### Abstract

It is a sad reflection of modern academia that code is often ignored after
publication – there is no academic ‘kudos’ for bug fixes / maintenance. Code
is often unavailable or, if available, contains bugs, is incomplete, or relies
on out-of-date / unavailable libraries. This has a significant impact on
reproducibility and general scientific progress. Neural Architecture Search
(NAS) is no exception to this, with some prior work in reproducibility.
However, we argue that these do not consider long-term reproducibility issues.
We therefore propose a checklist for long-term NAS reproducibility. We
evaluate our checklist against common NAS approaches along with proposing how
we can retrospectively make these approaches more long-term reproducible.

##  1 Introduction

The popularity of Deep Learning (DL) has increased significantly over the last
decade. It has enticed us with its ability to solve complex problems, such as
image classification, without the need for manual feature identification.
However, this has been at the cost of now not knowing the ‘best’ neural
architecture to use for a given problem.

Neural Architecture Search (NAS) is a promising field which offers greater
potential for finding the ‘best’ architecture for a given data set and problem
space – without needing to train all possible architectures and/or the need of
a deep learning expert. Historically, NAS has focused on high accuracy;
however, more recently, a move to allow a combination of multiple metrics
(e.g., accuracy, memory footprint, number of parameters) has emerged [1, 13,
5].

The potential for NAS was demonstrated by Zoph and Le [19] where they
outperformed all but one of the human-designed networks they compared against.
This potential was limited due to the number of GPU hours it took to find a
solution – which has become the primary driver for future NAS research. New
algorithms have reduced computation time – achieving orders of magnitude
improvement [20, 15, 12, 1, 13, 9, 2, 14, 7, 6]. However, as the authors move
on, their code has been left unmaintained. This is problematic, as it makes it
more difficult to use these codes for other data sets and problem spaces, and
for comparison with new approaches.

However, there is little incentive to spend time updating old code. While an
author could, every few years, spend time updating their previous projects to
work with the updated libraries (a task with no academic reward and eventually
all-time-consuming), they instead use this time to work on new research, which
is more likely to lead to citations. Often driven by the infamous academic
mantra of “Publish or Perish” [4]. This pressure can take away time academics
would otherwise use for activities that do not result in publications, such as
teaching students [17] or maintaining code. This neglect can leave codes which
are bug-ridden, lacking necessary documentation, or simply out-of-date.

These issues lead to codes that cannot be used without being at least
partially re-written. This problem hinders further research based on these
codes, removes the ability for others to use the approach for real-world
applications and prevents the purpose of the field, which is to replace
manually-designed architectures.

Some prior work has looked at the problem of reproducibility [11, 16] by
proposing a checklist to improve reproducibility of work at the time of
publication. This has focused on standards, making sure that the code is
complete and that all dependencies are resolvable. However, this only ensures
that the work is reproducible at the time of publication. If we seek to make
the work reproducible many years after publication, this brings in an
additional set of constraints, which need to be considered. These include the
long-term availability of dependencies for the code, how changes to libraries
effect the running of the code and the availability of a containerised version
of the software to encapsulate the code and environment to ensure a longer-
term reference used for validation and comparison.

In an ideal world, all authors would adhere to our checklist – including a
containerisation and ease of use. However, this is unlikely to happen for
legacy NAS, nor can we expect ‘usable’ systems from new developments. As such,
we propose the development of a Docker container approach for legacy NAS
approaches and a framework for NAS development separating the NAS approach
from core tasks such as dataset input and comparison.

##  2 Related Work

Accessibility of Software Artifacts Many pieces of work are missing software
artefacts, hosting dead links, or linking to bug-ridden code. Heumüller  _et
al_. show that of 604 reviewed papers that contain mention of software
artefacts between 2007 and 2017, only 289 (47.8%) shared links to these
artefacts, and only 163 (27%) were available. However, they do also show an
increase in software artefact availability during this period [8]. This may
suggest a more significant push for reproducibility in recent years.

This increase is due to several initiatives to encourage artefact sharing,
such as ACM badging, including three categories of badges: Artifacts
Evaluated, Artifacts Available, and Results Validated, awarded to papers based
on a set of criteria. While there is not enough evidence to suggest a cause-
and-effect relationship between these initiatives and the increase in software
artefact availability, there is enough to suggest these initiatives have
influenced this trend [3].

However, these initiatives do not include prolonged maintenance. Artefact
Available badges are awarded for the artefact having persistent storage and an
available link. The Artefact Evaluated is awarded for the quality of the
artefact at the time of the award. There are no criteria for bug fixing,
providing pseudocode, or ensuring usable code if the dependencies are no
longer available.

Reproducibility in NAS Even when code is made available, there are still
problems with reproducibility. We should be able to use virtualisation tools
and settable (random) seeds to reproduce the exact experimental conditions. Li
and Talwalker found that none of the twelve papers published between 2018-2019
that produced novel NAS works had included their experimental seeds [10],
highlighting historical threats to reproducibility in NAS.

Reproducibility Checklists Lindauer and Hutter highlight difficulties in
reproducing NAS algorithms. They have identified many pitfalls that authors
can fall into when publishing their work. They offer a checklist of best
practices authors can follow to reduce the chance of succumbing to these
problems [11]. However, the points raised in their work are for
reproducibility at the time of publication but does not encourage practices to
prolong reproducibility.

The machine learning
checklist111https://www.cs.mcgill.ca/~jpianeau/ReproducibilityChecklist.pdf by
Pineau et al. [16] is adopted by both ICML and NeurIPS. This checklist has
several sections covering all aspects of reproducibility. One of these
sections concerns related code and contains points regarding pre-trained
models, code availability, and good documentation practices. While this work
encourages reproducibility by asking for the code and a list of dependencies,
it does not require the code to be bug-free, and if one of those dependencies
becomes inaccessible, the code will no longer work.

While we agree with all points in both works, we believe they do not go far
enough regarding software artefacts. While they include points about ensuring
the code is released, there is little in making sure the code is usable and
future-proofed. Following all their checkpoints could still result in code
that does not work in a few years. Our checklist is designed to encourage
prolonged reproducibility, and hopefully become the basis of a unifying NAS
framework which will allow the works to be run into the future.

##  3 Reproducing Current Works

During our investigations into a range of common NAS algorithms[12, 15, 2,
18], we have found that it is not always a simple and easy process to get them
working. Some algorithms still require infeasibly large amounts of
computational power to complete such as the original NAS [19]. Others need
specific versions of libraries or code updating to run. 75% of the algorithms
we have attempted to get working have had some significant issues.

DARTS [12] for example, was one of the simplest to reproduce. However, they
state222https://github.com/quark0/darts a dependency PyTorch 0.3.1. This is
technically available, though only for MacOS333https://pytorch.org/get-
started/previous-versions. In order to mitigate this, certain sections of
DARTS were updated to PyTorch 1.0+ (specific changes on our GitLab
page444https://gitlab.com/D-Towers/nas-reproducibility). These are the sort of
long-term issues which are not as obvious at the time of publication.

ENAS [15] is a more difficult case. On its GitHub
page555https://github.com/melodyguan/enas, they note the Language Model
implementation is now hosted elsewhere. The Image Recognition implementation
remains in the repository but lacks a requirements list in the documentation.
It can be found, however, in an answered issue. There were also bugs in the
code. Community members have fixed these bugs in the issues section. However,
these fixes were not pushed to the main branch. Furthermore, ENAS needs a
particular version of the cudatoolkit and CuDNN libraries. One of which is
archived and needs an NVIDIA account to access, and then needs to be manually
installed.

Containerisation tools, such as Docker, can alleviate some of these problems.
A properly set-up Dockerfile will recreate the developer environment, so a
dependency list would not be required as Docker would handle this
automatically. Docker can also be used to save an image of a container,
providing a permanent working environment. While this increases the file size
considerably, it has several advantages of providing a full set of
dependencies, a known working environment and tool, along with removing the
issue of dependencies no longer being available.

Criteria |  Descriptor |  ENAS |  DARTS |  PC-DARTS |  DrNAS
---|---|---|---|---|---
Code Stability |  |  |  |  |
Seed Reporting |  The code should return the seed used in the logs/output. | ✗ | ✓ | ✓ | ✓
Seed Setting |  The seed should be able to be set. | ✗ | ✓ | ✓ | ✓
Bug-Free |  The code should be free of bugs which prevent sections of code from working. | ✗ | ✓ | ✓ | ✗i
Documentation |  |  |  |  |
Examples |  There should be examples of all the high-level commands. | ✓ | ✓ | ✓ | ✓
Argument Details |  The documentation should include details regarding additional arguments that can be applied to the high-level commands. | ✗ | ✗ | ✗ | ✗
Dependency List |  Environment specification indicating how to replicate the environment. | ✗ | ✓ | ✓ | ✓
Pipeline Instructions |  Instructions on how to search, train, and test a model from scratch. | ✗ii | ✗iii | ✗iii | ✗iii
Ease of Running |  |  |  |  |
Dependency Resilience |  Dependencies should be installed with the code. | ✗ | ✗ | ✗ | ✗
Executable |  The code can be executed from an executable file, such as a bash script. | ✓ | ✗ | ✗ | ✗
Intuitive Commands |  The commands should be indicative to their function. | ✓ | ✓ | ✓ | ✓
Standardisation |  |  |  |  |
Standard Phases |  Search, Train, and Test. | ✗ | ✓ | ✓ | ✗
Data Inclusive |  Data is retrieved from a directory named data that accepts common data formats. | ✗ | ✓ | ✓ | ✓
Standard Output |  The model is saved in an output folder in a form that is easily imported into other scripts. | ✗ | ✓ | ✓ | ✓
Future Proofing |  |  |  |  |
Accessible Pseudocode |  Pseudocode should be made available to allow recreation of the code if all other methods are no longer available. | ✗ | ✗ | ✗ | ✗
Containerised Environment |  A compressed file of the containerised image (such as tar), which allows the entire environment to be saved, dependencies included. | ✗ | ✗ | ✗ | ✗
Container Build File |  A file containing the required info for a container manager to build the environment and run the code. | ✗ | ✗ | ✗ | ✗
Source Code |  Source code should be publicly accessible to allow others to inspect and change local versions. | ✓ | ✓ | ✓ | ✓
Table 1: Evaluated NAS algorithms using our proposed checklist. iCode is
missing a ‘hp’, which needs to be manually entered. ii Does not mention that
you need to change the default architecture in the training file. iii Does not
explicitly explain how to train a found architecture.

##  4 Proposed Checklist

We propose a checklist containing five categories which combines desired
features we believe could help increase reproducibility in NAS. Some points in
the checklist may become superseded with the inclusion of others. This
redundancy should maintain reproducibility if some of the other aspects are
not met.

In comparison to the already-existing checklists [11, 16], we have focused on
aspects which we believe will help prolong reproducibility, with points such
as dependency resilience, which ensures the code will remain usable even if a
dependency is no loner available.

Table 1 shows our checklist applied to four common NAS approaches (ENAS [15],
DARTS [12], PC-DARTS [18] and DrNAS [2]). The evaluated approaches have not
been future-proofed well, as can be seen from the table. This is evidenced by
DARTS requiring a no longer accessible library. We are reproducing several NAS
algorithms into a working state in Docker containers. This should allow others
to build on our working containers with a working environment. Currently, we
have reproduced ENAS into a working Docker Container. These containers will be
available, along with any changes we made to the code, on our GitLab4.

##  5 Conclusion

In this paper, we have proposed a checklist that is designed to improve and
prolong the reproducibility of NAS algorithms. Our checklist covers several
areas which can negatively impact how reproducible an author’s work is. These
areas include code stability, documentation, ease of running, standardisation,
and future-proofing.

In particular, we identify the issue of inaccessible dependencies and suggest
containerisation tools as a solution, which allows a researcher’s working
environment to be saved as an image, including dependencies, which other
researchers can import to recreate exactly.

We are currently working on recreating the NAS approaches used in this paper
into Docker containers, which will be made available on our
GitLab666https://gitlab.com/D-Towers/nas-reproducibility. We have already
replicated ENAS into a docker container.

Our repository also includes instructions on how to reproduce environments of
each of the NAS approaches used in this paper, using the code from the
author’s repositories.

We will extend upon this work by developing a framework removing the burden of
reproducibility from the NAS developer, thus allowing them to focus on their
NAS approach while maintaining a strong level of reproducibility.

## References

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  * [2] Xiangning Chen, Ruochen Wang, Minhao Cheng, Xiaocheng Tang, and Cho-Jui Hsieh.  DrNAS: Dirichlet Neural Architecture Search.  arXiv:2006.10355 [cs, stat], Mar. 2021.  arXiv: 2006.10355 version: 4.
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  * [4] Mark De Rond and Alan N. Miller.  Publish or Perish: Bane or Boon of Academic Life?  Journal of Management Inquiry, 14(4):321–329, Dec. 2005.
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  * [7] Rob Geada, Dennis Prangle, and Andrew Stephen McGough.  Bonsai-Net: One-Shot Neural Architecture Search via Differentiable Pruners.  arXiv:2006.09264, June 2020.
  * [8] Robert Heumüller, Sebastian Nielebock, Jacob Krüger, and Frank Ortmeier.  Publish or perish, but do not forget your software artifacts.  Empirical Software Engineering, 25(6):4585–4616, Nov. 2020.
  * [9] Changlin Li, Jiefeng Peng, Liuchun Yuan, Guangrun Wang, Xiaodan Liang, Liang Lin, and Xiaojun Chang.  Block-Wisely Supervised Neural Architecture Search With Knowledge Distillation.  In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 1986–1995, Seattle, WA, USA, June 2020\. IEEE.
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# Electro-mechanical Casimir effect

Mikel Sanz<EMAIL_ADDRESS>Department of Physical Chemistry, University of
the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain  Witlef
Wieczorek  Department of Microtechnology and Nanoscience, Chalmers University
of Technology, Kemivägen 9, SE-41296 Göteborg, Sweden  Simon Gröblacher  Kavli
Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628CJ
Delft, The Netherlands  Enrique Solano  Department of Physical Chemistry,
University of the Basque Country UPV/EHU, Apartado 644, E-48080 Bilbao, Spain
IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, E-48013
Bilbao, Spain  Department of Physics, Shanghai University, 200444 Shanghai,
China

###### Abstract

The dynamical Casimir effect is an intriguing phenomenon in which photons are
generated from vacuum due to a non-adiabatic change in some boundary
conditions. In particular, it connects the motion of an accelerated mechanical
mirror to the generation of photons. While pioneering experiments
demonstrating this effect exist, a conclusive measurement involving a
mechanical generation is still missing. We show that a hybrid system
consisting of a piezoelectric mechanical resonator coupled to a
superconducting cavity may allow to electro-mechanically generate measurable
photons from vacuum, intrinsically associated to the dynamical Casimir effect.
Such an experiment may be achieved with current technology, based on film bulk
acoustic resonators directly coupled to a superconducting cavity. Our results
predict a measurable photon generation rate, which can be further increased
through additional improvements such as using superconducting metamaterials.

##  1 Introduction

Quantum mechanics predicts that virtual particles can emerge from vacuum. This
phenomenon, known as quantum fluctuations, is a cornerstone to explaining key
effects in nature, ranging from the anomalous magnetic moment of the electron
[1], to the enhancement in quantum transport phenomena [2], and the Lamb shift
of atomic spectra [3]. Another paramount example is the Casimir effect, which
results from a force between two separated conducting plates [4, 5, 6, 7]. G.
T. Moore [8] suggested the existence of a kinetic analogue to this phenomenon,
known as the dynamical Casimir effect (DCE). This effect, based on the fact
that a moving mirror modifies the mode structure of the electromagnetic vacuum
dynamically, is the result of a mismatch of vacuum modes in time. Indeed, if
the velocity of a mirror is much smaller than the speed of light, no
excitations emerge out of the vacuum, since the electromagnetic modes
adiabatically adapt to the changes. However, if the mirror experiences
relativistic motion, these changes occur non-adiabatically. Hence, the vacuum
state of the electromagnetic field can be excited by taking energy from the
moving mirror, resulting in the generation of real photons. It has been
suggested that, due to the large stress generated on any macroscopic material
moving with relativistic speed, the observation of the DCE might actually be
unrealistic [10, 9]. This challenge inspired various alternative proposals to
observe the DCE [11, 12, 13, 14, 15, 16, 10, 17, 18, 19, 20, 21, 22], e.g., by
employing nanomechanical resonators, modulating two-level systems inside a
cavity or modulating the electric boundary conditions of a superconducting
cavity; for an overview see, e.g., [9, 23, 24].

Only recently, DCE radiation has been experimentally observed [25] in
superconducting circuits by moving the electric boundary condition of a
superconducting cavity [20, 21] instead of the center of mass of a mirror. In
a similar experiment, DCE photons were created by modulating the effective
speed of light [26] making use of a superconducting metamaterial.
Additionally, relevant applications of these experiments, such as a robust
generation of multi-partite quantum correlations [20, 21, 27, 28, 29, 30],
were proposed, and recently observed [31], thus, highlighting the potential of
this effect beyond its fundamental interest. However, no experiment to date
has succeeded in generating photons out of vacuum using a moving mechanical
object in the spirit of the original DCE proposal.

In this letter, we propose an experiment consisting of a nanomechanical
resonator that is directly coupled to a high impedance superconducting cavity
(see Fig. 1) in order to create a measurable rate of electro-mechanically
generated DCE photons. Our proposal develops the idea to employ film bulk
acoustic resonators (FBAR) for DCE photon creation from Ref. [17] further by
combining it with the technology of superconducting circuits [20, 21, 32, 25].
Our calculations make use of recent advances in mechanical oscillators based
on FBAR technology using thin films of Al-AlN-Al [32]. The presence of a
superconducting cavity allows for a resonant enhancement of the photon rate
which surpasses the radiation from a single mirror by several orders of
magnitude [12, 13, 14]. We study the minimally required conditions to observe
a stable flux of photons resulting from the electro-mechanically amplified
quantum vacuum, concluding that such a measurement is feasible with current
technology. Finally, we propose technical improvements to further enhance the
mechanical photon production. Our work also paves the way to experimentally
test other fundamental relativistic effects with mechanical resonators, such
as the Unruh effect or the Hawking radiation [24]

Figure 1: Scheme of the coupling between an FBAR and a superconducting cavity.
(a) Artistic view representing an FBAR (top left), i.e., a resonator composed
of two superconducting layers sandwiching a piezoelectric material. The FBAR
terminates one side of a superconducting cavity that is capacitively coupled
to a semi-infinite transmission line (lower right). (b) Equivalent circuit of
the proposed implementation, which is composed of the mechanical resonator
(time-dependent capacitance $C(t)$ and voltage drive $V(t)$) and the
superconducting cavity (with $\mathcal{C}$ and $\mathcal{L}$ capacitance and
inductance per unit length $\Delta x$). We have employed the modified van
Dyke-Butterworth model to provide the equivalent lumped-element circuit
representation of the FBAR and model the cavity as an infinite set of LC
circuits which will be afterwards capacitively coupled via $C_{c}$ to a semi-
infinite waveguide.

##  2 Proposed implementation

The proposed setup consists of a piezoelectric FBAR resonator with resonance
frequency $\Omega/2\pi=4.20$ GHz (e.g. made from Al-AlN-Al) and a
superconducting cavity (e.g. made from Al) with length $d=33$ mm, which is
capacitively coupled to a semi-infinite transmission line for the read-out, as
shown in Fig. 1, all assumed to be operating at a temperature of $10\,$mK. To
match the resonance condition in an actual experimental implementation, we
note that the superconducting cavity can be made tunable in frequency by
incorporating a flux-biased SQUID [25, 33]. We have chosen values that closely
follow the parameters from Ref. [32], for details see Appendix B.4. An AC
voltage is applied to the superconducting plates of the mechanical resonator.
This voltage drive has two effects: (i) the piezoelectric material of the FBAR
converts the AC voltage into mechanical contraction and expansion, which
essentially leads to a change of the length of the superconducting cavity;
(ii) the voltage drive changes the potential of the cavity’s boundary. Both
effects result in the production of DCE photons.

###  2.1 FBAR modeling

We make use of the modified van Dyke-Butterworth model to construct the
equivalent lumped-element circuit representation of the FBAR [34, 35] (for
details see Appendix B). This circuit contains two parallel branches. The
first one consists of a mechanical capacitance $C_{m}$, mechanical inductance
$L_{m}$ and a resistor $R_{m}$ modeling the mechanical dissipation connected
in series, while the second one contains a resistor $R_{0}$ taking into
account the dielectric loss and, crucially, a geometrical capacitance $C(t)$,
which slightly changes in time when an AC voltage is applied to the plates of
the FBAR. Composing the impedances for both branches, $Z_{m}$ and $Z_{0}$, it
is straightforward to prove that the total impedance $Z\approx Z_{0}$, which,
to first order, can be reduced to $C(t)$ (see Appendix B).

Let us estimate the change in the capacitance $C(t)$ when an AC voltage
$V(t)=V_{\text{pp}}\cos(\omega t+\phi)$ is applied to the electrodes. An AC
voltage with an amplitude $V_{\mathrm{pp}}$ at the resonance frequency of the
mechanical resonator $\omega=\Omega$ results in a change of the inter-plate
distance $\Delta x(V_{\mathrm{pp}})\approx 1.7\cdot V_{\mathrm{pp}}$ nm/V (see
Appendix A). Crucial here is that the piezoelectric effect is enhanced by the
mechanical quality factor of the FBAR, assumed to be $Q=300$ [32]. Applying a
voltage of $V_{\mathrm{pp}}=500\mu$V thus results in $\Delta x=8.5\cdot
10^{-13}$m, which is more than five orders of magnitude smaller than the
thickness $t_{\text{AlN}}=3.5\cdot 10^{-7}$m of the piezoelectric layer. Thus,
we model the resulting mechanical contraction and expansion as harmonic. The
time-varying capacitance is then given as
$C(t)=\varepsilon_{\text{AlN}}A/[t_{\text{AlN}}+\Delta x\cos(\Omega t)]$, with
$A$ as area of the FBAR and $\varepsilon_{\text{AlN}}$ as dielectric constant
of AlN. We expand $C(t)$ for $\Delta x\ll t_{\text{AlN}}$ and finally obtain

$C(t)\approx C_{0}+\Delta C\cos(\Omega t)$ (1)

with $C_{0}=\varepsilon_{\text{AlN}}A/t_{\text{AlN}}$ and $\Delta C\approx
C_{0}\Delta x/t_{\text{AlN}}$.

###  2.2 Lumped-element circuit model

The Lagrangian describing the circuit of the FBAR connected to an open
transmission line (without the coupling capacitor $C_{c}$ shown in Fig. 1) can
be written as

$\displaystyle L$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\left[\frac{\delta
x\mathcal{C}}{2}\dot{\Phi}_{i+1}^{2}-\frac{1}{2\delta
x\mathcal{L}}(\Phi_{i+1}-\Phi_{i})^{2}\right]$ (2) $\displaystyle+$
$\displaystyle\frac{1}{2}C(t)(\dot{\Phi}_{0}-\dot{\Phi}_{v})^{2}+\frac{1}{2}C_{g}\dot{\Phi}_{0}^{2}.$

Here, $\mathcal{C}$ and $\mathcal{L}$ are the densities of capacitance and
inductance of a transmission line per unit length $\delta x$, respectively,
$C_{g}$ is the capacitive coupling of the FBAR to ground, which we will
discard in our analysis as it is much smaller than any other quantity
involved, $\Phi_{i}$ is the $i$-th node flux and $\dot{\Phi}_{v}=V$ is the
voltage of the AC source. In the continuous limit, the equation of motion
corresponding to $i=0$ is given by

$C(t)\ddot{\Phi}(0,t)+\dot{C}(t)\dot{\Phi}(0,t)-\frac{1}{\mathcal{L}}\frac{\partial\Phi(x,t)}{\partial
x}\Big{|}_{x=0}=F(t),$ (3)

with $F(t)=\frac{d}{dt}(\theta(t)C(t)V(t))$ being the electro-mechanical
source term, and $\theta(t)$ is the Heaviside step function. In order to solve
it, we follow a similar approach to Ref. [20, 21] and expand the field in the
Fourier components

$\displaystyle\Phi(x,t)=\sqrt{\frac{\hbar Z_{0}}{4\pi}}$
$\displaystyle\int_{0}^{\infty}d\omega\frac{1}{\sqrt{\omega}}\left[a_{\text{in}}(\omega)e^{-i(\omega
t-k_{\omega}x)}\right.$
$\displaystyle\left.+a_{\text{out}}(\omega)e^{-i(\omega
t+k_{\omega}x)}+H.c.\right],$

with $Z_{0}\approx 55$ $\Omega$. Equation (3) can now be written as

$\displaystyle\int_{0}^{\infty}d\omega$ $\displaystyle\sqrt{\frac{\hbar
Z_{0}}{4\pi|\omega|}}\Bigg{[}$
$\displaystyle+\left(-\omega^{2}C(t)-i\omega\dot{C}(t)-\frac{ik_{\omega}}{\mathcal{L}}\right)a_{\text{in}}(\omega)e^{-i\omega
t}$
$\displaystyle+\left(-\omega^{2}C(t)-i\omega\dot{C}(t)+\frac{ik_{\omega}}{\mathcal{L}}\right)a_{\text{out}}(\omega)e^{-i\omega
t}$ $\displaystyle+H.c.\Bigg{]}=F(t).$

By integrating over
$\int_{-\infty}^{\infty}dt\sqrt{|\omega^{\prime}|}e^{i\omega^{\prime}t}$, we
can see that in the case of a static capacitor (i.e. $C(t)=C_{0}$), it behaves
as a mirror placed at $x=-L_{\text{eff}}\approx\frac{C_{0}}{\mathcal{C}}$,
such that the effective length of the resonator shown in Fig. 1, which will be
introduced below, is $d_{\text{eff}}=d+L_{\text{eff}}$.

Resonances emerge due to the inelastic interaction of the photons with the
oscillating mirror, and the incoming modes $a_{\text{in}}(\omega)$ with
frequency $\omega$ are scattered elastically as $a_{\text{out}}(\omega)$ and
inelastically as $a_{\text{out}}(\omega+\Omega)$,
$a_{\text{out}}(\omega+2\Omega)$, … . Keeping only the first inelastic
resonances, as higher resonances have higher orders in $\Delta C$, and
assuming that $0<\omega\leq\Omega$, which is natural since the condition for
the rotating wave approximation holds around $\Omega\approx 2\omega^{\prime}$,
the output mode is given by

$\displaystyle a_{\text{out}}(\omega,L_{\text{eff}})$
$\displaystyle=h(\omega,\Omega)+a_{\text{in}}(\omega,L_{\text{eff}})$
$\displaystyle+S(\omega,\Omega+\omega)a_{\text{in}}(\Omega+\omega,L_{\text{eff}})$
$\displaystyle+S(\omega,\Omega-\omega)a_{\text{in}}^{\dagger}(\Omega-\omega,L_{\text{eff}}).$
(4)

Here, the operators $a_{\text{in}}$ and $a_{\text{out}}$ are defined in the
displaced position $x=L_{\text{eff}}$, $h(\omega,\Omega)$ is the identity in
operator space, as we have treated the source classically, and

$\displaystyle S(\omega^{\prime},\omega^{\prime\prime})$
$\displaystyle=-i\Delta
CZ_{0}\sqrt{|\omega^{\prime}||\omega^{\prime\prime}|}\theta(\omega^{\prime})\theta(\omega^{\prime\prime}),$
(5) $\displaystyle h(\omega,\Omega)$ $\displaystyle=-i\sqrt{\frac{4\pi
Z_{0}}{\hbar\omega}}\mathcal{F}(\omega,\Omega),$ (6)

where
$\mathcal{F}(\omega,\Omega)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}dtF(t)e^{i\omega
t}$ is the Fourier transform of $F(t)$.

The aforementioned result describes an oscillatory mirror coupled to a one-
dimensional open transmission line. The photon production due to the change in
the boundary conditions can be dramatically increased by introducing a cavity
with a well chosen resonance condition, as shown for example in Ref. [12, 13,
14]. In order to realize this, we will introduce a capacitor at a distance $d$
of the FBAR, as depicted in Fig. 1. We redefine the origin of the coordinates
in the coupling capacitance between the cavity and the line and hence the
harmonic mirror is placed at $x=d_{\text{eff}}=d+L_{\text{eff}}$, with $d$ the
length of the cavity for a static mirror. Therefore, the natural frequency of
the resonator is $\omega_{0}/2\pi=v/d_{\text{eff}}$ with $v$ the speed of
light in the superconducting material. The input-output relations connecting
the fields inside the cavity $\\{a_{\text{in}},a_{\text{out}}\\}$ to fields in
the transmission line $\\{b_{\text{in}},b_{\text{out}}\\}$ are given by

$\begin{pmatrix}a_{\text{in}}(\omega,0)\\\
a_{\text{out}}(\omega,0)\end{pmatrix}=\begin{pmatrix}\bar{\alpha}_{\omega}&\beta_{\omega}\\\
\bar{\beta}_{\omega}&\alpha_{\omega}\end{pmatrix}\,\begin{pmatrix}b_{\text{in}}(\omega,0)\\\
b_{\text{out}}(\omega,0)\end{pmatrix},$ (7)

with $\alpha_{\omega}=1+i\omega_{c}/2\omega$,
$\beta_{\omega}=i\omega_{c}/2\omega$ and the coupling rate
$\omega_{c}=1/C_{c}Z_{0}$. Notice that $\omega_{c}$ is a measure of how large
the coupling of the resonator to the line is. Indeed, for
$\omega_{c}\gg\omega$ it is completely decoupled, while $\omega_{c}=0$ means
that there is no cavity. Equation (7) holds for $x=0$ [20, 21], but as we want
to connect the fields in the line with the reflected fields on the oscillating
mirror, we must displace the cavity fields to $x=d_{\text{eff}}$ by using the
matrix
$\text{diag}(e^{ik_{\omega}d_{\text{eff}}},e^{-ik_{\omega}d_{\text{eff}}})$.
In order to compute the reflection coefficient $R_{\text{res}}(\omega)$ of the
cavity, we can consider that the inelastic scattering process is absent and
that only the reflection-transmission process remains, so
$a_{\text{in}}(\omega,d_{\text{eff}})=a_{\text{out}}(\omega,d_{\text{eff}})$.
Under this condition, we get

$R_{\text{res}}(\omega)=\frac{1+(1+\frac{2i\omega}{\omega_{c}})e^{2ik_{\omega}d_{\text{eff}}}}{(1-\frac{2i\omega}{\omega_{c}})+e^{2ik_{\omega}d_{\text{eff}}}}.$
(8)

Similarly, we also want to study the mode structure of the resonator. This
information is encoded in the function $A_{\text{res}}$, defined as
$a_{\text{out}}(\omega,d_{\text{eff}})=A_{\text{res}}(\omega)b_{\text{in}}(\omega,0)$,
as it contains the response of the resonator to any input signal coming from
the line,

$A_{\text{res}}(\omega)=\frac{(\frac{2i\omega}{\omega_{c}})e^{ik_{\omega}d_{\text{eff}}}}{(1-\frac{2i\omega}{\omega_{c}})+e^{-2ik_{\omega}d_{\text{eff}}}}.$
(9)

The resonance frequencies can be deduced from the denominator of Eq. (9) and,
as shown in Ref. [21], they are approximately the solutions of the
transcendental equation $\tan(2\pi\omega/\omega_{0})=\omega_{c}/\omega$. The
outgoing mode in the open transmission line in the presence of a driven FBAR,
keeping only the first order of $S(\omega^{\prime},\omega^{\prime\prime})$,
reads

$\displaystyle b_{\text{out}}(\omega)$
$\displaystyle=h_{\text{res}}(\omega,\Omega)+R_{\text{res}}(\omega)b_{\text{in}}(\omega)$
$\displaystyle+S_{1}^{\text{res}}(\omega,\Omega+\omega)b_{\text{in}}(\Omega+\omega)$
$\displaystyle+\bar{S}_{2}^{\text{res}}(\omega,\Omega-\omega)b_{\text{in}}^{\dagger}(\Omega-\omega),$
(10)

with

$\displaystyle h_{\text{res}}(\omega,\Omega)$
$\displaystyle=h(\omega,\Omega)\left[(1-\frac{2i\omega}{\omega_{c}})+e^{-2ik_{\omega}d_{\text{eff}}}\right]^{-1}$
$\displaystyle S_{1}^{\text{res}}(\omega^{\prime},\omega^{\prime\prime})$
$\displaystyle=S(\omega^{\prime},\omega^{\prime\prime})A_{\text{res}}(|\omega^{\prime}|)A_{\text{res}}(|\omega^{\prime\prime}|)$
$\displaystyle S_{2}^{\text{res}}(\omega^{\prime},\omega^{\prime\prime})$
$\displaystyle=S(\omega^{\prime},\omega^{\prime\prime})\bar{A}_{\text{res}}(|\omega^{\prime}|)A_{\text{res}}(|\omega^{\prime\prime}|).$

##  3 Electro-mechanical DCE photon rate

For an initial thermal state, the mean photon number
$n_{\text{out}}(\omega)=\langle
b_{\text{out}}^{\dagger}(\omega)b_{\text{out}}(\omega)\rangle_{T}$ is given by

$\displaystyle n_{\text{out}}(\omega)$ $\displaystyle=$
$\displaystyle\left|R_{\text{res}}(\omega)\right|^{2}n_{\text{in}}(\omega)$
(11) $\displaystyle+$
$\displaystyle\left|S_{1}^{\text{res}}(\omega,\Omega+\omega)\right|^{2}n_{\text{in}}(\Omega+\omega)$
$\displaystyle+$
$\displaystyle\left|S_{2}^{\text{res}}(\omega,\Omega-\omega)\right|^{2}\left[1+n_{\text{in}}(\Omega-\omega)\right]$
$\displaystyle+$ $\displaystyle|h_{\text{res}}(\omega,\Omega)|^{2},$

Figure 2: Electro-mechanical DCE photon production. We consider the generated
photons $n_{out}(\omega)$ in Hz per unit bandwidth versus observation
frequency $\omega$ in units of the mechanical frequency $\Omega$. Main panel
(using $V_{\mathrm{pp}}=500\mu$V, $Q=300$): The orange (light gray) line
depicts the total number of photons in the open transmission line that
originate from the DCE effect and thermal radiation. The green (dark gray)
line shows the photons only generated via the electro-mechanical DCE effect,
whereas the purple (dotted) line shows the thermal contribution. The vertical
gray lines depict the resonances of the superconducting cavity. The inset
shows an enhanced mechanical DCE photon production assuming an FBAR with a
higher mechanical quality factor ( $Q=3\cdot 10^{6}$ and using a driving
voltage of $V_{\mathrm{pp}}=5\mu$V). The additional blue (dashed) line shows
the DCE photons only created through the mechanical motion of the FBAR.

where $n_{\text{in}}(\omega)=\left[\exp(\hbar\omega/k_{B}T)-1\right]^{-1}$ is
the thermal photon occupation at temperature $T$. The photon production of
electro-mechanical Casimir origin is given by the term
$|S_{2}^{\text{res}}(\omega,\Omega-\omega)|^{2}+|h_{\text{res}}(\omega,\Omega)|^{2}$.
The total mean photon number $n_{\text{out}}(\omega)$ is plotted in Fig. 2.
For $\omega/\Omega=1/2$ we see that the Casimir photon production is of the
order of $10^{-2}$ photons/s/unit bandwidth for a driving voltage of
$V_{\text{pp}}=500~{}\mu$V. This is comparable to the photon production
predicted by [20, 21] and should be easily measurable with current technology
[36, 37]. The DCE part of the total photon number is well above the thermal
photon noise floor.

The non-adiabatic change in the boundary conditions is of electro-mechanical
nature, as the voltage driving the FBAR also produces a change in the
potential of the boundary, in addition to the mechanical movement of the
plate. Even though it can not be measured, one may wonder about the ratio of
the photon production which is purely due to the mechanical movement. In order
to calculate electrically generated photons, we consider the situation in
which the piezoelectric material is removed, i.e., $\Delta C=0$, while the
voltage source is still connected. By subtracting this from the total number
of generated photons with the same parameters as before, we estimate the
purely mechanical photon production to be $5\times 10^{-9}$ photons/s/unit
bandwidth. Therefore, only about one in every million generated photons can be
ascribed to mechanical motion alone. However, in our situation it is not
possible to experimentally distinguish the photons generated by changes in the
electrical boundary conditions from the ones generated by changes in the
mechanical boundary conditions. Consequently, we can only talk about electro-
mechanically produced photons.

Improving the quality factor of the FBAR will drastically change the picture.
Assuming a quality factor of $3\cdot 10^{6}$, i.e. 4 orders of magnitude
larger than in Ref. [32], and a smaller driving voltage of
$V_{\text{pp}}=5\,\mu$V results in $\Delta x=8.5\cdot 10^{-11}$ m. This high-Q
regime is in principle already experimentally accessible with other materials,
for example with GHz mechanical oscillators made of silicon [38]. The higher
mechanical quality factor and the lower driving voltage benefit the ratio of
DCE photons generated via mechanical compared to electrical origin, such that
both can be of equal magnitude, see inset of Fig. 2.

##  4 Discussion and Conclusion

Further improvements could be obtained by using a material with higher
piezoelectric coefficient, higher mechanical quality factor and larger
dielectric constant for the FBAR. The most promising approach is the use of
SQUID-based metamaterials for the resonator. Indeed, recent experiments with
metamaterials based on long arrays of Josephson junctions have demonstrated
[39, 40] an increase in the effective impedance $Z_{0}$ by two orders of
magnitude to $Z_{0}\approx 10^{4}$ Ohm. This was achieved while keeping the
total capacitance of the resonator approximately constant, which means that
the speed of light in the metamaterial $v=(\mathcal{C}Z_{0})^{-1}$ is reduced
by two orders of magnitude. Such a device would lead to an enhancement in
photon production by four orders of magnitude, from which we can again roughly
estimate the ratio of the mechanically produced photons with respect to the
electrically generated ones, which would improve by two orders of magnitude.
This results from the former scaling with $Z_{0}$, as shown in Eq. (5), while
the latter scaling as $Z_{0}^{1/2}$, as shown in Eq. (6). However, using
superconducting metamaterials could lead to other unwanted effects such as the
emergence of a finite-size lattice structure or the impedance mismatch between
the resonator and the transmission line. Incorporating these properly into our
proposal would require more detailed studies, which we leave as a future
direction.

Finally, we would like to highlight that there are other traces of the Casimir
effect related to quantum correlation functions. For instance, techniques
developed in recent accurate experiments to measure two-time correlation
functions in propagating quantum microwaves [41, 42, 43] can be used to
measure $g^{(2)}(\tau)$. Indeed, the photons which leak out of the cavity will
show a decaying behavior of these auto-correlations, however the formalism
developed in this letter is not suitable to calculate such effects, as we
treat both the capacitor and the voltage source as classical elements.

In summary, we have proposed an experiment consisting of a mechanical
resonator based on a FBAR directly coupled to a superconducting cavity, which
generates a resonance enhancement of the electro-mechanically generated
dynamical Casimir radiation. We calculate the stable flux of photons
proceeding from an electro-mechanically amplified quantum vacuum,
demonstrating that measuring an effect is within reach of current technology.
We also propose a heuristic method to estimate the fraction which can be
ascribed to the non-adiabatic change in the mechanical boundary conditions.
Further improvements can either be realized by using Josephson metamaterials
or a larger mechanical quality factor to enhance both the photon production
and the ratio of the purely mechanically generated photons. Our work also
paves the way towards experimentally tests of other fundamental relativistic
effects, such as the Unruh effect or the Hawking radiation, by properly
modifying the proposals in Ref. [24] employing mechanical resonators.

We would like to thank Adrian Parra, Simone Felicetti, Philipp Schmidt, Hans
Huebl, Nicola Roch, and Gary Steele for fruitful discussions and useful
insights. M.S. and E.S. are grateful for funding through the Spanish
MINECO/FEDER FIS2015-69983-P and Basque Government IT986-16. S.G. acknowledges
financial support from Foundation for Fundamental Research on Matter (FOM)
Projectruimte grants (15PR3210, 16PR1054), the European Research Council (ERC
StG Strong-Q), and the Netherlands Organisation for Scientific Research
(NWO/OCW), as part of the Frontiers of Nanoscience program, as well as through
a Vidi grant (016.159.369). W.W. acknowledges financial support by Chalmers
Excellence Initiative Nano.

##  Appendix A Piezo-electromechanical coupling

We base our proposal on an FBAR device of thickness $t$ and square surface
area $A=b^{2}$ with volume $V=A\cdot t$ and define the z-direction (subscript
$3$) along the thickness of the FBAR. A voltage is applied to the plates of
the FBAR, resulting in an electric field along the z-direction that is assumed
to be homogeneous. Due to piezoelectric coupling this voltage results in
mechanical strain $\epsilon$:

$\epsilon_{3j}=-d_{3j}E_{3}=-d_{3j}\frac{V_{3}}{t},$ (12)

where $d_{ij}$ is the piezoelectric coupling coefficient. This mechanical
strain has a geometric and stress-related effect, which are discussed in the
following.

###  A.1 Geometric effect

Strain leads to contraction and expansion of the mechanical device and, thus,
its dimensions change. For the $z$ direction, one obtains:

$\Delta z=\epsilon_{33}\cdot t=-d_{33}V_{3}.$ (13)

This dimensional change shifts the resonance frequency of the FBAR device.
Using $\Omega=2\pi v/2t$ ($v$ velocity of sound) as approximate formula for
the resonance frequency of an FBAR, one gets $\Delta\Omega/\Omega_{0}=-\Delta
t/t\approx-\epsilon_{33}$. Using (12), one arrives at

$\frac{\Delta\Omega}{\Omega_{0}}=d_{33}\frac{V_{3}}{t}.$ (14)

###  A.2 Stress-related effect

Mechanical strain $\epsilon$ results in mechanical stress $\sigma$ via the
relation $\sigma=E\epsilon$ ($E$ is Young’s modulus). Mechanical stress over
an area $A$ results in a force $F$ acting on the crystal surface
$F=\int_{A}\sigma dA$.

This force can change the resonance frequency of the device, if the crystal
structure could not relax along that direction. For example, this would be the
case for a doubly-clamped mechanical beam. However, the FBAR has no
fixed/clamped boundaries along the z-direction and, thus, can relax. Hence, we
neglect any shift in resonance frequency of the FBAR.

The force $F$ can nevertheless result in driving the mechanical amplitude of
the resonator as it acts as an additional source term in the dynamic equation.
The solution of a driven, damped harmonic oscillator in Fourier space is

$\tilde{x}(\omega)=\chi(\omega)\cdot\tilde{F}(\omega)/m,$ (15)

with the mechanical susceptibility
$\chi(\omega)=(\Omega^{2}-\omega^{2}-i\gamma\omega)^{-1}$ and the mass $m$ of
the resonator. A sinusoidal driving voltage
$V_{3}(t)=V_{\text{pp}}\cos{(\omega t+\phi)}$ results in a force

$\displaystyle F_{3}(t)$ $\displaystyle=$
$\displaystyle\int_{A}\sigma_{3}(t)dA=Ed_{33}\frac{V_{3}(t)}{t}\int_{A}dA$
(16) $\displaystyle=$ $\displaystyle Ed_{33}\frac{V}{t^{2}}V_{3}(t).$

Evaluating the mechanical response $\tilde{x}(\omega)$ on resonance
$\omega=\Omega$ yields

$|\tilde{x}(\Omega)|=\frac{Q}{\Omega^{2}}\frac{E}{\rho
t}d_{33}\frac{V_{\text{pp}}}{t}.$ (17)

This formula can be rewritten by using the FBARs resonance frequency
$\Omega=2\pi v/2t$ (with velocity of sound $v=\sqrt{K/\rho}$, bulk modulus
$K=E/(3(1-2\nu))$, Poisson’s ratio $\nu$, density $\rho$) to

$\displaystyle|\tilde{x}(\Omega)|$ $\displaystyle=$
$\displaystyle\frac{Q}{\pi^{2}}\left(3(1-2\nu)\right)d_{33}V_{\text{pp}}$ (18)
$\displaystyle=$ $\displaystyle\frac{Q}{\pi^{2}}\left(3(1-2\nu)\right)\Delta
z.$

This means that the driven response is by a factor of $\sim Q/\pi^{2}$ larger
than the geometric response $\Delta z$ alone.

##  Appendix B Proposed implementation and modeling

###  B.1 FBAR

We base our proposal on existing technology and employ experimental parameters
closely following Ref. [32]. The mechanical resonator is a piezoelectric FBAR,
made up of a heterostructure from Al-AlN-Al, whereby the Al is used to contact
the piezoelectric AlN. A specific device could have a total thickness of
approximately $1000$ nm, consisting of a $300$ nm SiO2 layer, two $150$ nm Al
electrodes surrounding a $350$ nm AlN film, with an average speed of sound of
$v_{s}=9100$ m/s. The device could be fabricated on a high-resistivity
silicon-on-insulator wafer. We drive the FBAR with an AC voltage source
$V(t)=V_{\text{pp}}\cos(\omega t+\phi)$ with $\omega=\Omega$, i.e., a driving
frequency equal to the fundamental mechanical frequency $\Omega$, and a phase
difference $\phi=\frac{\pi}{2}$.

###  B.2 Modified Butterworth-Van Dyke circuit

The Modified Butterworth-Van Dyke circuit (MBVD) [34] enables to extract the
necessary parameters for modeling a mechanical resonator as an equivalent
electrical circuit. This has been, e.g., used in Ref. [32] for calculating the
response of the FBAR made from Al/AlN/Al. Note that the MBVD is an
approximation of the Generalized Butterworth-Van Dyke circuit [44] for low
electro-mechanical coupling $k_{t}^{2}$.

The simplest model for a FBAR is derived from approximating the 3-port Mason
model to a 4 element circuit consisting of a plate capacitance $C_{0}$ in
parallel with a series $R_{m}-L_{m}-C_{m}$ circuit. In our case, the
capacitance $C_{0}\equiv C(t)$ actually changes in time when an AC driving is
applied, due to the change in the geometric structure. Indeed, mechanical
resonators consisting of a piezoelectric material sandwiched between
superconducting layers suffer from a geometrical change in their structure due
to the piezoelectric effect when a voltage is applied. Therefore, the value of
the electric elements describing the electric response of the resonator
depends on the applied voltage (note that our voltage source is a function of
time). For the usual applications of FBARs and other mechanical oscillators,
this dependence is negligible, since it is a tiny correction of the mean
value. However, this dependence has already been studied, for instance, in
Ref. [45], and references thereof. The aforementioned 4-element circuit has a
series resonance $\omega_{s}$ (given by $L_{m},C_{m}$ as
$\omega_{s}=1/\sqrt{L_{m}C_{m}}$) and a parallel resonance $\omega_{p}$ (set
by $C_{0}$ in series with $L_{m},C_{m}$ as
$(\omega_{p}/\omega_{s})^{2}=1+1/r$). The capacitance ratio $r$ is defined as
$r=C_{0}/C_{m}$. The electro-acoustic coupling constant $k_{t}^{2}$ can be
derived to be

$(k_{t})^{2}=\frac{\pi^{2}}{4}\frac{\omega_{s}}{\omega_{p}}\frac{\omega_{p}-\omega_{s}}{\omega_{p}}=\frac{\pi^{2}}{8}\frac{1}{r}\left(1-\frac{1}{r}\right)$
(19)

Figure 3: Equivalent circuit for the FBAR. We have employed the modified van
Dyke-Butterworth model to provide the equivalent lumped-element circuit
representation of the FBAR.

Adding a series resistor $R_{s}$ at the input to this model allows to account
for the electrode electrical loss. This is the so-called Butterworth-Van Dyke
circuit, which has 5 parameters: $C_{0},R_{s},R_{m},L_{m},C_{m}$. One can add
a sixth parameter, a resistor $R_{0}$ in series with $C_{0}$, accounting for
material loss. In Ref. [34], it is derived a set of equations to obtain the 6
parameters from measurable quantities: series and shunt resonant frequencies,
the effective quality factors $Q_{s0},Q_{p0}$, and the capacitance and
resistance far away from the resonances.

To model the FBAR we use experimental parameters closely following Ref. [32]:
$C_{m}=0.655$ fF, $L_{m}=1.043$ $\mu$H and $R_{m}=146$ Ohm for the mechanical
part, and $C_{0}=0.4$ pF (average value in time) and $R_{0}=8$ Ohm for the
geometric part. We would like to note that a similar FBAR was recently
demonstrated with $C_{0}=1.00$ pF [46]. This higher coupling capacitance would
allow increasing the DCE photon rate further. Taking into account that the
permittivity of AlN is $\varepsilon_{\text{AlN}}\approx 9.2\varepsilon_{0}=81$
pF/m and that the distance between the plates is $t_{\text{AlN}}=350$ nm, we
can model a parallel plate capacitor and estimate its area
$A=t_{\text{AlN}}C_{0}/\varepsilon\approx 7.7\cdot 10^{-10}~{}$m2.

In this work, we are primarily interested in calculating the DCE photon
production. Therefore, we simplify the full model to the elements containing
the most relevant information, as shown in Fig. 3. Taking the parameters
considered in Appendix B.4, then $Z_{m}=R_{m}+j(\Omega L_{m}-\frac{1}{\Omega
C_{m}})\approx 146+j2.8\cdot 10^{4}$ Ohm, while $Z_{0}=R_{0}-j\frac{1}{\Omega
C_{0}}\approx 8-j189$ Ohm. As
$Z_{\text{eq}}=\frac{Z_{0}Z_{m}}{Z_{0}+Z_{m}}\approx Z_{0}$, since
$|Z_{0}|\ll|Z_{m}|$ and the resistor is much smaller than the capacitor, then
our approximation follows.

In order to understand the connection between the mechanical properties
described in the Appendix A and the modified Butterworth-Van Dyke model, let
us explain how the mechanical $Q$-factor which appears in Eq. (17) is
described in terms of the electric elements of the circuit. This connection is
deeply related, as one may expect, to the resistances in the circuit and the
coupling of the losses of the circuit when coupled to other circuits. A
detailed theoretical and experimental analysis of this dependence may be found
in Ref. [34]. In this reference, it is proven that

$\frac{1}{Q}\approx\left(\frac{1}{Q_{s}}+\frac{1}{Q_{e}}\right)$ (20)

with $(Q_{s})^{-1}=\omega R_{m}C_{m}$, $(Q_{e})^{-1}=\omega R_{0}C_{m}$, and
$\omega$ the resonance frequency of the circuit. Hence, the quality factor is
indeed determined by the electric elements of the circuit.

###  B.3 Superconducting cavity

The superconducting cavity has a length of $d=3.3\cdot 10^{-2}$ m, with a
fundamental frequency of $\omega_{c,0}=2\pi\cdot v/d=2\pi\cdot 3.03$ GHz with
the speed of light in the superconducting material of $v=10^{8}$ m/s. It is
capacitively coupled to a semi-infinite transmission line with impedance
$Z_{0}\approx 55$ Ohm. The capacitor coupling the resonator to the
transmission line has a frequency $\omega_{c}/2\pi=(2\pi
Z_{0}C_{c})^{-1}\approx 2\pi\cdot 29.1$ GHz.

###  B.4 Parameters for the proposed implementation

The following table summarizes the experimental parameters used in this
proposal.

Parameter | Symbol | Value
---|---|---
Material properties AlN
Youngs modulus | $E$ |  $308$ GPa
density | $\rho$ |  $3230$ kg/m3
piezoelectric coupling [47, 48] | $d_{33}$ |  $5.1\cdot 10^{-12}$ m/V
Poissons ratio along {0001} | $\nu$ | $0.287$
average velocity of sound | $v$ |  $9100$ m/s [32]
permittivity | $\varepsilon_{\text{AlN}}$ | $9.2\varepsilon_{0}$
FBAR parameters
quality factor | $Q$ | $300$
resonance frequency | $\Omega$ |  $2\pi\cdot 4.2$ GHz
thickness AlN layer | $t_{\text{AlN}}$ |  $350$ nm
drive voltage | $V_{\text{pp}}$ |  $0.5$ mV
driven motional amplitude | $\Delta x(V_{\text{pp}})$ |  $1.7\cdot V_{\text{pp}}$ nm/V
Superconducting cavity parameters
length | $d$ |  $3.3\cdot 10^{-2}$ m
cavity frequency | $\omega_{c,0}$ |  $2\pi\cdot 3.03$ GHz
impedance | $Z_{0}$ |  $55$ Ohm
DCE capacitance | $C_{0}$ |  $0.4\cdot 10^{-12}$ F
transmission line coupling rate | $\omega_{c}$ |  $2\pi\cdot 29.1$ GHz
Further parameters
Temperature | $T$ |  $10$ mK

##  Appendix C DCE miscellanea

###  C.1 DCE photon rate and its relation to $v/c$

In the following, we will relate the photon rate from Eq. (11) to the commonly
used speed ratio $v/c$ in the DCE, where $v$ is the velocity of the mirror (in
our case the velocity of the vibrational motion of the FBAR) and $c$ is the
speed of light (in our case the speed in the superconducting material).

The mechanical photon production rate from Eq. (11) depends essentially on the
term $\langle
n_{\text{out}}(\omega)\rangle=|S_{2}^{\text{res}}(\omega,\Omega-\omega)|^{2}$,
which is a function of $S(\omega^{\prime},\omega^{\prime\prime})=-i\Delta
CZ_{0}\sqrt{|\omega^{\prime}||\omega^{\prime\prime}|}\theta(\omega^{\prime})\theta(\omega^{\prime\prime})$.
Therefore, $\langle n_{\text{out}}(\omega)\rangle\propto\Delta
C^{2}Z_{0}^{2}\omega(\Omega-\omega)$, which in resonance $\omega=\Omega/2$
gives $\langle n_{\text{out}}(\Omega/2)\rangle\propto\Delta
C^{2}Z_{0}^{2}\Omega^{2}/4$. Taking into account that $\Delta C\approx
C_{0}\Delta x/t_{\mathrm{AlN}}$ and that the maximal speed of the FBAR is
given by $v=\Delta x\cdot\Omega$, then $\langle
n_{\text{out}}(\Omega/2)\rangle\propto
C_{0}^{2}Z_{0}^{2}v^{2}/(4t_{\text{AlN}}^{2})$. Finally, the speed of light in
the material is given by $c=(\mathcal{C}Z_{0})^{-1}$, with $\mathcal{C}$ the
density of capacitance of the superconducting cavity, so

$\langle n_{\text{out}}(\Omega/2)\rangle\propto
C_{0}^{2}\mathcal{C}^{2}Z_{0}^{2}v^{2}/(4\mathcal{C}^{2}t_{\text{AlN}}^{2})=Qv^{2}/c^{2},$
(21)

with $Q=C_{0}^{2}/(4\mathcal{C}^{2}t_{\text{AlN}}^{2})$ is related with the
impedance mismatch between the FBAR and the superconducting cavity and,
consequently, with the quality factor of the resonator. This expression
coincides with Eq. (15) of Ref. [12] and Eq. (3) of Ref. [21] for photon
production in the presence of a cavity, and shows that, effectively, in the
context of the FBAR scheme, the $v/c$ ratio in photon production still holds.

For the sake of completeness, let us estimate the ratio of $v/c$ as follows.
The maximal velocity of the mechanical resonator on resonance is given as
$v=\Delta x\cdot\Omega$. The speed of light in the superconducting cavity is
$c=(\mathcal{C}Z_{0})^{-1}=1\cdot 10^{8}$ m/s. We obtain with
$\Omega=2\pi\cdot 4.2$GHz a ratio $v/c$ of $\sim 2\cdot 10^{-10}$ and $\sim
2\cdot 10^{-8}$ for $\Delta x=8.5\cdot 10^{-13}$ m (low mechanical Q) and
$\Delta x=8.5\cdot 10^{-11}$ m (high mechanical Q), respectively.

###  C.2 DCE as a parametric effect

In the following, we show by using a simplified model how the parametric
effect of the DCE emerges from a change in boundary conditions. To this end,
we use the Hamiltonian of a cavity ended by an oscillating mirror, with
$C_{T}=C+C_{0}$, the sum of the total capacitance of the cavity and the
capacitor. Therefore,

$H=\frac{1}{2C_{T}}q^{2}+\frac{1}{2L}\Psi^{2},$ (22)

which is the Hamiltonian corresponding to an $LC$ circuit. Let us assume that
the face of the resonator vibrates due to the phonons as a classical harmonic
oscillator of frequency $\Omega$ and amplitude $\Delta x$, so
$d(t)=d_{0}+\Delta x\cos(\Omega t)$ and the capacitance, that we assume given
by a coplanar capacitor, varies as

$\displaystyle\frac{1}{C_{T}(t)}$
$\displaystyle=\frac{1}{C+C_{0}(t)}=\frac{1}{C+\frac{\epsilon A}{d(t)}}$
$\displaystyle=\frac{1}{C+C_{0}(1-\frac{\Delta x}{d_{0}}\cos(\Omega t))}$
$\displaystyle=\frac{1}{C_{T}}+\frac{C_{0}\Delta x}{C_{T}^{2}d_{0}}\cos(\Omega
t)$ (23)

We can consider the term on the right hand side a perturbation, so the
Hamiltonian can be expressed by means of the same creation and annihilation
operators as

$H=\hbar\omega\left(a^{\dagger}a+\frac{1}{2}\right)-\frac{C_{0}\Delta
x}{2C_{T}^{2}d_{0}}\frac{\hbar\omega C_{T}}{2}\cos(\Omega
t)\left(a^{\dagger}-a\right)^{2}$

where we made use of the standard definition of creation and annihilation
operators for an $LC$ circuit $q=i\sqrt{\frac{\hbar\omega
C_{T}}{2}}(a^{\dagger}-a)$ and $\Psi=\sqrt{\frac{\hbar}{2\omega
C_{T}}}(a^{\dagger}+a)$, and $\omega=\sqrt{\frac{1}{LC_{T}}}$ is the frequency
of the superconducting circuit. By removing constants and using the rotating
wave approximation with $\Omega=2\omega$, one can rewrite the Hamiltonian in
the interaction picture as

$H=\frac{\hbar\omega}{8}\frac{C_{0}\Delta
x}{C_{T}d_{0}}\left[(a^{\dagger})^{2}+a^{2}\right],$ (24)

which is a squeezing Hamiltonian.

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