How precisely can the speed of light in vacuum and the metric of space-time be determined {\em in principle
Daniel Braun$^1$
Fabienne Schneiter$^1$
and Uwe R. Fischer$^2$
% The speed of light in vacuum is one of the most important and most % precisely measured natural constants. It The speed of light in vacuum, one of the most important and precisely measured natural constants, is fixed by convention to $c=299 792 458$ m/s. Advanced theories predict possible deviations from this universal value, or even quantum fluctuations of $c$. Combining arguments from quantum parameter estimation theory and classical general relativity, we here establish rigorously the existence of lower bounds on the precision to which the speed of light and the metric of space-time can be determined {\em in principle. They provide a novel perspective on the experimental falsifiability of predictions for the quantum fluctuations of space-time. %This provides a novel perspective on the quantum fluctuations of space-time that %, \col{as predicted by quantum gravity theories, %questions their meaning independent of an experimental procedure to measure them. %\col{their experimental falsifiability. % by only employing % these two firmly established physical theories.
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\title{How precisely can the speed of light in vacuum and the metric of
  space-time be determined {\em in principle}?} 
\author{Daniel Braun$^1$, Fabienne Schneiter$^1$, and Uwe R. Fischer$^2$}
\affiliation{$^1$Eberhard-Karls-Universit\"at T\"ubingen, Institut f\"ur Theoretische Physik, 72076 T\"ubingen, Germany\\
$^2$Seoul National University, Department of Physics and Astronomy \\  Center for Theoretical Physics, 08826 Seoul, Korea}

%\centerline{\today}
\begin{abstract}
% The speed of light in vacuum is one of the most important and most
% precisely measured natural constants.  It 
The speed of light in vacuum, one of the most important and
 precisely measured natural constants,
is fixed by convention to $c=299 792 458$ m/s. Advanced theories 
predict possible deviations from this universal value, or even quantum
fluctuations of $c$.  Combining arguments from quantum parameter estimation
theory and classical general relativity, we here establish rigorously the
existence of lower bounds on the precision to which the speed of light
and the metric of space-time can be determined {\em in principle}.
They provide a novel perspective on the experimental falsifiability of predictions for the quantum fluctuations of space-time.
%This provides a novel perspective on the quantum fluctuations of space-time that
%, \col{as predicted by quantum gravity theories,} 
%questions their meaning independent of an experimental procedure to measure them.
%\col{their experimental falsifiability.} 
% by only employing
% these two firmly established physical theories. 
 
\end{abstract}
%\pacs{03.67.-a, 03.67.Lx, 03.67.Mn }
\maketitle
%\begin{twocolumn}


{\em Introduction.}
It is generally accepted that the speed of light in vacuum $c$ is a
universal natural constant, isotropic, independent of frequency, and
independent 
of the motion of the inertial frame with respect to which it is
measured.  These properties have been experimentally demonstrated with
very high precision, e.g.~isotropy up to a
relative uncertainty of the order of $\sim 10^{-9}$
\cite{Scheithauer05},
and lie at the basis
of special relativity.  % The 
% speed of light enters as parameter in the Lorentz transformation and
% thus in the definition of causality and Lorentz invariance, which is
% believed to be a fundamental symmetry of Nature for all particle
% energies below the Planck energy  
% \cite{Weinberg05}.
By 1972, measurements of the speed of light  
became more precise 
than the definition of the meter, leading
in 1983 to the definition of the speed of light in vacuum $c=299
792 458$\,m/s.  % However,
% already in the 1950s Pauli suggested that 
% smearing out of the light-cone due to quantum gravitational effects
% might mend
% the divergence problems of quantum field theory
% \cite{Pauli56,Deser57,Isham71}. 
But attempts to quantize gravity have led to the concept of 
space-time as a fuzzy ``quantum foam'' on the
Planck length $l_{\rm Pl}=\sqrt{\hbar G/c^3}\simeq 1.62 \times 10^{-35}\,$m  
% due to quantum gravitational effects, i.e.~that the quantization 
% of space-time
\cite{snyder_quantized_1947,doplicher_quantum_1995} 
that implies an uncertainty or dispersion of $c$ % , and in particular an energy dependence
\cite{Hawking80,Ashtekar92,Ford95,Yu00}.
Experimental data based on gamma-ray bursts, pulsars,  and TeV-flares
from active galaxies imply upper 
bounds on deviations of $c$ %from its fixed value 
over cosmic distances 
\cite{Amelino-Camelia98,Ellis00,Biller99,Kaaret99,Amelino-Camelia14,vasileiou_planck-scale_2015}.   
Quantum fluctuations of $c$ were also proposed
%physical models of the due to the vacuum being composed 
due to virtual fermion-anti-fermion pairs, % in the quantum vacuum, 
leading to %the prediction of 
a scaling of the jitter of the arrival time of light pulses with
propagation distance \cite{Urban13,Leuchs13}. 
Satellite experiments are being planned to verify fundamental
space-time properties with unprecedented precision, such as the isotropy of $c$ and its  independence from the laboratory frame velocity \cite{Scheithauer05}. 
%\vspace*{-0.5em}

Here we establish how
precisely $c$ in a given region of space--time may be 
determined {\em in principle}, i.e.~independent of any technical
challenges.  Our approach is based
on the firmly 
established quantum parameter estimation theory (q-pet)
\cite{Helstrom1967,helstrom_quantum_1969,Braunstein94,braunstein_generalized_1996,wiseman_quantum_2009,Giovannetti04,giovannetti_quantum_2006,Paris09}
and general relativity (GR) in semiclassical approximation.  
Q-pet allows one to obtain a lower bound
on the precision with which a parameter $\theta$ 
may be estimated that
parametrizes a quantum state specified by  a density matrix $\rho(\theta)$. The
power of q-pet is due to the facts that {\em i.)} the bound is reachable
in the limit of a large number of measurements, and {\em ii.)} it is
optimized over all possible  
quantum mechanical measurements (positive operator valued measures,
POVM \cite{Peres93}) and all data-analysis schemes (unbiased estimator
functions). This so-called quantum Cram\'er-Rao bound
\cite{Helstrom1967,helstrom_quantum_1969,Braunstein94,braunstein_generalized_1996} 
becomes relevant once all technical
noise problems have been solved, and only the fundamental quantum
uncertainties remain.  It is the ultimate achievable lower bound on
the uncertainty with which any parameter can be measured.
%, and has as such become the accepted benchmark in the modern theory of
%quantum measurements. 
%It has been used to establish the ultimate
%sensitivity of interferometers, measurements of magnetic fields, mass,
%temperature, and chemical potential. 
Recently, the q-pet formalism was applied to the
measurement of parameters in relativistic quantum field
theory such as 
proper times and accelerations, the Unruh effect, gravitation, or the
estimation of 
the mass of a black hole \cite{Ahmadi14,Ahmadi14.2,Aspachs10,Downes11}.  In the
present work we go a step further by examining the back-action of the
quantum probe on the metric of space-time. Taking back-action into account
was proposed before
\cite{Ahluwalia94,Amelino-Camelia94,Ng00} but to the best of our knowledge we
combine for the first 
time modern q-pet %theory of optimal quantum parameter-estimation 
with a precise calculation of the back-action of the probe on the space-time
metric. 
We show that there is an optimal photon number at which the perturbation of
the space-time metric due to the probe equals the quantum uncertainty
of the measurement itself, establishing thus an ultimate lower bound
on how precisely $c$ and the metric can be determined {\em in
  principle}.  The fact that an uncertainty of $c$ implies an
uncertainty of the metric of space-time  is based on the generally
accepted premise that the parameter $c$ in GR (which 
also plays the role of the maximum speed of propagation of information
and the speed of gravitational waves) is 
identical to the speed of light in vacuum.  

%\vspace*{-0.5em}

{\em Quantum parameter estimation.}
Any {\em direct} measurement of the speed of light has to use a
light signal. Indirect measurements,
e.g.~through measuring the fine-structure constant, the electron charge and
Planck's constant, may need no light but do
not reflect the definition of $c$ as a speed and need an
elaborate theoretical framework for their interpretation. Any light signal
can be decomposed in modes of the electromagnetic (e.m.)~field
which are the fundamental dynamical objects  in quantum optics. 
Q-pet shows that with $m$ modes the
sensitivity can be improved at most by a factor $1/m$
\cite{giovannetti_quantum_2006}. Below we 
find that with at most $n$ photons in a single mode the best sensitivity
scales as $\propto 1/n$; one can thus achieve for given maximum photon
number $nm$  the same  
sensitivity scaling as $\propto 1/(nm)$  as with $m$ modes. In
\cite{giovannetti_quantum-enhanced_2001} the  
problems of positioning and clock synchronization were
analyzed.  They were reduced to measuring a travel time of a light
pulse with constant $c$, which is closely related to 
measuring $c$ for a known propagation distance.  Also there it was shown that the
best uncertainty in the arrival time of the pulse for a squeezed m-mode
state scales as $1/(nm)$. Furthermore, using the Margolus-Levitin
quantum speed limit theorem, it was argued in
\cite{giovannetti_quantum-enhanced_2001} that this is the optimal 
scaling possible for any state.
The scaling $\propto 1/\bar{n}$ for large average photon number
$\bar{n}$ was also obtained for phase estimation with two-mode 
squeezed light in \cite{anisimov_quantum_2010}.
As for relativistic effects, if we are 
interested in knowing $c$ in a given space-time region, they cannot be
diluted by using several modes in 
parallel in different space-regions or sequentially. 
We can thus restrict ourselves to studying a single mode.
For concreteness, we consider a cubic cavity with edges
of length $L$,  % in $x,y,z-$directions, 
and perfectly reflecting walls or symmetric boundary conditions.  
% Differently
% shaped cavities or different boundary conditions lead
% only to different numerical prefactors in our final results. \\

Maxwell's equations in vacuum with appropriate boundary conditions impose quantized
modes with wave vectors $\bk$ that are independent of 
$c$, whereas the frequency $\omega=c|\bk|$.
Obtaining the best 
possible precision of $c$ 
is thus equivalent to the optimal frequency measurement of a harmonic oscillator, for which
the quantum Cram\'er-Rao bound was calculated
% in the context of optimal mass
% sensing with a  nano-mechanical resonator
in \cite{Braun11.2}.
The smallest $\delta \omega/\omega$, and
hence %in the present context 
smallest $\delta c/c$ for fixed
maximum excitation $2n$ and for $\tau=\omega t\gg 1$, is achieved with
the optimal state
$|\psi_{\rm opt}\rangle=(|0\rangle+|2n\rangle)/\sqrt{2}$. In a single
measurement, it leads to a minimal 
$c$-uncertainty %in $c$ given by 
\begin{equation}
  \label{eq:opt}
  \frac{\delta
    c}{c}\simeq \frac{1}{2\tau n}\,.
\end{equation}

For existing measurements with large $n$, coherent states are  
more relevant than the optimal state. A coherent state 
with amplitude %complex label 
$\alpha$ at time $t=0$, $|\psi_{\rm coh}\rangle =|\alpha\rangle$,
%($a|\alpha\rangle=\alpha|\alpha\rangle$) 
evolves according to
$\alpha(t)=\alpha e^{-i\omega t}$  \cite{Scully97} and leads to 
%a minimal uncertainty 
\begin{equation}\label{eq:dccoh}
  \frac{\delta
    c}{c}=\frac{1}{2}\frac{1}{|(\frac{1}{2}+n)\sin^2\tau+n\tau(\tau+\sin(2\tau))|^{1/2}}
  \simeq  \frac{1}{2\tau\sqrt{n}}\,,
\end{equation}
where the last equality is for large $\tau=\omega t$ and large average
photon number $n=\alpha^2$ ($\tau^2\alpha\gg 1$) \cite{Braun11.2}. 

%\section{General relativistic effects}
From these results one is tempted to conclude that ${\delta c}/{c}$ can be made 
arbitrarily small by increasing $n$. % , as $\delta c/c$ decreases as
% $1/n$ (for $\ket{\psi_{\rm opt}}$), or $1/\sqrt{n}$ (for the
% coherent state)
However, the 
energy-momentum tensor
increases $\propto n$ for $n\gg 1$, and will at some point perturb itself
the metric of space-time.  
We argue that the ultimate sensitivity is reached when the general
relativistic modification
of % the speed of light % due to the light signal itself 
space-time 
becomes comparable to the minimal quantum uncertainty of the measurement.
This leads to a finite optimal number of photons, and a finite optimal
sensitivity. Increasing the photon number even more will modify
space-time 
to a point where one cannot speak of light
propagation in vacuum anymore. In principle one may re-calculate from
the measured value using GR what
the speed of light in vacuum {\em would} be, but this is a counterfactual
reasoning and not a direct measurement of $c$. On the other 
hand, 
reducing the photon number would 
increase the quantum noise. The situation is very similar to the
optimization of the photon number in LIGO-type gravitational wave
interferometers, where one balances photon-shot noise against
radiation pressure noise
\cite{Caves81,abadie_gravitational_2011,demkowicz-dobrzanski_fundamental_2013}.
However,  
%the effect considered here is much more fundamental: 
whereas radiation pressure noise is 
specific to the measurement instrument, 
%(Weber bars, e.g., did not suffer from such a limitation), 
in our case the properties of space-time itself and thus the very meaning of light propagation in vacuum are
affected when increasing the photon number further, and 
this effect is unavoidable.

% While there is still no agreement on a theory of quantum gravity, 
The gravitational effects sought here are
well in the %\qq took out "classical" as we say
                      %"semiclassical" below
regime where Einstein's field equations are valid: 
Firstly, we consider light at
wavelengths $\lambda$ and structures of the energy-momentum tensor on
scales much larger than the Planck-length (e.g.~$\lambda=500$\,nm and a
standard (lossy) cavity of size $L=1$\,km).  % If needed, one may restrict the
% theory even to wavelengths larger than a few $\mu$m over which
% Newtonian gravity 
% has been tested. 
Secondly, we consider light fields 
of very large intensity and effects linear in the perturbation of the metric, for which the
energy-momentum tensor should be  
well approximated by its quantum mechanical
expectation value \cite{FORD1982238}. % for the electro-magnetic fields.  
It is the effect
of this average energy-momentum tensor on space-time that we calculate 
%in general relativity 
and compare to the minimal uncertainty of $c$ obtained from q-pet, not the
fluctuations of space-time themselves.
% introduced by the quantum
% fluctuations of the e.m. field
The former is
established on the solid ground of general relativity, %in its range
                                %of validity, 
whereas the latter would require a quantum gravity theory to make
reliable predictions. The  
quantum fluctuations that we {\em are} interested in here are those of
light probing the space-time, which are reliably described by quantum optics. 
%describing these fluctuations is well established.   
% the latter once more require a quantum gravity theory.  
%Therefore, the quantities we compare can be described 
%in the firmly established realms of quantum optics and
%general relativity: 
Our results therefore rely only on well-tested theories, 
in distinction to predictions of the fluctuations of space-time
obtained by various theories of quantum gravity. 



  
{\em Perturbation of metric.} The modification of the metric of
space-time is found from the weak  
field limit of the Einstein field equations, 
where  the metric tensor is given by
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, i.e.~the flat Minkowski metric 
% $(-c^2,1,1,1)$ in terms of $t,x,y,z$ (or 
$\eta_{\mu\nu}={\rm
  diag}(-1,1,1,1)$ (in terms of $ct,x,y,z$)
plus a small perturbation, %on top of ${g_0}_{\mu\nu}$, 
$|h_{\mu\nu}|\ll 1$.
Einstein's  equations yield a wave
equation for the trace inverse, $\bar{h}^{\mu\nu}=h^{\mu\nu}-\frac{1}{2}\eta^{\mu\nu}\eta^{\alpha\beta}
h_{\beta\alpha}$, 
\begin{equation}
  \label{eq:weakfield}
\square \bar{h}^{\mu\nu}=-16\pi \frac{G}{c^4} T^{\mu\nu} \,,
\end{equation}
%where $G$ is the gravitational constant (
where the (flat space-time) Lorenz gauge (FLG) condition
%\bea
$\bar h^{\mu\nu}{}_{,\nu}=0$ is used; see eq.~18.8b in
\cite{MTW}. %\label{Lorenz}
%\ea
The energy-momentum tensor $T^{\mu\nu}$ of the e.m.~field reads \cite{MTW}
\begin{eqnarray}
  \label{eq:tmn}
  T^{00}&=&\frac{1}{2}(\epsilon_0\bE^2+\mu_0\bH^2), \quad 
T^{0i}=T^{i0}=\frac{1}{c}(\bE\times \bH)_i\nn
T^{ij}&=&-\left(\epsilon_0E_iE_j+\mu_0H_iH_j\right)+T^{00}\delta_{ij}\,,
\end{eqnarray}
where $i,j\in\{1,2,3\}=\{x,y,z\}$. We use the
q.m.~expectation value of $T^{\mu\nu}$ as source term in
(\ref{eq:weakfield}) for the (011) and the (01M) modes
($k_i=l_i\pi/L$, $l_x=0,l_y=1$, and $l_z=1$ or $l_z=M$, respectively;
$\Omega_l=c|\bk|$, and $V=L^3$). This ``semiclassical approximation''
is justified if one is interested only in effects to first order in
$h_{\mu\nu}$ \cite{FORD1982238}. For $\ket{\psi_{\rm
    opt}}$ with $n\gg 1$, the  
solution of (\ref{eq:weakfield}) for the (011) mode reads  ($\bxi=\pi\bx/L$)
\begin{eqnarray}
  \bar h^{\mu\nu}(\bxi)&=& {\mathcal P} \int_0^\pi\int_0^\pi
d\eta'd\zeta'I(\xi,\eta-\eta',\zeta-\zeta')t^{\mu\nu}(\eta',\zeta')\,,\nonumber\\  
  \label{eq:A}
  {\mathcal P}&=&\frac{4n}{\pi}\kappa ,\quad\quad
  \kappa=\left(\frac{l_{\rm 
        Pl}}{L}\right)^2,\\ 
  I(\xi,\eta,\zeta)&=&\ln\left(\frac{\xi+\sqrt{\xi^2+\eta^2+\zeta^2} 
    }{\xi-\pi+\sqrt{(\xi-\pi)^2+\eta^2+\zeta^2
    }}\right)\,,
\end{eqnarray}
with %the Planck length and 
dimensionless trigonometric functions $t^{\mu\nu}:=
T^{\mu\nu}/(n\hbar\Omega_l/V)$ of order one inside the cavity, 
and zero outside \cite{suppl}. $T\indices{^\mu_\mu}=0$ for the
e.m.~field \cite{LandauLifschitz87}, hence $h\indices{^\mu_\mu}=0$ and 
$h^{\mu\nu}=\bar{h}^{\mu\nu}$. 

The deviations of $\bar h^{\mu\nu}$ in (\ref{eq:A})
from FLG are of second order
in $h$ and can be neglected \cite{Eddington}.  
For $\ket{\psi_{\rm coh}}$, $\bar{h}^{\mu\nu}$ is
the same as for $\ket{\psi_{\rm opt}}$
plus retarded oscillation on top of it, with an amplitude of the same
order.  We therefore restrict the analysis to the
time-independent part. For the $(01M)$ 
mode, and $n,M\gg 1$, only ${h}^{00}$
and ${h}^{33}$ are non-negligible, 
\begin{eqnarray}
%\begin{align}
  \label{eq:h00}
 {h}^{00}={h}^{33}\simeq 4{\mathcal P} M \int_0^\pi\int_0^\pi \!
d\eta'd\zeta'I(\xi,\eta-\eta',\zeta-\zeta')\sin^2\eta' .\nonumber
\end{eqnarray}
%\end{align}
% In the supplemental information 
% %\cite{See the supplemental material for an extended discussion}, 
% we calculate the quantum mechanical expectation value
% of $T^{\mu\nu}$ for a $(01M)$ mode and the
% quantum mechanical states considered above, and then solve Einstein's
% field equations  with $T^{\mu\nu}$ as source term. %\qq hab das wieder
                                %reingetan, um klarzustellen, dass wir
                                %nicht selbst-konsistent l"osen; und
                                %refs in sup mat.
From the geodesic 
condition $ds^2=g_{\mu\nu}dx^\mu dx^\nu=0$, the local modification of the speed of
light 
\begin{eqnarray}
  \label{eq:dc2}
  \delta c(x)/c = -\frac12 (h_{00} + h_{11})
\end{eqnarray}
 is obtained for the (011) mode, with similar expressions for $\delta
 c(y)$ and $\delta c(z)$ (see also Fig.1 in \cite{suppl}).
For the $(01M)$ mode with $n,M\gg 1$, 
$
\delta c(x)/c =\delta c(y)/c =-\frac12 h_{00}\,,\,\,\delta c(z)/c=2  \delta c(x)/c\,.
$
One may object that according to the
equivalence principle one could always find a
coordinate system (CS) in which $c(x)=c(y)=c(z)=c$, and that
by the definition of $c$ one {\em should} go to the free falling
CS for measuring $c$, where $c$ is always the same.  However, one has
to distinguish between the 
universal constant $c$ and the experimental value $c_{\rm exp}$
obtained in measurements of it.  The most
direct experimental definition of $c$,  $c_{\rm exp}=\Delta x/ \Delta t$,
where $\Delta x$ is the 
distance that a light signal travels in time $\Delta t$ 
implies that for any finite $\Delta x$ 
the measurement is non-local, which precludes transforming the
discussed GR effect away by a local transformation. Moreover, the
cavity cannot be free falling in the gravitational field of the light
it contains, as it carries that light with it. A time delay can be
measured with a single clock by passing the light through a beam
splitter (BS), reflecting it on a mirror and sending it back to the
BS. The two passes through the BS trigger start/stop of the clock. The
clock measures its eigentime, $d \tau=  
\sqrt{-g_{00}} dt$. For a rigid arrangement of BS and mirror,
with a distance measured by rigid measuring rods (gravitational forces
due to the enclosed light much smaller than the e.m.~forces
that determine their shape and arrangement), $\Delta x$ remains
unchanged.  In the limit
$R\gg L$ ($R$ = typical radius of 
curvature of space time), the
experimentally found value $c_{\rm 
  exp}(x)= \Delta x/\Delta \tau\simeq
dx/d\tau=c(x)/\sqrt{-g_{00}}$ is then directly related to the
coordinate speed $c(x)$ determined above.  This gives $\delta c_{\rm
  exp}/c=-h_{00}-h_{11}/2$ (or $-h_{00}$) for the (011) mode (or (01M)
mode with $M\gg 1$),
respectively, where $\delta c_{\rm
  exp}(x):=  c_{\rm  exp}(x)-c $ can be
 considered a systematic error in the determination of $c$. 

\begin{table*}[hbt]
%\begin{table*}[t]
%  \begin{tabular}
  \centering
  \begin{tabular}{|c|c|c|c|c||c|c|c|}
\hline
&\pbox{3cm}{optimal state\\ (lossless)}&
\pbox{3cm}{optimal state\\ (lossy cavity)}& 
\pbox{3cm}{coherent state\\ (lossless)}&
\pbox{3cm}{coherent state\\ (lossy cavity)}&
Ref.\cite{Ng00}&
% \pbox{3cm}{Ref.\cite{amelino-camelia_gravity-wave_1999}\\ eq.(3)}& 
% \pbox{3cm}{Ref.\cite{amelino-camelia_gravity-wave_1999}\\
%   eq.(5)}\\\hline
 eq.(3) in Ref.\cite{amelino-camelia_gravity-wave_1999}& 
eq.(5) in Ref.\cite{amelino-camelia_gravity-wave_1999}
\\\hline
$\delta c/c$&
$\frac{l_{\rm Pl}}{(LcT)^{1/2}}$&
$\frac{l_{\rm Pl}}{LF^{1/2}}$&
$\left(\frac{l_{\rm Pl}^2\lambda}{L(cT)^2}\right)^{\frac{1}{3}}$&
$\left(\frac{l_{\rm Pl}^2\lambda}{L^3 F^2}\right)^{\frac{1}{3}}$&
$\left(\frac{l_{\rm Pl}}{L}\right)^{2/3}$&
$\frac{(l_{\rm QG}cT)^{1/2}}{L}$&
$\frac{(l_{\rm QG}^2cT)^{1/3}}{L}$\\
\hline
estimate&
$10^{-38}$&
$10^{-40}$&
$10^{-29}$& 
$10^{-31}$&
$10^{-26}$&
$10^{-18}$& 
$10^{-25}$\\\hline
$n_{\rm opt}$& 
$\frac{\lambda}{l_{\rm Pl}}\sqrt{\frac{L}{c T}}$&
$\frac{\lambda}{l_{\rm Pl}\sqrt{F}}$&
$\left(\frac{L \lambda^2}{c T l_{\rm Pl}^2}\right)^{2/3}$&
$\left(\frac{\lambda^2}{l_{\rm Pl}^2F}\right)^{2/3}$&&&\\\hline
%\pbox{3cm}{Numerical\\estimate}&
estimate &
$ 10^{28}$&
% $10^{0}$ for $ \lambda=l_{\rm Pl}$}&
$10^{26}$&
$10^{37}$&$10^{35}$&
&&\\\hline  
  \end{tabular}
  \caption{Scaling of the minimal uncertainty $\delta c/c$ for
    different quantum states and 
    cavity combinations, compared to $\delta L/L$ of three different
    quantum gravity 
    predictions (last three columns, where $l_{\rm QG}$ is expected 
    to be of the order of the Planck length).
Numerical estimates (second line for $\delta c/c$ and last line for
$n_{\rm opt}$) are for $\lambda=500$\,nm and $L=1000$\,m, and
$F=10000$ (lossy cavity, with $T=LF/c$); $T=L/c$ for the
lossless case for comparison with $\delta L/L$ bounds.
  \vspace*{-1em} }
  \label{tab:scal}
  \end{table*} 
  %\end{tabular}
%\end{widetext}

Since q-pet was based on the uncertainties 
$\delta\omega$, we also
compare q-pet and GR based on the GR shift of the
cavity resonance frequencies
by solving the e.m.~wave
equation in the entire cavity with mirrors at 
$0,x_L$ and symmetric boundary conditions (SBC),  
$A^\mu(0,y,z)=A^\mu(x_L,y,z)$ (and correspondingly for the other
directions).  The unperturbed single modes are plane waves  
%\begin{equation}  \label{eq:Apbc}
 $A^3(t,x,y,z)=\left({\hbar\omega}/{(2\epsilon_0
     V)}\right)^{1/2}
     (e^{ik(x-ct)}a+{\rm h.c.})$\,,
%\end{equation}
$A^\mu=0$ for $\mu\in\{0,1,2\}$, and $k:=k_0= k_1>0$. This leads to
$T^{\mu\nu}=-\hbar\omega/(2\epsilon_0V)\langle
(a\,e^{ik(x-ct)}+{\rm h.c.})^2\rangle$ for
$(\mu,\nu)\in\{00,01,10,11\}$ inside the cavity, and $T^{\mu\nu}=0$ else or
outside. For $\ket{\psi_{\rm opt}}$,
$T^{\mu\nu}$ is time-independent, and for $\ket{\psi_{\rm coh}}$ we once
more consider only the time-independent part. Then, 
%corrections %to the metric tensor are then are 
$h^{\mu\nu}(\bxi)=\epsilon(\bxi)$ for 
$(\mu,\nu)\in\{00,01,10,11\}$ and $h^{\mu\nu}=0$ else, where
\begin{equation}
  \label{eq:fx}
  \epsilon(\bxi) :=2 {\mathcal
  P}M\int_0^\pi\int_0^\pi d\eta'd\zeta'I(\xi,\eta-\eta',\zeta-\zeta')\,.
\end{equation}
The wave equation describing the propagation of light in curved
space-time reads
$
  \nabla_\beta F^{\alpha\beta}=0
$
(see 22.17a in \cite{MTW}),  % Different explicit forms of this
% system of differential equations result by considering the covariant
% or contravariant form, and making various other choices
% such as gauge conditions, and expansion of $g$ in its covariant or
% contravariant form.  
with $F^{\alpha\beta}=g^{\alpha\mu}g^{\beta\nu}(A_{\nu,\mu}-A_{\mu,\nu})$.
Using FLG for $A$ and $h$, and $h\indices{^\nu_\nu}=0$, we obtain to first order 
in  $\epsilon$ 
 \begin{eqnarray}
   \label{eq:Awavefin}
  0=-A\indices{^{\alpha,\nu}_\nu}+(h\indices{^{\alpha}_{\mu,\nu}}-h\indices{^\alpha_{\nu,\mu}})A^{\mu,\nu} \,.
 \end{eqnarray}
Eq.\,\eqref{eq:Awavefin} is solved exactly by the original plane wave
%\eqref{eq:Apbc}, 
despite the changed metric, as the $\epsilon$-correction in $A^3$ is $\propto
(\partial_x+\partial_{ct})^2$.  
However, for a rigid
cavity the quantization of the $k$ vector and hence the frequency of the plane wave 
can be modified, depending on how ``length of cavity''
is defined in curved space-time. If $L$ is measured via a 
light-signal, $L$ is given by $L=\int_0^{x_L}
dl=\int_0^{x_L}\sqrt{dx^a dx^b \gamma_{ab}}\simeq
x_L+\delta(x_L)/2$, where $\delta(x_L):=
\int_0^{x_L}\epsilon(\bx)dx$ and the time-independent spatial metric
$\gamma_{ab}=g_{ab}-g_{0a}g_{0b}/g_{00}\simeq
g_{ab}=\eta_{ab}+h_{ab}+{\cal O}(\epsilon^2)$, $a,b\in
\{1,2,3\}$ (eq.(84.7) in 
\cite{LandauLifschitz87}).  
 Hence, $x_L\simeq L-\delta(x_L)/2\simeq
L-\delta(L)/2$. SBC % at a given time $t$ 
thus imply
$e^{ik(L-\delta(L)/2)}=1$, and lead to a shift of the cavity resonance frequency
of the original plane wave mode, $\delta \omega/\omega=\delta
k/k=\delta(L)/(2L)=\overline{\epsilon(\bx)}/2:=
\frac{1}{2L}\int_0^L\epsilon(\bx)dx$. This is a global property
that cannot be 
transformed away by local %coordinate 
transformations. However, if $L$
is defined as measured through rigid measurement rods, $x_L=L$ remains
unchanged, and hence also the cavity frequency. 

The existence of a mode with unchanged dispersion relation suggests
that judging whether the vacuum may still be 
considered as such based on the change of a single mode frequency can
be insufficient. In such a case, the change of the metric can normally
still be probed using other modes. In the example 
above the frequencies of modes propagating in different directions,
e.g.~$A^3\propto \exp(i k(x+ct))$, are modified locally  by a 
relative amount of order $\epsilon(\bx)$, as can be shown by solving
\eqref{eq:Awavefin} in eikonal approximation. 

Hence, up to
numerical prefactors of order 1, both the systematic error $\delta
c_{\rm exp}$ in the local measurement of $c$
and the global modification as judged by the shift of a cavity
resonance, possibly in another 
                         mode, scale as  
\begin{equation}
  \label{eq:dcGR}
  \frac{\delta c}{c}\sim 
  -\kappa\,n\,M\,
\end{equation}
for $n,M\gg 1$. With this, we can now obtain the
smallest possible uncertainty with which $c$ can be determined.

{\em Minimal uncertainty.}
For $\ket{\psi_{\rm opt}}$, equating (\ref{eq:opt}) and the absolute value of (\ref{eq:dcGR}) 
leads with $M\sim L/\lambda$ 
to an optimal photon number $n_{\rm opt}\sim (\lambda/l_{\rm
  Pl}) \sqrt{{L}/{(c T)}}$, and a minimal %uncertainty
\begin{equation}
  \label{eq:dcopt}
  \frac{\delta c}{c}\sim\frac{l_{\rm
      Pl}}{(c\,T\,L)^{1/2}}\,, 
\end{equation}
{\em independent of frequency}: the gain in quantum
mechanical sensitivity due to longer dimensionless evolution time for
more energetic photons is exactly cancelled by the increased
perturbation of the metric. % A truly fundamental lower
% bound on this uncertainty is obtained by assuming for $L$ the size of
% the visible Universe today all the way to the surface of last
% scattering and 
% for $T$ the lifetime of the Universe since the recombination epoch,
% about 300,000 years after the Big Bang. With $T\simeq 13.8\times 10^9$
% years and $L\simeq 93\times 10^9$ light years, we find
% $\delta c/c\sim 10^{-62}$. 
% % This may be considered to be the ultimate minimal
% % uncertainty of the speed of light. % in our visible part of the
% % Universe. % (see {\em Discussion}).   

In an experiment, the measurement time is
bound by the finite photon-storage time of the photons in the
cavity. 
 While obtaining optimal bounds including photon loss requires mixed
 state q-pet \cite{escher_general_2011,Kolodynski10}, 
 %it is known that 
 the sensitivity cannot be better than that obtained from the pure
 states from which the state is mixed \cite{Braun10}.
For a cavity of length $L$ and finesse $F$, 
$T=L
F/(\pi c)$. 
This leads to an optimal number
of photons {\em independent of the length of the cavity}, $n\sim
\lambda/(l_{\rm Pl} F^{1/2})$. For numerical estimates we use in the
following a standard situation: visible light with $\lambda=500$\,nm, a
finesse $F=10000$, and $L=1000$\,m.  The optimal $n$ for the optimal 
state is then 
$n\sim 10^{26}$, and the minimal uncertainty 
$\delta c/c 
\sim l_{\rm Pl}/(L F^{1/2}) \sim 10^{-40}$. 

For $\ket{\psi_{\rm coh}}$, equating 
(\ref{eq:dccoh}) and (\ref{eq:dcGR}) 
leads to % an optimal photon number 
$n_{\rm opt}
\sim(L\lambda^2/(l_{\rm Pl}^2 c T))^{2/3}$, and a minimal uncertainty 
\begin{equation}
  \label{eq:dcoptF1}
  \frac{\delta c}{c}\sim
  \left(\frac{l_{\rm Pl}^2\lambda }{L(c\,T)^2}\right)^{1/3}\,. 
\end{equation}
For a cavity with finesse $F$, the length of the cavity is again 
%becomes
irrelevant for the optimal photon number, $n_{\rm opt}\sim
\left({\lambda}/{l_{\rm Pl}}\right)^{4/3}/F^{2/3}$, and
%The minimal uncertainty of $c$ reads 
%\begin{equation}\label{eq:dcoptF}
$  {\delta c}/{c}\sim {l_{\rm Pl}^{2/3}\lambda^{1/3}}/({LF^{2/3}})\,. 
      $
%\end{equation}
Contrary to $\ket{\psi_{\rm opt}}$, the minimal uncertainty
depends here on the wavelength.  In principle, $\delta c/c$ could
therefore be
smaller for $\ket{\psi_{\rm coh}}$ than for
$\ket{\psi_{\rm opt}}$, but only for wavelengths
$\lambda<l_{\rm Pl}\sqrt{c T/L}$ in lossless cavities, and for  
$\lambda< l_{\rm Pl} \sqrt{F}$ in cavities with finesse $F$, 
which are outside the validity of the theory.
For the lossy cavity considered, the optimal coherent state photon
number %for $\ket{\psi_{\rm coh}}$  
is $n\sim 10^{35}$ and $\delta c/c\gtrsim
10^{-31}$, demonstrating the superiority of $\ket{\psi_{\rm opt}}$.  


{\em Comparison with quantum gravity predictions.}
The minimal uncertainties of $c$ and hence the metric of flat
space-time that we have derived   
are reminiscent
of ideas about the fuzziness of space-time on the
Planck scale, their different physical meaning not withstanding.  
In particular, the finite value of $\delta c/c$ translates, in
experiments where a length $L$ is measured through $L=cT$, to
fluctuations of $\delta L/L$ and can be compared to the corresponding
fluctuations predicted 
by different quantum gravity candidates. 
For a coherent state in a lossless cavity and fixed
measurement time $T$, our lower bound of $\delta c/c$ scales as
$1/L^{1/3}$, and as $1/L$  if 
$T$ is given by $T\sim L\,F/c$ for a  
cavity of finite finesse $F$.  For the optimal
state the scaling is as $1/L^{1/2}$ for fixed $T$, and as $1/L$ for
$T\sim L\,F/c$.   
Remarkably, the bound 
  (\ref{eq:dcoptF1}) is, for $L=cT$, just a factor $(\lambda/L)^{1/3}$
  smaller than the relative fluctuations $\delta L/L\sim (l_{\rm
    Pl}/L)^{2/3}$ of a distance $L$ predicted in \cite{Ng00}. %  due to
  % the   quantum foaminess of space-time.
  In particular, it scales only
  as $l_{\rm Pl}^{2/3}$, {\em not} as $l_{\rm Pl}$. The same scaling
  is given in eq.(5) of 
  \cite{amelino-camelia_gravity-wave_1999}, assuming the ``quantum
  gravity'' length scale $l_{\rm QG}\sim l_{\rm Pl}$. 

Our results % These
  % identical scalings suggest 
imply  that {\em one should not think of these quantum 
  fluctuations of space-time as
  existing independently of the measurement devices that probe them},  
 % (and in particular with a fixed length-scale), 
  but rather as something that can only be defined in conjunction with
  them.  This is in 
  line with the modern theory of quantum measurement, where the
  possible measurement results do not only depend on the quantum
  system, 
but
  also on the   quantum probe and its initial quantum state.
 % (which is the origin of the more general POVM measurements).  
  Interestingly, the scaling $\delta c/c\propto l_{\rm Pl}^{2/3}$ corresponds to 
  the use of $\ket{\psi_{\rm coh}}$ 
  i.e.~a classical light beam, whereas the
  optimal quantum state leads to a substantially reduced uncertainty,
  $\delta c/c\propto l_{\rm Pl}$.  In
  \cite{amelino-camelia_gravity-wave_1999},  
 other estimates of
  length fluctuations were discussed, one of them scaling as $\delta L/L
  \sim (l_{\rm QG}\,c\,T)^{1/2}/L$, based on the requirement that a clock
  should not turn into a black hole. 
Bounds on $l_{\rm QG}$ were obtained from experimental
  data from Caltech's 40\,m interferometer 
  \cite{abramovici_improved_1996}. Our bounds can serve 
  for judging the falsifiability of different quantum gravity
  theories: Predictions of fluctuations smaller than those given by
  our bounds can never be falsified
  through direct measurement {\em as a matter of principle}.  
Fluctuations of $\delta L /L$ of the order $l_{\rm Pl}/L$ cannot be verified by
measurements with light in a 
coherent state, but would be accessible to ``quantum enhanced
measurements'' using the optimal quantum state of light.  The
fluctuations predicted in 
\cite{Urban13} are well above our bounds for any cavity of realistic
size. Table \ref{tab:scal} 
  summarizes our predictions and compares them with %those in
  Refs.\,\cite{Ng00,amelino-camelia_gravity-wave_1999}. 
  


% The photon numbers required, %to see the effect 
% using $\ket{\psi_{\rm opt}}$, are about six orders of magnitude away from the optimal photon number
% in LIGO  of order $10^{20}$ (assuming a finesse $F$ of order 10000),
% where, however, a coherent state is used \cite{Brif99}. 
% For the latter of the order of $10^{35}$ photons are needed before one
% hits the corresponding fundamental bound. This adds weight to the
% proposition that gravitational wave observatories may become
% useful in a not so far future for testing quantum gravity effects
% \cite{amelino-camelia_gravity-wave_1999,Ng00,amelino-camelia_gravity-wave_2000}.
% All optimal photon 
% numbers scale with a positive power of $\lambda$.  
% For both optimal
% and coherent state and $L\sim c T$, the optimal photon number is of
% order one for $\lambda\sim l_{\rm Pl}$ without photon loss
% (or $\lambda\sim l_{\rm Pl}\sqrt{F}$ in a cavity of finite finesse $F$), 
% which thus sets the minimal 
% wavelength for an optimal measurement. 
% Optimal and coherent state lead in that case to the
% same uncertainty $\delta c/c$. Note, however, that for such short
% wavelengths our theory might break down, as there one {\em does} enter
% in the realm of 
% quantum gravity.  There is no need for  such short
% wavelengths, however: The minimal uncertainty of $c$ is
% independent of $\lambda$ for the optimal 
% state, and one can stay with macrosopic wavelengths.
% where both quantum optics and general relativity work reliably. 

We neglected  uncertainties in $c$ due to the expansion of the Universe
\cite{kopeikin_optical_2015}, non--inertial observers, local
gravitation potentials e.g.~from Earth or a (stochastic) gravitational-wave background  
\cite{Sorge,sabin_phonon_2014,meacher_mock_2015}, 
and quantum fluctuations of the mirror positions.  
In the quantum foam picture, the latter are
of $\ord (l_{\rm Pl})$ %of the order of the Planck length 
and should give an uncertainty $\delta c/c\sim l_{\rm Pl}/L$, comparable to our 
smallest $\delta c/c$ %of our uncertainties 
obtained with $\ket{\psi_{\rm opt}}$. 
%More generally, s
Such additional sources of the modification of the speed of
light will only increase the uncertainty of $\delta c/c$, and therefore not
invalidate our lower bounds.  
 
% {\em Summary.} We have established lower bounds to the precision
% for $c$ measurements that shed a new light on the presumed quantum
% foaminess of space-time: Fluctuations of space time have only physical meaning
% in conjunction with the way they are probed and the quantum
% properties of the probe determine the scaling of the smallest
% measurable fluctuations with
% the Planck length. Remarkably, this measurement-based approach
% predicts uncertainties of space-time metric comparable to those of
% several quantum-gravity candidates, while only relying on established
% quantum optics and relativity in their domain of validity.



{\em Acknowledgments.} We thank Kostas Kokkotas, Nils Schopohl, and Claus
Zimmermann  for discussions and a critical reading of the manuscript.
The research of URF was supported by the NRF of 
Korea, grant No.~2014R1A2A2A01006535. 


\bibliography{ReSubQCR64PRL}

\vspace*{20em}

\pagebreak

\widetext
\begin{center}
\textbf{\large Supplemental Material}
\end{center}
%%%%%%%%%% Merge with supplemental materials %%%%%%%%%%
%%%%%%%%%% Prefix a "S" to all equations, figures, tables and reset the counter %%%%%%%%%%
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{page}{1}
\renewcommand{\theequation}{S\arabic{equation}}
\renewcommand{\thefigure}{S\arabic{figure}}

%\begin{widetext} 

The vector  potential of the e.m.~field in the cavity in Coulomb gauge 
$\bA(\br,t)=\Upsilon q(t) 
\bv(\br)$, where $\Upsilon$ is a constant, $q(t)$ the time dependent
amplitude, and $\bv(\br)$ the mode function,
with components
  \begin{eqnarray}
    \label{eq:modes}
    v_x&=&{\cal N}e_x\cos k_xx\sin k_y y\sin k_z z\nn
    v_y&=&{\cal N}e_y\sin k_xx\cos k_y y\sin k_z z\nn
    v_z&=&{\cal N}e_z\sin k_xx\sin k_y y\cos k_z z\,.
  \end{eqnarray}
The polarization vector $\vec{e}=(e_x,e_y,e_z)$ is normalized to
length one, and has to be orthogonal to the $\bk$-vector
$\bk=(k_x,k_y,k_z)$, where $k_i=l_i\pi/L$, and
$l_i\in\mathbb{N}_0$, and at most one of three given $l_i$ can be zero. 
Therefore, there are two polarization directions
(transverse modes) for each $\bk$ vector, with the exception of cases
where one of the $l_i=0$, where only one polarization is possible. 
The request that the modes be orthonormal,
\begin{equation}
  \label{eq:norm}
  \int d^3r \bv_l(\br)\cdot\bv_{l'}(\br)=\delta_{l,l'}
\end{equation}
leads to ${\cal N}=\sqrt{8/V}$, and we can define the mode-volume
$V_l=V/8$.  Note that the index $l$ stands here for both the discrete
$\bk$ vector and the polarization direction (1,2). Finally, we choose
$\Upsilon=1/\sqrt{\epsilon_0}$, such that
\begin{eqnarray}
  \label{eq:AEH}
  \bA(\br,t)&=&\sum_l\frac{1}{\sqrt{\epsilon_0}}q_l(t)v_l(\br)\nn
  \bE(\br,t)&=-&\sum_l\frac{1}{\sqrt{\epsilon_0}}\dot{q}_l(t)v_l(\br)\nn
  \bH(\br,t)&=&\sum_l\frac{1}{\mu_0\sqrt{\epsilon_0}}q_l(t)\nabla\times
  v_l(\br)\,.
\end{eqnarray}
After quantization, the amplitudes $q_l$ become the quadrature operators
of a harmonic oscillator,
$q_l=\sqrt{\frac{\hbar}{2\Omega_l}}(a_l+a_l^\dagger)$,
$\dot{q}_l=p_l=\frac{1}{i}\sqrt{\frac{\hbar
    \Omega_l}{2}}(a_l-a_l^\dagger)$, where $\Omega_l=|\bk_l|c$. 
In the semiclassical approach the 
energy-momentum tensor for a single mode with mode function $\bv$ is given by
the quantum mechanical expectation value \cite{Deser57,MTW},
\begin{eqnarray}
  \label{eq:tmn2}
  T^{00}&=&\frac{\hbar \Omega}{4}(-\langle
  (a-a^\dagger)^2\rangle
  \bv^2+\frac{1}{k^2}\langle(a+a^\dagger)^2\rangle(\nabla\times\bv)^2)\nn
T^{0i}&=&\frac{i\hbar\Omega}{2k}(\langle a^2\rangle-\langle a^{\dagger
  2}\rangle)(\bv\times(\nabla\times\bv))_i\nn 
T^{ij}&=&\frac{\hbar\Omega}{2}\left(\langle(a-a^\dagger
  )^2\rangle
  v_iv_j-\langle(a+a^\dagger)^2\rangle\frac{1}{k^2}(\nabla\times \bv)_i(\nabla\times \bv)_j\right) +T^{00}\delta_{ij}\,,
\end{eqnarray}
where $k^2=\bk^2$, and we have used the symmetrized form
$(q\dot{q}+\dot{q}q)/2$ of the quantum mechanical operators for the
$T^{0i}$ components. 


For a $(01M)$ 
mode, $l_x=0,l_y=1,l_z=M$ dictates $\vec{e}=(1,0,0)$ as unique possible polarization.  
For $M=1$, the frequency $\Omega_l=\sqrt{2}\pi c/L$, and
%\begin{multline}
\bea
{\bm v} &=& \sqrt{\frac8V} \sin(\pi y/L ) \sin (\pi z /L) {\bm e}_x ,\nn % \quad 
\nabla\times{\bm v} &=& \sqrt{\frac8V}\frac\pi L \sin(\pi y/L ) \cos
(\pi z /L) {\bm e}_y - \sqrt{\frac8V}\frac\pi L \cos(\pi y/L ) \sin
(\pi z /L) {\bm e}_z \,.
\ea
%\end{multline}
For $\ket{\psi_{\rm opt}}$
with $n\gg 1$, and neglecting terms of order ${\cal O}(n^0)$ (all
other terms are of order $n$), we find that for the fundamental $(011)$ mode the 
only 
non-vanishing components of $T^{\mu\nu}$ can
be expressed in terms of four functions,
\begin{equation}  \label{eq:tmn3}
 T^{\mu\nu}=n\frac{\hbar\Omega_l}{V}t^{\mu\nu}
\end{equation}
with the dimensionless tensor components 
$t^{00}(\eta,\zeta)=f_1(\eta,\zeta)$,
$t^{11}(\eta,\zeta)=f_2(\eta,\zeta)$,
$t^{22}(\eta,\zeta)=f_3(\eta,\zeta)$,
$t^{33}(\eta,\zeta)=\tilde f_3(\eta,\zeta) = f_3(\zeta,\eta)$,
$t^{23}(\eta,\zeta)=t^{32}(\eta,\zeta)=f_4(\eta,\zeta)$, and
\begin{eqnarray}
  \label{eq:3fs}
f_1(\eta,\zeta)&=&2-\cos(2\eta)-\cos(2\zeta)\nn 
f_2(\eta,\zeta)&=&\cos(2\eta)+\cos(2\zeta)-2\cos(2\eta)\cos(2\zeta)\nn 
f_3(\eta,\zeta)&=&\frac{1}{2}(2-4\cos(2\zeta)+2\cos(2\zeta)\cos(2\eta))\nn
f_4(\eta,\zeta)&=&\sin(2\eta)\,\sin(2\zeta)\,,
\end{eqnarray}
where we write $x,y,z$ in units of $L/\pi$, $\xi=x\pi/L$,
$\eta=y\pi/L$, $\zeta=z\pi/L$, and thus $\xi,\eta,\zeta\in 
[0,\pi]$. Outside the cavity $T^{\mu\nu}$ vanishes. 
For this state the field equations are solved with a time-independent
metric. The wave 
equation reduces to the Poisson equation, 
\bea
\Delta \bar h^{\mu\nu} = -16\pi \frac{G}{c^4} T^{\mu\nu}\,. 
\ea
The solution is obtained by integrating the
inhomogeneity $T^{\mu\nu}$ over with the Green's function of the
Poisson equation, i.e. 
\begin{eqnarray}
  \bar h^{\mu\nu}&=&\frac{4G}{c^4}\int\frac{T^{\mu\nu}(\bx')}{|\bx-\bx'|}d^3x'\nn
&=& {\mathcal P} \int_0^\pi\int_0^\pi
d\eta'd\zeta'I(\xi,\eta-\eta',\zeta-\zeta')t^{\mu\nu}(\eta',\zeta')\,,\label{eq:hmnfin}  
\end{eqnarray}
where the parameter $\mathcal P$ is given by  
\begin{equation}
  \label{eq:Asuppl}
  {\mathcal P}=4\frac{n\hbar G}{\pi c^3L^2}=\frac{4n}{\pi}\kappa\mbox{,\quad\quad
  }\kappa=\left(\frac{l_{\rm 
        Pl}}{L}\right)^2 \,.
\end{equation}
The integral kernel reads
\begin{equation}
  \label{eq:I}
  I(\xi,\eta,\zeta)=\ln\left(\frac{\xi+\sqrt{\xi^2+\eta^2+\zeta^2} 
    }{\xi-\pi+\sqrt{(\xi-\pi)^2+\eta^2+\zeta^2
    }}\right)\,.
\end{equation}
Numerical evaluation of the two remaining integrals in
Eq.\,(\ref{eq:hmnfin}) shows that they are of order one inside the
cavity, and decay rapidly outside, as is required by the boundary
conditions of a 
flat metric far from the cavity. 

For $\ket{\psi_{\rm coh}}$, we have to consider the full retarded solution
of the wave equation 
according to 
\begin{eqnarray}
  \bar
  h^{\mu\nu}&=&\frac{4G}{c^4}\int\frac{T^{\mu\nu}(t-|\bx-\bx'|/c,\bx')}{|\bx-\bx'|}
  d^3x'.  
\end{eqnarray}
For example, the $yz$ component reads
\bea
\bar h^{yz}&=& 
\bar h^{yz}_{\rm opt}+
\frac{4n\hbar G \Omega}{c^4} \int d\xi' d\eta' d\zeta' 
\frac{\sin[2\omega(t-|\bx-\bx'|/c)]  \sin (2\eta') \sin (2\zeta')}{ |\bx-\bx'| }.
\ea 
This metric element is thus the solution of  $\ket{\psi_{\rm opt}}$ \eqref{eq:hmnfin}
plus some retarded oscillation on top of it, which is of the same
order.  In the following we will therefore restrict our analysis to the
time-independent part given by  $\ket{\psi_{\rm opt}}$.

For the $(01M)$ 
mode, with $M>1$, $l_x=0,l_y=1,l_z=M$, the
general expressions for $T^{\mu\nu}$ are more complicated, but for
 $\ket{\psi_{\rm opt}}$ with $n\gg 2$, and in
the limit of $M\gg 1$, we have
$T^{00}=T^{33}=4n(\hbar\Omega/V)\sin^2\eta$,
$T^{11}=-T^{22}=4n(\hbar\Omega/V)\sin^2\eta\cos(2M\zeta)$.
Corrections are of
order $1/M$.  All other tensor elements of $T$ vanish to order
$M^0$.  The rapidly oscillating term $\cos(2M\zeta)$ in $T^{11}$,
$T^{22}$ leads to a rapid decay of
$\bar{h}^{11}$ and $\bar{h}^{22}$ as function of $M$. Numerics indicates that
the decay is roughly as $1/M$ for 
fixed $(\xi,\eta,\zeta$), including the factor $M$ that is gained due
to the prefactor $\Omega\propto M$ for large $M$.   This means that
for large $n$ and $M$, only 
$T^{00}=T^{33}$ are non-negligible, with
\begin{eqnarray}
  \label{eq:h00Suppl}
 \bar{h}^{00}&=&\bar{h}^{33}\simeq {\mathcal P} M
  \tilde{h}(\xi,\eta,\zeta)\nn
\tilde{h}(\xi,\eta,\zeta)&:=& 4\int_0^\pi\int_0^\pi
d\eta'd\zeta'I(\xi,\eta-\eta',\zeta-\zeta')\sin^2\eta'\,,
\end{eqnarray}
where the dimensionless function $\tilde h(\xi,\eta,\zeta)$ is once more of
order 1 inside the cavity and falls off rapidly outside. So using a
higher mode has the effect of reducing the perturbation of the metric
essentially to two diagonal elements of the metric tensors, but
increases the perturbation by a factor equal to the mode-index
$M$. 

In all cases, the amplitude of the space-time perturbation
due to the e.m.~field in the cavity scales as
\begin{equation}
  \label{eq:gres}
  \overline{h}^{\mu\nu}\sim \left(\frac{l_{\rm Pl}}L\right)^2 \,n\,M\,,
\end{equation}
proportional to the number of photons $n$ in the cavity, the mode
index $M$, 
and the squared ratio $l_{\rm Pl}/L$ of Planck length $l_{\rm
  Pl}\simeq 1.62 \times 10^{-35}\,$m and size $L$ of the cavity. The
expression remains 
valid for the fundamental mode with $M=1$. 

We note that throughout our analysis we tacitly assume that the photon densities in the cavity are
small enough and the cavity sufficiently large, such that we stay well below the critical
(electric) field strength $E_c =
m_e^2c^3/(e\hbar)=1.3\times10^{18}$\,V/m,  where $m_e$ is the mass of
the electron, beyond which nonlinear corrections to Maxwellian
electrodynamics  
 due to polarization of the quantum vacuum become important
\cite{EulerHeisenberg}. 
This condition may be translated into a minimal
cavity size $L$ using an energy density $\ord (\hbar c n M/L^4)$ and a
critical energy density $\ord(\epsilon_0 E_c^2)$. We obtain that  
$L\gg (\hbar^{3/4} e^{1/2} \epsilon_0^{-1/4}m_e^{-1}c^{-5/4}) (n M)^{1/4}=
(2.1\times 10^{-13}\,$m)$(n M)^{1/4}$ for linear
electrodynamics in the cavity to hold. For the two types of cavities
considered 
and all combinations of $n_{\rm opt}$ %optimal photon numbers and mode index 
and $M$, the lower bound on $L$ is satisfied by the
cavity sizes considered.  



From $h_{\mu\nu}$ we now
calculate a local measure of the modification of the speed of light
defined through the geodesics of the modified metric.
%, and a global one, expressed as shift of the cavity resonance frequency.


%\subsection{Local change of speed of light}
A finite $h_{\mu\nu}$ leads to a new line element 
\begin{equation}
  \label{eq:ds2n}  
  ds^2= - (1- h_{00}) c^2 dt^2 + (1+h_{ii}) (dx^i)^2 
  +2h_{23}dy dz 
\end{equation}
where the metric elements are, for the (011) mode,
\bea
h_{00} &=& \frac12 {\mathcal P}(g_1+ g_2+g_3+\tilde g_3), \quad h_{11}=\frac12
{\mathcal P}(g_1+g_2-g_3-\tilde g_3),\nn
\quad h_{22}&=&\frac12{\mathcal P}(g_1-g_2 +g_3-\tilde g_3), 
\quad h_{33}=\frac12{\mathcal P}(g_1-g_2 +\tilde g_3-g_3),
\quad h_{23}={\mathcal P}g_4 \label{eq:hij}
\ea
with the definitions, cf. \eqref{eq:hmnfin},
\bea\label{defgi}
g_i &=& \int_0^\pi\int_0^\pi
d\eta'd\zeta'I(\xi,\eta-\eta',\zeta-\zeta')f_i (\eta',\zeta') .\nn 
\tilde g_i &=& \int_0^\pi\int_0^\pi
d\eta'd\zeta'I(\xi,\eta-\eta',\zeta-\zeta') \tilde f_i (\eta',\zeta') 
\ea
For the $(01M)$ mode we have $h_{00} =h_{33}$ with $h_{00}$ given by
(\ref{eq:hij})
% $ =2 AM(g_1+g_2+g_3+\tilde g_3)$ 
 whereas $h_{\mu\nu}$ vanishes for all other
values of $\mu$,$\nu$. 
The light ray trajectories are determined through the geodesic
condition $ds^2=0$. The 
speed of light in $x^1$-direction (meaning all other $dx^j=0,\, j \neq
1$, i.e.~locally straight paths along $x^1=x$) is then $c(x)=c
\sqrt{(1-h_{00})/(1+h_{11})}$, and correspondingly for the other directions.
The relative change of the coordinate speed of light in $x^i$-direction
then reads, for the (011) mode with $n\gg 1$,
\begin{eqnarray}
  \label{eq:dc2suppl}
  \delta c(x)/c = -\frac12 (h_{00} + h_{11}) =
  -\frac12{\mathcal P}(g_1+g_2) %:= -{\mathcal P} g_5    
\nn
  \delta c (y)/c = - \frac12 (h_{00} + h_{22}) =-\frac12 {\mathcal P} (g_1+g_3)
% := -{\mathcal P} g_6  
\nn   
  \delta c (z)/c =- \frac12 (h_{00} + h_{33}) = -\frac12 {\mathcal P} (g_1+
  \tilde g_3)\,,
%:= -{\mathcal P} \tilde g_6 
%  \frac{\delta c}{c}\simeq 2\phi-\frac{h_+}{2}
\end{eqnarray}
For the $(01M)$ mode with $n,M\gg 1$, 
\bea
\delta c(x)/c &=&\delta c(y)/c 
=-\frac12 h_{00} = -{\mathcal P}M(g_1+g_2+g_3+\tilde{g}_3)/4,\quad\delta c(z)/c=2
\delta c(x)/c\,,
\eea 
where the equalities in terms of the $g_i,\tilde g_i$ are for 
$\ket{\psi_{\rm opt}}$.  

\begin{figure}[t]
\centering
%\includegraphics[width=0.5\columnwidth]{fig1.png}
\includegraphics[width=0.5\columnwidth]{fig1_Supplement.pdf}
\caption{Relative change of the local coordinate speed of light  in
  x-direction as function of dimensionless coordinates 
  $\eta,\zeta$ at $\xi=1.5$ in units of ${\mathcal P}=(4n/\pi)\kappa$ with
  $\kappa=(l_{\rm Pl}/L)^2$ (see Eq.\,(\ref{eq:Asuppl})) for the (011) mode. The
  cavity extends from 0 to $\pi$ in these units.
}
\label{fig.dc}
\end{figure}


In Fig.\,\ref{fig.dc},  we plot the relative coordinate change of the speed of
light in $x-$direction for the (011) mode. We see that up to position
dependent functions of order 1 the 
relative change of speed of light is given by Eq.\,(5) in the main text. %(\ref{eq:dcGR}). 
Very
similar plots are obtained for other directions.






\end{document}
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  {pages} {3439--3443} (\bibinfo {year} {1994})}\BibitemShut {NoStop}%
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  {Braunstein}, \citenamefont {Caves},\ and\ \citenamefont
  {Milburn}}]{braunstein_generalized_1996}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Samuel~L.}\
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  {135--173} (\bibinfo {year} {1996})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Wiseman}\ and\ \citenamefont
  {Milburn}(2009)}]{wiseman_quantum_2009}%
  \BibitemOpen
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  {2009})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Giovannetti}\ \emph {et~al.}(2004)\citenamefont
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  \BibitemOpen
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\bibitem [{\citenamefont {Giovannetti}\ \emph {et~al.}(2006)\citenamefont
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  \BibitemOpen
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {M.~G.~A.}\
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  \BibitemOpen
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  {NoStop}%
\bibitem [{\citenamefont {Ahmadi}\ \emph
  {et~al.}(2014{\natexlab{a}})\citenamefont {Ahmadi}, \citenamefont {Bruschi},
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  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Mehdi}\ \bibnamefont
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  {Fuentes}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Relativistic
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  technologies},}\ }\href {\doibase 10.1038/srep04996} {\bibfield  {journal}
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\bibitem [{\citenamefont {Ahmadi}\ \emph
  {et~al.}(2014{\natexlab{b}})\citenamefont {Ahmadi}, \citenamefont {Bruschi},\
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  \BibitemOpen
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\bibitem [{\citenamefont {Aspachs}\ \emph {et~al.}(2010)\citenamefont
  {Aspachs}, \citenamefont {Adesso},\ and\ \citenamefont
  {Fuentes}}]{Aspachs10}%
  \BibitemOpen
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  {Fuentes}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Optimal
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\bibitem [{\citenamefont {Downes}\ \emph {et~al.}(2011)\citenamefont {Downes},
  \citenamefont {Milburn},\ and\ \citenamefont {Caves}}]{Downes11}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {T.~G.}\ \bibnamefont
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  {journal} {\bibinfo  {journal} {{arXiv}:1108.5220}\ } (\bibinfo {year}
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\bibitem [{\citenamefont {Ahluwalia}(1994)}]{Ahluwalia94}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {D.~V.}\ \bibnamefont
  {Ahluwalia}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Quantum
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  10.1016/0370-2693(94)90622-X} {\bibfield  {journal} {\bibinfo  {journal}
  {Physics Letters B}\ }\textbf {\bibinfo {volume} {339}},\ \bibinfo {pages}
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\bibitem [{\citenamefont {Amelino-Camelia}(1994)}]{Amelino-Camelia94}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Giovanni}\
  \bibnamefont {Amelino-Camelia}},\ }\bibfield  {title} {\enquote {\bibinfo
  {title} {Limits on the measurability of space-time distances in the
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  {journal} {\bibinfo  {journal} {Mod. Phys. Lett. A}\ }\textbf {\bibinfo
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\bibitem [{\citenamefont {Ng}\ and\ \citenamefont {Dam}(2000)}]{Ng00}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Y.~Jack}\ \bibnamefont
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  }\bibfield  {title} {\enquote {\bibinfo {title} {Measuring the foaminess of
  space-time with gravity-wave interferometers},}\ }\href {\doibase
  10.1023/A:1003745212871} {\bibfield  {journal} {\bibinfo  {journal}
  {Foundations of Physics}\ }\textbf {\bibinfo {volume} {30}},\ \bibinfo
  {pages} {795--805} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Giovannetti}\ \emph {et~al.}(2001)\citenamefont
  {Giovannetti}, \citenamefont {Lloyd},\ and\ \citenamefont
  {Maccone}}]{giovannetti_quantum-enhanced_2001}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Vittorio}\
  \bibnamefont {Giovannetti}}, \bibinfo {author} {\bibfnamefont {Seth}\
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  \bibnamefont {Maccone}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
  {Quantum-enhanced positioning and clock synchronization},}\ }\href@noop {}
  {\bibfield  {journal} {\bibinfo  {journal} {Nature}\ }\textbf {\bibinfo
  {volume} {412}},\ \bibinfo {pages} {417--419} (\bibinfo {year}
  {2001})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Anisimov}\ \emph {et~al.}(2010)\citenamefont
  {Anisimov}, \citenamefont {Raterman}, \citenamefont {Chiruvelli},
  \citenamefont {Plick}, \citenamefont {Huver}, \citenamefont {Lee},\ and\
  \citenamefont {Dowling}}]{anisimov_quantum_2010}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Petr~M.}\ \bibnamefont
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  {Chiruvelli}}, \bibinfo {author} {\bibfnamefont {William~N.}\ \bibnamefont
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  \bibinfo {author} {\bibfnamefont {Hwang}\ \bibnamefont {Lee}}, \ and\
  \bibinfo {author} {\bibfnamefont {Jonathan~P.}\ \bibnamefont {Dowling}},\
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  {Heisenberg} {Limit}},}\ }\href@noop {} {\bibfield  {journal} {\bibinfo
  {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {104}},\ \bibinfo
  {pages} {103602} (\bibinfo {year} {2010})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Braun}(2011)}]{Braun11.2}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont
  {Braun}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Ultimate quantum
  bounds on mass measurements with a nano-mechanical resonator},}\ }\href@noop
  {} {\bibfield  {journal} {\bibinfo  {journal} {Europhys. Lett.}\ }\textbf
  {\bibinfo {volume} {94}},\ \bibinfo {pages} {68007} (\bibinfo {year}
  {2011})},\ \bibinfo {note} {{Erratum}: {Europhys.~Lett.}~{\bf 99}, 49901
  (2012)}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Scully}\ and\ \citenamefont
  {Zubairy}(1997)}]{Scully97}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {M.~O.}\ \bibnamefont
  {Scully}}\ and\ \bibinfo {author} {\bibfnamefont {M.~S.}\ \bibnamefont
  {Zubairy}},\ }\href@noop {} {\emph {\bibinfo {title} {Quantum Optics}}}\
  (\bibinfo  {publisher} {Cambridge University Press},\ \bibinfo {address}
  {Cambridge, UK},\ \bibinfo {year} {1997})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Caves}(1981)}]{Caves81}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Carlton~M.}\
  \bibnamefont {Caves}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
  {Quantum-mechanical noise in an interferometer},}\ }\href@noop {} {\bibfield
  {journal} {\bibinfo  {journal} {Phys. Rev. D}\ }\textbf {\bibinfo {volume}
  {23}},\ \bibinfo {pages} {1693--1708} (\bibinfo {year} {1981})}\BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {\relax{LIGO
  Collaboration}}(2011)}]{abadie_gravitational_2011}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibnamefont {\relax{LIGO
  Collaboration}}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {A
  gravitational wave observatory operating beyond the quantum shot-noise
  limit},}\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Nature
  Physics}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} {962--965}
  (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Demkowicz-Dobrza\ifmmode~\acute{n}\else
  \'{n}\fi{}ski}\ \emph {et~al.}(2013)\citenamefont
  {Demkowicz-Dobrza\ifmmode~\acute{n}\else \'{n}\fi{}ski}, \citenamefont
  {Banaszek},\ and\ \citenamefont
  {Schnabel}}]{demkowicz-dobrzanski_fundamental_2013}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Rafa\l{}}\
  \bibnamefont {Demkowicz-Dobrza\ifmmode~\acute{n}\else \'{n}\fi{}ski}},
  \bibinfo {author} {\bibfnamefont {Konrad}\ \bibnamefont {Banaszek}}, \ and\
  \bibinfo {author} {\bibfnamefont {Roman}\ \bibnamefont {Schnabel}},\
  }\bibfield  {title} {\enquote {\bibinfo {title} {{Fundamental quantum
  interferometry bound for the squeezed-light-enhanced gravitational wave
  detector GEO 600}},}\ }\href {\doibase 10.1103/PhysRevA.88.041802} {\bibfield
   {journal} {\bibinfo  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
  {88}},\ \bibinfo {pages} {041802} (\bibinfo {year} {2013})}\BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {Ford}(1982)}]{FORD1982238}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {L.~H.}\ \bibnamefont
  {Ford}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Gravitational
  radiation by quantum systems},}\ }\href {\doibase
  http://dx.doi.org/10.1016/0003-4916(82)90115-4} {\bibfield  {journal}
  {\bibinfo  {journal} {Annals of Physics}\ }\textbf {\bibinfo {volume}
  {144}},\ \bibinfo {pages} {238--248} (\bibinfo {year} {1982})}\BibitemShut
  {NoStop}%
\bibitem [{\citenamefont {Misner}\ \emph {et~al.}(1973)\citenamefont {Misner},
  \citenamefont {Thorne},\ and\ \citenamefont {Wheeler}}]{MTW}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Charles~W.}\
  \bibnamefont {Misner}}, \bibinfo {author} {\bibfnamefont {Kip~S.}\
  \bibnamefont {Thorne}}, \ and\ \bibinfo {author} {\bibfnamefont
  {John~Archibald}\ \bibnamefont {Wheeler}},\ }\href@noop {} {\emph {\bibinfo
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  {1973})\BibitemShut {NoStop}%
\bibitem [{sup()}]{suppl}%
  \BibitemOpen
  \href@noop {} {}\bibinfo {note} {See the supplemental material for a detailed
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  change of $c$ due to the geodesic condition $ds^2=0$.}\BibitemShut {Stop}%
\bibitem [{\citenamefont {Landau}\ and\ \citenamefont
  {Lifschitz}(1987)}]{LandauLifschitz87}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
  {Landau}}\ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
  {Lifschitz}},\ }\href@noop {} {\emph {\bibinfo {title} {Klassische
  Feldtheorie}}},\ \bibinfo {series} {Lehrbuch der Theoretischen Physik (German
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  {Akademie-Verlag Berlin},\ \bibinfo {address} {Berlin},\ \bibinfo {year}
  {1987})\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Eddington}(1920)}]{Eddington}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.~S.}\ \bibnamefont
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  Press},\ \bibinfo {year} {1920})\BibitemShut {NoStop}%
\bibitem [{\citenamefont
  {Amelino-Camelia}(1999)}]{amelino-camelia_gravity-wave_1999}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Giovanni}\
  \bibnamefont {Amelino-Camelia}},\ }\bibfield  {title} {\enquote {\bibinfo
  {title} {Gravity-wave interferometers as quantum-gravity detectors},}\ }\href
  {\doibase 10.1038/18377} {\bibfield  {journal} {\bibinfo  {journal} {Nature}\
  }\textbf {\bibinfo {volume} {398}},\ \bibinfo {pages} {216--218} (\bibinfo
  {year} {1999})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Escher}\ \emph {et~al.}(2011)\citenamefont {Escher},
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  {Davidovich}}]{escher_general_2011}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {B.~M.}\ \bibnamefont
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  {pages} {406--411} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ko\l{}ody\ifmmode~\acute{n}\else \'{n}\fi{}ski}\ and\
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  \'{n}\fi{}ski}(2010)}]{Kolodynski10}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Jan}\ \bibnamefont
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  {\bibfnamefont {Rafa\l{}}\ \bibnamefont
  {Demkowicz-Dobrza\ifmmode~\acute{n}\else \'{n}\fi{}ski}},\ }\bibfield
  {title} {\enquote {\bibinfo {title} {Phase estimation without a priori phase
  knowledge in the presence of loss},}\ }\href {\doibase
  10.1103/PhysRevA.82.053804} {\bibfield  {journal} {\bibinfo  {journal} {Phys.
  Rev. A}\ }\textbf {\bibinfo {volume} {82}},\ \bibinfo {pages} {053804}
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\bibitem [{\citenamefont {Braun}(2010)}]{Braun10}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Daniel}\ \bibnamefont
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  {\bibinfo  {journal} {Eur. Phys. J. D}\ }\textbf {\bibinfo {volume} {59}},\
  \bibinfo {pages} {521--523} (\bibinfo {year} {2010})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Abramovici}\ \emph {et~al.}(1996)\citenamefont
  {Abramovici}, \citenamefont {Althouse}, \citenamefont {Camp}, \citenamefont
  {Durance}, \citenamefont {Giaime}, \citenamefont {Gillespie}, \citenamefont
  {Kawamura}, \citenamefont {Kuhnert}, \citenamefont {Lyons}, \citenamefont
  {Raab}, \citenamefont {Savage~Jr.}, \citenamefont {Shoemaker}, \citenamefont
  {Sievers}, \citenamefont {Spero}, \citenamefont {Vogt}, \citenamefont
  {Weiss}, \citenamefont {Whitcomb},\ and\ \citenamefont
  {Zucker}}]{abramovici_improved_1996}%
  \BibitemOpen
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  {\bibfnamefont {S.}~\bibnamefont {Kawamura}}, \bibinfo {author}
  {\bibfnamefont {A.}~\bibnamefont {Kuhnert}}, \bibinfo {author} {\bibfnamefont
  {T.}~\bibnamefont {Lyons}}, \bibinfo {author} {\bibfnamefont {F.~J.}\
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  {\bibfnamefont {R.}~\bibnamefont {Weiss}}, \bibinfo {author} {\bibfnamefont
  {S.}~\bibnamefont {Whitcomb}}, \ and\ \bibinfo {author} {\bibfnamefont
  {M.}~\bibnamefont {Zucker}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
  {Improved sensitivity in a gravitational wave interferometer and implications
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  {journal} {\bibinfo  {journal} {Physics Letters A}\ }\textbf {\bibinfo
  {volume} {218}},\ \bibinfo {pages} {157--163} (\bibinfo {year}
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\bibitem [{\citenamefont {Kopeikin}(2015)}]{kopeikin_optical_2015}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Sergei~M.}\
  \bibnamefont {Kopeikin}},\ }\bibfield  {title} {\enquote {\bibinfo {title}
  {Optical cavity resonator in an expanding universe},}\ }\href@noop {}
  {\bibfield  {journal} {\bibinfo  {journal} {Gen. Relativ. Gravit.}\ }\textbf
  {\bibinfo {volume} {47}},\ \bibinfo {pages} {1--18} (\bibinfo {year}
  {2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sorge}(2000)}]{Sorge}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Francesco}\
  \bibnamefont {Sorge}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Do
  gravitational waves create particles?}}\ }\href
  {http://stacks.iop.org/0264-9381/17/i=22/a=306} {\bibfield  {journal}
  {\bibinfo  {journal} {Classical and Quantum Gravity}\ }\textbf {\bibinfo
  {volume} {17}},\ \bibinfo {pages} {4655} (\bibinfo {year}
  {2000})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Sab{\'i}n}\ \emph {et~al.}(2014)\citenamefont
  {Sab{\'i}n}, \citenamefont {Bruschi}, \citenamefont {Ahmadi},\ and\
  \citenamefont {Fuentes}}]{sabin_phonon_2014}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Carlos}\ \bibnamefont
  {Sab{\'i}n}}, \bibinfo {author} {\bibfnamefont {David~Edward}\ \bibnamefont
  {Bruschi}}, \bibinfo {author} {\bibfnamefont {Mehdi}\ \bibnamefont {Ahmadi}},
  \ and\ \bibinfo {author} {\bibfnamefont {Ivette}\ \bibnamefont {Fuentes}},\
  }\bibfield  {title} {\enquote {\bibinfo {title} {Phonon creation by
  gravitational waves},}\ }\href@noop {} {\bibfield  {journal} {\bibinfo
  {journal} {New Journal of Physics}\ }\textbf {\bibinfo {volume} {16}},\
  \bibinfo {pages} {085003} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Meacher}\ \emph {et~al.}(2015)\citenamefont
  {Meacher}, \citenamefont {Coughlin}, \citenamefont {Morris}, \citenamefont
  {Regimbau}, \citenamefont {Christensen}, \citenamefont {Kandhasamy},
  \citenamefont {Mandic}, \citenamefont {Romano},\ and\ \citenamefont
  {Thrane}}]{meacher_mock_2015}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {Duncan}\ \bibnamefont
  {Meacher}}, \bibinfo {author} {\bibfnamefont {Michael}\ \bibnamefont
  {Coughlin}}, \bibinfo {author} {\bibfnamefont {Sean}\ \bibnamefont {Morris}},
  \bibinfo {author} {\bibfnamefont {Tania}\ \bibnamefont {Regimbau}}, \bibinfo
  {author} {\bibfnamefont {Nelson}\ \bibnamefont {Christensen}}, \bibinfo
  {author} {\bibfnamefont {Shivaraj}\ \bibnamefont {Kandhasamy}}, \bibinfo
  {author} {\bibfnamefont {Vuk}\ \bibnamefont {Mandic}}, \bibinfo {author}
  {\bibfnamefont {Joseph~D.}\ \bibnamefont {Romano}}, \ and\ \bibinfo {author}
  {\bibfnamefont {Eric}\ \bibnamefont {Thrane}},\ }\bibfield  {title} {\enquote
  {\bibinfo {title} {Mock data and science challenge for detecting an
  astrophysical stochastic gravitational-wave background with {Advanced} {LIGO}
  and {Advanced} {Virgo}},}\ }\href@noop {} {\bibfield  {journal} {\bibinfo
  {journal} {Phys. Rev. D}\ }\textbf {\bibinfo {volume} {92}},\ \bibinfo
  {pages} {063002} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Deser}(1957)}]{Deser57}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
  {Deser}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {General
  relativity and the divergence problem in quantum field theory},}\ }\href@noop
  {} {\bibfield  {journal} {\bibinfo  {journal} {Rev. Mod. Phys.}\ }\textbf
  {\bibinfo {volume} {29}},\ \bibinfo {pages} {417--423} (\bibinfo {year}
  {1957})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Heisenberg}\ and\ \citenamefont
  {Euler}(1936)}]{EulerHeisenberg}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
  {Heisenberg}}\ and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
  {Euler}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Folgerungen aus
  der {Diracschen Theorie des Positrons}},}\ }\href@noop {} {\bibfield
  {journal} {\bibinfo  {journal} {Z.~Phys.}\ }\textbf {\bibinfo {volume}
  {98}},\ \bibinfo {pages} {714} (\bibinfo {year} {1936})}\BibitemShut
  {NoStop}%
\end{thebibliography}%
