text:
\section{Introduction}

The term ``matter wave'' is generally associated with the wave properties exhibited by massive particles having momentum $p$, which are characterized by the de Broglie  wavelength relationship $\lambda = h/p$, where $h$ is Planck's constant \cite{zubairy.2020}.  Whereas a wave is a spatially extended entity, the classical view of a particle ascribes to it a well-defined position.  Quantum mechanics dictates that the localization of a particle to with a span of $\delta x$ requires a spread of particle momenta $\delta p$ consistent with Heisenberg's uncertainty principle, $\delta x \delta p \geq \hbar/2$, where $\hbar$ is the reduced Planck's constant.  A particle said to be propagating would furthermore have momentum characterized by $\left|\delta p/p \right| \ll 1$.

Here we will use the term ``de Broglie matter wave,'' or simply ``de Broglie wave,'' to refer to the wave characteristics associated with a flux of identical massive particles having fixed energy $E_{\rm{v}}$ and a given spectral width. A de Broglie wave can  transmit power, do useful work, and be utilized in sensing applications, for example.  In particular, however, the spectral components of a de Broglie matter wave have no specific relationship to each other: each particle is independent of the other particles comprising the entire flux.  Because the individual particles are associated with a spectral width, we understand that an interaction such as a collision will generally result in a distribution of energy outcomes.

A view of matter that is different from the customary de Broglie wave becomes apparent as one seeks to identify states in which interactions lead to a definite energy change.  In particular, given a free particle having vacuum energy eigenvalue $E_{\rm{v}}$ (and therefore it is not a localized particle), due to an interaction it subsequently bares energy $E_{\rm{v}}'=\hbar(\omega_{\rm{v}}+\omega_0)$ that is also an energy eigenvalue. This leads  to the notion of a single-mode matter-wave field identified by its frequency $\omega_0$.  In the presence of this field a particle can absorb an additional $\hbar \omega_0$ amount of energy, or return to its original vacuum state by releasing that amount of energy back into the field.  We will show that in a classical limit this field is associated with a pair of propagating oscillatory fields:
\begin{eqnarray}\label{Eq:OscillatoryFields}
     & \label{Eq:MatterEField} \textbf{F}\left(x,t\right)=\textbf{F}_0 \sin\left(k x - \omega_0  t\right), \\
     &\label{Eq:MatterBField} \textbf{ G}\left(x,t\right)=\textbf{G}_0 \sin\left(k x - \omega_0  t\right),
\end{eqnarray}
which are analogous to the electric and magnetic fields $\textbf E $ and $\textbf B$ associated with waves, say, produced by an electronic microwave oscillator, though they do not propagate at the speed of light. As unintuitive as electric and magnetic matter analogs might be, they are inescapable consequences of the single-mode formalism.  In any case, the analogy between electromagnetic and matter waves proves to be generally fruitful. We thus introduce the term ``Maxwell matter wave,'' or sometimes simply ``Maxwell wave,'' to refer to classical states of the matter-wave field.

We originally introduced Maxwell matter waves in \cite{Anderson.2021}, there referred to as ``classically coherent matter waves,'' as the emission generated by an atomtronic transistor oscillator \cite{Caliga.2012nlo, Caliga.2016i7o, Dinardo.2018}. For the sake of this work we do not explain how a matter-wave oscillator functions; we simply assume that such an instrument exists.  Instead we wish to establish that Maxwell matter waves are consistent with quantum mechanics and discuss some of their elementary aspects.

The oscillator involves atomic ensembles having a temperature $T \lesssim \hbar \omega_0 /k_B$, where $k_B$ is Boltzmann's constant, so that the energy associated with the emitted particles exceeds their thermal energy. Such low temperatures, on the order of $50$nK in the experimental work of \cite{Dinardo.2018},  is also a regime in which the de Broglie aspects of matter will be manifest.  One might say that de Broglie waves are the essence of the wave-particle duality of quantum mechanics in which one thinks classically in terms of particles that prove to exhibit wave-like behavior under the laws of quantum mechanics. Yet the Maxwell matter waves expressed by the fields Eqs. (\ref{Eq:MatterEField}) and (\ref{Eq:MatterBField}) are clearly classical and clearly more wave-like than they are particle-like.  Maxwell waves provide a complementary view of matter within the wave-particle duality: classical waves that prove to exhibit particle-like behavior under the laws of quantum mechanics.  We will reflect below on the two views of matter in the context of Bose-Einstein condensation.

Louis de Broglie tendered his hypothesis regarding the wave properties of matter in 1924 — nearly 100 years ago — and it was validated by the experiments of Thomson \cite{THOMSON.1927} and of Davisson and Germer \cite{davisson.1927b} in 1927. These early diffraction experiments and many validations since have placed de Broglie's momentum-wavelength relationship on firm ground. We shall establish here that the massive particles associated with Maxwell matter waves do not exhibit the expected de Broglie relationship.  The conclusion might seem to contradict de Broglie's hypothesis, if not quantum mechanics more fundamentally, but it does not. It is merely surprising.  We unravel the apparent inconsistency in the Remarks section.

While the distinction between de Broglie and Maxwell matter waves is meaningful, the practical utility of Maxwell matter waves provides our primary motivation here.  In that context our purpose is to lay some ground work by cultivating the analogy between Maxwell matter waves and electromagnetic waves.  By doing so one makes available to the matter-wave domain the tremendously extensive and powerful set of tools that have been developed for the electromagnetics domain.  We shall apply such tools to an example inertial sensing application. Drawing out the analogy also enables a deeper understanding of the nature of Maxwell matter waves and their distinction from de Broglie waves.

Our approach starts with a classical description of matter in terms of particles, transitions to a quantum description which interprets particles as waves, then comes full circle to a classical description of matter in terms of coherent state excitations of a matter-wave field.

\section{Classical Particle Description}
The classical-to-quantum transition begins with a classical derivation of the Hamiltonian, so we will start by considering a particle having mass $m$ propagating in the $x$-direction.  Charged masses repel or attract according to whether they have same or opposite charge signs.  Neutral identical particles also typically repel or attract; in the ultracold domain they interact as though they are spherical.  Let us leverage the analogy by subjecting our particle to a vector potential; the conjecture that it leads to a valid single-mode picture is to be validated later by the outcome.  We choose a vector potential that points as well as travels in the $x$-direction:
\begin{equation}\label{Eq:VectorPotential}
    \textbf{A} = \hat{x}A_0\cos(k x - \omega_0 t).
\end{equation}
The Lagrangian $\mathcal L$ for a particle is given by the difference of its kinetic and potential energies, $T$ and $V$:
\begin{equation}
    \mathcal L = T - V,
\end{equation}
in which the kinetic energy is:
\begin{equation}
    T= \frac{1}{2}m \dot{x}^2,
\end{equation}
and the potential energy:
\begin{equation}
    V = m\left(\frac{\omega_0}{k}-\dot x \right) A_0\cos(k x-\omega_0 t).
\end{equation}
The Lagrangian is determined from the energies:
\begin{equation}
    \mathcal L
     =\frac{1}{2}m \dot{x}^2 + m(\dot x-\frac{\omega_0}{k})  {A}_0 \cos(k x - \omega_0 t).
\end{equation}
The canonical momentum $p$ is determined from the Lagrangian:
\begin{equation}
    p=\frac{\partial \mathcal L}{\partial \dot{x}}=m \dot{x} + mA_0 \cos(k x-\omega_0 t).
\end{equation}
The Hamiltonian is then:
\begin{equation}
      H=p\dot{x}-\mathcal L = \frac{1}{2}m \dot{x}^2 + \frac{m\omega_0}{k} A_0\cos(k x-\omega_0 t),
\end{equation}
or:
\begin{equation}
\begin{split}
    H &= \frac{1}{2m} \left|p-m A_0\cos(k x-\omega_0 t)\right|^2   \\
    &+ \frac{m\omega_0}{k} A_0\cos(k x-\omega_0 t).
\end{split}
\end{equation}
Now Hamilton's equations of motion for the position and momentum give,
\begin{equation}
    \dot{x}=\frac{\partial H}{\partial p}=\frac{p}{m}-A_0 \cos (k x-\omega_0 t),
\end{equation}
\begin{equation}
\begin{split}
    \dot{p}=-\frac{\partial H}{\partial x}&=(m\omega_0-p k) A_0 \sin (k x-\omega_0 t)\\
    &+2k mA_0^2\cos(k x-\omega_0 t)\sin(k x-\omega_0 t).
\end{split}
\end{equation}
Note that in the limit of small vector potential, $A_0 \ll p/m$, the canonical momentum is constant provided:
\begin{equation}\label{Eq:GroupVelocity}
    v_{\rm{v}}\equiv \frac{\omega_0}{k}=\frac{p}{m}.
\end{equation}
We comment that the atomtronic oscillator inherently fulfills this condition by the nature of its dynamics. Therefore the system's canonical momentum is nearly time independent, $p=p_{\rm{v}}$, and the system Hamiltonian is simply that of a free particle and of the field separately:
\begin{equation}\label{Eq:Hamiltonian}
    H=\frac{p_{\rm{v}}^2}{2m}+\frac{1}{2}mA_0^2\cos^2(k x-\omega_0 t).
\end{equation}
The particle's kinetic momentum $P$, however, oscillates in time and space around its mean value:
\begin{equation}
    P\equiv m \dot{x} = p_{\rm{v}}-mA_0\cos(k x-\omega_0 t).
\end{equation}

\section{Quantum Mechanical Description}
With the Hamiltonian in hand, the transition from the classical to the quantum regime is relatively straightforward, replacing Hamilton's equations for position and momentum with Schrodinger's for the wave function $\psi$, $\hat{\mathcal H}\psi = E \psi$.  The Hamiltonian, Eq. (\ref{Eq:Hamiltonian}), looks to have the form of a particle subject to a moving optical lattice, the solution to which can be expressed in terms of Bloch functions.  The difference lies in the dispersion relation from Eq. (\ref{Eq:GroupVelocity}), $k=\omega_0/v_{\rm{v}}$. That is, the ``lattice'' is tied to the particle, rather than to an independent frame of reference.

In absence of the vector potential, the time-independent Schrodinger equation admits the fundamental de Broglie matter wave: the single-mode plane wave solutions for a wave function —solutions that conceptually occupy all of space.
\begin{equation}\label{Eq:SingleMode}
    \psi = \frac{1}{\sqrt{N}}e^{i(k_{\rm{v}} x - \omega_{\rm{v}}t)},
\end{equation}
for which, as above, $E_{\rm{v}}=p_{\rm{v}}^2/2m=\hbar \omega_{\rm{v}}$, $k_{\rm{v}} = p_{\rm{v}}/\hbar$ and $N$ is an appropriate normalization factor.

The vector potential field must also be quantized, which can be done using the same method as for electromagnetic field \cite{Scully.1997}. In so doing we introduce of a set of creation and annihilation operators $\{\hat{a}^{\dagger} (\omega_0), \hat{a}(\omega_0) \}$.  These operators act on a single mode of the matter-wave field, raising or lowering the energy by $E_{\rm{m}}=\hbar \omega_0$. Notably, they do not create and destroy massive particles. The packets of energy, dubbed ``matterons'' in Ref. \cite{Anderson.2021}, are the matter-wave analogs of the photon.  The vector potential field, Eq.  (\ref{Eq:VectorPotential}), is a coherent state, that is, an eigenstate of the annihilation operator $\hat{a}$.  The state is conventionally written as $|\alpha\rangle$:
\begin{equation}
    \hat{a}|\alpha\rangle = \alpha |\alpha\rangle .
\end{equation}
As such, the vector field does not have a well-defined number of matterons or energy,
\begin{equation}\label{Eq:MeanEnergy}
    \langle E_{\alpha} \rangle = \left(|\alpha|^2+ \frac{1}{2} \right) \hbar \omega_0;
\end{equation}
however, the excited matterwave is associated with exactly one matteron, so that each particle itself has a well-defined vacuum energy, $E_{\rm{v}}'=\hbar (\omega_{\rm{v}}+\omega_0)$.  Such definite energy was our objective.  Notice that the energy associated with the particle's canonical momentum $p_{\rm{v}}$ is the same as that of the particle in absence of the field, while the additional $\hbar \omega_0$ can be viewed as internal energy associated with the dipole motion of the particle, which itself is associated with momentum having magnitude $\hbar |k|$.

As our focus is on their classical properties, here we defer important aspects concerning the zero-point energy $\hbar \omega_0/2$ and spontaneous phenomena associated with the matter-wave field to a later section. For related reasons, we also sidestep questions about multi-matteron excitation associated with a single massive particle.

The coherent state character of the vector potential is embossed on the state of the matterwave field; it seems thus fitting to refer the result as a ``coherent excitation of the matter-wave field''. This brings us full circle and enables us to treat an ensemble of particles classically as we do in the following sections, but as a wave rather than as a localized propagating entity.  It is interesting to note, in fact, that a localized, particle-like, entity can be constructed either as a superposition of de Broglie waves associated with a spectrum of energies $\hbar\delta \omega_{\rm{v}}$, or as a superposition of Maxwell matter waves having a spectrum of excitation energies $\hbar \delta \omega_0$. In the latter case, however, energy uncertainly is reflected through the uncertainty in the number of particles.

\section{Matter wave Maxwell analogs}
The atomtronic oscillator emits a constant flux of particles we identify as a Maxwell matter wave. We derived the mechanics with the introduction of a vector potential, the matter analog of the vector potential utilized in electromagnetics.  Evidently, though perhaps less intuitively, we can equally well describe the behavior in terms of matter analogs to electric and magnetic fields.  While we have borrowed the electromagnetic nomenclature with the symbol ``$\textbf {A}$'' for the vector potential, we use different symbols for the electric and magnetic field analogs to underscore the fact that matterwaves do not involve electric charges and currents:
\begin{equation}\label{Eq:Efield}
    \textbf{F} =- \frac{\partial \textbf{A}}{\partial t},
\end{equation}
\begin{equation}\label{Eq:Bfield}
    \textbf{G}=\nabla \times \textbf{A} ,
\end{equation}
from which we substantiate our assertion of Eq's (\ref{Eq:MatterEField}) and (\ref{Eq:MatterBField}). These fields propagate at the group velocity of the particle.  Indeed, it seems that the fields are intimately tied to the particles, and thus we can describe the physics either in terms of the fields or in terms of the particles.  The latter context is very much like the fields associated with the carriers in an electrical circuit; namely, instead of the fields we can equivalently work with the matter analogs of electric potential and electric current.  We will thus refer to the analogs as ``potential'' and ``current:''
\begin{eqnarray}\label{Eq:OscillatoryWaves}
     & \label{Eq:PotentialWavess} {\mathcal V}\left(x,t\right)={\mathcal V}_0 \cos\left(k x - \omega_0  t\right), \\
     &\label{Eq:CurrentWave} \mathcal I\left(x,t\right)={\mathcal I}_0 \cos\left(k x - \omega_0  t\right).
\end{eqnarray}

Generally the fields are suited to a matter-wave optics context while the $\mathcal{V}$-$\mathcal{I}$ treatment is particularly appropriate in an atomtronic circuit context. In experiments, particles are typically manipulated and confined in wave-guide structures.  In this regime the physics is indeed much like that of microwave circuitry.  Writing equations in terms of potentials and currents allows one to use the familiar mathematical tools for analyzing matter-wave circuits such as transmission line theory \cite{Gonzalez.1997}.  More generally, Kirchhoff's laws and the concepts of resistance and reactance, are all at play in atomtronic circuit analysis. The SI units associated with matter-wave quantities can simply be found by substituting ``kilogram'', or ``kg'', for every occurrence of ``Coulomb'', or ``C''. Thus electric potential as J/C becomes matter potential J/kg and electrical capacitance as C$^2$/J becomes matter capacitance kg$^2$/J.

The potential and current amplitudes are related through a characteristic impedance ${Z}$\cite{Gonzalez.1997}:
\begin{equation}
	{\mathcal V}_0 =   Z{\mathcal I}_0,
\end{equation}
where
\begin{equation}
	Z=n^2 \frac{ \hbar} {m^2}\equiv n^2 Z_0 .
\end{equation}
Here $n$ plays the role of a refractive index:
\begin{equation}
	n \equiv \sqrt{\frac{ \omega_{\rm{0}} }{\omega_{\rm{v}}}},
\end{equation}
where recall that $E_{\rm{v}}\equiv \hbar \omega_{\rm{v}}$ is the vacuum energy of individual particles associated with the wave. While it is uncommon in optics to have an refractive index less than unity, here it is easily conceivable to have $1/n\gg1$.

As we here look to highlight the classical nature of Maxwell matter waves, we take the natural impedance $Z_0$ to be the fundamental entity describing wave properties rather than Planck's constant and substitute $\hbar=m^2 Z_0$. As evident from Eq.  (\ref{Eq:GroupVelocity}) the waves propagate at the group velocity of a deBroglie wave:
\begin{equation}\label{wavevelocity}
	v_{a}=\frac{1}{n}\sqrt {2 m \omega_0 Z_0 } \equiv \frac{v_0}{n} ,
\end{equation}
and are thus associated with the wave number:
\begin{equation}\label{Eq:wavenumber}
	k=n\sqrt{\frac{ \omega_0  }{2 m Z_0}}\equiv n k_0	.
\end{equation}

Though superfluous for our present purposes, we can complete the matter-wave field analogy with electromagnetic fields by defining matter-wave permeability and permittivity:
\begin{equation}\label{Eq:permeability}
\begin{split}
     & \mu_0 \rightarrow \upsilon_0 \equiv \sqrt{\frac{ Z_0}{2 m \omega_0}}, \\
     & \mu \rightarrow \upsilon = n^3\upsilon_0 .
\end{split}
\end{equation}
\begin{equation}\label{Eq:CurrentField}
\begin{split}
     &\epsilon_0 \rightarrow \xi_0 \equiv \frac{1}{Z_0\sqrt{2 m \omega_0 Z_0}}, \\
     &\epsilon \rightarrow \xi =  \xi_0/n.
 \end{split}
\end{equation}
Table \ref{Table:1} provides a list of analogs between matter-wave quantities and electromagnetic quantities.

The matter-wave fields satisfy:
\begin{equation}\label{Eq:MWMaxwell}
     \nabla \times \textbf{F} = -\frac{\partial \textbf{G}}{\partial t},
\end{equation}
\begin{equation}
    \nabla \times \textbf{G} = \upsilon \xi \frac{\partial \textbf{F}}{\partial t} ,
\end{equation}
from which it follows they also satisfy a familiar wave equation:
\begin{equation}
    \frac{\partial^2 \textbf{F}}{\partial t^2}=\frac{1}{\upsilon \xi}\frac{\partial^2 \textbf{F}}{\partial x^2},
\end{equation}
and similarly for $\textbf{G}$ (as well as $\mathcal V$ and $\mathcal I$). We recognize that the wave's phase velocity $1/\sqrt{\upsilon \xi}$ is the particle's group velocity.  Note, too, that the Maxwell wave phase and group velocities are equal.

\begin{table}[ht]
    \centering
    \setlength{\tabcolsep}{9pt}
    \begin{tabular}{ l c c l }
         EM Element & EM & MW &  SI Units\\
        \hline
        Vector potential & $\textbf {A}$ & $ \textbf {A}$ & $\rm{m} \cdot \rm{s}^{-1}$\\
        Electric field & $\textbf {E}$ & $\textbf {F}$ & $\rm{m} \cdot \rm{s}^{-2}$ \\
        Magnetic field & $\textbf {B}$ & $\textbf {G}$ & $\rm{s}^{-1}$ \\
        Electric potential & $\mathcal V$ & $\mathcal V$ & $\rm{m}^2 \cdot \rm{s}^{-2}$ \\
        Current & $\mathcal I$ & $\mathcal I$ & $\rm{kg} \cdot \rm{s}^{-1}$ \\
        Impedance & Z & Z & $\rm{m}^2 \cdot \rm{kg}^{-1} \cdot \rm{s}^{-1}$ \\
        Capacitance & C & C & $\rm{kg}\cdot \rm{s}^2 \cdot \rm{m}^{-2} $ \\
        Inductance & L & L & $\rm{m}^2 \cdot \rm{kg}^{-1} $ \\
        Permittivity & $\epsilon$ & $\xi$ & $\rm{kg} \cdot \rm{s}^2 \cdot \rm{m}^{-3}  $ \\
        Permeability & $\mu$ & $\upsilon$ & $\rm{m} \cdot \rm{kg}^{-1}  $ \\
        \hline
    \end{tabular}
    \caption{Matter-wave (MW) analogs to electromagnetic (EM) quantities and corresponding SI units}
    \label{Table:1}
\end{table}

It is useful to appreciate that the current wave, Eq.  (\ref{Eq:CurrentWave}) does not imply a time-varying particle flux, any more, say, than the electromagnetic field of a laser beam implies a time-varying photon flux.  The wave amplitudes are determined from Eq.  (\ref{Eq:MeanEnergy}), or equivalently through the (experimentally measurable) particle flux $I_{\rm{v}}$ given in units of particles/s:
\begin{equation}\label{Eq:particleflux}
	{\mathcal I}_0 =\frac{m}{n}\sqrt{2\omega_0 I_{\rm{v}}}.
\end{equation}

\begin{figure}[t]
\includegraphics[width=\columnwidth]{Step.png}
\caption{\label{fig:Step} The reflection and transmission coefficients $R$ ad $T$ of a Maxwell matter wave from a step change in refractive index yields the same result as that of a de Broglie wave from a step potential.  The oscillation frequency $\omega_0$ as well as the particle energy $E=\hbar \omega_{\rm{v}}$ remain constant while the wavelength decreases in the higher index region.    }
\end{figure}

\section{Properties}
Let us contrast the nature of the Maxwell wave with the wave associated with a particle having energy $E_{\rm{v}}$.  The latter has wavenumber:
\begin{equation}\label{Eq:deBroglieWavenumber}
    k_{\rm{v}}=\sqrt{\frac{2m\omega_{\rm{v}}}{\hbar}}=\sqrt{\frac{2\omega_{\rm{v}}}{m Z_0}}=\frac{2}{n} k_0.
\end{equation}
The distinction is clear, since the de Broglie wavenumber increases with increasing particle energy, yet decreases for the Maxwell matter wave case, Eqn. (\ref{Eq:wavenumber}).  While the conclusion might seem to contradict de Broglie's hypothesis that a particle's wavelength is inversely proportional to its momentum, we consider the elementary quantum mechanics problem of reflection from a step potential of height $U$, as illustrated in FIG. \ref{fig:Step}. The refractive index for the Maxwell waves may be generalized to:
\begin{equation}\label{Eq:GeneralizedIndex}
	n(U) \equiv \sqrt{\frac{ \omega_{\rm{0}} }{\omega_{\rm{v}}}}\left(1-\frac{U}{m^2\omega_{\rm{v}}Z_0}\right)^{-1/2},
\end{equation}
whereas that for the de Broglie waves is:
\begin{equation}
    \tilde{n}(U)=\frac{1}{n(U)}.
\end{equation}
Amplitude reflection and transmission of a wave from a potential step is given by a well-known pair of formulas \cite{Hecht.2017}; for the Maxwell matter wave:
\begin{equation}
\begin{split}
     & r=\frac{n_1-n_2}{n_1+n_2}, \\
     & t=\frac{2n_1}{n_1+n_2},
 \end{split}
\end{equation}
for which we have taken $n_1=n(0)$ and $n_2=n(U_0)$.  Using the same formulas for the de Broglie wave we find:
\begin{equation}
\begin{split}
     & \tilde{r}=-r ,\\
     & \tilde{t}=\frac{n_2}{n_1} t,
 \end{split}
\end{equation}
The transmitted and reflected particle flux in the two cases are identical:
\begin{equation}
\begin{split}
     & R\equiv |r|^2=\tilde{R} ,\\
     & T\equiv |t|^2=\tilde{T}.
 \end{split}
\end{equation}
We have indicated that the Maxwell matter fields are associated with a constant flux $I_{\rm{v}}$ of mono-energetic particles, thus the Maxwell and de Broglie treatments of the interface are consistent.  The conclusions are very different, however, when propagation involves multiple paths for which the coherence properties of the wave come into play.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{Mach_Zehnder.png}
\caption{\label{fig:MachZehnder} Interference of matter waves in a Mach-Zehnder interferometer will reveal the distinction between de Broglie and Maxwell matter waves.  The ``fringe'' spacing, i.e. the change required in the path length difference $\Delta L$ to cause the detected particle flux to change from a maximum to zero in the two cases will differ by a factor of $n^2/2$ }.
\end{figure}

Consider the Mach-Zehnder interferometer of FIG. \ref{fig:MachZehnder}.  For atomic particles it is conceptually straightforward to implement mirrors and beamsplitters utilizing A.C. Stark shifts induced with narrowly-focused knife-shaped laser beams tuned to the blue side of an atomic resonance. One arm of our interferometer incorporates an adjustable path length and there is a particle detector at the output of each port.  The properties of the interferometer are well known \cite{Hecht.2017}:  For identical arm lengths one port will be ``dark'' meaning no particles are detected while the other will be ``bright'' meaning the entire input atom flux will be detected.  The question is: by how much does the path length difference need to be changed to reverse the dark and bright ports?  Taking the case that $\omega_{\rm{v}} \gg \omega_0$ the de Broglie result would indicate a much smaller path length difference than the Maxwell wave result, by a factor of $n^2/2$.  The two calculations yield different predictions for the particle count as a function of path length difference, so Maxwell and de Broglie waves are definitively distinct.

\section{Resonant Matter-Wave Interferometry}
The temporal coherence of the Maxwell matter waves accommodates the matter-wave analog of the optical resonator, and in particular enables resonant matter-wave interferometry.  FIG. \ref{fig:FabryPerot} illustrates a notional system in which a matter wave is injected into a resonator formed with a pair of matter-wave reflectors having reflectively $R$. To simplify the discussion we suppose that the flux is sufficiently low that particle interactions can be ignored.  Feedback to the matter-wave generator adjusts the oscillator frequency to maintain its frequency on resonance \cite{Black.2001}.  The treatment of the matter-wave resonator follows identically that for the optical case: the mirrors are separated by a distance $L$ chosen to be an integer number $N$ of half-wavelengths of the Maxwell wave, $k L=n k_0L = \pi N$, to produce a resonating matter-wave interferometer. Converting the resonant wavelength to a frequency,
\begin{equation}
    \omega_{\rm{N}}=N\frac{\pi v_{\rm{v}}}{L}.
\end{equation}
A measure of the quality of a resonator is given by the finesse \cite{Hecht.2017}:
\begin{equation}
    F=\pi \frac{\sqrt{R}}{1-R},
\end{equation}
from which one can determine a resonator linewidth:
\begin{equation}
    \delta \omega = \frac{\pi v_{\rm{v}}}{ L F}.
\end{equation}

\begin{figure} [t]
\includegraphics[width=\columnwidth]{Fabry_Perot.png}
\caption{\label{fig:FabryPerot} Notional scheme of a resonant matterwave interferometer for sensing acceleration is comprised of a pair of matterwave ``mirrors''.  The drive frequency of an atomtronic oscillator is servo-controlled to maintain resonance using a matter-wave version of Pound-Drever-Hall feedback arrangement \cite{Black.2001}.  The presence of an acceleration induces a resonant frequency shift.    }
\end{figure}

Consider now the the presence of an inertial force, let us say causing an acceleration $a$ along the resonator axis as shown in FIG. \ref{fig:FabryPerot}: in the frame of the resonator the particle energy varies during propagation in the resonator. The acceleration can be treated as a spatially-dependent refractive index that induces an apparent change in the path length:
\begin{equation}
   n(x)=n\left( 1 + \frac{ax}{m\omega_{\rm{v}}Z_0}\right)^{-1/2}.
\end{equation}
Mimicking its optical analog we refer to the optical path length:
\begin{equation}
    L_{\rm{eff}}=n \int_{0}^{L} \left( 1 + \frac{ax}{m\omega_{\rm{v}}Z_0}\right)^{-1/2} dx .
\end{equation}
Taking the acceleration to be small:
\begin{equation}
    L_{\rm{eff}} \simeq n L\left(1-\frac{L}{4m\omega_{\rm{v}}Z_0}a\right).
\end{equation}
The acceleration $a$ can be determined by locking the oscillator frequency to on particular resonant mode, $\omega_0 =\omega_{\rm{N}}$ and  measuring its frequency shift $\Delta \omega_{\rm{N}}\equiv \omega_{\rm{N}}^\prime - \omega_{\rm{N}}$,
\begin{equation}
    \Delta \omega_{\rm{N}} = \kappa a,
\end{equation}
for which the scale factor
\begin{equation}
    \kappa=\frac{\pi N }{2 v_{\rm{v}}} .
\end{equation}
Associated with the finite linewidth of the resonator will be a characteristic acceleration resolution of the interferometer:
\begin{equation}
    a_{\rm{res}} = \frac{2\pi }{ F  }\frac{v_{\rm{v}}^3}{\omega_0 L^2} .
\end{equation}
The key result here is that the acceleration signal corresponds to a frequency shift of the resonator, and the acceleration resolution is determined by the resonator finesse, which can be large. Setting, for example, $v_{\rm{v}} = 0.01$m/s, $L=0.01$m, $\omega_0 = 2 \pi \times 1$kHz, and $F=100$, yields $a_{\rm{res}}=10^{-7}$m/s${}^2$. Given the small size, this compares favorably with cold atom interferometers when provided with even a modest atom flux, say, of $10^3$ particles/second assuming particle shot-noise limited performance \cite{Berman.1997}.

The matter-wave resonator, like its optical counterpart illuminated with appropriate laser light, relies on the temporal coherence of the Maxwell matter wave.  Resonance can viewed as a compounding of constructive interference, versus the lack of it, between the wave that reflects off of the input mirror and that which has been transmitted from the built up, stored matter wave, bouncing between the two mirrors. Resonant systems find many uses in the optical and more generally electromagnetic domain.  We expect the same can be true for resonant matter-wave systems.

\section{The Atom Analog of the Laser}
Bose-Einstein condensation takes place when a sufficiently high density of particles is brought to temperatures near absolute zero \cite{Pethick.2002}.   The Bose-Einstein condensate (BEC) itself is an ensemble of particles in thermal equilibrium. Quantum mechanically the BEC is a number state, meaning a state having a fixed number particles.  At zero temperature the particles comprising a BEC are described by a common localized wave function having a well-defined energy.  While the coherence of a BEC was debated at the time \cite{Castin.1997}, Ketterle and collaborators' unequivocal demonstration of matter-wave interference laid to rest any doubt about the quantum coherence ascribable to a condensate \cite{Andrews.1997}. In principle one knows everything about the position and momentum of the particles, and also the ensemble of particles as a whole, that the laws of quantum mechanics allow. In this respect the BEC is the epitome of the deBroglie perspective in which matter is comprised of particles whose wave-like character is imposed by quantum mechanics.

From the time of its first demonstration in 1995 \cite{Anderson.1995bec,Davis.1995} the BEC was hailed as the atom analog of the laser \cite{Kleppner1997}. A laser is a nonthermal equilibrium system that belongs to the general class of driven oscillators, conceptually a close cousin to electronic transistor oscillators: both produce classically coherent electromagnetic fields whose mechanics is governed by Maxwell's equations. Indeed the particle aspects of light articulated by Planck and by Einstein marked the birth of quantum theory.  Whereas a BEC is comprised of a localized ensemble of particles having a definite number, a coherent state is nonlocal, does not have a well-defined number of particles or energy, nor a particle wave function from which one can determine the probability of finding a particle at a given position at a given time \cite{Scully.1997}.  The atomtronic transistor circuit that produces Maxwell matter waves is also a driven oscillator and its emission is also a coherent state, but of matter rather than light, with its particle-like aspects attributed to matterons as the analog to photons.  So while there are aspects of a BEC that resemble the laser, we now understand that observations of de Broglie matter-wave interference of BECs is distinct from the laser-like interference of Maxwell matter waves.

\section{Interactions}
That one can leverage the formalism afforded by Maxwell equations is a compelling aspect of a matter-as-wave classical perspective.  One still faces the duality of quantum mechanics, however, and in the context of matter interactions the particle view of matter appears to provide a more natural picture. Yet the matter-as-wave picture was motivated in the first place by the search for a representation in which particles could interact and yet remain in energy eigenstates; the search led us to the notion of a matteron as the analog of the photon. In contrast to photons, matterons can evidently interact.

Consider first the interaction between counter-propagating but otherwise identical Maxwell matter waves.  The resonator discussed in the previous section serves as a illustrative model in which to consider such a case.   Elastic events for which the energy carried by the waves remains unchanged can be treated in a mean-field context.  The energy density associated with the particle flux $I_{\rm{v}}$ of the forward-propagating wave is:
\begin{equation}
    u = \hbar k  \frac{I_{\rm{v}}}{\mathcal{A}} ,
\end{equation}
where $\mathcal{A}$ is an effective cross-sectional area of the matter wave.  For the simple case in which the particle interaction is characterized by a scattering length $a_{\rm{s}}$, particles associated with the backward-traveling wave experience a mean-field interaction energy:
\begin{equation}
	H_{\rm{int}} =  8\pi\frac{a_{\rm{s}} u}{k_0^2} .
\end{equation}
The impact of the interaction can be cast in terms of an refractive index shift $\delta n$, following from  Eq.  (\ref{Eq:GeneralizedIndex}):
\begin{equation}
    \delta n \simeq  4 \pi n^4\frac{a_{\rm{s}}}{k_0}\frac{I_{\rm{v}}}{A}.
\end{equation}
The significance for resonant interferometry is that the index shift causes a corresponding shift of the resonant frequency.

In addition to elastic interactions, one can also anticipate spontaneous parametric interactions arising from matteron exchange between the counter-propagating beams. The simplest spontaneous interaction generates two new waves, one having an increased energy by $\hbar \omega_0$ and the other decreased by the same amount, and correspondingly a momentum increase of $\hbar k$ for the one wave and decrease by the same amount for the other.  The question of momentum conservation can be addressed through the refractive indexes $n_+, n_-$ corresponding to the waves having acquired or lost a matteron, respectively:
\begin{equation}\label{Eq:Interaction}
    n_{\pm} =  \frac{n}{\sqrt{1 \pm n^2}} \simeq n \left( 1\mp  \frac{1}{2} n^2 \right),
\end{equation}
where we have taken the case $\omega_{\rm {v}} \gg \omega_0$. The change in the canonical momentum can be found from Eq. (\ref{Eq:deBroglieWavenumber}):
\begin{equation}
    \Delta p_{\rm{v}} = 2 \left(\frac{1}{n_+}-\frac{1}{n_-}\right) \hbar k_0 \simeq 2 \hbar  k.
\end{equation}
Thus we see that the change in canonical momentum compensates for the change in momentum due to the matteron exchange.

The example of spontaneous interactions can highlight several important aspects of the Maxwell matter wave formalism.  It is clear, for example, that matteron number is not conserved:  In the above example one matteron was removed from each initial wave to create two new waves. Yet each of the initial waves is an eigenstate of an annihilation operator, i.e. a coherent state, and so the mean number of matterons does not change. Moreover, the process does not involve the creation or annihilation of mass - the state of matter is merely altered.

Finally and importantly, the presence of parametric interactions suggests the possibility of parametric amplifiers and oscillators, which are systems that are familiar particularly within  optics \cite{Saleh.1991}.  Indeed, this brings us full circle in providing some insight toward the construction of a matter-wave oscillator, the output of which is a Maxwell matter wave.

\section{Remarks}
We have developed a framework in which matter is viewed classically in terms of waves rather than particles. Within this framework, matter can be treated using analogs to Maxwell's equations and surrounding formalism. The presence of matter analogs of electric and magnetic fields is non-intuitive: Maxwell unified the two seemly different phenomena of electricity and magnetism to be recognized as one phenomenon, that of electromagnetism.  Maxwell's electromagnetic waves have an existence of their own, yet viewed in the context of matter, electromagnetism is simply a description of the interaction among charged massive particles.  Reflecting on our earlier comments, neutral atoms such as $^{87}$Rb repel each other, as do same-sign charges.  The repulsive forces can be cast in terms of chemical potential energy, which becomes large as the particle density becomes high, as in a BEC \cite{Pethick.2002}. The atomtronic oscillator \cite{Dinardo.2018,Anderson.2021} incorporates a harmonic atomic potential having frequency $\omega_0$ and develops a time- and space- varying potential in a self-organized fashion (as one might also expect from a parametric oscillator): this is the physical origin of the forces that underlie the introduction of the vector potential, Eq. (\ref{Eq:VectorPotential}). The analogy between electromagnetic and matter interactions is a faithful one in many respects: given a matter vector potential, the electric field and magnetic field analogs mathematically follow.

The alternative to de Broglie's view of matter sought to identify states in which particle interactions preserved their energy-eigenstate character. This led to the notion of  single-modes of the matter-wave field. The previous section illustrated that coherent state excitations of these modes lead to a picture in which the particle interactions do preserve their eigenstate character, even though the waves themselves are not energy eigenstates.     The aspect that is fundamentally new to the theoretical foundation is the presence of a matter-wave field, associated with which are matterons as massless particles having energy $E_{\rm{m}}=\hbar \omega_0$ and momentum $p_{\rm{m}}= \hbar k$.  Implicit also in the formalism is the presence of matter awaiting excitation, so to speak, since the matterons do not create or annihilate mass.

The distinction between Maxwell and de Broglie matter waves associated with the same particle energy is highlighted by the substantially different outcomes of interferometric measurements.  This difference can be easily understood starting from the classical mechanics treatment: the particle center of mass is oscillating along the direction of propagation at frequency $\omega_0$.  Surely the center of mass must arrive from each of the two arms at the same time at the output port to produce constructive interference.  The quantum mechanical particle has no center of mass \textit{per se}, but its momentum nevertheless oscillates with the same period. Constructive interference can only take place when the interferometer arm length difference corresponds to a propagation time difference equal to an integral number of those periods.

Maxwell matter waves have significant practical implications.  Among them is the possibility of using the waves in conjunction with resonant structures, as already described.  More subtle is the role of interference in simple matter-wave analogs of optical elements, such as a diffraction grating. Such subtleties and other aspects of Maxwell matter waves can be addressed with the well-developed framework of electromagnetics, appropriately translated to the matter-wave domain.

Matteron interaction lends both a complexity and a richness to the wave picture of matter, especially since the strength of interactions can be controlled.  In the case of atoms one can utilize electromagnetic fields to tune interactions, utilizing for example Feshbach resonances \cite{Chin.2010} and Rydberg interactions \cite{Shaffer.2018} to suppress or enhance the interaction strength. Matteron interactions highlight the particle character of Maxwell matter waves and potentially makes uniquely quantum-mechanical aspects such as squeezing and entanglement substantially accessible and controllable.

\begin{acknowledgments}
The author is grateful to D. Gu{\'e}ry-Odelin, S. Du, and M. Saffman for valuable feedback and discussion.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
The redshift determination for blazars is often difficult as the bright non-thermal emission of the jet easily hides the host galaxy.
For blazars detected in the VHE range, this measurement is very important as it allows to model the imprint of the Extragalactic
Background Light (EBL) on their emission constraining models of both.

\section{Observations and data reduction}
We briefly present the results of observations\footnote{Based on data collected at the European Southern Observatory under programs P086.B-0.135(A) in 2010 and P088.B-0485(A) in 2011.}
of eight sources with the X-Shooter spectrograph \cite{verne_2011},
which is a single-object medium resolution ($\lambda$/$\Delta$$\lambda$ = 5000 -- 10000) \'echelle spectrograph operating on the VLT.
The main characteristic of X-Shooter is its unprecedented simultaneous wavelength range from 300 nm to 2400 nm, obtained by splitting the light in three arms: UVB ($\lambda$ = 0.3 -- 0.56 $\mu$m),
VIS ($\lambda$ = 0.55 -- 1.02 $\mu$m), and NIR ($\lambda$ = 1.00 -- 2.40 $\mu$m). For each source, the observations consist of a set of different exposures of 600 seconds each.
We processed the spectra using version 1.3 of the X-Shooter data reduction pipeline \cite{goldo_2006}.
The reduction involves usual steps, including bias subtraction, flat field division with background estimation and wavelength calibration, all performed using day-time calibration frames.

\section{Results}

\begin{figure}
 \includegraphics[width=0.9\textwidth]{figure_8pads_crop2}
  \caption{
Selected regions where features (indicated by dashed vertical lines)  were searched for and/or identified for five VHE blazars:
PKS~0447--439 (A \& B), KUV~00311--1938 (C \& D), PKS~0301--243 (E \& F), BZB~J0238--3116 (G) and BZB~J0816--1311 (H). See text for explanations.
}
\label{features}
\end{figure}

{\bf PKS~0447--439}:
this is one of the brightest blazars dectected by Fermi-LAT \cite{Abdo_2009} and was recently discovered as a VHE emitter by H.E.S.S. \cite{zech_2011}.
Its redshift is still uncertain, with a proposed value z=0.205 \cite{perlman_1998} based on a very weak spectral feature identified as the CaII doublet,
consistent with different non-spectroscopic estimations (\cite{landt_2008}, \cite{zech_2011}, \cite{prandini_2012}).
Conversely, \cite{landt_2012} put a very high lower limit of z>1.246 on its redshift\footnote{
If confirmed, this surprising result would imply that our understanding of VHE $\gamma$-ray propagation is incorrect,
or that the EBL density is far below the value obtained from galaxy counts.
}, based on weak absorption lines at 6280 \AA\ interpreted as the redshifted MgII 2800 \AA\ doublet.

The spectrum obtained in this work is strongly dominated by the emission of the nucleus and, despite
favourable resolution and signal-to-noise levels, does not show any extragalactic spectral line.
The very good signal-to-noise level allows the examination of the weak absorption feature present around 6280 \AA\ (black continuous line in Fig. \ref{features}A), where atmospheric
molecular absorption is expected from $\mathrm{O_2}$ (indeed, this feature is seen in all our sources, see for example Figs. \ref{features}D and \ref{features}F).
The expected transmission rate -- smeared to the resolution of our data -- is shown in the top of the same figure.
We see that the observed feature and the expected absorption by $\mathrm{O_2}$, once normalized to the flux level of the source (dashed line), match
very well, both in shape and wavelength position. This clearly demonstrates that the association of the 6280 \AA\ feature with the Mg II doublet by \cite{landt_2012}
is not correct\footnote{
In a measurement obtained after this conference, a similar conclusion has been proposed by \cite{fumagalli_2012}.}.
Besides, no absorption is seen where the CaII doublet is expected for a redshift of 0.205 as proposed by \cite{perlman_1998}, see Fig. \ref{features}B,
although it should be noted that this region is dominated by the non-thermal emission of the nucleus.

{\bf KUV~00311--1938}: identified as a BL Lac by \cite{bauer_2000} and recently detected by H.E.S.S. \cite{Bech_2012}, this source is considered as probably the farthest
BL Lac detected at VHE, based on a tentative redshift of $z$=0.61 proposed by \cite{pirano_2007}.
From a careful examination of our spectrum, we do not confirm this tentative redshift: as shown in Fig. \ref{features}D, there is no indication of CaII absorption
at $z$=0.61 (see dashed vertical lines). However, a MgII doublet of narrow lines is clearly identified in the UVB arm around 4215 \AA\, which corresponds to a redshift
of $z$=0.506 (see Fig. \ref{features}C). This is the only extragalactic feature in our spectrum, it could be due to an intervening system unrelated to the blazar or to
the blazar itself, giving a clear lower limit to the redshift of KUV~00311--1938: $z\ge$0.506.

{\bf PKS~0301--243}: this source has first been identified as a blazar by \cite{impey_1988}, later discovered at high energies by Fermi-LAT \cite{Abdo_2009}, and recently by H.E.S.S. \cite{wout_2012}.
The best determination of its redshift \cite{falomo_2000} was based on the plausible identification of a single weak emission line with [OIII] 5007 \AA.
We confirm this result and propose a more precise determination of the redshift at $z$=0.266, given the clear detection of the emission lines [OII] 3727.3 \AA\ in the UVB arm
and [OIII] 5007 \AA\ in the VIS arm (see Figs. \ref{features}E and \ref{features}F, where the dashed lines show the expected positions of these emission lines for $z=$0.266).
Note that due to an overlap with the atmospheric $\mathrm{O_2}$ absorption feature, one of the [OIII] doublet lines is not clearly detected.

{\bf VHE blazar candidates}: we studied also the spectra of five other blazars, which may be considered as good candidates for TeV emission based on their favourable levels of flux density at
1.4 GHz ($F_{R}$), their X-ray flux between 0.1 and 2.4 keV ($F_{X}$) and/or the hardness of their spectral slope $\mathrm{\Gamma_{Fermi}}$ as seen by Fermi-LAT. The corresponding numbers and the
redshift search results are summarized in Table \ref{ztab}. As an illustration, the identified lines for BZB~J0238--3116 and BZB~J0816--1311 are shown in Figs. \ref{features}G
and \ref{features}H, respectively.
In the case of RBS~334, only a tentative redshift is proposed, as the identified lines are found at the overlap between two arms (UVB and VIS), where we are more subject to systematic errors.

\begin{table}[h]
\begin{tabular}{llllll}
\hline
Source         &  $F_{R}$ & $F_{X}$ & $\Gamma_{Fermi}$ & $z$ & Comments\\
\hline
PKS~0447--439   &  0.73 & 0.025 & 1.86 & -- & $z\geq$1.26 invalidated, 0.205 not confirmed \\
KUV~00311--1938 &  0.04 & 0.026 & 1.76 & $\geq$0.506 & $z=$0.61 not confirmed\\
PKS~0301--243   &  1.43 & 0.018 & 1.94 & 0.266 & [OII] and  [OIII] emission lines \\
\hline
BZB~J0238--3116 &  0.15 & 0.016 & 1.85 & 0.233 & CaII doublet, $H_{\delta}$ and CaIG absorption lines \\
BZB~J0543--5532 &  0.08 & 0.028 & 1.74 & 0.273 & CaII doublet and NaID absorption lines\\
BZB~J0505+0415 &  0.34 & 0.009 & 2.15 & 0.424 & Mg, CaFe and NaID absorption lines\\
BZB~J0816--1311 &  0.14 & 0.027 & 1.80 & $\ge$ 0.288 & Three MgII doublets at different redshifts\\
RBS~334        &  0.05 & 0.007 & 1.56 & 0.411(?) & Indication of the CaII doublet\\
\hline
\end{tabular}
\caption{Radio and X-ray fluxes (relative to corresponding values for PKS~2155--304) and Fermi-LAT spectral slope as reported in BZCat. The redshifts are from this work. See text for details.}
\label{ztab}
\end{table}

\section{Conclusions}
We demonstrate that the lower limit proposed for PKS~0447--439 at $z\ge$1.246 by \cite{landt_2012} is not correct, and find no evidence of the feature used by \cite{perlman_1998} to propose $z=$0.205.
We place a clear lower limit at z$\geq$0.506 to the redshift of KUV~00311--1938, but do not see evidence for the feature used by \cite{pirano_2007} to propose the tentative redshift $z=$0.61.
We confirm the redshift proposed by \cite{falomo_2000} for PKS~0301--243 and improve its precision at $z=$0.266.
Finally, for a set of five blazars considered as good candidates for TeV emission, we give three firmly determined redshifts, a tentative redshift and a lower limit.
We encourage their observation by current VHE $\gamma$-ray observatories.

\begin{theacknowledgments}
S. Pita wishes to thank the ESO staff and in particular C. Martayan for their help in performing the observations.
\end{theacknowledgments}

\bibliographystyle{aipproc}


text:
\section{Introduction}

Recent advances in string theories suggest that  a special
11-dimension theory (dubbed as M theory) \cite{mtheory} may be the theory of
everything.  Impacts of M theory on our present world can be studied with
compactification of the 11 dimensions down to our $3+1$ dimensions.  The
path of compactification is, however, not unique.  One popular path is to first
compactify the 11 dimensions down to 5 dimensions \cite{5d}.  In this
5-dimensional world, the standard model particles live on a brane ($3+1$ dim)
while there are other fields, like gravity and super Yang-Mill fields,
live in the bulk.  A novel mechanism to break the supersymmetry (SUSY)
is offered
in this picture, in which a hidden sector lives on another brane in this
5-dimensional world.  This brane is entirely separated from the standard
model (SM) brane.
While SUSY is broken in this hidden brane by any means, the SUSY breaking
is communicated to the SM brane via the interactions of the fields in
the bulk.  Large number of studies in this area have been proposed
\cite{msusy}.
Apart from the above, radical ideas like TeV scale string theories
were also proposed \cite{tevstring}.

Inspired by string theories a simple but probably workable solution to
the gauge hierarchy was recently
proposed by Arkani-Hamed, Dimopoulos and Dvali (ADD)
\cite{theory}.  They assumed the space is $4+n$ dimensional, with the
SM particles living on a brane.  While the electromagnetic, strong,
and weak forces are confined to this brane, gravity can also propagate
in the extra dimensions.  To solve the gauge hierarchy problem they
proposed the ``new'' Planck scale $M_S$ is of the order of
TeV in this picture with
the extra dimensions of a very large size $R$.
The usual Planck scale
$M_G=1/\sqrt{G_N} \sim 1.22 \times 10^{19}$ GeV is related to this effective
Planck scale $M_S$ by using Gauss's law:
\begin{equation}
R^n \, M_S^{n+2} \sim M_G^2 \;.
\end{equation}
For $n=1$ it gives a large value for $R$, which is already ruled out
by gravitational
experiments.  On the other hand, $n=2$ gives $R \alt 1$ mm, which is
in the margin beyond the reach of present gravitational experiments.

The graviton including its excitations in the extra dimensions
can couple to the SM particles on the brane with a strength of
$1/M_S$ at short distances
(instead of $1/M_G$), and thus the gravitation interaction
becomes comparable in strength to weak interaction at TeV scale.  Hence, it
can give rise to a number of phenomenological activities testable at existing
and future colliders \cite{wells,shrock,han,mira,collider}.
So far, studies show that there
are two categories of signals: direct and indirect.  The indirect signal
refers to exchanges of gravitons in the intermediate states, while direct
refers to production or associated production of gravitons in the final
state.
Since gravitons interact weakly with detectors, they will escape detection
and give rise to missing energies.  Thus, the logical signal to search for
would be the associated production of gravitons with other SM particles.
At $e^+ e^-$ colliders, the best signals would be the associated production
of graviton with a $Z$ boson, a photon, or a fermion pair.
The production of graviton and photon at LEPII
has been studied in Ref. \cite{mira}.
In this work, we shall study the associated production of graviton with
a $Z$ boson or a photon at $e^+ e^-$ colliders of energies from LEPII
to about 1.5 TeV.

The detection of photon will be direct while for the
$Z$ boson its decay products, a lepton pair or a quark pair, need to be
detected.  The branching ratio of the $Z$ boson into visible products is
about 80\% (the decay of $Z$ into neutrinos
will not be useful because the final state would then be all missing
energies.)  Therefore, the signature will be a lepton pair or a quark pair
of the $Z$ mass with a large missing energy (or a photon with missing energy
for $\gamma G$ production.)
In contrary to the background, the
recoil mass spectrum will not show a particular resonance because the
mass spectrum of the graviton excitations is almost a continuous one.
Recent analyses were performed by L3, ALEPH, and DELPHI \cite{aleph}
at LEPII in search for $ZH$ production with the Higgs boson decaying into
invisible particles.  This signature is similar to our signal of interest.
Thus, we shall also use these data to constrain the new cut-off scale $M_S$.
In order to have a handle on the feasibility of our signal, we choose
$ZH$ production as a bench-mark for comparison with the cross section of
$ZG$.
The major background comes from $e^+ e^- \to Z\;(\gamma)\; \nu_i \bar\nu_i
\;(i=e, \mu,\tau)$, of which $\nu_\mu$ and $\nu_\tau$ come mainly from
$ZZ\;(\gamma  Z)$ pair production while $\nu_e$ also has contributions from
$W$-exchange diagrams.

The organization of the paper is as follows.  In the next section, we shall
give the details of the calculations. In Sec. III, we shall give numerical
results and derive possible limits that can be reached by the NLC.
In Sec. IV, we use the data from LEPII on the search for $ZH$ production with
the Higgs boson decaying invisibly and the data on the production of
a single photon
with missing energy to constrain the cut-off scale.
We shall then conclude in Sec. V.

\section{Calculations}

We concentrate on the spin-2 component of the Kaluza-Klein (KK)
states, which are the excited modes of  graviton in the extra dimensions.
The spin-1 and spin-0 components are less interesting
phenomenologically. We follow the convention  in Ref.\cite{han}.
There are four contributing Feynman diagrams for the process $e^+ e^- \to
Z G$: one 4-point vertex diagram and the other three diagrams are obtained
by attaching the graviton to each leg of $e^+$, $e^-$, and $Z$.
The amplitudes for $e^-(p_1) e^+(p_2) \to Z(k_1) G(k_2)$  are given by:
\begin{eqnarray}
{\cal M}_1 &=& \frac{g \kappa}{2 \cos\theta_W}\, \frac{1}{s - m_Z^2} \,
\bar v(p_2) \gamma_\beta (g_v- g_a\gamma^5) u(p_1) \;
\left( \eta^{\alpha\beta} - \frac{ (k_1+k_2)^\alpha (k_1+k_2)^\beta }{m_Z^2}
\right )
\nonumber \\
&\times & \biggr [
 - 2 k_1\cdot k_2 \, \epsilon^\nu (k_1) \epsilon_{\alpha\nu}(k_2)
+ 2 \epsilon^\nu (k_1) \epsilon_{\mu\nu}(k_2) k_1^\mu {k_1}_\alpha \nonumber \\
&&- 2 \epsilon_\alpha (k_1) \epsilon_{\mu\nu}(k_2) k_1^\mu k_1^\nu
+ 2 k_2\cdot \epsilon(k_1) \epsilon_{\alpha\mu}(k_2) k_1^\mu
            \biggr ] \;, \\
{\cal M}_2 &=& \frac{g \kappa}{2 \cos\theta_W}\, \frac{1}{(k_2-p_2)^2}
\bar v(p_2) \gamma^\mu (\overlay{/}{k}_2 - \overlay{/}{p}_2 )
\overlay{/}{\epsilon}(k_1) (g_v - g_a \gamma^5) u(p_1) \; p_2^\nu \;
\epsilon_{\mu\nu}(k_2) \;, \\
{\cal M}_3 &=& -\frac{g \kappa}{2 \cos\theta_W}\, \frac{1}{(p_1-k_2)^2}
\bar v(p_2) \overlay{/}{\epsilon}(k_1) (g_v - g_a \gamma^5) \,
(\overlay{/}{p}_1 - \overlay{/}{k}_2 )
\gamma^\mu u(p_1) \; p_1^\nu \; \epsilon_{\mu\nu}(k_2)
\; \\
{\cal M}_4 &=& \frac{g \kappa}{2 \cos\theta_W} \; \bar v(p_2) \gamma^\nu
(g_v - g_a \gamma^5) \,u(p_1) \; \epsilon^\mu(W) \, \epsilon_{\mu\nu}(G) \;.
\end{eqnarray}
where $\kappa =\sqrt{16\pi G_N}$, $g_v$ and $g_a$ are the vector and
axial-vector coupling of $Z$ to electron.

We have used the REDUCE program
to evaluate the square of the sum of  amplitudes.
The spins of the $Z$ boson and the graviton are summed using these formulas:
\begin{eqnarray}
\sum_s \epsilon^s_\mu(k_1) \, \epsilon^{s*}_\nu(k_1) &=&
 -\eta_{\mu\nu} + \frac{ {k_1}_\mu  {k_1}_\nu}{ m_Z^2} \;, \nonumber \\
\sum_s \epsilon^s_{\mu\nu}(k_2) \, \epsilon^{s*}_{\rho\sigma}(k_2) &=&
 \hbox{$1\over2$} B_{\mu\nu,\rho\sigma}(k_2) \; \;,
\end{eqnarray}
where $B_{\mu\nu,\rho\sigma}(k_2)$ is given by\cite{han}
$$
B_{\mu\nu,\rho\sigma}(k_2)=
P_{\mu\rho,\nu\sigma}(k_2)+P_{\mu\sigma,\rho\nu}(k_2)
-\hbox{$2\over3$}P_{\mu\nu,\rho\sigma}(k_2) \ ,
$$
\begin{equation}
P_{\mu\nu,\rho\sigma}(k_2)=(\eta_{\mu\nu}-k_{2\mu} k_{2\nu}/m^2)
                           (\eta_{\rho\sigma}-k_{2\rho} k_{2\sigma}/m^2)
\ .
\end{equation}
Here $m$ denotes the mass of the KK state.
The spin-averaged amplitude squared for the process
$e^+ e^- \to ZG$ is given by
\begin{eqnarray}
\overline{\sum} |{\cal M}|^2 &=& \frac{g^2 \kappa^2 (g_v^2 + g_a^2)}
{48 \cos^2 \theta_W u^2 t^2
(s-m_Z^2)^2} \; \Biggr \{
8 m_Z^6 t u [ 3 m^2 ( m^2 -t-u) + 4 t u ] \nonumber \\
&& + 2 m_Z^4 tu [ 27 m^6 - 42 m^4 (t+u) + 15 m^2 ( t^2 + u^2) + 80 m^2 t u
  - 28 ( t^2  u + t  u^2 ) ] \nonumber \\
&& + m_Z^2 [ 3m^8 ( -t^2 -u^2 + 12 t u)
       +6 m^6 (t^3 - 12( t^2 u + t u^2) +u^3 )
       +3 m^4 (-t^4 +14 t^3 u +62 t^2 u^2 + 14 t u^3 - u^4 ) \nonumber \\
&& \qquad       +6 m^2 ( -t^4 u -23(t^3 u^2 + t^2 u^3) -t u^4)
       + 36 (t^4 u^2 + t^2 u^4) + 52 t^3 u^3  ] \nonumber \\
&& + 3ut ( -m^2 +t+u) [ -m^4 +m^2(t+u) -4tu][ 2m^4 -2 m^2 (t+u) +t^2 +u^2 ]
 \;\Biggr \} \;.
\end{eqnarray}
To obtain the total cross section we have to sum over all discrete KK states
with $m_k = 2\pi k/R$ for all $m_k$ below $\sqrt{s} - m_Z$.
Since the mass spacing of these KK states is much smaller than any physical
scales in the problem, it is convenient to convert the discrete sum on $k$
to an integral over $m^2$ as follows:
\begin{equation}
\sum_k \Rightarrow \int dm^2 \; \frac{R^n m^{n-2} }{ (4\pi)^{n/2} \Gamma(n/2)}
\;.
\end{equation}
The size $R$ of the extra dimension, the scale $M_S$ and $G_N$ are
related by
\footnote{The definition of the cut-off scale $M_S$ (we follow Ref. \cite{han})
is different from the cut-off scale $M$ of Ref. \cite{mira}.  $M_S$ is
related to the $M$ by $M_S^4 = 4M^4$ for $n=2$.
}
\begin{equation}
G_NR^nM_S^{n+2}=(4\pi)^{n\over2}\Gamma(n/2) \;.
\end{equation}

For the similar process $e^+ e^- \to \gamma G$ we can reproduce the
expression given in Ref. \cite{mira}
(for unpolarized case):
\begin{eqnarray}
\frac{d\sigma}{d \cos\theta} &=& \frac{\pi \alpha G_N}{ 4 \left( 1-
\frac{m^2}{s} \right )} \; \biggr[ (1+\cos^2\theta) \left( 1 +
(\frac{m^2}{s})^4 \right ) \nonumber \\
&&+ \left( \frac{ 1-3\cos^2\theta +4\cos^4\theta}{1-\cos^2\theta} \right )
\, \frac{m^2}{s} \, \left( 1+ (\frac{m^2}{s})^2 \right ) +
6 \cos^2\theta (\frac{m^2}{s})^2 \biggr ] \;.
\end{eqnarray}

\section{Cross sections and Distributions}

\subsection{$e^+ e^- \to ZG$}

We start with the result for the associated production of graviton with
the $Z$.
The total cross sections for the signal ($e^+ e^- \to ZG$) and background
($e^+ e^- \to Z \nu \bar \nu$) versus $\sqrt{s}$ for $n=2$ and $M_S=2.5,4$
TeV with an angular cut $|\cos \theta_Z|<0.8$
are shown in Fig. \ref{fig1}, where we
also show the bench-mark process $e^+ e^- \to ZH$.
The angular cut, though is not necessary, can help reducing the background.
The background contains three flavors of neutrinos.
The $Z\nu_\mu \bar \nu_\mu$ and
$Z\nu_\tau \bar \nu_\tau$ production mainly comes from $ZZ$ production and
decreases with $\sqrt{s}$, while
$Z \nu_e \bar \nu_e$ can also come from the
$t-$channel $W$-exchange diagrams and so increases with $\sqrt{s}$.
The signal cross section increases with $\sqrt{s}$ because more KK levels
contribute to the cross section.
Due to phase space suppression the signal cross section is rather small at
low $\sqrt{s}$.  Only
until $\sqrt{s}$ reaches at least 0.5 TeV does the signal become significant
relative to the background.

We look more closely at a particular $\sqrt{s}$ and examine kinematic
distributions and see if we can find some ways to improve the
signal-to-background ratio.  We choose $\sqrt{s}=1$ TeV, $n=2$, and
$M_S=2.5$ TeV.
The $ZZ$ production is rather back-to-back while the $ZG$ is
less  back-to-back, and that is why we imposed a cut on the angle of the
$Z$ boson, which will not hurt the signal too much.
In reality, the $Z$ boson actually
decays visibly into either a pair of quarks or leptons with a branching
ratio of $0.8$.  Experimental reconstruction of the $Z$ boson is excellent
and so in this study we only impose a smearing on the $Z$ momentum to
approximate the decay.
We have used an energy resolution of $\delta E/E = 0.2/\sqrt{E}$
for the $Z$ boson, which gives approximately a 15--20 GeV spread of
the reconstructed $Z$ mass.  In Fig. \ref{fig2} we use $M_S=2.5$ TeV.  In
Ref. \cite{wells}, the effective theory with a cut-off at $M_S$ remains
valid up to a few times of $M_S$ as long as unitarity is concerned.
Perhaps, the gravitation interaction already becomes strong at or below the
scale $M_S$.

\begin{table}[t]
\caption{\label{table1}
The signal $S$, background $B$, signal-to-background ratio
$S/B$, and the significance $S \sqrt{{\cal L}}/\sqrt{B}$ for  $e^+ e^- \to
ZG$ at $e^+ e^-$ colliders of various $\sqrt{s}$ with a luminosity ${\cal L}$
of 50 fb$^{-1}$ for $n=2$ and $M_S=2.5$ TeV.
Cuts of $|\cos\theta_Z|<0.8$ and $M_{\rm recoil} >200$ GeV are imposed.}
\medskip
\begin{tabular}{|ccccc|}
$\sqrt{s}$ (TeV) & $S$ (fb) & $B$ (fb)& $S/B$ & $S\sqrt{\cal L}/\sqrt{B}$ \\
\hline
0.5   & 13.1   & 179 & 0.073 &  6.9  \\
0.75  & 48.9   & 334 & 0.15  &  18.9 \\
1.0   & 115    & 452 & 0.25  &  38.1 \\
1.25  & 217    & 543 & 0.40  &  65.8 \\
1.5   & 360    & 613 & 0.59  &  103 \\
\end{tabular}
\end{table}

Another useful variable is the recoil mass, $M_{\rm recoil}$, which is
defined as
\begin{equation}
M_{\rm recoil} = \biggr[ s - 2 \sqrt{s} E_Z + m_Z^2({\rm recons.})
  \biggr]^{1/2} \;.
\end{equation}
In Fig. \ref{fig2}(a) we compare the recoil mass spectrum of the signal with
that of the background.  Obviously, a part of the background comes from
$ZZ$ production and, therefore, the visible $Z$ boson recoiled against
the another $Z$ boson.  It explains a peak around 90 GeV.  The spectrum for
the signal, on the other hand, does not show any peak structure.  This
is a characteristics of the continuous mass spectrum of the KK levels.
A cut of
\begin{equation}
M_{\rm recoil} > 200  \;\;{\rm GeV}
\end{equation}
can remove the $ZZ$ background.  However, the rest of the recoil mass
spectra for the signal and background look very much alike.
Another useful distribution is the transverse momentum of the visible $Z$
boson.  The background falls more rapidly than the signal, which means we
can impose a cut on the transverse momentum to increase the
signal-to-background ratio: see Fig. \ref{fig2}(b).  From Fig. \ref{fig2}(b)
a cut of about 150 GeV may help  but, however, it also cuts away about half
of the signal.  Therefore, we do not impose any cuts on $p_{T_Z}$.

\begin{table}[t]
\caption{\label{table2}
Table showing the limit of $M_S$ that can be reached at $e^+ e^-$
colliders of various $\sqrt{s}$.
The signal $S$, $S/B$, and the significance $S \sqrt{{\cal L}}/\sqrt{B}$
are also shown.  A luminosity ${\cal L}$ of 50 fb$^{-1}$ is assumed.
Part (a) requires both the significance larger than 5 and $S/B >0.1$ for
discovery while part (b) only requires significance larger than 5.
Cuts of $|\cos\theta_Z|<0.8$ and $M_{\rm recoil} >200$ GeV
are imposed.}
\medskip
\begin{tabular}{|ccccc|}
&&&&\\
\multicolumn{5}{|c|}{(a) With $S/B>0.1$ and $S\sqrt{\cal L}/\sqrt{B} >5$ }\\
&&&&\\
$\sqrt{s}$ (TeV) & $M_S$ limit (TeV) &  $S$ (fb) & $S/B$  &
$S\sqrt{\cal L}/ \sqrt{B}$  \\
\hline
0.5   & 2.3   & 17.9 & 0.1  &  9.5  \\
0.75  & 2.8   & 33.4 & 0.1  &  12.9 \\
1.0   & 3.2   & 45.2 & 0.1  &  15.0 \\
1.25  & 3.5   & 54.3 & 0.1  &  16.5 \\
1.5   & 3.9   & 61.3 & 0.1  &  17.5 \\
\hline
\hline
&&&&\\
\multicolumn{5}{|c|}{(b) With $S\sqrt{\cal L}/\sqrt{B} >5$} \\
&&&&\\
\hline
0.5   & 2.7   & 9.5  & 0.053  &  5 \\
0.75  & 3.5   & 12.9 & 0.039  &  5 \\
1.0   & 4.2   & 15.0 & 0.033  &  5 \\
1.25  & 4.8   & 16.5 & 0.030  &  5 \\
1.5   & 5.3   & 17.5 & 0.029  &  5 \\
\end{tabular}
\end{table}

We show the cross sections for the signal $S$ and the background $B$, the
signal-to-background ratio $S/B$, and the significance
$S\sqrt{\cal L}/\sqrt{B}$ at $e^+ e^-$ colliders of energies from 0.5 to 1.5
TeV, with the choice of $n=2$ and $M_S=2.5$ TeV, in Table \ref{table1}.
The nominal yearly luminosity
at these next-linear colliders is of the order of 50 fb$^{-1}$.  With such a
luminosity a decent amount of signal with a large significance is achievable
for $n=2$ and $M_S=2.5$ TeV.  Since the cross section for the signal scales
as $1/M_S^4$ for $n=2$ and $1/M_S^6$ for $n=4$, and even higher power of
$1/M_S$ for larger $n$, so the signal cross section drops rapidly with $n$ or
$M_S$.  For $n>2$ the signal cross section in Table \ref{table1},
being further down by some orders of magnitudes, becomes phenomenologically
uninteresting.  Thus, we concentrate on the case $n=2$.
In Table \ref{table2}, we show the limit on the cut-off scale $M_S$ that
can be obtained by requiring both the $S/B>0.1$ and significance larger than
5.  The limit ranges from about 2.3 TeV to about 4 TeV.
If we only require the significance of the signal larger than 5 to set the
limit, the limit on $M_S$ is slightly better, ranging from 2.7 TeV to about
5.3 TeV.  However, in this case  the signal-to-background ratio
becomes too small.

There are two  other important backgrounds, which are
$e^+ e^- \to W^+ W^-$ and $W^\pm\ell^\mp \nu$ production when the $W$ boson(s)
decays
leptonically.  If the invariant mass of the lepton pair falls within the
$Z$ mass region,
such events will look like a $Z$ boson with missing energies.
Fortunately, these two backgrounds are reducible if (i) we use only
the hadronic decay of the $Z$ boson in our signal, or (ii) restrict the
reconstructed $Z$ mass to a narrower range.
The semi-leptonic decay mode of $WW \to q \bar q' \ell \nu$ will be
relevant as a true background to $ZG$ only if the measurement of the
$q\bar q'$ invariant mass falls within the $Z$ mass region and the lepton
$\ell$ is missing down the beam pipe.
These two conditions  reduce the $W^+ W^-, W^\pm \ell^\mp \nu$ backgrounds
to a negligible level.  Thus, the remaining concern is the leptonic decay
mode of $W^+ W^-$ and $W^\pm \ell^\mp \nu$ if we include the leptonic mode of
$Z$ in our signal.  We performed a calculation of $e^+ e^- \to
W^\pm \ell^\mp \nu$ with $W^\pm \to \ell^\pm \nu \;(\ell =e,\mu)$, and
impose the requirement that the invariant mass of $\ell^+ \ell^-$ falls
within the $Z$ mass region (80--120 GeV) and impose also the other cuts:
$|\cos \theta_Z|<0.8$, $M_{\rm recoil}>200$ GeV.
We found that these $W^+ W^-, W^\pm \ell^\mp \nu$ backgrounds are only
about 5\% of the $Z\nu\bar \nu$ background at $\sqrt{s}=0.5$ TeV and
continuously decreases to 0.5\% at $\sqrt{s}=1$ TeV and finally less than
0.1\% at $\sqrt{s}=1.5$ TeV.  We can, therefore, safely ignore these
backgrounds.

\subsection{$e^+ e^- \to \gamma G$}

Here we repeat the study on the process $e^+ e^- \to \gamma G$ with the
background $e^+ e^- \to \gamma \nu \bar \nu$.  The cross sections for the
signal and background versus $\sqrt{s}$ for $n=2$ and $M_S=2.5$ TeV
with $|\cos\theta_\gamma|<0.9 \;(0.8)$ and $E_\gamma>10$ GeV are shown in
Fig. ~\ref{fig3}.  The signature of the event is a single photon with
missing energy in the final state.  For this signature the signal cross
section easily becomes larger than the background cross section when
$\sqrt{s}$ reaches  about 1 TeV.  Again, we show the recoil mass and
photon transverse momentum spectrum for the signal and the background.
They show similar characteristics as the $ZG$ channel.
In table \ref{table3}, we show the signal and background cross sections for
$n=2$ and $M_S=2.5$ TeV at energies from 189 GeV to 1.5 TeV.  At LEPII 189 GeV
energy, the signal is only about 3.2\% of the background and has a
significance of only 0.65.  Therefore, any effect of the
new gravity interactions with $M_S$ of the order of
2.5 TeV is unnoticeable.  Finally, in table \ref{table4}
we show the limit of $M_S$ that can be obtained for $\sqrt{s}=189$ GeV to
1.5 TeV, by requiring both $S/B>0.1$ and the significance larger than 5.
The limit that can be obtained at LEPII is only about 1.5 TeV while at
$\sqrt{s}=1.5$ TeV it can reach up to about 5.6 TeV.

So overall, the limit on the cut-off scale $M_S$ ranges from about 1.5
TeV to 5.6 TeV for $\sqrt{s}$ from LEPII energy to 1.5 TeV at
$e^+ e^-$ colliders using the associated production of graviton with a
$Z$ boson or a photon. Such a small increase in the limit is due to the
scaling of the cross section with high powers of $1/M_S$.

\begin{table}[t]
\caption{\label{table3}
The signal $S$, background $B$, signal-to-background ratio
$S/B$, and the significance $S \sqrt{{\cal L}}/\sqrt{B}$ for  $e^+ e^- \to
\gamma G$ at $e^+ e^-$ colliders of various $\sqrt{s}$ and a
luminosity ${\cal L}$ of 50 fb$^{-1}$ (0.5 fb$^{-1}$ at LEPII) for $n=2$ and
$M_S=2.5$ TeV.
Cuts of $|\cos\theta_\gamma|<0.9$, $E_\gamma > 10$ GeV,
and $M_{\rm recoil} >200$ GeV (120 GeV at
LEPII and $\sqrt{s}=0.25$ TeV) are imposed.}
\medskip
\begin{tabular}{|ccccc|}
$\sqrt{s}$ (TeV) & $S$ (fb) & $B$ (fb)& $S/B$ & $S\sqrt{\cal L}/\sqrt{B}$ \\
\hline
0.189 & 26.2   & 815 & 0.032  &  0.65  \\
0.25  & 59.5   & 972 & 0.061  &  13.5  \\
0.5   & 352    & 1406 & 0.25  &  66.5  \\
0.75  & 955    & 1652 & 0.58  &  166 \\
1.0   & 1898   & 1791 & 1.06  &  317 \\
1.25  & 3211   & 1878 & 1.71  &  524 \\
1.5   & 4912   & 1939 & 2.53  &  789 \\
\end{tabular}
\end{table}

\begin{table}[t]
\caption{\label{table4}
Table showing the limit of $M_S$ that can be reached at
$e^+ e^-$ colliders of various $\sqrt{s}$
for the process $e^+ e^- \to \gamma G$.
The signal $S$, $S/B$, and the significance $S \sqrt{{\cal L}}/\sqrt{B}$
are also shown.  A luminosity ${\cal L}$ of 50 fb$^{-1}$ (0.5 fb$^{-1}$
at LEPII) is assumed.
It requires both the significance larger than 5 and $S/B >0.1$ for
discovery. Cuts of $|\cos\theta_\gamma|<0.9$, $E_\gamma > 10$ GeV,
and $M_{\rm recoil} >200$ GeV
(120 GeV at LEPII and $\sqrt{s}=0.25$ TeV) are imposed.}
\medskip
\begin{tabular}{|ccccc|}
&&&&\\
\multicolumn{5}{|c|}{(a) With $S/B>0.1$ and $S\sqrt{\cal L}/\sqrt{B} >5$ }\\
&&&&\\
$\sqrt{s}$ (TeV) & $M_S$ limit (TeV) &  $S$ (fb) & $S/B$  &
$S\sqrt{\cal L}/ \sqrt{B}$  \\
\hline
0.189 & 1.5   & 202  & 0.25  &  5  \\
0.25  & 2.2   & 97.3 & 0.1  &  22.0  \\
0.5   & 3.1   & 141 & 0.1  &  26.6  \\
0.75  & 3.9   & 165 & 0.1  &  28.7 \\
1.0   & 4.5   & 179 & 0.1  &  29.9 \\
1.25  & 5.1   & 188 & 0.1  &  30.6 \\
1.5   & 5.6   & 194 & 0.1  &  31.1
\end{tabular}
\end{table}

\section{Comparison with Existing data}

\subsection{$e^+ e^- \to ZG$}

As in the last section, we put in the cross section for the process
$e^+ e^- \to ZH$ as a comparison to the signal of $e^+ e^- \to ZG$.  These
two processes share the same signature when the Higgs boson decays
invisibly \cite{higgs}. Recently,  L3, ALEPH, and DELPHI \cite{aleph}
have searched for the invisibly decaying Higgs in association with a $Z$
boson.  We shall
use their  data to constrain the cut-off scale $M_S$.

The best limit on $M_H$ with the Higgs boson decaying invisibly was obtained by
ALEPH \cite{aleph} in their recent analysis at LEPII with
$\sqrt{s}=189$ GeV.  The 95\%CL lower limit on $M_H$ is 87.5 GeV (only
preliminary) assuming the Higgs boson is produced with the SM strength
in the $ZH$ production and the Higgs boson decays 100\% invisibly.
We calculate the corresponding 95\%CL upper limit on the cross section
$\sigma(e^+ e^- \to ZH)= 0.55$ pb in the leading-order.
With the following two approximations we can apply this limit directly to
$e^+ e^- \to ZG$:  (i)
since the reported selection efficiencies are rather stable for a
fairly wide range of $M_H$ (we refer to the similar analysis done at
$\sqrt{s}=181-184$ GeV, the details at $\sqrt{s}=189$ GeV are not available),
we can treat the limit of $\sigma(e^+ e^- \to ZH)=0.55$ pb as roughly
constant over a wide range of $M_H$. (ii)
The signal of $e^+ e^- \to ZG$ has the same signature as
$e^+ e^- \to ZH$ and so they should have similar efficiencies.
We can then apply this cross section limit to the signal of $ZG$ and we
obtain the 95\%CL limit on $M_S$:
\begin{equation}
M_S \agt 515\;{\rm GeV}\;.
\end{equation}
This estimate is only a rough estimate but should be sufficient for our
purpose.  Even if there were a large difference between the selection
efficiencies for $ZG$ and $ZH$ (which is not likely), the change in
$M_S$ would still be small, because the cross section scales as $1/M_S^4$ for
$n=2$.

\subsection{$e^+ e^- \to \gamma G$}

The LEP Collaborations \cite{lep}
have been searching for single-photon events with
missing energies.  This is an interesting signature for a number of new
physics, including supersymmetry, which are (i) $e^+ e^- \to XY, X \to
Y\gamma$, (ii) $e^+ e^- \to \tilde{G} \tilde{G} \gamma$, and (iii) $e^+ e^-
\to \tilde{G} \tilde{\chi}^0_2, \tilde{\chi}^0_2 \to \tilde{\chi}^0_1 \gamma$.
Their limits on the single-photon cross section are rather model-dependent
because the detection efficiencies depend on model parameters.
Since in our case it is very difficult to fully simulate the experimental
conditions, we simply use their 95\%CL upper limits on production cross
sections.  It means we assume that the detection efficiencies for our graviton
signal are within the ranges of these experiments.
We show the limits on the cut-off scale $M_S$ in table \ref{table6}.
In the table, we can see that the limits reported by the LEP Collaborations
have rather wide ranges, simply because of the wide range of the detection
efficiencies.  It justifies our assumption that the efficiencies of
our graviton signal are easily within the ranges of these LEP experiments.
The limits we obtain are from about 1.2 to 2.2 TeV
(it is consistent with the value of $M=1.2$ TeV obtained in Ref. \cite{mira},
please see the footnote.)

\begin{table}[t]
\caption{\label{table6}
The range of the 95\%CL upper limit on the single photon production
cross section at $\sqrt{s}=183$ GeV (the data by DELPHI is preliminary at
$\sqrt{s}=189$ GeV)  given by the LEP Collaborations and the corresponding
limits of the cut-off scale obtained from these measurements.}
\medskip
\begin{tabular}{cc}
95\%CL limit of single-photon cross section  &  95\%CL limit on $M_S$ \\
\hline
 & \\
L3 & \\
$E\gamma>5$ GeV, $|\cos\theta_\gamma|<0.97$, $\sigma_{95} \sim 0.1-0.5$ pb &
$\sim 1.5 - 2.2$ TeV \\
\hline
& \\
ALEPH & \\
$p_{T_\gamma}>0.0375 \sqrt{s}$, $|\cos\theta_\gamma|<0.95$,
$\sigma_{95} \sim 0.1-0.6$ pb &
$\sim 1.2 - 1.9$ TeV \\
\hline
&\\
OPAL & \\
$x_T=p_{T_\gamma}/E_{\rm beam}>0.05$, $|\cos\theta_\gamma|<0.966$,
$\sigma_{95} \sim 0.075-0.8$ pb &
$ \sim 1.2 - 2.2$ TeV \\
\hline
&\\
DELPHI & \\
$x_\gamma=E_\gamma/E_{\rm beam}>0.06$, $45^\circ<\theta_\gamma<135^\circ$,
$\sigma_{95} \sim 0.3-0.4$ pb &
$\sim 1.25 - 1.35 $ TeV
\end{tabular}
\end{table}

\section{Conclusions}

Excess signals of missing energy events in the process
$e^+e^- \to Z(\gamma) +\overlay{/}{E}$ can provide a useful test for
the low scale gravity with extra space dimension compactified at the size
of mm. This suggests a TeV $e^+e^-$ collider can investigate the
direct graviton production and study the early unification.
Using the available data at LEPII a limit of $M_S \agt 515$ GeV is obtained
in the $ZG$ channel
while $M_S \agt 1.2-2.2$ TeV is obtained in the $\gamma G$ channel.
For the future $e^+ e^-$ linear colliders of energies $0.25-1.5$ TeV and
a luminosity of 50 fb$^{-1}$ $M_S$ can be probed up to about  5.6 TeV.
Although some other non-accelerator physics, e.g., cooling of supernova
\cite{theory},
may give a much stronger constraint on the cut-off scale $M_S$,
the collider signatures, including those considered in this paper,
would provide independent tests for the new gravity interactions.

\section*{\bf Acknowledgments}
This research was supported in part by the U.S.~Department of Energy under
Grants Nos. DE-FG02-84ER40173 and DE-FG03-91ER40674 and
by the Davis Institute for High Energy Physics.
K.C. wants to thank the National Center for Theoretical Sciences in Taiwan
where part of this work was done.