# The quantum optics of gravitational waves

Luca Abrahão<EMAIL_ADDRESS>Department of Physics, Pontifical
Catholic University of Rio de Janeiro, Rio de Janeiro 22451-900, Brazil
Francesco Coradeschi<EMAIL_ADDRESS>Istituto del Consiglio Nazionale
delle Ricerche, OVI, Italy  Antonia Micol Frassino
<EMAIL_ADDRESS>Departament de Física Quàntica i Astrofísica,
Institut de Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès 1,
E-08028 Barcelona, Spain  Thiago Guerreiro<EMAIL_ADDRESS>Department of
Physics, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro
22451-900, Brazil  Jennifer Rittenhouse West<EMAIL_ADDRESS>Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Enrico Junior Schioppa
<EMAIL_ADDRESS>Via della libertà 36, 73025 Martano
(LE), Italy

###### Abstract

By utilizing quantum optics techniques, we examine the characteristics of a
quantum gravitational wave (GW) signature at interferometers. In particular,
we study the problem by analyzing the equations of motion of a GW interacting
with an idealized interferometer. Using this method, we reconstruct the
classical GW signal from a representation of the quantum version of an _almost
classical_ monochromatic wave (a single-mode coherent state), then we discuss
the experimental signatures of some specific, more general quantum states. We
calculate the observables that could be used at future interferometers to
probe possible quantum states carried by the gravitational waves.

##  1 Introduction

Our understanding of gravity is simply and elegantly expressed through the
theory of general relativity (GR). GR is a well-defined _classical_ (as in,
non-quantum) theory and its study offers numerous fruitful avenues of research
in various domains, ranging from cosmology to astrophysics to tabletop
experiments. GR may of course also be viewed as the low energy limit of a
_quantum_ field theory. While a full-blown quantum theory of gravity is as yet
unavailable, it is possible to give an _effective_ quantum description of GR
by treating the spacetime metric as a classical background, and perturbatively
quantizing fluctuations around it [1, 2]. The resulting effective theory is a
quantum field theory of gravitons in curved spacetime, and its non-
renormalizability is harmless at energies much below the Planck scale. Such an
effective description may arise, for instance, as the intermediate energy
limit of a more complete theory. The detection of gravitational waves – long
awaited theoretically and difficult to achieve experimentally – has been a
huge scientific achievement. In the same vein, a natural next step is to
search for experimental or astrophysical observations that can provide
definitive proof of (or exclude) the existence of gravitons.

Recently, there has been an increasing interest in combining characteristic
features of gravity with aspects of quantum optics. Some interesting
applications that would benefit from this union would be the detection of
gravitational waves using cavities [3, 4, 5, 6, 7, 8, 9, 10] or,
alternatively, the analysis of possible quantum effects in the weak gravity
regime [11, 12, 13]. While the former points in the direction of increasing
sensitivity, the latter focuses more on the phenomenological advantages of
considering such experimental setups.

From the phenomenological point of view, a significant benefit of combining
quantum optics and gravity is the possibility of experimentally probing the
existence of the graviton as the mediator of the gravitational force. Indeed,
while empirical evidence supports the existence of quantum particles, such as
photons, gluons, W/Z bosons, and Higgs bosons, which are known to give rise to
all known interactions according to the Standard Model, gravitons have not
been directly observed (and have even been argued to be in principle non-
observable [14]), even though they are likely to be an ingredient of any
quantum theory of gravity.

As already mentioned, the quantum field description of an interacting spin-2
particle, having GR as a classical limit, has been long well-known. Even
though such a theory only works phenomenologically at low energy, a great deal
of knowledge can be gained from just focusing on such a low-energy, weak-
gravity (perturbative) quantum regime 111See also the detailed discussion
regarding the definition of “perturbative quantum gravity” presented in [15]..
From a practical point of view, this means that it makes sense to explore the
possibility that weak gravitational phenomena, such as gravitational waves far
from their sources, may have a quantum nature. Observations of gravitational
waves by the LIGO-Virgo-KAGRA collaboration support classical general
relativity, and quantum corrections, if any, are assumed to be negligible
because graviton shot noise is suppressed by the large occupancy number of
gravitons in a detectable wave [16, 17]. However, if quantum effects
(negligible or not) are present, the argument for their smallness is only
reliable if the gravitational wave is in a mostly classical state, such as a
coherent state. For such states, it is unlikely that any quantum behaviour
will be observed in the near future; however this conclusion does not apply,
at least straightforwardly, to more general quantum states.

Recently, considerable effort has been devoted to understanding possible
signatures at LIGO-like experiments of hypothetical quantum states of gravity
that have no classical analog [16, 17, 18, 19, 20] (see also [21, 15, 22]).
These signatures can be fruitfully investigated by using an effective field
theory description of gravity and using the formal tools of quantum optics.
The picture that comes out gives a powerful perspective on some fundamental
aspects of the possible phenomenology of quantum gravity.

We start by treating both gravity and the instrument one uses to detect it as
quantum objects. We thus describe the dynamics of the system by a Hamiltonian
operator of the form

$\hat{H}=\hat{H}_{0}+\hat{H}_{int}\,.$ (1)

where

$\hat{H}_{0}=\hat{H}_{g}+\hat{H}_{d}\,,$ (2)

gives the dynamics of pure gravity (subscript $g$) and of the detector
(subscript $d$). The details of $\hat{H}_{0}$ are not important for our
discussion, beyond treating gravity as a quantum theory: the states we will
consider are quantum superpositions of plane gravitational wave, that is,
states of many gravitons of well-defined energy. The most interesting
information lives instead in the interaction term $\hat{H}_{int}$, the form of
which is very specific to the system we are studying. In other words, even
though we are going to make use of the formal tools of quantum optics, which
are generic no matter the specific form of $\hat{H}_{int}$, it is precisely
the specific form of $\hat{H}_{int}$ that carries the differences between
electromagnetism and gravity. This will allow us to draw conclusions that
specifically pertain to the quantum weak regime of the gravitational field.

Let us now briefly outline the way it is possible to arrive at these
conclusions with details to follow in the next sections. As a first step, we
formally prepare the gravitational part of our GW-detector system in some
specific state $\ket{\Psi\left(t=0\right)}$ (we will consider several
different set-ups for $\Psi$), then we choose a quantum observable
$\hat{\mathcal{O}}$ and compute the outcome of (classical) measurements
yielding expectation values of the form

$\mathcal{O}^{n}\left(t\right)=\matrixelement{\Psi}{\hat{U}^{\dagger}\left(t\right)\hat{\mathcal{O}}^{n}\hat{U}\left(t\right)}{\Psi}\,,$
(3)

where $U$ is the time evolution operator

$\hat{U}\left(t\right)=e^{-\frac{i}{\hbar}\hat{H}t}\,,$ (4)

whose specific form for our GW-detector setup is known from previous results
[18, 23]. Calculating expectation values for different values of $n$, we can
reconstruct the probability distribution for observable $\hat{\mathcal{O}}$ on
state $\Psi$ in some detail (mean value, variance
$\left[\Delta\mathcal{O}\left(t\right)\right]^{2}=\mathcal{O}^{2}\left(t\right)-\left[\mathcal{O}\left(t\right)\right]^{2}$,
and so on); we show that the moments of certain observables (in particular,
the cavity’s electric field $\mathcal{E}$) can depend strongly on the choice
of $\Psi$, meaning that measurements of $\mathcal{O}^{n}$ can discriminate
between different states $\Psi$ and – potentially – probe whether $\Psi$
possesses inherently quantum features that can’t be mimicked by a purely
classical GW.

The outline of the paper is the following. In Section 2, we make use of a few
important previous results [24, 25] to reduce the complicated starting forms
of $\hat{H}_{d}$ and $\hat{H}_{int}$ to simple expressions. Moving forward, we
will focus on examining the changes in observables when the gravitational
waveform $\ket{\Psi_{g}\left(t=0\right)}$ is prepared in specifically selected
states in the remaining half of the Hilbert space.

In Section 3, we test our approach by applying the equations of motion derived
in Sec. 2 to some example cases. First we examine the simplest allowed states
for the gravitational field: the vacuum and the single-mode coherent state.
Although the former does show some interesting quantum effects, they
ultimately seem too small to be measured at present-day experiments. On the
other hand, the latter serves to validate our program by providing the
expected signal from a classical analog of a single mode coherent state: a
classical plane monochromatic gravitational wave. Additional validation comes
from Sec. 3.3, where we show how the thermal gravitational background, when
treated as an ensemble of mixed states, produces gravitational-induced
decoherence [19]. We thus recover known results from previous work [26].

In Section 4, we comment on the effect of GW fluctuations on electric fields,
and how these can be used to obtain information on the GW state.

In Section 5, we study hypothetical collective quantum states of gravitational
waves with no classic analogue, and analyze their detection at an
interferometer. We find that the gravitational _squeezed vacuum_ gives an
effect on the _noise_ of the intereferometer [16, 18], and that gravitational
_squeezed coherent_ states affect its _signal_ , with some room left for such
an effect being very significant [19]. Needless to say, these conclusions
remain hypothetical before we actually do see such a signal in an experiment.
Nonetheless, and this is the topic of Section 5.1, we give some heuristic
arguments in favour of the existence of squeezed states in gravity by noting
similarities and differences with the case of electromagnetism.

Finally, in Section 6 we give our conclusions.

##  2 Dynamics of the system

We will be working within the framework of linearized gravity, employing a
flat metric background to enable us to consider weak gravity far from sources.
With this choice, the quadratic part of the Einstein-Hilbert action (in
vacuum) in the harmonic gauge $\partial_{\mu}h^{\mu\nu}=0$ reduces to

$S_{EH}=\frac{c^{4}}{32\pi G}\int
d^{4}x\left(\frac{1}{2}\partial_{\mu}h_{\alpha\beta}\partial^{\mu}h^{\alpha\beta}-\frac{1}{4}\partial_{\mu}h\partial^{\mu}h\right)\,,$
(5)

where, as usual, the field $h^{\mu\nu}$ represents the small perturbations of
the otherwise flat metric $\eta_{\mu\nu}$, and $h=\eta^{\mu\nu}h_{\mu\nu}$ is
contracted by the flat metric tensor. We neglect higher order terms (that is,
gravitational self-interactions) throughout, as their impact is negligible in
GWs far from their source.

In the transverse traceless (TT) gauge, we expand the field into Fourier
components as

$h_{ij}^{TT}\left(t,\bm{x}\right)=\int\frac{d^{3}\bm{k}}{\sqrt{(2\pi)^{3}}}\epsilon_{ij}^{\lambda}\left(\bm{k}\right)h_{\lambda}\left(t,\bm{k}\right)e^{i\bm{k}\cdot\bm{x}}\,,$
(6)

where the $\epsilon_{ij}^{\lambda}\left(\bm{k}\right)$ are the tensors for the
two polarization states $\lambda=+,\times$, satisfying the due conditions of
orthonormality
($\epsilon_{ij}^{\lambda}\epsilon_{jk}^{\lambda^{\prime}}=\delta_{ik}\delta^{\lambda\lambda^{\prime}}$),
transverseness ($\epsilon^{\lambda}_{ij}k^{j}=0$) and tracelessness
($\text{Tr}\left[\epsilon^{\lambda}_{ij}\right]=0$). Notice that Greek indices
have become Latin indices, as the time components of the field in the TT gauge
are null ($h_{0\mu}=0$).

With the field expressed in this form, we can execute canonical quantization
by rewriting the field to operators. We promote the Fourier coefficients to
annihilation and creation operators as follows

$h_{\lambda}\left(t,\bm{k}\right)\quad\rightarrow\quad\hat{\mathfrak{b}}^{\lambda}_{\bm{k}}\,,$
(7)
$h^{*}_{\lambda}\left(t,\bm{k}\right)\quad\rightarrow\quad{\hat{\mathfrak{b}}^{\lambda\dagger}_{\bm{k}}}\,,$
(8)

which obey the standard commutation relations (from here on we work in units
where $\hbar=1$),

$\left[\hat{\mathfrak{b}}^{\lambda}_{\bm{k}},{\hat{\mathfrak{b}}^{\lambda^{\prime}\dagger}_{\bm{k}^{\prime}}}\right]=\delta_{\lambda\lambda^{\prime}}\delta^{(3)}\left(\bm{k},\bm{k^{\prime}}\right)\,.$
(9)

The classical field now gets promoted to a quantum field operator and we can
write explicitly

$\hat{h}_{ij}\left(t,\bm{x}\right)=\int\frac{d^{3}\bm{k}}{\sqrt{(2\pi)^{3}}}\left(\sqrt{\frac{8\pi
G}{k}}\epsilon_{ij}^{\lambda}\left(\bm{k}\right)\hat{\mathfrak{b}}^{\lambda}_{\bm{k}}e^{i\left(\bm{k}\cdot\bm{x}-\Omega_{k}t\right)}+h.c.\right)\,.$
(10)

Equation (10) concludes the description of the setup we will be using for the
gravitational part of our problem; we will mostly consider single-mode metric
perturbations in the following, $\sim h_{\mu\nu}e^{ikx}$, corresponding to
planar waves of well-defined frequency. This does not limit the scope of our
calculations since we will be able to express any potential initial (quantum)
state of gravity as a superposition of plane waves.

As far as the detector is concerned, we want to model a GW interferometer.
Arvanitaki and Geraci [27] have shown that already a single-mode Fabry-Pérot
cavity is sensitive to gravitational waves. This can be achieved either by
inserting a nanosphere in the setup [27], or by letting one of the two mirrors
be free of moving, as described by Buonanno and Chen in [24], and by Pang and
Chen in [25]. Let us narrow our focus to the second case of study. Pang and
Chen have demonstrated, by making realistic assumptions (see [25]), that a
complete model of a GW interferometer (including power recycling and signal
recycling mirrors) can be mapped to a single Fabry-Pérot cavity where one
mirror is fixed and the other is free to move. This simplifies the complexity
of the interferometer, and we can work with a single cavity of length $L_{0}$
as the only degree of freedom to describe our model detector. When a GW of
strain $h$ passes through such a cavity perpendicularly to its axis, its
length changes in the following way:

$L_{0}\quad\rightarrow\quad L_{0}\left(1+\frac{1}{2}h\right)\,.$ (11)

This can be seen as a “gravitomechanical” coupling between the GW and the
detector, much like an optomechanical coupling between the electromagnetic
field and a mechanical oscillator [28].

Instead of working with the GW coupled to the detector’s mirror, one can move
to a perspective in which the GW couples directly to the cavity’s
electromagnetic field (the laser beam). When the cavity is stretched, its
resonance frequency changes accordingly as

$\omega_{0}=\frac{n\pi}{L_{0}}\quad\rightarrow\quad\omega=\omega_{0}\,\frac{n\pi}{L_{0}\left(1+\frac{1}{2}h\right)}\,,$
(12)

which can be expanded as

$\omega=\omega_{0}\left(1-\frac{1}{2}h+\mathcal{O}\left(h^{2}\right)\right)\,.$
(13)

The induced frequency shift can be interpreted as producing an effective
coupling between the GW and the electromagnetic field inside the cavity. It
turns out, following [18, 29], that for a $+$ polarized GW propagating in the
$z$ direction perpendicularly to the cavity axis ($x$ direction), and
satisfying $k_{x}L_{0}\ll 1$, such a coupling is represented by an interaction
Hamiltonian of the form

$\hat{H}_{\text{GW}}^{\text{int}}=-\frac{\omega_{0}}{4}\hat{a}^{\dagger}\hat{a}\int\frac{d^{3}\bm{k}}{\sqrt{(2\pi)^{3}}}\left(\sqrt{\frac{8\pi
G}{k}}\hat{\mathfrak{b}}_{\bm{k}}+\text{h.c.}\right)\,.$ (14)

In this expression, $\hat{a}$ and $\hat{a}^{\dagger}$ are the annihilation and
creation operators of the cavity field, which we take to be in a single mode
state for simplicity, while the operators $\hat{\mathfrak{b}}$ are defined in
(7). Notice that, having fixed the polarization of the GW, the $\lambda$ index
has dropped.

Following a procedure which is standard in quantum optics [30], we introduce a
quantization volume $V$ to define a dimensionless quantity
$\hat{b}_{\bm{k}}=\hat{\mathfrak{b}}_{\bm{k}}/\sqrt{V}$, and transform the
continuous integral in Eq. (14) into its discretized version [18]

$\hat{H}_{\text{GW}}^{\text{int}}=-\frac{\omega_{0}}{4}\hat{a}^{\dagger}\hat{a}\sum_{\bm{k}}\left(\sqrt{\frac{8\pi
G}{Vk}}\hat{b}_{\bm{k}}+\text{h.c.}\right)\,.$ (15)

Let us now define, respectively, the single graviton strain $f_{\bm{k}}$, the
opto-gravitational coupling constant $g_{\bm{k}}$, and the dimensionless
coupling $q_{\bm{k}}$, in the following way:

$f_{\bm{k}}=\sqrt{\frac{8\pi G}{Vk}},\quad
g_{\bm{k}}=\frac{\omega_{0}f_{\bm{k}}}{4},\quad
q_{\bm{k}}=\frac{g_{\bm{k}}}{\Omega_{k}}\,.$ (16)

Here, $\bm{k}$ represents the GW frequency for the mode $\bm{k}$, where
$|\bm{k}|=\Omega_{k}$. Since $q_{\bm{k}}$ is a small number by definition, we
will treat it as a perturbative parameter. With these definitions, the
Hamiltonian for the complete system, including the GW, the cavity field, and
their effective interaction, can be defined as

$\hat{H}=\hat{H}_{0}+\hat{H}_{\text{GW}}^{\text{int}}\,,$ (17)

with the free Hamiltonian given by

$\hat{H}_{0}=\omega\hat{a}^{\dagger}\hat{a}+\sum_{\bm{k}}\Omega_{k}\hat{b}_{\bm{k}}^{\dagger}\hat{b}_{\bm{k}}\,,$
(18)

and the interaction Hamiltonian further reduced to

$\hat{H}_{\text{GW}}^{\text{int}}=-\hat{a}^{\dagger}\hat{a}\sum_{\bm{k}}q_{\bm{k}}\Omega_{\bm{k}}\left(\hat{b}_{\bm{k}}+\hat{b}^{\dagger}_{\bm{k}}\right)\,.$
(19)

The derivation of the explicit form of the time evolution operator for the
interaction term (19) is a lengthy but straightforward calculation, using an
approach which is standard in quantum optics. As reported in [18, 23], we can
express the result for a single mode $\bm{k}$ (omitting the index for the sake
of readability) as

$\hat{U}\left(t\right)\ket{\Psi\left(t\right)}=e^{q\hat{a}^{\dagger}\hat{a}\left[\eta\left(t\right)\hat{b}-\eta^{*}\left(t\right)\hat{b}^{\dagger}\right]}e^{iB\left(t\right)\left(\hat{a}^{\dagger}\hat{a}\right)^{2}}\ket{\Psi\left(t\right)}\,,$
(20)

where the time evolution is contained in the definition of

$\displaystyle\eta\left(t\right)$ $\displaystyle=$ $\displaystyle
1-e^{-it}\,,$ (21) $\displaystyle\eta^{*}\left(t\right)$ $\displaystyle=$
$\displaystyle 1-e^{it}\,,$ (22) $\displaystyle B\left(t\right)$
$\displaystyle=$ $\displaystyle q^{2}\left(t-\sin\left(t\right)\right)\,,$
(23)

and the time-evolving state that appears in (20) is defined as
$\ket{\Psi\left(t\right)}=e^{-i\hat{b}^{\dagger}\hat{b}t}\ket{\Psi}$.

##  3 GW state reconstruction

When examining how a GW interacts with an optical cavity, the most suitable
observable is the electric field operator that characterizes the cavity
field’s state

$\hat{\mathcal{E}}=\sqrt{\frac{\omega}{V_{c}}}\frac{\hat{a}+\hat{a}^{\dagger}}{\sqrt{2}}\,.$
(24)

This is indeed the physical quantity that one measures at an interferometer to
produce a detectable signal. Note that here $V_{c}$ denotes the cavity mode
volume.

Now that we have defined the operator, we can proceed to calculate the matrix
elements as outlined in Eq. (3). Our focus will be on the mean value,
specifically when $n=1$. The classical GW signal at an interferometer is
sensed as a variation of the phase of the field quadrature:

$\mathcal{E}\quad\rightarrow\quad\mathcal{E}e^{i\phi}\,,$ (25)

where $\phi$ changes in time. For example, this variation of the phase
produces typical chirp-like signatures observed for binary merger events. Any
result we find that produces a departure of $\phi$ from its classical
behavior, namely which has the form

$\mathcal{E}\quad\rightarrow\quad\mathcal{E}e^{i\left(\phi+\delta\phi\right)}\,,$
(26)

is interpreted as an effect on the _signal_. Contrariwise, if we find a
modification of the form

$\mathcal{E}\quad\rightarrow\quad\mathcal{E}+\epsilon\,,$ (27)

then we are witnessing an effect on the _noise_.

To begin with, we observe that the expression for the time evolution operator
(20) includes an exponential term in $q^{2}$ (the one defined as
$B\left(t\right)$ in (23)). However, since the terms in $q$ is dominant, we
can disregard this term for now (although we will revisit it in Sec. 3.3).
Based on this assumption, we can use the time evolution of the electric field
operator to obtain the following result

$\mathcal{E}\left(t\right)=\sqrt{\frac{\omega}{V_{c}}}\left(\frac{\matrixelement{\Psi\left(t\right)}{\hat{\mathcal{D}}\left[q\eta\left(t\right)\right]\hat{a}}{\Psi\left(t\right)}+h.c.}{\sqrt{2}}\right)\,,$
(28)

where we have defined the operator in parenthesis as

$\hat{\mathcal{D}}\left[q\eta\left(t\right)\right]=e^{q\hat{a}^{\dagger}\hat{a}\left[\eta\left(t\right)\hat{b}-\eta^{*}\left(t\right)\hat{b}^{\dagger}\right]}\,.$
(29)

This operator acts on the gravitational field as a _displacement operator_
whose amplitude is proportional to the optical field’s intensity.

Let us proceed with the selection of specific states within the Hilbert space,
commencing with the detector component. To a very good approximation, the
electromagnetic field inside the cavity is a monochromatic wave at frequency
$\omega$. From the quantum point of view, we can model it as a single mode
coherent state $\ket{\alpha}$ for some complex number $\alpha$. With this
hypothesis, we can easily trace out the detector component of the state
$\ket{\Psi\left(t\right)}$, and we are left with

$\displaystyle\mathcal{E}\left(t\right)=\sqrt{\frac{\omega}{V_{c}}}\left(\frac{\alpha\matrixelement{\Psi_{g}\left(t\right)}{\hat{\mathcal{D}}\left[q\eta\left(t\right)\right]}{\Psi_{g}\left(t\right)}+c.c.}{\sqrt{2}}\right)\,,$
(30)

where now the (quantum) GW wavefunction $\ket{\Psi_{g}}$ enters into play
independently.

###  3.1 Vacuum state

Now, let us consider the possibility of preparing GW states in specific
quantum states, with the simplest one being the vacuum state. The vacuum state
for the generic mode $\bm{k}$ (that is, a state with no gravitons of energy
$\bm{k}$) can be written as:

$\ket{\Psi_{g}}=\ket{0_{\bm{k}}\left(t\right)}=e^{-i\hat{b}^{\dagger}_{\bm{k}}\hat{b}_{\bm{k}}\Omega_{k}t}\ket{0_{\bm{k}}}=\ket{0}\,,$
(31)

where we are following the standard harmonic oscillator convention of writing
the eigenstates of the number operator $a_{\bm{k}}^{\dagger}a_{\bm{k}}$ as
$\ket{n_{\bm{k}}}$, and $\ket{0}$ is the state with no gravitons at all.
Therefore, to obtain the mean field we must evaluate the following expression:

$\displaystyle\mathcal{E}\left(t\right)=\sqrt{\frac{\omega}{V}}\left(\frac{\alpha\prod_{\bm{k}}\matrixelement{0}{\hat{\mathcal{D}}\left[q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}+c.c.}{\sqrt{2}}\right)\,.$
(32)

The matrix element is easily calculated by considering that the vacuum can be
seen as the coherent state with $\alpha=0$. The displacement operator acting
on such a state gives

$\hat{\mathcal{D}}\left[q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]\ket{0}=\ket{q_{\bm{k}}\eta\left(\Omega_{k}t\right)}\,.$
(33)

Using the normalization condition for coherent states,222 The normalization
condition is that given two coherent states characterized by complex numbers
$\alpha$ and $\beta$, one has
$\innerproduct{\alpha}{\beta}=e^{-\frac{1}{2}\left(\left|\alpha\right|^{2}+\left|\beta\right|^{2}-\alpha^{*}\beta-\alpha\beta^{*}\right)}\,.$
(34) we can rewrite the previous expression as

$\displaystyle\matrixelement{0}{\hat{\mathcal{D}}\left[q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}=\innerproduct{0}{q_{\bm{k}}\eta\left(\Omega_{k}t\right)}=e^{-\frac{1}{2}q^{2}_{\bm{k}}\left|\eta\left(\Omega_{k}t\right)\right|^{2}}\,,$
(35)

and thus define

$D\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\prod_{\bm{k}}\matrixelement{0}{\hat{\mathcal{D}}\left[q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}=e^{-\frac{1}{2}\sum_{\bm{k}}q^{2}_{\bm{k}}\left|\eta\left(\Omega_{k}t\right)\right|^{2}}\,.$
(36)

This quantity (36) was calculated in [18] by introducing both an infrared and
an ultraviolet cutoff to avoid divergence and by noting that by simple algebra

$\left|\eta\left(\Omega_{k}t\right)\right|^{2}=2\left[1-\cos\left(\Omega_{k}t\right)\right]\,.$
(37)

The final result for the the mean field, in the case of a vacuum state, is

$\mathcal{E}\left(t\right)=\sqrt{\frac{2\omega}{V_{c}}}D\Re{\alpha}\,.$ (38)

We can express this as equation (27), which represents the effect of the
detector’s noise. Essentially, we have determined how the gravitational vacuum
affects the interferometer’s sensitivity curve. However, it’s important to
note that this impact is orders of magnitude below any reasonably achievable
sensitivity limit [16, 18]. In fact, it’s even lower than the theoretical
quantum thermal noise of gravity across a wide range of frequencies, as
demonstrated in [12]. Ultimately, this effect is impractical to measure.

When analyzed more closely, however, we do find an interesting side result. We
performed this calculation after we had neglected the subdominant $q^{2}$ term
in equation (20). Had we retained it, we would have arrived to the surprising
conclusion that the gravitational vacuum induces squeezing of the cavity field
[18]. Once again, after plugging in the right numbers, this effect turns out
to be practically unmeasurable. However, in general, we find that,
unsurprisingly, to achieve a measurable effect in an experiment where gravity
couples to optical observables, one needs to start from gravity modes
populated with a large mean number of gravitons, as we will see in the
following.

###  3.2 Coherent state

The simplest state in which we can collect a large mean number of gravitons
together is a coherent state. A calculation similar to the one we performed to
arrive at equation (38) can be carried out to derive the effect on the
cavity’s field quadrature of a single mode coherent state of gravity. We write
such state as

$\ket{\Psi_{g}}=\begin{cases}\ket{he^{i\Omega_{GW}t}}&\text{if }k=k_{GW}\,,\\\
\ket{0}&\text{otherwise}\,.\end{cases}$ (39)

Here $h$ is real (we set the phase to zero, for simplicity) and is indeed a
large number linked to the population of the mode. Now we must calculate

$\mathcal{E}\left(t\right)=\sqrt{\frac{\omega}{V_{c}}}\left(\frac{\alpha\matrixelement{he^{i\Omega_{GW}t}}{\hat{\mathcal{D}}\left[q_{GW}\eta\left(\Omega_{GW}t\right)\right]}{he^{i\Omega_{GW}t}}\prod_{\bm{k}\neq\bm{k}_{GW}}\matrixelement{0}{\hat{\mathcal{D}}\left[q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}+c.c.}{\sqrt{2}}\right)\,.$
(40)

When evaluating the $\bm{k}\neq\bm{k}_{GW}$ product, we would obtain a product
of terms in $\sim e^{\frac{1}{2}q^{2}_{\bm{k}}}$ which - because of the small
value of $q_{\bm{k}}$ \- are all of order 1, much in the same way as we
calculated expression (36). This means we can neglect all but

$\displaystyle\mathcal{E}\left(t\right)\sim\sqrt{\frac{\omega}{V_{c}}}\left(\frac{\alpha\matrixelement{he^{i\Omega_{GW}t}}{\hat{\mathcal{D}}\left[q_{GW}\eta\left(\Omega_{GW}t\right)\right]}{he^{i\Omega_{GW}t}}+c.c.}{\sqrt{2}}\right)\,.$
(41)

Again, the full calculation is straightforward, arriving at

$\mathcal{E}\left(t\right)\sim e^{i2qh\sin\Omega t}\,.$ (42)

This expression is of the form (26), and it tells us that we are measuring a
signal oscillating in phase with the GW. We have thus recovered the classical
GW signal from a representation of the quantum analog of a classical
monochromatic wave: the single-mode coherent state. This gives us a first
validation check of our program.

###  3.3 GW-induced decoherence

In Sec. 3, we initially neglected the $q^{2}$ terms in the time evolution
operator (20). This choice was justified as these terms give rise to effects,
such as the aforementioned squeezing of the cavity field induced by the
gravitational quantum vacuum, that is by far dominated by effects that are
linear in $q$. However, it is worth paying some more attention to such
$q$-quadratic terms, as it turns out they produce gravity-induced decoherence.
Although the effect is weak and difficult to measure, it serves as a secondary
validation check by linking our setup to established results.

To see this, let us now repeat the calculations presented in Sec. 3 but this
time by preparing our system as an electromagnetic (EM) qubit interacting with
the gravitational vacuum

$\ket{\Psi\left(0\right)}=\frac{\ket{0}_{\text{EM}}+\ket{N}_{\text{EM}}}{\sqrt{2}}\otimes\ket{0}_{\text{GW}}$
(43)

where, once again, $\ket{N}_{\text{EM}}$ ($\ket{N}_{\text{GW}}$) denotes a
state with $N$ photons (gravitons). We can now perform the following three
steps:

  1. 1.

Evolve the state using the simplified time evolution operator as expressed in
equation (29);

  2. 2.

write the total density matrix $\rho\left(t\right)$ associated to the time-
evolving state;

  3. 3.

calculate the density matrix $\rho_{EM}\left(t\right)$ associated to the EM
subsystem, by tracing out the GW degrees of freedom.

When carrying out the calculations, we arrive at the following denity matrix
[19]:

$\rho_{\text{EM}}\left(t\right)=\begin{pmatrix}\frac{1}{2}&\rho_{01}\\\
\rho_{01}^{*}&\frac{1}{2}\end{pmatrix}\,,$ (44)

where

$\rho_{01}=\innerproduct{0}{qN\eta}=e^{-\frac{1}{2}q^{2}N^{2}\left|\eta\right|^{2}}\,.$
(45)

The presence of such time dependent off-diagonal terms (via the time
dependence of $\eta\left(t\right)$) shows indeed that the $q^{2}$ component of
the time evolution operator is inducing decoherence.

This can be taken farther. By replacing the vacuum $\ket{0}_{\text{GW}}$ in
equation (43) with a GW single mode coherent state $\ket{\alpha}_{\text{GW}}$,
and repeating the same calculations, we obtain an EM density matrix of the
same form as (44), but now with

$\rho_{01}=e^{-\frac{1}{2}\left[q^{2}N^{2}\left|\eta\right|^{2}+qN\left(\eta^{*}\alpha-\eta\alpha^{*}\right)\right]}\,.$
(46)

When we extend it to a single mode GWs in a thermal state, we obtain

$\rho_{01}=e^{-\frac{1}{2}q^{2}N^{2}\left|\eta\right|^{2}\left(1+\overline{n}\right)}$
(47)

where $\overline{n}$ is the mean number of gravitons333 To arrive at equation
(47), one should remember that the density matrix of a thermal state with mean
number of gravitons $\overline{n}$, can be related to the continuum of
coherent states $\ket{\alpha}$ as
$\rho=\int\frac{d^{2}\alpha}{\pi\overline{n}}e^{-\frac{\left|\alpha\right|^{2}}{\overline{n}}}\ket{\alpha}\bra{\alpha}\,.$
(48) . This result is useful in that it finally allows us to consider an
ensemble of modes in thermal states. In such a case, we would need to
reintroduce the state index $\bm{k}$, and calculate

$\displaystyle\prod_{\bm{k}}e^{-\frac{1}{2}q_{\bm{k}}^{2}N^{2}\left|\eta{\bm{k}}\right|^{2}\left(1+\overline{n}_{\bm{k}}\right)}=e^{-\frac{1}{2}N^{2}\sum_{\bm{k}}q_{\bm{k}}^{2}\left|\eta{\bm{k}}\right|^{2}\left(1+\overline{n}_{\bm{k}}\right)}$
(49)

For large temperatures $T$, one approximates $(1+\overline{n}_{\bm{k}})$ with
$\overline{n}_{\bm{k}}$, and $T$ simply counts the number of gravitons in each
mode as

$\overline{n}_{\bm{k}}=\frac{k_{B}T}{\Omega_{\bm{k}}}\,.$ (50)

where $k_{B}$ is Boltzmann’s constant. Using the explicit forms of
$q_{\bm{k}}$ and $\eta_{\bm{k}}$, and averaging over the Bose-Einstein
distribution (see [19]), we arrive at an expression of the form

$\rho_{01}\approx e^{-\Gamma t}\,,$ (51)

where

$\Gamma\propto k_{B}T\left(\dfrac{\Delta E}{E_{\rm pl}}\right)^{2}\,,$ (52)

and $\Delta E=N\omega$ is the energy of the state $\ket{N}_{\text{EM}}$, in
accordance with previous results on gravitational-induced decoherence [26].

##  4 GW-induced electric field fluctuations

Before we continue discussing other gravity wave states, it is important to
comment on the practicality of measuring deviations from the classical theory
with elecrtomagnetic probes. In a previous work [19], we showed that the
measurement problem for our quantum gravitational states can be stated in
terms of photon-number tomography of the optical mode that interacts with the
wave. In particular, we showed that if the GW state is Gaussian, it can be
reconstructed from experimentally accessible data that can be measured from
non-classical (yet macroscopic) observables. Here, we would like to point out
that information on the GW states can also be obtained from field (homodyne)
measurements, which is more practical than a photon-number resolving
measurement experiments.

Reconstruction of the second moments of a general GW state $|\Psi\rangle$ can
be achieved by measuring expectation values of the form
$\langle\Psi|\mathcal{D}(nq\eta(t))|\Psi\rangle$, where $n$ is an integer
[19]. General reconstruction of the first and second moments can be achieved
if we measure these expectation values for $n=1,2,3$. For coherent states,
this can be done by measuring the first three moments of the electric field.

The variance of the field is
$\Delta\mathcal{E}=\langle\mathcal{E}^{2}\rangle-\langle\mathcal{E}\rangle^{2}$
and we compute $\langle\mathcal{E}^{2}\rangle$ by tracing out the detector
component. Noticing that

$\mathcal{E}^{2}(t)=\frac{\omega}{2V_{c}}\left(2a^{\dagger}a-1+a^{2}\mathcal{D}(-2q\eta(t))+\left(a^{\dagger}\right)^{2}\mathcal{D}^{*}(-2q\eta(t)\right),$
(53)

we find that for general states,

$\langle\mathcal{E}^{2}(t)\rangle=\frac{\omega}{2V_{c}}\left(|\alpha|^{2}-1+\alpha^{2}\matrixelement{\Psi_{g}(t)}{\hat{\mathcal{D}}(2q\eta(t)}{\Psi_{g}(t)}+c.c.\right)\,.$
(54)

Assuming that the gravitational wave is initially a vacuum state, we have

$\langle\mathcal{E}^{2}\rangle=\frac{\omega}{2V_{c}}\left(\alpha^{2}\matrixelement{0}{\hat{\mathcal{D}}\left[2q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}+{\alpha^{*}}^{2}\matrixelement{0}{\hat{\mathcal{D}}^{*}\left[2q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}-1+|\alpha|^{2}\right)\,.$
(55)

Analogously,

$\matrixelement{0}{\hat{\mathcal{D}}\left[2q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}=\innerproduct{0}{2q_{\bm{k}}\eta\left(\Omega_{k}t\right)}=e^{-{2}q^{2}_{\bm{k}}\left|\eta\left(\Omega_{k}t\right)\right|^{2}}\,,$
(56)

and we can define the quantity,

$D_{2}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\prod_{\bm{k}}\matrixelement{0}{\hat{\mathcal{D}}\left[2q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}=e^{-2\sum_{\bm{k}}q^{2}_{\bm{k}}\left|\eta\left(\Omega_{k}t\right)\right|^{2}}\,.$
(57)

Using the previous equations, the mean square value of the electric field
interacting with the GW (55) becomes

$\langle\mathcal{E}^{2}\rangle=\frac{\omega}{2V_{c}}\left[2D_{2}\real(\alpha^{2})-1+|\alpha|^{2}\right]\,.$
(58)

As was shown in [19], to determine the second order correlation functions of
the GW, we need to evaluate terms proportional up to
$\matrixelement{0}{\hat{\mathcal{D}}\left[3q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}$.
In order to achieve that, we need to go up to the third order moment, the
skewness ($\Delta s$), defined as

$\Delta
s=\langle\mathcal{E}^{3}\rangle-\langle\mathcal{E}\rangle\langle\mathcal{E}^{2}\rangle+2\langle\mathcal{E}\rangle^{3}\,.$
(59)

Note that all the terms in the above definition have already been computed,
except for $\langle\mathcal{E}^{3}\rangle$. We now turn our attention to this
particular term. Notice that

$\displaystyle\mathcal{E}^{3}(t)$ $\displaystyle=$
$\displaystyle\left(\frac{\omega}{2V_{c}}\right)^{3/2}\left\\{a^{3}\mathcal{D}[-3q\eta(t)]+(2a^{\dagger}aa-3a)\mathcal{D}[-q\eta(t)]\right.$
(60)
$\displaystyle\left.+(2a^{\dagger}a^{\dagger}a-3a^{\dagger})\mathcal{D}^{*}[-q\eta(t)]+(a^{\dagger})^{3}\mathcal{D}^{*}[-3q\eta(t)]\right\\}\,.$

For the initial vacuum state we find

$\langle\mathcal{E}^{3}(t)\rangle=\left(\frac{\omega}{2V_{c}}\right)^{3/2}\left[\alpha^{3}\matrixelement{0}{\hat{\mathcal{D}}\left[3q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}+\left(|\alpha|^{2}\alpha-3\alpha\right)\matrixelement{0}{\hat{\mathcal{D}}\left[q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}+c.c\right],$
(61)

and

$\langle\mathcal{E}^{3}\rangle=\left(\frac{\omega}{2V_{c}}\right)^{3/2}\left[2D_{3}\real(\alpha^{3})-2D\real(\alpha)(3-|\alpha|^{2})\right]\,,$
(62)

where $D_{3}$ is defined as

$D_{3}\stackrel{{\scriptstyle\mathclap{\tiny\mbox{def}}}}{{=}}\prod_{\bm{k}}\matrixelement{0}{\hat{\mathcal{D}}\left[3q_{\bm{k}}\eta\left(\Omega_{k}t\right)\right]}{0}=e^{-\frac{9}{2}\sum_{\bm{k}}q^{2}_{\bm{k}}\left|\eta\left(\Omega_{k}t\right)\right|^{2}}\,.$
(63)

With this, we see that the quantities $D,D_{2}$ and $D_{3}$ can be obtained
from measurements of the first three moments of the electric field, which in
turn can be measured via homodyne detection. In possession of these
quantities, we can then reconstruct the first and second moments of GW vacuum
fluctuations. This calculation can easily be extended to the case of a
coherent sate. GW-induced electric field fluctuations can also be calculated
for other states following the recipe introduced above, although in general
perfect state reconstruction cannot be achieved from measurements of the
electric field moments alone. These fluctuations could, however, lead to
interesting signatures [31].

##  5 Squeezed gravity

So far, we have considered the vacuum, coherent states, and thermal states for
the gravity modes. While interesting theoretical insight can be gained by
studying these cases, none of them will yield measurable quantum effect under
our assumptions. It is still interesting (and, indeed, a required sanity check
of our approach) that in the case of a highly populated coherent state, we
recover the classical signal even though we started from a completely quantum
picture.

The natural next step in our investigation is asking the question: can any
quantum states of gravity exist that have no classical analog and have a
chance of yielding a detectable effect?

We give a tentative answer to this question by drawing from quantum optics
experience and investigating what happens if we _assume_ that gravity can live
in a _squeezed state_ – the analogue of squeezed states of light that are
routinely produced in optics laboratories. Before going on, we stress that
this assumption implies the existence of some mechanism that does put gravity
into such a state, which is as yet unknown for GWs in the LIGO band. To be
more precise, there is at least one candidate in the category: an established
consensus exists on the hypothesis that inflation might indeed have squeezed
gravity at primordial times; however, once again – much like in the cases of
vacuum corrections or gravitational decoherence – this effect leads to a weak
signal [12]. Nonetheless, calculating the potential signature of a
gravitational squeezed vacuum on our model detector is instructive. After all,
we cannot a priori exclude mechanisms, other than inflation, that could
produce squeezing (see section 5.1 for more detailed discussion on this
point).

Let us thus prepare gravity in a squeezed state, which we may model as a mode
of the form

$\ket{\Psi_{g}}=\ket{\beta e^{2i\Omega
t}}=\hat{S}\left(\beta\right)\ket{0}\,,$ (64)

where the complex number $\beta$ is the squeezing parameter and
$\hat{S}\left(\beta\right)$ is the squeezing operator, as defined in
textbooks. Now the matrix element for the mean electric field (28) contains
terms of the form

$\alpha\matrixelement{0}{\hat{S}^{\dagger}\left(\beta\right)\hat{\mathcal{D}}\left[q\eta\left(t\right)\right]\hat{S}\left(\beta\right)}{0}\,,$
(65)

and after some calculations, which were slightly more involved but nonetheless
straightforward, we arrive at our result:

$\mathcal{E}\left(t\right)\sim
2\alpha\left[1-8q^{2}e^{2\left|\beta\right|}\sin^{4}\left(\frac{\Omega
t}{2}\right)\right]\,.$ (66)

This term is of the form of equation (27) and it thus tells us that we found
an effect on the _noise_ : a squeezed gravitational vacuum would manifest
itself at an interferometer as an additional, oscillating term, in the noise
spectrum of the instrument [16, 17, 18]. The most interesting part is that the
amplitude of such an oscillating term contains an exponential factor. Such a
factor could behave as an enhancement term to the noise, depending on the
magnitude of the squeezing parameter $\beta$. The magnitude of $\beta$, in
turn, depends on the details of the source dynamics, which at this moment we
cannot foresee. Nonetheless, if mechanisms exist in nature that would produce
such an exponentially enhanced effect, we cannot exclude that future, more
sensitive detectors could actually see it. In the subsequent section, we shall
delve deeper into this topic.

Something even more interesting happens when we prepare gravity in a
_squeezed-coherent_ state. For a single mode, this would mean

$\ket{\Psi_{g}}=\hat{S}\left(\beta\right)\hat{\mathcal{D}}\left(he^{i\Omega_{GW}t}\right)\ket{0}\,,$
(67)

where we have used the same notation as in eq. (39). Such a state would
represent a squeezed quantum gravitational wave mode propagating from the
source to the detector. The electromagnetic analog would be a squeezed laser
beam, which we are able to generate and propagate in a laboratory.

After calculating the electric field matrix element as usual, for the first
time we find an effect of the type described by equation (26) which deviates
from the classical behavior. Specifically, after rewriting $\beta=re^{i\xi}$,
we get

$\leavevmode\nobreak\ \delta\phi=2hq\left[\sin\left(\Omega
t\right)\cosh\left(r\right)+\sin\left(2\xi-\Omega
t\right)\sinh\left(r\right)-\sin\left(2\xi\right)\sinh\left(r\right)\right]$
(68)

Thus, an exponentially enhanced or suppressed effect, but this time on the
_signal_. Once again, the magnitude of the effect depends on the dynamics at
the source, which at this point remains unknown. Nonetheless, equation (68)
clearly shows how a purely quantum effect (squeezing) involving a state with a
macroscopically high number of gravitons $\left|h\right|$, would produce a
signal which can be exponentially enhanced – even to order one – and can thus
be detectable with current or near future technology [19].

###  5.1 Are squeezed gravitational waves produced in nature?

In the framework in which we are working, where GR is seen as a classical
limit of an intermediate-energy effective quantum field theory of
(self)-interacting gravitons, squeezed gravitational waves definitely exist
_theoretically_ , that is, they are allowed states in the Hilbert space of the
theory.

The question however remains open on whether there are any realistic
astrophysical sources that could produce GWs with a sizable (i.e. potentially
measurable) amount of squeezing. While the aforementioned hypothesis on the
squeezing of the relic gravitational background induced by inflation seems to
be widely accepted, it is predicted to be too small to be observed at
gravitational interferometers. Drawing on our experience on quantum optics, we
can outline two basic conditions that we can expect to be met in order to have
measurable squeezing in a physical process:

  1. 1.

The process should involve states characterized by a high – macroscopic –
occupancy number;

  2. 2.

The resulting state should be capable of propagating (ideally) undisturbed
from source to detector.

Both conditions are naturally met by the processes that produce the
gravitational waves that we are able to observe. First of all, there is no
doubt that mergers of black holes (or really any other sources of high-
intensity GWs) involve macroscopic number of gravitons (assuming, of course,
that gravitons do exist). Furthermore, gravity is naturally weak-interacting
at low energy, which means a GW basically stops interacting as soon as it
leaves its source, traveling the distance to the detector nearly undisturbed.
Note that this contrasts sharply with the behaviour of squeezed sources in
electromagnetism: in quantum optics laboratories, squeezed beams of light are
commonly produced by using intense laser beams, which are prone to losing
coherence because of the high probability of interaction with any medium
present in the laboratory. Transporting the quantum state of a laser beam
(e.g. a squeezed coherent state) over long distances from source to detector
is thus challenging, and can only be achieved with great effort in a
laboratory. Gravitational waves can be expected to be free of this problem.

However, even if merger events (or other sources of strong GWs) do satisfy the
minimum requirements to be candidate producers of squeezed gravitational
waves, it is a challenge to understand if they _actually_ produce such states.
Answering this question appears to involve the theoretical treatment of
quantum effects at strong GR regimes, which is beyond our current capacities.
However, while we are not (yet) capable of providing a formal argument in
favour of the production of squeezed GWs at mergers and similar events (but
for some discussion on the topic, see [19]), we would still like to argue,
more heuristically, that this possibility does deserve further investigation.
Let us think once again of the analogy with quantum optics. Squeezed states of
light are produced in the lab by making intense laser beams interact with
anisotropic crystals. In fact, the mechanism that turns a coherent laser beam
into a squeezed coherent laser beam involves the interaction of intense light
with a highly non-linear optical medium. As electromagnetism is a linear
theory at the classical level (nonlinear effects are only manifest in the
quantum regime), producing squeezed light is in a sense an “exotic” process,
in that it needs well-controlled laboratory conditions and is not found
spontaneously in nature. Compare this with the case of gravity. Contrary to
Maxwell’s equations, Einstein’s equations are already nonlinear at the
classical level, and nonlinear effects affect phenomena taking place in strong
regimes [32, 33]. This means that squeezing of macroscopic gravitational waves
might well be a natural effect, provided the source is strong enough – which
is definitely true in the case of mergers. Strong nonlinear astronomical
sources, perhaps those already known to emit GWs, seem therefore reasonable
candidates to investigate. A recent step in this direction has been proposed
in [20] where nonlinear effects present in black hole’s ringdown [32, 33, 34,
35, 36, 37] have been considered.

##  6 Conclusions

In this paper, we have shown explicitly how to make use of quantum optics in
order to derive phenomenological results in quantum gravity in the weak
gravity regime. We focused on the treatment of the problem from the point of
view of the equations of motion and applied the result to a model GW
interferometer interacting with a few possible quantum states of gravity. We
have examined various quantum states ranging from the basic vacuum state to
the coherent state, and ultimately concluded with an evaluation of squeezed
states. Among the ones we evaluated, _squeezed-coherent_ gravitational waves
have proven to be the most promising candidates for providing potentially
detectable quantum aspects of gravity. The findings of Sec. 5.1, however
basic, together with the results reported in Sec. 5, show how a squeezed
coherent GW could produce an effect on the _signal_ of a GW interferometer,
and that such an effect has the potential of being of order 1. This indicates
to us that further research on the topic - especially regarding the existence
of possible sources - has promise and is worth pursuing.

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