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\begin{document}

%\title{Quantum principle of sensing gravitational waves: From the zero-point spacetime fluctuations to the cosmological stochastic background}

\title{Quantum principle of sensing gravitational waves: From the zero-point fluctuations to the cosmological stochastic background of spacetime}

\author{Diego A. Qui\~nones}
\email{pydaqo@leeds.ac.uk}
\affiliation{School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom}


\author{Teodora Oniga}
\email{t.oniga@abdn.ac.uk}
\affiliation{Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom}

\author{Benjamin T. H. Varcoe}
\email{b.varcoe@leeds.ac.uk}
\affiliation{School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom}

\author{Charles H.-T. Wang}
\email{c.wang@abdn.ac.uk}
\affiliation{Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom}

%\date{}


\begin{abstract}
We carry out a theoretical investigation on the collective dynamics of an ensemble of correlated atoms, subject to both vacuum fluctuations of spacetime and stochastic gravitational waves. A general approach is taken with the derivation of a quantum master equation capable of describing arbitrary confined nonrelativistic matter systems in an open quantum gravitational environment.
It enables us to relate the spectral function for gravitational waves and the distribution function for quantum gravitational fluctuations and to indeed introduce a new spectral function for the zero-point fluctuations of spacetime.
The formulation is applied to two-level Rydberg-like identical bosonic atoms in a cavity, leading to a gravitational transition mechanism through certain quadrupole moment operators. For a large number $N$ of such atoms, we find their equilibrium state to satisfy the Boltzmann distribution. The overall relaxation rate before reaching equilibrium is found to scale collectively with $N$. However, we are able to identify certain states whose decay and excitation rates with stochastic gravitational waves and vacuum spacetime fluctuations amplify more significantly with a factor of $N^2$.
Using such favourable states as a means of measuring both conventional stochastic gravitational waves and novel zero-point spacetime fluctuations,
we determine the theoretical lower bounds for the respective spectral functions.
Finally, we discuss the implications of our findings on future observations of gravitational waves of a wider spectral window than currently accessible.
Especially, the possible sensing of the zero-point fluctuations of spacetime could provide an opportunity to generate initial evidence and further guidance of quantum gravity.
\end{abstract}

% Research Areas: Quantum gravity, Open quantum systems & decoherence

\maketitle

\section{Introduction}
\label{sec:intr}

Recently there has been considerable interest in experimenting with gravity using quantum instrument\refs{Amelino2013, Pfister2016}, to better explore its theory, predictions, and applications in uncontested regimes\refs{Wang2006a, Kiefer2013, Oniga2016a}. This trend has been established through laboratory measurements of the gravitational acceleration by ultra cold atom interferometers with accuracies exceeding that achievable by conventional gravimeters\refs{Muller2008}. The realization of such quantum gravity and gravity gradient sensors as portable devices would herald a new level of quantum applications in engineering and technology\refs{Carraz2014}.

For physicists, it has been hoped that major progress across a range of areas could be enabled by the availability of precision quantum gravity sensors. These include testing of the equivalence principle as a foundation of general relativity in the quantum domain\refs{gauge2009}, confronting macroscopic quantum behaviour with a microgravity environment\refs{Muntinga2013}, and assessing a long range effect such as gravitational time dilation on quantum coherence\refs{Pikovski2015}. Above all, quantum experiments may promise fundamental breakthroughs by offering a unique opportunity of accessing phenomena from quantum gravity and Planck scale physics that would otherwise have required an inconceivable $10^{19}$ GeV GUT-scale particle collider\refs{Amelino2000, Wang2006, Lamine2006, Everitt2013}.

There are a number of counterintuitive effects unavailable in the classical domain to justify novelties and advantages of quantum measurements. For Bell's inequality related experiments\refs{PPT1}, quantum entanglement and nonlocality are the key. More relevant to precision measurements, in the case of atom interferometry metrology, the short matter wavelength underpins a fine scale resolution. When atom interferometry is applied to gravity measurements including gradiometry and spacetime fluctuations, the slowness of the motion for cold atoms contributes to a longer interaction time to build up effects in terms of phase shift due to acceleration and decoherence due to granulation respectively\refs{gauge2009}.
%However, it is not without the ability to control and manipulate the quantum dynamics of atoms and photons that these quantum techniques can be exploited for precision measurements.
%Quantum correlations are responsible for the collective quantum behaviours for

For quantum systems with a large number of correlated particles, nontrivial collective behaviours can arise\refs{Oniga2016b, Oniga2016c}. An early remarkable example shows up in the Hanbury Brown and Twiss's intensity interferometry\refs{HBT1956}. Collective behaviours can give rise to significant amplifications of quantum effects associated with a large number of correlated particles, as exemplified in Dicke's seminal superradiance\refs{Dicke1954}. In this paper, we will analyze previously unexploited collective quantum interactions between a large number of particles with gravitational waves that may be relevant for their detections beyond the existing observation windows e.g. in the frequency domain of fundamental, astronomical and cosmological interest\refs{LIGO2009, LIGO2016b, GWC2016, SGWB2016}.

In particular, we consider stochastic gravitational waves of higher frequencies than the existing detection range, that may have a primordial origin and carry imprints of post-inflation structure formation processes\refs{LIGO2009, kiefer2000, Easther2006, Oniga2016d}. The nature and possible observation of such stochastic gravitational waves have attracted significant recent attention since the ground breaking detection of gravitational waves by LIGO and Virgo Collaborations\refs{LIGO2016} with anticipation of broader gravitational wave astronomy and cosmology\refs{GWC2016, Schutz1999, Sesana2016}.
%We consider new gravitational interaction with matter at quantum and atomic levels for the following physically significant reasons.
In what follows, we present a generic theoretical analysis without assuming a specific distribution function $\ngw(\omega)$ and the equivalent spectral function $\Omega_\gw(f)$ for gravitational waves apart from being Gaussian, which is physically reasonable \refs{Allen1996}, and the type of atoms apart from being bosonic so as to have large occupation number states and the resulting collective dynamics.

For possible future implementation considerations, we have in mind Rydberg atoms\refs{Gallagher2008, Saffman2010}, in view of a number of exceptional physical characteristics they possess: (i) Their large principal quantum number $n$ means that the effective mass quadrupole moment, which provides external gravitational wave coupling and scales with $n^4$, can be greatly enhanced. (ii) These atoms are very stable having extremely long life times of the internal transitions up to the order of one second allowing us to neglect them in many practical situations\refs{Branden2010}. (iii) By placing Rydberg atoms in a high-$Q$ cavity with controlled boundary conditions, undesirable transitions may be further suppressed\refs{Varcoe2006}. The presence of the cavity may also change the quantum electrodynamics of the atom to modify the interaction between the nucleus and the electron so as to increase the quadrupole interaction while suppressing the atom-atom dipole interaction as well\refs{Marrocco1998}. (iv) Increasingly large numbers of Rydberg atoms with high principal quantum numbers $n>10^2$ have been produced in laboratories\refs{Ye2013}.
By virtue of these properties, we will hypothesize for simplicity a setup of $N$ identical free 2-level Rydberg atoms with a quadrupole moment transition frequency $f=\omega/2\pi$ used to evaluate the distribution function $\ngw(\omega)$ and the equivalent spectral function $\Omega_\gw(f)$ for the stochastic gravitational waves. Further refinement is possible to reduce the present level of idealization but will be deferred to future work.

The investigations of Rydberg atoms as a many-body system have often featured in the recent literature\refs{Browaeys2013} with their collective quantum dynamics receiving current research interest both theoretically\refs{Jaksch2000, Lukin2001, Olmos2010, Lee2012} and experimentally\refs{Gaetan2009, Wilk2010, Heidemann2007, Dudin2012, Mourachko1998, Schauss2012, Kumar2016}.
%Here we assume EM transition rates are so low they can be neglected.
%In the interaction picture, we consider the transitions between atomic states in a stochastic gravitational wave back ground with a Gaussian distribution function $\ngw(\omega)$ \cite{Allen1996}.
The effect of that curved metric have on hydrogen-like Rydberg systems have been considered in a number of works\refs{Fischbach1981, Fischert1994, Marques2002, Olmo2008, Zhao2007}, in which the calculations show a shift in the energy spectrum. Gravity as been proposed to induce decoherence in atomic states\refs{Blencowe2013, Oniga2016a, Quinones2016}, making the analysis of the evolution of the atomic state a potential method for the detection of gravitational fluctuation\refs{Parker1980, Parker1982, Leen1983, Pinto1995}. However, these effects have also been found to be generally too small to be measured in practice.

Here, rather than considering the gravitational interactions for individual systems, we investigate the coherent quantum interactions for a large number of correlated atoms with gravitational waves. We find that the resulting collective interactions can lead to amplified transition rate of states with a quadrupole moment in a fashion akin to superradiance\refs{Dicke1954}. This collective amplification therefore adds to the application of Rydberg atoms, which are already more sensitive to spacetime perturbations \cite{Pinto1993,Pinto1995}, as a basis for a possible future detection scheme for gravitational waves of a wider spectral window, especially towards higher frequencies than currently accessible, which may be of fundamental and cosmological importance.

%Rydberg states are highly excited atomic states with a large principal quantum number, close to the ionization threshold (for a detailed review, see \Ref{Gallagher2008}).

%A detailed review of the applications of Rydberg atoms to quantum information processing has been written by \Ref{Saffman2010}

\section{Gravitational master equation for confined nonrelativistic systems}
\label{sec:mas}


In the interaction picture with relativistic units where $c=1$, the general gravitational master equation for  matter that may be interacting and relativistic has recently been obtained in \Ref{Oniga2016a} to be
\begin{eqnarray}
\dot\rho
&=&
-\frac{i}{\hbar} [H_\intt, \rho]
+\DD\rho
\label{maseqn}
\end{eqnarray}
with $H_\intt$ describing matter interaction with itself or other fields not part of the gravitational environment,  and the non-Markovian quantum dissipator
\begin{eqnarray}
\DD\rho
&=&
-\frac{8\pi G}{\hbar}
\int\! \frac{\dd^3 k}{2(2\pi)^3k}
\,\times
\nppp&&
\hspace{-10pt}
\Big \{
\int_{0}^{t} \!\dd t'
e^{-i k (t - t')}
\big(
[
\tau^\dag_{ij} (\kk, t),\,
\tau_{ij}(\kk, t') \rho
]
\nppp&&
\hspace{-10pt}
\,+\,\ngw(\omega_k)\,
[
\tau^\dag_{ij} (\kk, t),\,
[
\tau_{ij}(\kk, t'),\,
\rho
]]
\big)
+\hc
\Big\}
\label{diss}
\end{eqnarray}
describing in general nonunitary statistical quantum evolution of the matter system as a result of dissipative iterations with the fluctuating gravitational  environment \refs{Oniga2016d, Oniga2016a}.
%
Since typical compact nonrelativistic systems have spatial extensions much smaller than the effectively coupled gravitational wavelength\refs{Flanagan2005, MTW1973}, we have $e^{-i \kk\cdot\rr}\approx 1$ and therefore
\begin{eqnarray}
\tau_{ij}(\kk, t)
&=&
\int\!
\tau_{ij}(\rr, t)\,
e^{-i \kk\cdot\rr}\,\dd^3x
\nppp
&\approx&
\frac12
\ddot\Idash_{ij}^\TT(\kk,t)
\label{ITT1}
\end{eqnarray}
where
\begin{eqnarray}
\Idash_{ij}^\TT(\kk,t)
&=&
P_{ijkl}(\kk)\Idash_{kl}(t)
\label{ITT2}
\end{eqnarray}
and
\begin{eqnarray}
\Idash_{ij}(t)
&=&
\frac{1}{3}\,\int\dd^3x\,Q_{ij}(\rr)\,T^{00}(\rr,t)
\label{qij2}
\end{eqnarray}
in terms of the transverse-traceless (TT) projection operator $P_{ijkl}$\refs{Flanagan2005, Oniga2016a} and the quadrupole moment tensor
\begin{eqnarray}
Q_{ij}
&=&
3x^i x^j - \delta_{ij}\, r^2 .
\label{Qij}
\end{eqnarray}

%
It is useful to introduce the frequency domain reduced quadrupole moments $\Idash_{ij}(\omega)$ for some positive frequencies $\omega$ by writing
\begin{eqnarray}
\Idash_{ij}(t)
&=&
\sum_{\omega}
\Idash_{ij}(\omega)\, e^{-i\omega t} + \Idash_{ij}^\dag(\omega)\, e^{i\omega t} .
\label{ITT3}
\end{eqnarray}
%
Let us consider the following integral in \Eq{diss}
\begin{eqnarray}
\lefteqn{
\int_{0}^{t} \!\dd t'
e^{-i k (t - t')}
\tau_{ij}(\kk, t')
}
\nppp
&=&
-\sum_{\omega}\frac{\omega^2}{2}\,\Idash_{ij}^\TT(\omega,\kk)
\int_{0}^{t} \!\dd t'
e^{-i k (t - t')}
e^{-i\omega t'}
\nppp&&
-\sum_{\omega}\frac{\omega^2}{2}\,\Idash_{ij}^{\TT\dag}(\omega,\kk)
\int_{0}^{t} \!\dd t'
e^{-i k (t - t')}
e^{i\omega t'}
\label{iet}
\end{eqnarray}
which is nonlocal in time representing the non-Markov memory effect.

To apply the Markov approximation, valid for physical time scales where short-term transient memories are damped way\refs{Breuer2002}, we change the time variable from $t'$ to $s$ using $t'=t-s$ in above to get
\begin{eqnarray*}
%\lefteqn
\int_{0}^{t} \!\dd t'
e^{-i k (t - t')}
e^{\pm i\omega t'}
&\approx&
e^{\pm i\omega t}
\int_{0}^{\infty} \!\dd s\,
e^{-i (k\pm\omega) s}
\nppp
&=&
e^{\pm i\omega t}\big[\,
\pi \delta(k\pm\omega) - i \,\text{P} \frac{1}{k\pm\omega}
\,\big]
\end{eqnarray*}
where the Sokhotski-Plemelj theorem (see \Eq{fma} below) has been used in the last step.

Using the above and neglecting the the Cauchy principal values, as they do not contribute to quantum decoherence and dissipation, we see that \Eq{iet} becomes
\begin{eqnarray}
\lefteqn{
\int_{0}^{t} \!\dd t'
e^{-i k (t - t')}
\tau_{ij}(\kk, t')
}
\nppp
&=&
-\sum_{\omega}
\frac{\pi\omega^2}{2}\,\Idash_{ij}^\TT(\omega,\kk)\,
e^{-i\omega t} \delta(k-\omega)
\nppp&&
-\sum_{\omega}
\frac{\pi\omega^2}{2}\,\Idash_{ij}^{\TT\dag}(\omega,\kk)\,
e^{i\omega t} \delta(k+\omega)
\label{iet2}
\end{eqnarray}
where the second term does not contribute as $k \ge 0$ and $\omega > 0$.
Using \Eqs{ITT1} and \eqref{ITT3}, we have
\begin{eqnarray}
%\hspace{-40pt}
\lefteqn{
\tau_{ij}^\dag(\kk, t)
\approx
\frac12
\ddot\Idash_{ij}^{\TT\dag}(\kk,t)
}
\nppp
&=&
-\frac{\omega^2}2\big[
\Idash_{ij}^\TT(\omega,\kk)\, e^{-i\omega t}
+
\Idash_{ij}^{\TT\dag}(\omega,\kk)\, e^{i\omega t}
\big] .
\label{ITT4}
\end{eqnarray}
Substituting \Eqq{iet2}{ITT4} into the dissipator~\eqref{diss} and applying the rotating wave approximation we have
\begin{eqnarray}
\DD\rho
&=&
-\frac{G\omega^5}{8\pi \hbar}
\int\!{\dd\Omega(\kk)}
%\times\,
%\nppp&&
\Big \{
\big(
[
\Idash_{ij}^{\TT\dag}(\omega,\kk),\,
\Idash_{ij}^\TT(\omega,\kk)
\rho
]
\nppp&&
\hspace{-10pt}
+\ngw(\omega)\,
[
\Idash_{ij}^{\TT\dag}(\omega,\kk),\,
[
\Idash_{ij}^\TT(\omega,\kk),\,
\rho
]]
\big)
+\hc
\Big\}
\nppp
&=&
\frac{G\omega^5}{4\pi \hbar}
\int\!{\dd\Omega(\kk)}
P_{ijkl}(\kk)\times
\nppp&&
\hspace{-10pt}
\Big\{
(1+\ngw(\omega))
\big[
\Idash_{ij}(\omega)\, \rho\, \Idash_{kl}^\dag(\omega)
%\nppp&&
-\frac12
\big\{\Idash_{ij}^\dag(\omega) \Idash_{kl}(\omega), \rho\big\}
\big]
\nppp&&
\hspace{-10pt}
+\ngw(\omega)
\big[
\Idash_{ij}^\dag(\omega)\, \rho\, \Idash_{kl}(\omega)
-\frac12
\big\{\Idash_{ij}(\omega) \Idash_{kl}^\dag(\omega), \rho\big\}
\big]\!
\Big\}
%\nppp
\label{diss2}
\end{eqnarray}
summing over $\omega>0$ and $i,j,k,l=1,2,3$.
With the aid of the formulae~\cite{MTW1973}:
\begin{eqnarray*}
\int\! k_i k_j \,\dd\Omega
&=&
\frac{4\pi}3\,k^2 \delta_{ij}
\nppp
\int\! k_i k_j k_k k_l \,\dd\Omega
&=&
\frac{4\pi}{15}\,k^2
(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})
\end{eqnarray*}
we can calculate that
\begin{eqnarray}
\!\!\int\! P_{ijkl}(\kk) \,\dd\Omega
&=&
\frac{4\pi}{15}
(3\delta_{ik}\delta_{jl}+3\delta_{il}\delta_{jk}-2\delta_{ij}\delta_{kl})
\label{iPijkl} .
\end{eqnarray}
Substituting \Eq{iPijkl} into \Eq{diss2} and using the tracelessness of $\Idash_{ij}$, we obtain the following general dissipator
\begin{eqnarray}
\DD\rho
&=&
\frac{2G\omega^5}{5\hbar}
\,\times
\nppp&&
\hspace{-15pt}
\Big\{(1+\ngw(\omega))
\big[
\Idash_{ij}(\omega)\, \rho\, \Idash_{ij}^\dag(\omega)
-\frac12
\big\{\Idash_{ij}^\dag(\omega) \Idash_{ij}(\omega), \rho\big\}
\big]
\nppp&&
\hspace{-15pt}
+\ngw(\omega)
\big[
\Idash_{ij}^\dag(\omega)\, \rho\, \Idash_{ij}(\omega)
-\frac12
\big\{\Idash_{ij}(\omega) \Idash_{ij}^\dag(\omega), \rho\big\}
\big]
\Big\}
%\nppp
\label{diss3}
\end{eqnarray}
of the Lindblad form for any nonrelativistic matter systems with frequency domain reduced quadrupole moments $\Idash_{ij}(\omega)$.
For particles in an isotropic harmonic potential with $\ngw(\omega)=0$, \Eq{diss3} recovers the corresponding dissipator discussed in \Ref{Oniga2016d}.



\section{Gravitationally induced transitions of quantum states}
\label{sec:trans}

The theoretical framework established above is applied in this section to the quantum gravitational decoherence and radiation of a real scalar field $\phi$ with mass $m$ and the associated inverse reduced Compton wavelength $\mu = m/\hbar$, subject to an external potential $\nu(\rr)$ described by the Lagrangian density
\begin{eqnarray}
\LL
&=&
-
\frac12\,g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}
-\Big(\frac{1}{2}+\nu\Big)\mu^2
\phi^2.
\label{LL}
\end{eqnarray}

To consider the nonrelativistic dynamics of the scalar field representing nearly Newtonian particles we assume the potential energy to be much less than the rest mass energy so that $\nu \ll 1$ and the total energy density is given approximately by
\begin{eqnarray}
%T^{00}=\dot{\phi}^2
T^{00}=\frac{1}{2}\big(\dot{\phi}^2+\mu^2\phi^2\big) .
\label{T00}
\end{eqnarray}


The unperturbed part of Eq.~\eqref{LL} satisfies quantum field equation
\begin{eqnarray}
\ddot\phi
=
\nabla^2\phi
-
(1+2\nu)\mu^2\phi
\label{hseqa}
\end{eqnarray}
having solutions of the convenient form
\begin{eqnarray}
\phi
=
\sqrt{\frac{\hbar}{2\omega_{n}}}\,a_{n}
\psi_n(\rr)\, e^{-i\omega_n t} + \hc
\label{hPhi}
\end{eqnarray}
with some operators $a_{n}$ and orthogonal complex functions $\psi_n(\rr)$.
Hence Eq.~\eqref{hseqa} reduces formally to the time-independent Schr\"odinger equation
\begin{eqnarray}
-\frac{\hbar^2}{2m}\nabla^2\psi_n + V\psi_n
=
E_n \psi_n
\label{hschreq}
\end{eqnarray}
where $V=m\,\nu(\rr)$ and
\begin{eqnarray}
E_n
=
\frac{1}{2m}(\hbar^2\omega_n^2-m^2)
\label{wmE}
\end{eqnarray}
represent the potential and eigen energies respectively.

Let us consider the hydrogen-like atom potential energy $V(r)$
%\begin{eqnarray}
%V
%?=
%\frac{e^2}{r}
%\end{eqnarray}
In the nonrelativistic domain, Eq.~\eqref{hPhi} becomes
\begin{eqnarray*}
\phi
&=&
%\sum_{\nn}
\frac{\hbar}{\sqrt{2m}}\,
\big(
a_{\nn} e^{-i\omega_{\nn} t}
+
a_{\nn}^\dag  e^{i\omega_{\nn} t}
\big)
\psi_{\nn}(\rr)
%\label{3hphi}
\end{eqnarray*}
using the multiple indices $\nn=(n,l,m)$ and the hydrogen atom wavefunctions
\begin{eqnarray*}
\psi_{\nn}(\rr)
&=&
R_{nl}(r)Y_l^m(\theta,\phi)
\end{eqnarray*}
with
\begin{eqnarray*}
\omega_{\nn}
&=&
\mu + E_{\nn}/\hbar
%\label{3nome}
\end{eqnarray*}
arising from  the nonrelativistic limit of Eq.~\eqref{wmE}, where the corresponding ladder operators $a_{\nn}$ and $a_{\nn}^\dag$ are annihilation and creation operators respectively.
From \Eqq{qij2}{T00} we have
\begin{eqnarray}
\hspace{-12pt}
\Idash_{ij}
&=&
\frac{m}{3}\!\!\!\!\!
\sum_{\substack{{\nn,\nn'}\\{(\omega_{\nn}>\omega_{\nn'})}}}\!\!\!
\Bra{\nn'}Q_{ij}\Ket{\nn}\,
a_{\nn'}^\dag a_{\nn}\,
e^{-i(\omega_{\nn}-\omega_{\nn'}) t}
+\hc
\label{Iw}
\end{eqnarray}
where any irrelevant time-independent part with $\omega_{\nn}=\omega_{\nn'}$ have been left out and for simplicity we have adopted the quantum mechanics styled notation
\begin{eqnarray}
\Bra{\nn'}Q_{ij}\Ket{\nn}
&=&
\int\dd^3x\,
Q_{ij}
\psi_{\nn'}(\rr)
\psi_{\nn}(\rr)
\end{eqnarray}
where Dirac's bra-kets are denoted by $\Bra{\cdot}$ and $\Ket{\cdot}$, as opposed to $\bra{\cdot}$ and $\ket{\cdot}$ reserved for the main quantum states of our present discussion based on quantum field theory.

%
Comparing \Eq{ITT3} and \Eq{Iw} we see that
\begin{eqnarray}
\Idash_{ij}(\omega_{\nn}-\omega_{\nn'})
&=&
\frac{m}{3}
\Bra{\nn'}Q_{ij}\Ket{\nn}\,
a_{\nn'}^\dag a_{\nn}
\label{Iw2a}
\ppp
\Idash_{ij}^\dag(\omega_{\nn}-\omega_{\nn'})
&=&
\frac{m}{3}
\Bra{\nn'}Q_{ij}\Ket{\nn}^*\,
a_{\nn}^\dag a_{\nn'}
\label{Iw2b}
\end{eqnarray}
for $\omega_{\nn} > \omega_{\nn'}$.

%\subsection{A single particle}

Let us apply the above general setup to a 2-level atom model with states
$\ket{1} = \ket{n_1,l_1,m_1}$ and $\ket{2} = \ket{n_2,l_2,m_2}$ having respective eigen energies $E_1 = \hbar\omega_1$ and  $E_2 = \hbar\omega_2$ and the transitional frequency $\omega_0=\omega_2-\omega_1>0$.
It is useful to denote these two states by
\begin{eqnarray*}
\ket{1}
=
a_{1}^\dag \ket{0}
,\quad
\ket{2}
=
a_{2}^\dag \ket{0}
\end{eqnarray*}
with the ladder operators
\begin{eqnarray*}
\sigma^{+}
&=&
\ket{2}\bra{1}
=
a_{2}^\dag a_{1}
,\quad
\sigma^{-}
=
\ket{1}\bra{2}
=
a_{1}^\dag a_{2} .
\end{eqnarray*}
%
Introducing the transition quadrupole moment
\begin{eqnarray}
q_{ij}
&=&
\Bra{1} Q_{ij} \Ket{2}
\label{qij}
\end{eqnarray}
we can express \Eqq{Iw2a}{Iw2b} as
\begin{eqnarray}
\Idash_{ij}(\omega_0)
&=&
\frac{m}{3}\,q_{ij}\,a_{1}^\dag a_{2}=
\frac{m}{3}\,q_{ij}\,\sigma^-
\label{Iw3a}
\ppp
\Idash_{ij}^\dag(\omega_0)
&=&
\frac{m}{3}\,q_{ij}\,a_{2}^\dag a_{1}=
\frac{m}{3}\,q_{ij}^*\,\sigma^+ .
\label{Iw3b}
\end{eqnarray}
%
Substituting \Eqq{Iw3a}{Iw3b} into \Eq{diss3} we obtain the dissipator for the 2-level atom to be
\begin{eqnarray}
\DD\rho
&=&
\Gamma_0\,
\Big\{(1+\ngw(\omega_0))
\big[
\sigma^- \rho\, \sigma^+
-\frac12
\big\{\sigma^+ \sigma^-, \rho\big\}
\big]
\nppp&&
%\hspace{-15pt}
+\ngw(\omega_0)
\big[
\sigma^+ \rho\, \sigma^-
-\frac12
\big\{\sigma^- \sigma^+, \rho\big\}
\big]
\Big\}
%\nppp
\label{diss4}
\end{eqnarray}
where we have introduced the gravitational spontaneous emission rate
\begin{eqnarray}
\Gamma_0=
\frac{2 G m^2 \omega_0^5 q^2}{45\hbar}
\label{gg0}
\end{eqnarray}
in terms of the coupling coefficient related to the squared modulus of the quadrupole moment of the 2-level atom
\begin{eqnarray}
q^2 = q_{ij}^*q_{ij} .
\label{q2}
\end{eqnarray}

An alternative derivation of \Eq{diss4} based on a more conventional quantum mechanical treatment is given in the appendix \Sec{sec:diss}. Quantum field theory as the main approach of this work has the advantage of describing arbitrary number of bosonic and fermionic particles and their collective behaviours more naturally and effectively.

%\subsection{Many particles}
\section{Gravitational wave interactions with correlated 2-level atoms}
\label{sec:2level}

An $N$-particle 2-level state is given by
\begin{eqnarray}
\ket{N_1,N_2}
&=&
\frac{a_{1}^{\dag N_1}}{\sqrt{N_1!}}
\frac{a_{2}^{\dag N_2}}{\sqrt{N_2!}}
\ket{0}
\label{NN}
\end{eqnarray}
with $N_1+N_2=N$. There are $N+1$ such states, which can be represented with
\begin{eqnarray}
\ket{p}=\ket{N-p+1,p-1}
\label{pp}
\end{eqnarray}
for $p=1,2,\dots N+1$, so that $\ket{1}=\ket{N,0},\ket{2}={\ket{N-1,1}},\dots\ket{N+1}=\ket{0,N}$.
It follows from
\begin{eqnarray*}
\lefteqn{\hspace{-10pt}\bra{N_1',N_2'} a_{1}^\dag a_{2} \ket{N_1,N_2}}
%\nppp&=&
%\sqrt{(N_1+1)N_2}\,\braket{N_1',N_2'}{N_1+1,N_2-1}
\nppp&=&
\sqrt{(N_1+1)N_2}\,\delta_{N_1',N_1+1}\,\delta_{N_2',N_2-1}
\ppp
\lefteqn{\hspace{-10pt}\bra{N_1',N_2'} a_{2}^\dag a_{1} \ket{N_1,N_2}}
%\nppp&=&
%\sqrt{N_1(N_2+1)}\,\braket{N_1',N_2'}{N_1-1,N_2+1}
\nppp&=&
\sqrt{N_1(N_2+1)}\,\delta_{N_1',N_1-1}\,\delta_{N_2',N_2+1}
\end{eqnarray*}
that
\begin{eqnarray*}
\bra{p'} a_{1}^\dag a_{2} \ket{p}
%\nppp&=&
%\lefteqn{\bra{N-p'+1,p'-1} a_{1}^\dag a_{2} \ket{N-p+1,p-1}}
%\nppp&=&
%\sqrt{(N-p+2)(p-1)}\,\braket{N-p'+1,p'-1}{N-p+2,p-2}
%\nppp&=&
%\sqrt{(N-p+2)(p-1)}\,\delta_{N-p'+1,N-p+2}\,\delta_{p'-1,p-2}
%\nppp
&=&
\sqrt{(p-1)(N-p+2)}\;\delta_{p',p-1}
\label{p1}
\ppp
\bra{p'} a_{2}^\dag a_{1} \ket{p}
%\nppp&=&
%\bra{N-p'+1,p'-1}a_{2}^\dag a_{1}\ket{N-p+1,p-1}
%\nppp&=&
%\sqrt{(N-p+1) p}\,\braket{N-p'+1,p'-1}{N-p,p}
%\nppp&=&
%\sqrt{(N-p+1) p}\,\delta_{N-p'+1,N-p}\,\delta_{p'-1,p}
%\nppp
&=&
\sqrt{p\,(N-p+1)}\;\delta_{p',p+1} .
\label{p2}
\end{eqnarray*}
Using the above relations, we see that \Eqq{Iw2a}{Iw2b} become
\begin{eqnarray}
\Idash_{ij}(\omega_0)
&=&
\frac{m}{3}\,q_{ij}\,\Sigma^-
\label{Iw4a}
\ppp
\Idash_{ij}^\dag(\omega_0)
&=&
\frac{m}{3}\,q_{ij}^*\Sigma^+
\label{Iw4b}
\end{eqnarray}
using the $(N+1)$-dimensional ladder operators
\begin{eqnarray}
\Sigma^+_{p' p}
=
\Sigma^-_{p\, p'}
&=&
\sqrt{p\,(N-p+1)}\;\delta_{p',p+1} .
\end{eqnarray}
%
Substituting \Eqq{Iw4a}{Iw4b} into \Eq{diss3}, we obtain the dissipator for the correlated $N$ 2-level atoms
\begin{eqnarray}
\DD\rho
&=&
\Gamma_0\,
\Big\{(1+\ngw(\omega_0))
\big[
\Sigma^- \rho\, \Sigma^+
-\frac12
\big\{\Sigma^+ \Sigma^-, \rho\big\}
\big]
\nppp&&
%\hspace{-15pt}
+\ngw(\omega_0)
\big[
\Sigma^+ \rho\, \Sigma^-
-\frac12
\big\{\Sigma^- \Sigma^+, \rho\big\}
\big]
\Big\} .
\label{diss5}
\end{eqnarray}
For $N=1$, we have the reduction $\Sigma^\pm = \sigma^\pm$. However, in general we see that
\begin{eqnarray}
\Sigma^\pm_{p\pm1 p}\approx \frac{N}{4},
\quad\hbox{ for }
p \approx \frac{N}{2}
\end{eqnarray}
with a large $N$, providing an $N^2$ amplification for \Eq{diss5} compared with \Eq{diss4}.



Consider a diagonal density matrix
\begin{eqnarray}
\rho_{p q}
&=&
\rho_p\delta_{p,q}
\label{dia}
\end{eqnarray}
with the normalization
\begin{eqnarray}
\sum_{p=1}^{N+1}
\rho_p
&=&
1
\label{norm}
\end{eqnarray}
from which we can calculate the following:
\begin{eqnarray*}
(\Sigma^+\Sigma^-\rho)_{p q}
&=&
(p-1)\,(N-p+2)\,\rho_p\,\delta_{p,q}
\nppp
(\Sigma^-\Sigma^+\rho)_{p q}
&=&
p\,(N-p+1)\,\rho_{p}\,\delta_{p,q}
\nppp
(\Sigma^-\rho\,\Sigma^+)_{p q}
&=&
p\,(N-p+1)\,\rho_{p+1}\,\delta_{p,q}
\nppp
(\Sigma^+\rho\,\Sigma^-)_{p q}
&=&
(p-1)\,(N-p+2)\,\rho_{p-1}\,\delta_{p,q}
\nppp
(\rho\,\Sigma^-\Sigma^+)_{p q}
&=&
p\,(N-p+1)\,\rho_{p}\,\delta_{p,q}
\nppp
(\rho\,\Sigma^+\Sigma^-)_{p q}
&=&
(p-1)\,(N-p+2)\,\rho_p\,\delta_{p,q}
\end{eqnarray*}
Substituting the above relations into \Eq{diss5} we have
\begin{eqnarray}
(\DD\rho)_{p q}
&=&
\Gamma_0\,D_p\,\delta_{p,q}
\label{diss6}
\end{eqnarray}
where
\begin{eqnarray}
D_p
&=&
(1+\ngw(\omega_0))
p\,(N-p+1)\,\rho_{p+1}
\nppp&&
-(1+\ngw(\omega_0))
(p-1)\,(N-p+2)\,\rho_p
\nppp&&
%\hspace{-15pt}
+\ngw(\omega_0)
(p-1)\,(N-p+2)\,\rho_{p-1}
\nppp&&
-\ngw(\omega_0)
p\,(N-p+1)\,\rho_{p} .
\label{fp}
\end{eqnarray}
The trace of \Eq{diss6} vanishes
\begin{eqnarray}
\tr(\DD\rho)
&=&
\sum_{p=1}^{N+1}
(\DD\rho)_{p p}
=0
\label{trdiss6}
\end{eqnarray}
as required for the consistency of \Eq{norm} under time evolution.
%
Denoting by $\rho(s)=N\rho_p$, $D(s)=N D_p$ using ${s=(p-1)/N}$ and ${\epsilon=1/N}$ with
${0 \le s \le 1}$,
%Then $p=Ns+1$.
we see that \Eq{fp} becomes
\begin{eqnarray*}
\lefteqn{D(s+\epsilon)}
\nppp
&=&
\big[
(s+\epsilon)\,(1-s)\,\rho(s+2\epsilon)
\nppp&&
-
s\,(1-(s-\epsilon))\,\rho(s+\epsilon)
\big]/\epsilon^2
\nppp&&
+
\ngw(\omega_0)
\big[
(s+\epsilon)\,(1-s)\,(\rho(s+2\epsilon)-\rho(s+\epsilon))
\nppp&&
-
s\,(1-(s-\epsilon))\,(\rho(s+\epsilon)-\rho(s))
\big]/\epsilon^2 .
\end{eqnarray*}
For a large particle number $N \gg 1$ and hence $\epsilon \ll 1$, the above tends to the continuous limit
\begin{eqnarray}
%\hspace{-5pt}
D(s)
=
\big\{
s(1-s)
\big[N\rho(s)
+
\ngw(\omega_0)
\rho'(s)
\big]
\big\}'
\label{fs}
\end{eqnarray}
where ${}' = \partial_s$. Then the discrete dissipator\eq{diss5} becomes the following continuous ``diffuser'':
\begin{eqnarray}
%\hspace{-15pt}
\DD\rho(s)
&=&
%\nppp&&
%\hspace{-25pt}
\Gamma_0
\big\{
s(1-s)
\big[N\rho(s)
+
\ngw(\omega_0)
\rho'(s)
\big]
\big\}'
\label{diss7}
\end{eqnarray}
with possible collective amplification by a factor up to $N^2$ represented by $' \sim N$ and $'' \sim N^2$.
In terms of \Eq{fs}, the traceless condition\eq{trdiss6} then becomes
\begin{eqnarray}
\int_0^1 \! D(s) \,\dd s
&=&
0 .
\label{trdiss7}
\end{eqnarray}
For a stationary finite solution with $\DD\rho=0$ and $\ngw(\omega_0) > 0$, \Eq{diss7} reduces to
\begin{eqnarray}
\ngw(\omega_0)
\rho'(s)
=
-N\rho(s)
\label{d7z2}
\end{eqnarray}
yielding
\begin{eqnarray}
\rho(s)
=
\frac{e^{-s/w}}{w(1-e^{-1/w})}
\label{rs}
\end{eqnarray}
where $w=\ngw(\omega_0)/N$. \Eq{rs} takes the form of the Boltzmann distribution representing an equilibrium state at temperature $T_0$ so that
\begin{eqnarray}
\frac{s}{w} = \frac{E(s)}{k_B T_0}
\label{rs2}
\end{eqnarray}
where $E(s)=s N \hbar \omega_0$ is the energy of the $N$-atom system in the continuous limit. Indeed, if the stochastic gravitational waves are themselves in thermal equilibrium satisfying the Planck distribution
\begin{eqnarray}
\ngw(\omega)=\frac1{e^{\hbar \omega/k_B T}-1}
\label{Pdistgw}
\end{eqnarray}
at temperature $T$, and if
$k_B T > \hbar \omega_0$,  then \Eq{Pdistgw} gives
$1/\ngw(\omega_\gw)\approx N \hbar \omega_0/k_B T_\gw$, and so \Eq{rs2} implies $T_0\approx T$ meaning the atomic system reaches the thermal equilibrium as the gravitational waves.
%, with $k_B T > \hbar \omega_0$ describing a physically reasonable thermal quantum state.
However, here we also consider more general stochastic gravitational wave spectrum with $\ngw(\omega)$ not restricted to the Planck distribution\eq{Pdistgw}.
See \Fig{fig1} for illustrative numerical examples of dynamical relaxations into the stationary states described above.


\begin{figure}
\vspace{10pt}
\includegraphics[width=1\linewidth]{SGW_fig1a.eps}
\\
\vspace{10pt}
\includegraphics[width=1\linewidth]{SGW_fig1b.eps}
\caption{Simulations with $N$ atoms using the master equation\eq{maseqn} with the quantum dissipator\eq{diss5} for the correlated 2-level atoms and negligible matter interaction $H_\intt$.
From \Eq{diss7} the relaxation time is approximately given by
$\tau_N=(N\Gamma_0)^{-1}$.
The initial state of the forms\eqq{NN}{pp} at $t=0$ is given by $\ket{0,N}=\ket{p=N+1}$ with all $N$ atoms occupying the excited atomic level. During the dynamical evolution, the density matrix remains diagonal of the form\eq{dia} and relaxes into an inverse exponential profile represented by \Eq{rs}, yielding the asymptotic height of
$\rho_{p=1}=[\ngw(\omega_0)(1-e^{-N/\ngw(\omega_0)})]^{-1}$.
The stochastic gravitational wave distribution function evaluated at the atomic transition frequency $\omega_0$ is chosen to be $\ngw(\omega_0)=1$.
Accordingly, in plot (a) with $N=100$ atoms, the initial state corresponding to the spike of height one at $(p,\Gamma_0 t)=(101,0)$ relaxes into an inverse exponential profile peaked at
$(p,\Gamma_0 t)=(1,0.1)$ on time scale of $\tau_N=0.01/\Gamma_0$.
Similarly, in plot (b) with $N=200$ atoms, the initial state corresponding to the spike of height one at $(p,\Gamma_0 t)=(201,0)$ relaxes into an inverse exponential profile similarly peaked at
$(p,\Gamma_0 t)=(1,0.1)$ but on a halved time scale of $\tau_N=0.005/\Gamma_0$.
}
\label{fig1}
\end{figure}




\section{Gravitational wave interactions with correlated Rydberg-like atoms}
\label{sec:gw}


\begin{figure}
\vspace{10pt}
\includegraphics[width=1\linewidth]{SGW_fig2a.eps}
\\
\vspace{10pt}
\includegraphics[width=1\linewidth]{SGW_fig2b.eps}
\caption{Simulations of short-time collective spontaneous delays due to vacuum fluctuations of spacetime without additional stochastic gravitational waves so that $\ngw(\omega_0)=0$.
%
In plot (a) with $N=100$ atoms, the initial state corresponds to the spike of height one at $(p, t)=(50,0)$. As shown, it delays into adjacent lower states with a relaxation time $\tau_\vac=4/({N^2\Gamma_0})=0.0004/\Gamma_0$ according to \Eqq{ggvac}{tt}.
%
In plot (b) with $N=200$ atoms, the initial state corresponds to the spike of height one at $(p, t)=(100,0)$. For a doubled $N$, this state delays into adjacent lower states with a quarter of the preceding relaxation time with
$\tau_\vac=4/({N^2\Gamma_0})=0.0001/\Gamma_0$.}
\label{fig2}
\end{figure}

\begin{figure}
\vspace{10pt}
\includegraphics[width=1\linewidth]{SGW_fig3a.eps}
\\
\vspace{10pt}
\includegraphics[width=1\linewidth]{SGW_fig3b.eps}
\caption{Simulations of short-time collective transitions due to vacuum fluctuations of spacetime and additional stochastic gravitational waves with a chosen $\ngw(\omega_0)=1$.
%
In plot (a) with $N=100$ atoms, the initial state corresponds to the spike of height one at $(p, t)=(50,0)$. While this state delays similar to the descriptions in \Fig{fig2}, its adjacent higher state with $p=51$ is excited with an initial time scale of $\tau_\gw=4/({N^2\Gamma_0}\ngw(\omega_0))=0.0004/\Gamma_0$ according to \Eqq{gggw}{tt}.
%
In plot (b) with $N=200$ atoms, the initial state corresponds to the spike of height one at $(p, t)=(100,0)$. For a doubled $N$, 
its adjacent higher state with $p=101$ is excited with a quarter of the preceding initial time scale with $\tau_\gw=4/({N^2\Gamma_0}\ngw(\omega_0))=0.0001/\Gamma_0$.
}
\label{fig3}
\end{figure}

In this section, we would like to related the above theoretical description to physically relevant values.
To this end, let us first related the distribution function $\ngw(\omega)$ to the spectral function $\Omega_\gw(f)$ for gravitational waves.

The gravitational wave energy density
\begin{eqnarray*}
\rho_\gw
&=&
\frac1{32\pi G}
\langle
\dot{h}_{ij}\dot{h}_{ij}
\rangle
%\label{ugw}
\end{eqnarray*}
is related to $\ngw(k)$ by \cite{Schafer1981}:
\begin{eqnarray*}
\rho_\gw
&=&
\frac{\hbar}{\pi^2}
\int_0^\infty
\dd \omega\, \omega^3 \ngw(\omega) .
\label{ugw2}
\end{eqnarray*}
On the other hand, the energy density spectral function $\rho_\gw(\omega)$ of the gravitational waves is also given by
\begin{eqnarray*}
\dd \rho_\gw
&=&
\rho_\gw(\omega)\,\dd\omega
\label{uw}
\end{eqnarray*}
Comparing  the above two relations, we see that
\begin{eqnarray*}
\rho_\gw(\omega)
&=&
\frac{\hbar}{\pi^2}\, \omega^3 \ngw(\omega) .
%\label{uw2}
\end{eqnarray*}
Using the wave frequency $f=\omega/\,2\pi$, Planck constant $h=2\pi\hbar$,
and
$
\dd \rho_\gw
=
\rho_\gw(f)\,\dd f
$,
we can write the above as
\begin{eqnarray*}
\rho_\gw(f)
&=&
8\pi h\, f^3 \ngw(\omega) .
%\label{uw2a}
\end{eqnarray*}
Then by using the critical energy density $\rho_\text{c}$ in cosmology, we see that the dimensionless spectral function $\Omega_\gw(f)$ for gravitational waves using is given by
\begin{eqnarray}
\Omega_\gw(f)
&=&
\frac1{\rho_\text{c}}
\frac{\dd \rho_\gw}{\dd \ln f}
=
\frac{8\pi h}{\rho_\text{c}}\, f^4 \ngw(\omega) .
\label{Omf}
\end{eqnarray}
To recover $\ngw(\omega)$ from given $\Omega_\gw(f)$, we can invert \Eq{Omf} to get
\begin{eqnarray}
\ngw(\omega)
&=&
\frac{\rho_\text{c}}{8\pi h}
\frac{\Omega_\gw(f)}{f^4}
\label{Omfa}
\end{eqnarray}
which is the sought relation between the distribution function $\ngw(\omega)$ and spectral function $\Omega_\gw(f)$.

%%%%%%%%%%%%%%%%%%%%%%%%%

Let us now consider an estimate of the single atom transition rate $\Gamma_0$.
An example is provided by Rydberg states
$\ket{1} = \ket{n_1,l_1,m_1}$ and $\ket{2} = \ket{n_2,l_2,m_2}$
with $l_1=3$, $l_2=5$ and $m_1=m_2=$ 0, implying $m=0$ for the related nonzero 3-$j$ symbols as described in the appendix \Sec{sec:3j}.
It follows that
\begin{eqnarray*}
\sqrt{\frac{(2l_1+1)(2l+1)(2l_2+1)}{4\pi}}
=
\sqrt{\frac{385}{4\pi}}
\end{eqnarray*}
and
\begin{eqnarray*}
\begin{pmatrix}
  l_1 \;& l \;& l_2 \\
  0   \;& 0 \;& 0
\end{pmatrix}
&=&
\begin{pmatrix}
  l_1 \;& l \;& l_2 \\
  -m_1\;& m \;& m_2
\end{pmatrix}
\nppp
&=&
\begin{pmatrix}
  3 \;& 2 \;& 5 \\
  0 \;& 0 \;& 0
\end{pmatrix}
=
-\sqrt{\frac{10}{231}} .
\end{eqnarray*}
%
With the shorthand
\begin{eqnarray*}
R^2=
\Bra{R_{n_1,l_1}}r^2\Ket{R_{n_2,l_2}}
\end{eqnarray*}
Eq. \eqref{1Y2} gives
\begin{eqnarray}
\Bra{1} r^2 Y_2^0(\theta,\phi) \Ket{2}
&=&
\frac{10}{231}\sqrt{\frac{385}{4\pi}}\,R^2 .
\label{1Y2a}
\end{eqnarray}
%
Using \Eqq{qij}{1Y2a}, we see that Eqs.~\eqref{Q11}--\eqref{Q23} then yield
\begin{eqnarray*}
q_{11}
&=&
q_{22}
=
-\frac12\,q_{33}
=
-
\frac{10}{231}\sqrt{77}\,R^2
\label{q11}
\ppp
q_{12}
&=&
q_{13}
=
q_{23}
=
0 .
\end{eqnarray*}
Substituting these values into \Eq{q2}, we get
\begin{eqnarray*}
q^2 = \frac{200}{231} R^4 .
%\label{q2a}
\end{eqnarray*}
%
For a representative principal quantum number $n$, since the mass quadrupole moment scales with $n^4$, the above relation can be approximated by
\begin{eqnarray*}
q^2 = n^4 a_0^4
%\label{q2b}
\end{eqnarray*}
in terms of the Bohr radius $a_0$.
Then the gravitational spontaneous emission rate for a single atom given by \Eq{gg0}, with the speed of light $c$ restored,  reads
\begin{eqnarray}
\Gamma_0 =
\frac{2\,G m^2 n^4 a_0^4\, \omega_0^5}{45\,\hbar\, c^5} .
\label{gg0a}
\end{eqnarray}

As an order-of-magnitude estimate, let us take $h f_0$ to be a fraction of the Rydberg atom energy at level $n$, say,
\begin{eqnarray}
f_0 = 0.1 \times \frac{E_\text{R}}{n^2}
\label{f0ER}
\end{eqnarray}
where $E_\text{R}$ is the Rydberg energy. This is a reasonable estimate, for example, when considering the Rydberg states above with $n \approx n_1$ or $n_2$ which are respectively 55 or 57 using the following eigen energies:
\begin{eqnarray*}
E_{55\text F} &=& -9.07294 \times 10^{-22} \;\text{J},
\\
E_{55\text H} &=& -10.7575 \times 10^{-22} \;\text{J},
\\
E_{57\text F} &=& -8.37551 \times 10^{-22} \;\text{J},
\\
E_{57\text H} &=& -9.86176 \times 10^{-22} \;\text{J}.
\end{eqnarray*}
The resulting atomic transition frequency is close to the Cosmic Microwave Background value at around
\begin{eqnarray*}
f_0 = 100 \text{ GHz} .
%\label{f0}
\end{eqnarray*}
On applying the above to \Eq{gg0a} with $n=56$ and $m$ being the electron mass, we obtain a small single atom transition rate to be
\begin{eqnarray*}
\Gamma_0
=
10^{-56} \;\text{sec}^{-1} .
%\label{gg0b}
\end{eqnarray*}

To take advantage of an $N^2$ amplification factor for the transition rate discussed in Sec.~\ref{sec:2level}, we will take as initial state a spike-like profile specified by
\begin{eqnarray}
\rho_p
=
\delta_{p,p_0}
\label{drp}
\end{eqnarray}
with $1\le p_0 \le N+1$, then \Eq{fp} becomes
\begin{eqnarray}
D_p
&=&
(1+\ngw(\omega_0))
p\,(N-p+1)\,\delta_{p+1,p_0}
\nppp&&
-(1+\ngw(\omega_0))
(p-1)\,(N-p+2)\,\delta_{p,p_0}
\nppp&&
%\hspace{-15pt}
+\ngw(\omega_0)
(p-1)\,(N-p+2)\,\delta_{p-1,p_0}
\nppp&&
-\ngw(\omega_0)
p\,(N-p+1)\,\delta_{p,p_0} .
\label{fp2}
\end{eqnarray}
%
The transition rate of initial state follows as
\begin{eqnarray}
D_{p_0}
&=&
-(1+\ngw(\omega_0))
(p_0-1)\,(N-p_0+2)
\nppp&&
-\ngw(\omega_0)
p_0\,(N-p_0+1)
\label{fp2a}
\end{eqnarray}
whereas the transition rate of the state above initial state follows as
\begin{eqnarray}
D_{p_0+1}
&=&
\ngw(\omega_0)
p_0\,(N-p_0+1) .
\label{fp2b}
\end{eqnarray}
%
For a large $N$, to maximize the absolute values of \Eqq{fp2a}{fp2b}, we choose $p_0\approx N/2$ then we have
\begin{eqnarray}
D_{p_0}
&=&
-\frac{N^2}{4}(1+2\ngw(\omega_0))
\label{fp2a0}
\end{eqnarray}
and
\begin{eqnarray}
D_{p_0+1}
&=&
\frac{N^2}{4}
\ngw(\omega_0)
\label{fp2b0}
\end{eqnarray}
to the leading order in $N$.
The short-time evolutions of such ``favourable states'' under gravitational waves and spacetime fluctuations are illustrated in \Figg{fig2}{fig3}.

Using \Eqqqq{diss6}{gg0a}{fp2a0}{fp2b0}, it is convenient to we introduce the effective scale transition rates
\begin{eqnarray}
\Gamma_\gw
=
\frac{N^2}{4}\,
\Gamma_0
\ngw(\omega_0)
\label{gggw}
\ppp
\Gamma_\vac
=
\frac{N^2}{4}\,
\Gamma_0\nvac(\omega_0)
\label{ggvac}
\end{eqnarray}
and the associated relaxation times
\begin{eqnarray}
\tau_\gw
=
\frac1{\Gamma_\gw}
,\quad
\tau_\vac
=
\frac1{\Gamma_\vac}
\label{tt}
\end{eqnarray}
due respectively to stochastic gravitational waves and vacuum fluctuations, in terms of the constant vacuum distribution function
\begin{eqnarray}
\nvac(\omega)=1 .
\label{nvac}
\end{eqnarray}



%\begin{eqnarray}
%\ngw(\omega_0)
%=
%\frac{4}{N^2 \Gamma_0 \tau_\gw}
%\end{eqnarray}

\Eqq{Omf}{nvac} allow us to give an analogous spectral function for vacuum fluctuations as a quartic function of frequency to be
\begin{eqnarray}
\Omega_\vac(f)
&=&
\frac{8\pi h f^4}{c^3 \rho_\text{c}}
\label{Omvac}
\end{eqnarray}
with the speed of light $c$ restored.

Given $n$ and $N$, we find from \Eqqqqq{Omf}{gg0a}{f0ER}{gggw}{tt} the corresponding frequency
\begin{eqnarray}
f
&=&
3.3\times10^{14}\,n^{-2}\;\Hz
\label{ff0}
\end{eqnarray}
and the spectral function for gravitational waves to be
\begin{eqnarray}
\Omega(f)
&=&
1.1\times10^{55}\,n^{-2} N^{-2} (\tau/\sec)^{-1}
\label{Omf2}
\end{eqnarray}
which applies to $\Omega=\Omega_\gw$ and $\Omega_\vac$ for given $\tau=\tau_\gw$ and $\tau_\vac$ for the stochastic gravitational waves and vacuum spacetime fluctuations respectively. Subject to further experimental constraints to be investigated further, \Eq{Omf2} provides theoretical lower bounds on their spectral functions for detection, as described in \Fig{fig4}.





\begin{figure}
\vspace{10pt}
\includegraphics[width=1\linewidth]{SGW_fig4.eps}
\caption{
Theoretical lower bounds for the spectral functions of gravitational waves $\Omega(f)$ determined by \Eq{Omf2} that could be detected using an ensemble of correleted 2-level Rydberg-like atoms given their total atom number $N$, excitation principal quantum number $n$, and measurement time $\tau$. For conventional stochastic gravitational waves, we have $\Omega(f)=\Omega_\gw(f)$ with $\tau=\tau_\gw$, whereas for zero-point, i.e. vacuum, meric fluctuations, we have $\Omega(f)=\Omega_\vac(f)$ with $\tau=\tau_\vac$ introduced in this work by \Eqq{tt}{Omvac}. In both cases, we choose $\tau=1$ sec as a physically reasonable overall transition time scale for Rydberg atoms in generating the above diagram. As shown, the atomic transition frequency $f$ is related to the quantum number $n$ by \Eq{ff0}. The lower bounds for $\Omega(f)$ given by \Eq{Omf2} for five selected total atom numbers $N$ are plotted with inclined solid lines. It suggests, for example, measuring the zero-point fluctuations of spacetime would require $10^{28}$ atoms with $n=50$ at $f=100$ GHz approximatly. For comparison, we have superimposed the expected spectral functions of stochastic gravitational waves from various sources such as inflation and that may be detected by different methods such as LIGO from \Ref{LIGO2009} and its references.
}
\label{fig4}
\end{figure}



\section{Summary and conclusion}

Recent developments in gravitational and quantum physics have not only enriched applications with emerging prospects but also escalated the quest for a unified theory of quantum gravity that in turn has important bearing on the structure and evolution of the universe. The growing availabilities for quantum matter exemplified by Rydberg atoms with higher excitations, more correlations, better controls, and in larger quantities promise valuable tools in this scientific endeavour. In this work, we have carried out new theoretical analysis describing how a large number of correlated atoms could interact and sense gravitational waves of fundamental and cosmological origins to go beyond the physical capacities of conventional detectors.

In \Sec{sec:mas}, we have furthered the theoretical foundation of how general confined nonrelativistic matter interacts with an open gravitational environment, relevant to, for example, a system of atoms in a cavity attracting much recent attention. Specifically, we have provided a general master equation through \Eq{diss3} with negligible self-interactions to focus on the collective interactions between such a matter system and gravitational waves in the environment due to both vacuum and stochastic fluctuations.
The theoretical formulation is applied in \Sec{sec:trans} to a two-level atom, that allows us to derive specific transition mechanisms through the quadrupole moment operator with respect to the two atomic states given by \Eq{diss4}. In particular, we have derived a formula for the gravitational transition rate $\Gamma_0$ in \Eq{gg0} for such a 2-level atom.

Unsurprisingly, the transition rate $\Gamma_0$ for a single atom is too small to be measured under typical laboratory conditions. In \Sec{sec:2level}, we have invoked the generality of the formalism developed in \Sec{sec:mas} to extend the interaction between a 2-level atom and gravity to accommodate $N$ such atoms. In this case, as illustrated in \Fig{fig1}, we find an overall factor of $N$ collective amplification of the transition rate from any initial state to the final equilibrium state satisfying the Boltzmann distribution in the large $N$ limit\eq{rs}. The temperature of this distribution can be identified, which we show to agree with the environmental gravitational temperature, if it exists, under reasonable physical conditions.


In \Sec{sec:gw}, we have established a relation between the spectral function $\Omega_\gw(f)$ for gravitational waves and distribution function $\ngw(\omega)$ for open quantum gravitational systems and introduce a new spectral function $\Omega_\vac(f)$ for the zero-point, i.e. quantum vacuum, fluctuations of spacetime.
Although we have introduced a spectral function for vacuum fluctuations of spacetime $\Omega_\vac(f)$ in \Eq{Omvac}, it should be emphasized that the detection of zero-point fluctuations of spacetime requires a quantum detector through decoherence as classical detectors have already decoherenced and so can no longer provide this function.
Indeed, the possibility of sensing the zero-point fluctuations of spacetime provides a unique physical characteristics for the collective quantum process which could lead to initial evidence and further guidance of quantum gravity.
For optimal measurements, we have identified favourable states whose decay and excitation rates with gravitational waves and spacetime fluctuations scale amplify with a factor of $N^2$, as described by \Eqq{gggw}{ggvac} and illustrated in \Figg{fig2}{fig3}.
Finally, we have determined the theoretical lower bounds given by \Eq{Omf2} for the spectral functions of gravitational waves that could in principle be detected using an ensemble of identical 2-level Rydberg-like atoms given their excitation states and total atom number as well as measurement time. Furthermore, the lower bound relation\eq{Omf2} applies to both conventional stochastic gravitational waves and novel zero-point spacetime fluctuations.

In this paper, we have reported a theoretical results within generic physical constraints, but not fully taken onboard experimental implementations, which are a subject of future work.
Nonetheless, apart from extending conceptual framework, building up theoretical tools and guiding possible viable detection of gravitational waves outside existing windows including zero-point fluctuations, our findings can already be used to rule out potential claims of effects that can be mapping to below our theoretical lower bounds on measurable gravitational wave spectral function values using quantum techniques, as noted in the caption of \Fig{fig4}, unless new inputs in addition to considerations adopted here are supplied.
In going forward, it would also be worth exploring the effects of the interaction Hamiltonian $H_\intt$ in \Eq{maseqn}, which we neglect at present for simplicity, to accommodate atom-atom interactions, external control of atoms, and interactions with regular and non-stochastic gravitational waves for their quantum sensing based our framework.

\acknowledgments

The authors are grateful for financial support to
the National Council for Science and Technology (CONACyT) (D.Q.),
the Carnegie Trust for the Universities of Scotland (T.O.), and the Cruickshank Trust and EPSRC GG-Top Project (C.W.).


\appendix

\section{Alternative derivation of the quantum dissipator of the 2-level atom}
\label{sec:diss}

The quantum mechanical interaction Hamiltonian between a mass $m$ and gravitational waves  \cite{Fischert1994, Parker1980} is given by
\begin{eqnarray}
H_I
&=&
\frac12 m R_{0i0j} r_i r_j .
\label{GWI}
\end{eqnarray}
Using the weak gravitational wave relation \cite{Flanagan2005}
\begin{eqnarray*}
R_{0i0j} = -\frac12 \ddot{h}^\TT_{ij}
%\label{R2h}
\end{eqnarray*}
we see that Eq. \eqref{GWI} becomes
%\begin{eqnarray*}
%H_I
%&=&
%-\frac14 m \ddot{h}^\TT_{ij} r_i r_j
%\nppp
%&=&
%-\frac1{12} m \ddot{h}^\TT_{ij}
%(3 r_i r_j - \delta_{ij} r^2)
%\end{eqnarray*}
%i.e.
\begin{eqnarray}
H_I
&=&
-\frac1{12} m Q_{ij} \,\ddot{h}^\TT_{ij}
\label{GWI2}
\end{eqnarray}
%analogous to Eq. (3.179) in \cite{Breuer2002},
where $Q_{ij}$ is the quadrupole moment tensor given by \Eq{Qij}.

It is useful to write \Eq{GWI2} in the standard form of the interaction Hamiltonian for open quantum systems~\cite{Breuer2002}
\begin{eqnarray}
H_I
&=&
\sum_\alpha
A_\alpha B_\alpha
\label{GWI3}
\end{eqnarray}
with the following identifications:
\begin{eqnarray}
\alpha&\to&(i,j)
\label{idal}
\ppp
A_\alpha&\to& A_{ij} = m  Q_{ij}
\label{idA}
\ppp
B_\alpha&\to& B_{ij} = -\frac1{12} \ddot{h}^\TT_{ij}
\label{idB}
\end{eqnarray}
yielding in our 2-level case the following Lindblad operators
\begin{eqnarray}
A_{ij}(\omega)
&=&
\Ket{1}\Bra{1} A_{ij} \Ket{2}\Bra{2}
=
m\, q_{ij}\, \sigma^{-}
\label{Ap}
\ppp
A_{ij}(-\omega)
&=&
\Ket{2}\Bra{2} A_{ij} \Ket{1}\Bra{1}
=
m\, q_{ij}^*\, \sigma^{+}
\label{Am}
\end{eqnarray}
using \Eq{qij} and
\begin{eqnarray*}
\sigma^{+}
&=&
\Ket{2}\Bra{1}
,\quad
\sigma^{-}
=
\Ket{1}\Bra{2} .
\end{eqnarray*}
Clearly $A_{ij}(\omega)$ satisfies
\begin{eqnarray}
A_{ij}^\dag(\omega)=A_{ij}(-\omega),\quad A_{kk}(\omega)=0
\label{Aw}
\end{eqnarray}
for $\omega\to\pm\omega$.

Using the rotating wave approximation, we have the following dissipator
\begin{eqnarray}
\DD\rho
&=&
\sum_{\omega\to\pm\omega}  \sum_{\alpha,\alpha'}
\Gamma_{\alpha,\alpha'}(\omega)\times
\nppp&&
\hspace{-15pt}
\left [A_{\alpha}(\omega) \rho A_{\alpha'}^\dag (\omega) - A_{\alpha'}^\dag (\omega) A_{\alpha}(\omega) \rho \right ] + \hc
\label{master2}
\end{eqnarray}
with the spectral correlation tensor:
\begin{eqnarray}
\Gamma_{\alpha,\alpha'}(\omega)
&=&
\frac{1}{\hbar^2} \int_0^{\infty} \dd s\,
e^{i \omega s} \langle{B_{\alpha}^\dag(s) B_{\alpha'}(0)} \rangle
\label{Gaa}
\end{eqnarray}
using a Gaussian stationary bath of gravitational waves.
This expression can be evaluated by using \Eq{idB} with the field expansion for $h_{ij}^\TT$.
To evaluate \Eq{Gaa} consider the  wave expansion for the gravitons:
\begin{eqnarray}
{h}^\TT_{ij}
=
\int\dd^3 k
\sqrt{\frac{G\hbar}{\pi^2 k}}\,
e_{ij}^\lambda(\kk)\,
a_{\kk}^\lambda\,
e^{ikx}
+
\hc
\label{gwexa}
\end{eqnarray}
with the relations
\begin{eqnarray*}
\big[
a_{\kk}^\lambda, a_{\kk'}^{\lambda'\dag}
\big]
&=&
\delta_{\lambda\lambda'}\delta(\kk, \kk')
\ppp
\big[
a_{\kk}^{\lambda\dag}, a_{\kk'}^{\lambda'\dag}
\big]
&=&
\big[
a_{\kk}^\lambda, a_{\kk'}^{\lambda'}
\big]
=0 .
\end{eqnarray*}

To proceed, consider the discrete momentum $\kb$ is expressed in terms of the Cartesian wave numbers $n_i = 1, 2, \cdots$ as follows:
%
\begin{eqnarray*}
\kb
&=&
\left(\frac{2 \pi n_1}{L}, \frac{2 \pi n_2}{L},\frac{2 \pi n_3}{L}\right)
%\label{}
\end{eqnarray*}
%
In order to write the field in a box of volume $V$ with discrete modes we will use:
%
\begin{eqnarray}
\int \dd^3 k \leftrightarrow \frac{(2 \pi)^3}{V} \sum_{\kb}
\label{ds}
\end{eqnarray}
%
with the modes inside the box normalized such that
%
\begin{eqnarray*}
\int_V \dd ^3 x \; e^{i \kb \rr} e^{- i \kb' \rr} = V \delta_{\kb \kb'} .
\end{eqnarray*}
%
Along with the following representation for the delta function in a finite volume:
\begin{eqnarray*}
\delta(\rr - \rr') = \frac{1}{V} \sum_{\kb} e^{i \kb (\rr - \rr')}
\end{eqnarray*}
The Dirac delta function maps to Kronecker delta as follows:
\begin{eqnarray*}
\delta(\kk-\kk')
&\leftrightarrow&
\frac{V}{(2 \pi)^3}\,\delta_{\kb \kb'} .
%\label{DDD3}
\end{eqnarray*}
Ladder operators for free fields are related by
\begin{eqnarray*}
a_{\kk}
&\leftrightarrow&
\sqrt{\frac{V}{(2 \pi)^{3}}}\; a_{\kb}
%\label{aak3}
\\
a_{\kk}^\dagger
&\leftrightarrow&
\sqrt{\frac{V}{(2 \pi)^{3}}}\; a_{\kb}^\dag
%\label{aan3}
\end{eqnarray*}
so that
\begin{eqnarray*}
[a_{\kk}, a_{\kk'}^\dagger]
&=&
\delta(\kk-\kk')
%\label{aakn3}
\nppp
[a_{\kb}, a_{\kb}^\dagger]
&=&
\delta_{\kb \kb'} .
%\label{aank3}
\end{eqnarray*}
Then the discrete version of \eqref{gwexa} takes the form
\begin{eqnarray}
{h}^\TT_{ij}
&=&
\sum_{\kb}\sqrt{\frac{8\pi G \hbar}{V k}}\,
e_{ij}^\lambda(\kb)\,
a_{\kb}^\lambda\,
e^{ikx}
+
\hc
\label{gwexb}
\end{eqnarray}
For an equilibrium environment \cite{Breuer2002}, we have
\begin{eqnarray*}
\langle
a_{\kb}^{\lambda\dag} a_{\kb'}^{\lambda'}
\rangle
&=&
\delta_{\lambda\lambda'}\delta_{\kb, \kb'}\,\ngw(k)
\ppp
\langle
a_{\kb}^\lambda a_{\kb'}^{\lambda'\dag}
\rangle
&=&
\delta_{\lambda\lambda'}\delta_{\kb, \kb'}(1+\ngw(k))
\ppp
\langle
a_{\kb}^{\lambda\dag} a_{\kb'}^{\lambda'\dag}
\rangle
&=&
\langle
a_{\kb}^\lambda a_{\kb'}^{\lambda'}
\rangle
=0 .
\end{eqnarray*}
for some distribution function $\ngw(k)$ with the quantum ensemble average $\av{\cdot}$.
Then using Eqs.~\eqref{idB}, \eqref{gwexb}, and the above, Eq. \eqref{Gaa} becomes
\begin{eqnarray*}
\Gamma_{ij,kl}(\omega)
&=&
\frac{\pi G k^3}{9\hbar V}\,
\sum_{\kb}
P_{ijkl}(\kb)\times
\nppp&&\hspace{-30pt}
\int_0^{\infty} \dd s\,[
(1+\ngw(k))\,e^{-i(k-\omega)s}
+
\ngw(k)\,e^{i(k+\omega)s}
]
\end{eqnarray*}
Then by using \Eq{ds}, the above returns to the continuous momentum case as follows
\begin{eqnarray*}
\Gamma_{ij,kl}(\omega)
&=&
\frac{G}{72\pi^2\hbar}
\int_0^\infty \dd k \,k^5
\int \dd \Omega \,
P_{ijkl}(\kk)\times
\nppp&&\hspace{-35pt}
\int_0^{\infty} \dd s\,[
(1+\ngw(k))\,e^{-i(k-\omega)s}
+
\ngw(k)\,e^{i(k+\omega)s}
] .
\end{eqnarray*}
Substituting the relation\eq{iPijkl} into the above we have
\begin{eqnarray*}
\Gamma_{ij,kl}(\omega)
&=&
\frac{G(3\delta_{ik}\delta_{jl}+3\delta_{il}\delta_{jk}-2\delta_{ij}\delta_{kl})}
{270\pi\hbar}
\int_0^\infty \!\dd k \,k^5
\times
\nppp&&\hspace{-30pt}
\int_0^{\infty} \!\!\dd s [
(1+\ngw(k))\,e^{i(\omega-k)s}
+
\ngw(k)\,e^{i(\omega+k)s}
]
\end{eqnarray*}
Using the the Sokhotski-Plemelj theorem~\cite{Breuer2002}
\begin{equation}
\int_0^{\infty} \dd s\, e^{-i \epsilon s} = \pi \delta(\epsilon) - i \text{P} \frac{1}{\epsilon}
\label{fma}
\end{equation}
where P denotes the Cauchy principal value, the above becomes
\begin{eqnarray*}
\Gamma_{ij,kl}(\omega)
&=&
\frac{G(3\delta_{ik}\delta_{jl}+3\delta_{il}\delta_{jk}-2\delta_{ij}\delta_{kl})}
{270\pi\hbar}\times
\nppp&&
\hspace{-30pt}
\int_0^\infty \dd k \,k^5
\Big[
(1+\ngw(k))
\Big(
\pi \delta(\omega-k) + i\text{P}\frac{1}{\omega - k }
\Big)
\nppp&&
\hspace{-30pt}
+
\ngw(k)
\Big(
\pi \delta(\omega+k) + i\text{P}\frac{1}{\omega + k }
\Big)
\Big] .
%\label{Gamal3}
\end{eqnarray*}
Neglecting P terms as they do not contribute to quantum decoherence and dissipation, we see that the above reduces to
\begin{eqnarray*}
\Gamma_{ij,kl}(\omega)
&=&
\frac{G(3\delta_{ik}\delta_{jl}+3\delta_{il}\delta_{jk}-2\delta_{ij}\delta_{kl})}
{270\hbar}\int_0^\infty \!\dd k
\times
\nppp&&
\hspace{-20pt}
k^5\Big[
(1+\ngw(k))\delta(\omega-k)
+
\ngw(k)\delta(\omega+k)
\Big]
%\label{Gamal4}
\end{eqnarray*}
yielding
\begin{eqnarray*}
\Gamma_{ij,kl}(\omega)
&=&
\frac{G\omega^5}{270\hbar}\times
\nppp&&
(3\delta_{ik}\delta_{jl}+3\delta_{il}\delta_{jk}-2\delta_{ij}\delta_{kl})
(1+\ngw(\omega))
%\label{Gamal4a}
\nppp
\Gamma_{ij,kl}(-\omega)
&=&
\frac{G\omega^5}{270\hbar}
(3\delta_{ik}\delta_{jl}+3\delta_{il}\delta_{jk}-2\delta_{ij}\delta_{kl})
\ngw(\omega) .
%\label{Gamal4b}
\end{eqnarray*}
Substituting these two relations into the dissipator \eqref{master2} and using \eqref{Aw} we have:
\begin{eqnarray*}
\DD\rho
&=&
\frac{G\omega^5}{90\hbar}
\sum_{ij,kl}
(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})
\times
\nppp&&
\Big\{
(1+\ngw(\omega))
\big[
A_{ij}(\omega) \rho A_{kl}^\dag (\omega)
-
A_{kl}^\dag (\omega) A_{ij}(\omega) \rho
\nppp&&
+
A_{kl}(\omega) \rho A_{ij}^\dag (\omega)
-
\rho A_{ij}^\dag(\omega) A_{kl} (\omega)
\big]
\nppp&&
+
\ngw(\omega)
\big[
A_{ij}^\dag(\omega) \rho A_{kl} (\omega)
-
A_{kl}(\omega) A_{ij}^\dag(\omega) \rho
\nppp&&
+
A_{kl}^\dag (\omega) \rho A_{ij}(\omega)
-
\rho A_{ij}(\omega) A_{kl}^\dag(\omega)
\big]
\Big\}
\end{eqnarray*}
which yields the following dissipator of the Lindblad form:
\begin{eqnarray*}
\DD\rho
&=&
\frac{2\,G\omega^5}{45\hbar}
\sum_{ij}
\Big\{
(1+\ngw(\omega))\times
\\&&\hspace{-10pt}
\big [
A_{ij}(\omega) \rho A_{ij}^\dag (\omega)
-
\frac12 \big\{A_{ij}^\dag (\omega) A_{ij}(\omega),\, \rho\big\}
\big ]
\nppp&&\hspace{-10pt}
+
\ngw(\omega)
\big[
A_{ij}^\dag(\omega) \rho A_{ij} (\omega)
-
\frac12 \big\{A_{ij}(\omega) A_{ij}^\dag(\omega),\, \rho\big\}
\big]
\Big\} .
%\label{master3}
\end{eqnarray*}
Finally, by using \Eqq{Ap}{Am}, we can further simplify the above to the same form as \Eq{diss4}.


\section{Evaluation of the transition quadrupole moment}
\label{sec:3j}

To evaluate $q_{ij}$ in \Eq{qij} let us consider the components of $Q_{ij}$ given by \Eq{Qij}, noting that $Q_{ij}$ has 5 independent components for being traceless.
For $l=2$ we can express $Q_{ij}$ in terms of the spherical harmonics $Y_{l=2}^{m}(\theta,\phi)$ as follows:
\begin{eqnarray}
Q_{11}
&=&
\sqrt{\frac{6\pi}5}\, r^2
(Y_2^{-2}+Y_2^{2})
%\nppp&&
-
\sqrt{\frac{4\pi}{5}}\, r^2\,
Y_2^{0}
\label{Q11}
\ppp
Q_{22}
&=&
-\sqrt{\frac{6\pi}5}\, r^2
(Y_2^{-2}+Y_2^{2})
%\nppp&&
-
\sqrt{\frac{4\pi}{5}}\, r^2\,
Y_2^{0}
\ppp
Q_{33}
&=&
\sqrt{\frac{16\pi}{5}}\, r^2\,
Y_2^{0}
\end{eqnarray}
and
\begin{eqnarray}
Q_{12}
&=&
i\sqrt{\frac{6\pi}5}\, r^2
(Y_2^{-2}-Y_2^{2})
\ppp
Q_{13}
&=&
\sqrt{\frac{6\pi}5}\, r^2
(Y_2^{-1}-Y_2^{1})
\ppp
Q_{23}
&=&
i\sqrt{\frac{6\pi}5}\, r^2
(Y_2^{-1}+Y_2^{1})
\label{Q23} .
\end{eqnarray}
Denoting by
\begin{eqnarray*}
\Bra{R_{n_1,l_1}}D(r)\Ket{R_{n_2,l_2}}
=
\int_{0}^\infty\!\dd r\,r^2 D(r) R_{n_1,l_1}(r) R_{n_2,l_2}(r)
\end{eqnarray*}
we can therefore evaluate the matrix elements of $Q_{ij}$ in terms of Wigner's 3-$j$ symbols as follows:
\begin{eqnarray}
\lefteqn{\Bra{n_1,l_1,m_1} r^2 Y_l^m(\theta,\phi) \Ket{n_2,l_2,m_2}}
\nppp
&=&
\Bra{R_{n_1,l_1}}r^2\Ket{R_{n_2,l_2}}
(-1)^{m_1}
\nppp&&\times\;
\sqrt{\frac{(2l_1+1)(2l+1)(2l_2+1)}{4\pi}}
\nppp&&\times\;
\begin{pmatrix}
  l_1 \;& l \;& l_2 \\
  0   \;& 0 \;& 0
\end{pmatrix}
\begin{pmatrix}
  l_1 \;& l \;& l_2 \\
  -m_1\;& m \;& m_2
\end{pmatrix} .
\label{1Y2}
\end{eqnarray}






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%%%%%%%%%%%*********************************************************************

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\end{thebibliography}

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