# Detection of Gravitational Waves using Parametric Resonance in Bose-Einstein
Condensates

Matthew P. G. Robbins<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Waterloo, Waterloo ON, Canada, N2L 3G1  Waterloo
Centre for Astrophysics, University of Waterloo, Waterloo ON, Canada, N2L 3G1
Perimeter Institute for Theoretical Physics, 31 Caroline Street North,
Waterloo ON, Canada, N2L 2Y5  Niayesh Afshordi<EMAIL_ADDRESS>Department of
Physics and Astronomy, University of Waterloo, Waterloo ON, Canada, N2L 3G1
Waterloo Centre for Astrophysics, University of Waterloo, Waterloo ON, Canada,
N2L 3G1  Perimeter Institute for Theoretical Physics, 31 Caroline Street
North, Waterloo ON, Canada, N2L 2Y5  Alan O. Jamison<EMAIL_ADDRESS>Department of Physics and Astronomy, University of Waterloo, Waterloo ON,
Canada, N2L 3G1  Institute for Quantum Computing, University of Waterloo,
Waterloo ON, Canada, N2L 3G1  Robert B. Mann<EMAIL_ADDRESS>Department of
Physics and Astronomy, University of Waterloo, Waterloo ON, Canada, N2L 3G1
Waterloo Centre for Astrophysics, University of Waterloo, Waterloo ON, Canada,
N2L 3G1  Perimeter Institute for Theoretical Physics, 31 Caroline Street
North, Waterloo ON, Canada, N2L 2Y5  Institute for Quantum Computing,
University of Waterloo, Waterloo ON, Canada, N2L 3G1

###### Abstract

An interesting proposal for detecting gravitational waves involves quantum
metrology of Bose-Einstein condensates (BECs). We consider a forced modulation
of the BEC trap, whose frequency matches that of an incoming continuous
gravitational wave. The trap modulation induces parametric resonance in the
BEC, which in turn enhances sensitivity of the BEC to gravitational waves. We
find that such a BEC detector could potentially be used to detect
gravitational waves across several orders of magnitude in frequency, with the
sensitivity depending on the speed of sound, size of the condensate, and
frequency of the phonons. We outline a possible BEC experiment and discuss the
current technological limitations. We also comment on the potential noise
sources as well as what is necessary for such a detector to become feasible.

##  1 Introduction

The recent detection of gravitational waves is revolutionizing our
understanding of the Universe. Though our current observations have only
detected black holes and neutron stars with LIGO/Virgo [1], it is expected
that future gravitational wave detectors may also see phenomena such as
supernovae, extreme-mass inspirals, and a stochastic gravitational wave
background [2, 3, 4].

One of the challenges of gravitational wave astronomy is that current
detectors are sensitive to only a small range of frequencies. Constructing a
multitude of detectors (both terrestrial and in space) is thus required to
cover the entire gravitational wave spectrum [5, 6]. Much effort has been
exerted into designing detectors for sub-kHz sources, such as the Laser
Interferometer Space Antenna (LISA) (0.1 mHz-1 Hz) [7], the Einstein Telescope
($1-10,000$ Hz) [8, 9] and pulsar timing arrays like the International Pulsar
Timing Array and European Pulsar Timing Array ($\sim$nHz) [10, 11]. Atom
interferometers have also been proposed to detect gravitational waves in the
millihertz to decahertz regimes [12, 13, 14]. Though the Einstein Telescope is
proposed to be sensitive up to 10 kHz, its optimal sensitivity will occur
around the same frequency range as LIGO. Additional gravitational wave
detector technologies have also been proposed, including a superfluid detector
for continuous gravitational waves (0.1-1.5 kHz) [15] and a satellite-based
cold atom interferometer for mid-frequency gravitational waves (30 mHz-10 Hz)
[16].

In the frequency regime probed by LIGO, Virgo, and KAGRA (above 100 Hz and
below 1 kHz), the most common sources are binary black holes, binary neutron
stars, and black hole-neutron star mergers [17]. In the kHz regime, it is
predicted that transient sources include lower-mass black holes and neutron
star mergers at frequencies outside the range that LIGO can currently observe
(up to several kHz) [18, 19, 20]. Depending on the model, there could
potentially even be primordial black holes in the kHz domain [21]. In the
continuous regime, there may be magnetars (0.5-2 kHz) [22], neutron
stars/pulars (tens-hundreds of Hz) [23, 24], and boson clouds (extending into
the kHz regime and above) [25].

One intriguing proposal [26] suggests using a uniform (non-rotating) Bose-
Einstein condensate (BEC) as a gravitational wave detector. By noting that an
incoming gravitational wave is able to create phonons within the condensate
and comparing the state of the phonons before and after the gravitational wave
interacts with the BEC, it may be possible to gain knowledge about the
amplitude of the gravitational wave. Though a BEC is a quantum system, it is
not possible to use the usual quantum mechanical method of doing measurements
(projective-valued measurements, positive operator-valued measurements, etc.)
to estimate the amplitude $\epsilon$ of the gravitational wave as the
amplitude is not an operator observable (assuming the wave is classical).
Instead, the quantum Fisher information $H_{\epsilon}$ can be used, which
describes how much quantum information can be obtained from a single
measurement of an arbitrary parameter in a quantum system. By doing $M$
measurements of the amplitude of the gravitational wave (for example, using
quantum dots or phonon evaporation [26]), the quantum Cramer-Rao bound states
that the resultant uncertainty in the measurement of $\epsilon$ is
$\braket{(\Delta\epsilon)^{2}}\geq\frac{1}{MH_{\epsilon}}$ [27]. Increasing
either the number of measurements of the system or the quantum Fisher
information obtained from each measurement therefore improves sensitivity to
gravitational wave detection.

However, this proposal faces significant challenges. We have previously
pointed out that, while a BEC could in principle be used to detect
gravitational waves, it is necessary to squeeze the initial state of the BEC
phonons well beyond current experimental capabilities [28]. Furthermore, the
natural decay of the BEC by Beliaev damping limits the sensitivity, especially
at the higher frequencies of interest. A somewhat stronger critique [29]
argued that non-uniform condensates are required, as otherwise the phonons
produced are not within the realm of detectability, though it should be noted
that the critique assumed a condensate without any boundary conditions,
whereas the original proposal [26] considered a hard-wall condensate.

We investigate here the possibility of improving the sensitivity of a BEC
detector by modulating the trap size to induce a parametric resonance with the
BEC phonons, which then boosts their sensitivity to gravitational waves. In
contrast to previous studies [26, 28], we find that parametric resonance
causes the BEC to be more sensitive to gravitational waves at lower
frequencies than at higher frequencies, though detection of higher frequency
gravitational waves may still be possible. The optimum sensitivity depends
upon the parameters of the condensate such as its length, speed of sound and
phonon frequency.

The outline of the subsequent sections is as follows: In Section 2, we review
the theory behind BECs in curved spacetime and solve the equations of motion
of the phonons as well as the Bogoliubov coefficients. We also discuss the
effect of parametric resonance and non-linearities in our model. In Section 3,
we then show how quantum Fisher information is used to determine the
sensitivity to gravitational waves. Next, in Section 4, we discuss how Beliaev
damping will limit the sensitivity of the BEC to gravitational waves at high
frequencies. In Section 5, we discuss the possible sources that a BEC could
observe as well as consider the experimental feasibility of the proposal,
comment on expected noise sources, and discuss what may be necessary to make
this proposal a reality. Section 6 summarizes our conclusions and presents
possible directions for future work.

##  2 Bose-Einstein condensates in curved spacetimes

###  2.1 Equation of Motion

We first derive the equations of motion and Bogoliubov coefficients for BEC
phonons in a curved spacetime. Note that such derivations have also been done
in [26, 30, 31]. The Lagrangian for a BEC in a curved spacetime is

$\displaystyle\mathcal{L}=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi^{\dagger}-M^{2}(x^{i},t)|\phi|^{2}-U(|\phi|^{2}),$
(1)

where $g_{\mu\nu}$ is the metric, $\phi$ is the field, and
$U(|\phi|^{2})=\lambda|\phi|^{4}>0$ is the interaction potential. The quantity
$M^{2}(x^{i},t)=\frac{m^{2}c^{2}}{\hbar^{2}}+V(x^{i},t)$ describes a time-
dependent effective mass of the atoms in the condensate, where we take

$\displaystyle
V(x^{i},t)=V_{0}\prod_{i=1}^{3}\Theta\left[\frac{x^{i}}{\alpha(t)}\left(\frac{x^{i}}{\alpha(t)}-L\right)\right]$
(2)

and $V_{0}$ is a constant. This simple, separable effective potential models
the effect of trap boundaries oscillating at frequency $\Omega_{B}$, which in
turn act to cause the length of the condensate to oscillate. The quantity
$\alpha(t)=1+a\sin\Omega_{B}t$ describes the fluctuations of the walls of the
trap with $a=\frac{\delta L}{L}\ll 1$ the variation of the box potential
length.

Given the definition $\phi=\hat{\phi}e^{i\chi}$ with real $\hat{\phi}$ and
$\chi$, and assuming a slowly varying amplitude, i.e. that
$(\partial\hat{\phi})^{2}\ll\hat{\phi}^{2}(\partial\chi)^{2}$, we can
extremize $\mathcal{L}$ with respect to $\hat{\phi}$ to find:

$\displaystyle\hat{\phi}^{2}=\frac{1}{2\lambda}\left[\partial^{\mu}\chi\partial_{\mu}\chi-m^{2}-V(x^{i},t)\right]\
,$ (3)

and so

$\displaystyle\mathcal{L}=\frac{1}{2\lambda}\left[g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-\frac{m^{2}c^{2}}{\hbar^{2}}-V(x^{i},t)\right]^{2}\
.$ (4)

To study the effects of a gravitational wave, we introduce the coordinate
system $\bar{x}^{i}=\frac{x^{i}}{\alpha}$ and $ct=x^{0}=\bar{x}^{0}$. In the
$\bar{x}^{\mu}$ coordinate system, the line element becomes

$\displaystyle
ds^{2}=\left(g_{00}+g_{ij}\bar{x}^{i}\bar{x}^{j}\dot{\alpha}^{2}\right)(d\bar{x}^{0})^{2}+2g_{ij}\bar{x}^{j}\alpha\dot{\alpha}d\bar{x}^{0}d\bar{x}^{i}+g_{ij}\alpha^{2}d\bar{x}^{i}d\bar{x}^{j}\
,$ (5)

where we are using a $(+,-,-,-)$ signature. Let
$\chi=\frac{mct}{\hbar}+\pi(x^{\mu})$, where $\pi(x^{\mu})\in\Re$ is a pseudo-
Goldstone boson representing the acoustic perturbation (phonon) field of the
condensate. These phonons can be represented as fluctuations in the phase
[32]. Setting $g_{ij}=\eta_{ij}+h_{ij}$, with $[h_{ij}]\ll 1$, the action of
the system becomes

$\displaystyle
S=\frac{1}{2\lambda}\int\left[A\dot{\pi}^{2}/c^{2}+C^{i}\bar{\partial}_{i}\pi+D^{i}\dot{\pi}\bar{\partial}_{i}\pi/c+E^{ij}\bar{\partial}_{i}\pi\bar{\partial}_{j}\pi\right]\sqrt{-\bar{g}}d^{4}x\
,$ (6)

where $h_{ij}$ describes the distortion of spacetime due to the gravitational
wave and

$\displaystyle A/c^{2}$
$\displaystyle=\frac{1}{c^{2}}\left[-2V-2\frac{m^{2}c^{2}}{\hbar^{2}}+\frac{6\kappa^{2}}{\hbar^{2}c^{2}}\right]+\mathcal{O}\left(\frac{\Omega_{B}^{2}L^{2}}{c^{2}}\right)\
,$ $\displaystyle C_{i}/c$
$\displaystyle=\frac{a\Omega_{B}}{c}\left[-\frac{4cm^{2}\kappa}{\hbar^{3}}+\frac{4\kappa^{3}}{\hbar^{3}c^{3}}-\frac{4V\kappa}{\hbar
c}\right]g_{ij}\bar{x}^{j}\cos\Omega_{B}t+\mathcal{O}\left(\frac{\Omega_{B}^{2}L^{2}}{c^{2}}\right)\
,$ $\displaystyle D_{i}/c$
$\displaystyle=\frac{a\Omega_{B}}{c}\left[-4V-\frac{4c^{2}m^{2}}{\hbar^{2}}+\frac{12\kappa^{2}}{\hbar^{2}c^{2}}\right]g_{ij}\bar{x}^{j}\cos\Omega_{B}t+\mathcal{O}\left(\frac{\Omega_{B}^{2}L^{2}}{c^{2}}\right)\
,$ $\displaystyle E_{ij}/c^{2}$
$\displaystyle=\frac{1}{c^{2}}\left[-2V-\frac{2m^{2}c^{2}}{\hbar^{2}}+\frac{2\kappa^{2}}{\hbar^{2}c^{2}}\right]g_{ij}\left(1+2a\sin\Omega_{B}t\right)+\mathcal{O}\left(\frac{\Omega_{B}^{2}L^{2}}{c^{2}}\right)\
,$

as $x^{i}\lesssim L$. We denote differentiating with respect to $\bar{x}^{i}$
as $\bar{\partial}_{i}$ and assume that $\pi$ vanishes at the boundary.

As $V$ consists of Heaviside functions, we can treat the cases of $V=0$ and
$V=V_{0}$ separately. The only difference in what follows will be the value of
the speed of sound. Therefore, the equation of motion to leading order in $a$
is

$\displaystyle\ddot{\pi}+c_{s}^{2}g^{ij}(1+2a\sin\Omega_{B}t)\bar{\partial}_{i}\bar{\partial}_{j}\pi=0\
,$ (7)

where

$\displaystyle
c_{s}^{2}=\frac{-V_{0}-\frac{m^{2}c^{2}}{\hbar^{2}}+\frac{\kappa^{2}}{\hbar^{2}c^{2}}}{-V_{0}-\frac{m^{2}c^{2}}{\hbar^{2}}+\frac{3\kappa^{2}}{\hbar^{2}c^{2}}}c^{2}$
(8)

is the speed of sound, assuming a linear dispersion relation for the phonons.
Setting $\pi=e^{i\vec{k}\cdot\bar{x}}\psi(t)$, we have

$\displaystyle\ddot{\psi}+c_{s}^{2}g_{ij}(1+2a\sin\Omega_{B}t)k^{i}k^{j}\psi=0\
,$ (9)

where we have neglected terms
$\mathcal{O}\left(\frac{\Omega_{B}^{2}L^{2}}{c^{2}}\right)$ and higher. A
derivation of the dispersion relation for the BEC for $V_{0}\neq 0$ is similar
to that of the $V=0$ case [28], with $m$ becoming a time-dependent effective
mass, such that
$\frac{m^{2}c^{2}}{\hbar^{2}}\to\frac{m^{2}c^{2}}{\hbar^{2}}+V_{0}$. In the
limit of $V_{0}>2m\mu/\hbar^{2}$, the phonons cannot propagate outside the
walls, and thus we can approximately use Dirichlet boundary conditions.

By working in the TT-gauge and assuming no cross-polarization for simplicity,

$\displaystyle h_{\mu\nu}=\begin{pmatrix}0&0&0&0\\\ 0&h_{+}(t)&0&0\\\
0&0&-h_{+}(t)&0\\\ 0&0&0&0\end{pmatrix}\ .$ (10)

Taking $h_{+}(t)=\epsilon\sin\Omega t$, where $\epsilon$ is the amplitude of
the continuous gravitational wave and $\Omega$ is its frequency, the equation
of motion (9) becomes

$\displaystyle\ddot{\psi}+\omega^{2}(1+2a\sin\Omega_{B}t)(1+\tilde{\epsilon}\sin\Omega
t)\psi=0\ ,$ (11)

where $\tilde{\epsilon}=\frac{(k_{x}^{2}-k_{y}^{2})}{|k|^{2}}\epsilon$.

Let $\omega_{1}(t)^{2}=\omega^{2}(1+\tilde{\epsilon}\sin\Omega
t+2a\sin\Omega_{B}t+2a\tilde{\epsilon}\sin\Omega t\sin\Omega_{B}t)$. As we
want to consider the gravitational wave on resonance with the magnetic field,
we will take $\Omega_{B}=\Omega$. Expanding to first-order in $\epsilon$ and
second-order in $a$ yields $\omega_{1}(t)\approx\omega+a\omega\sin(\Omega t)$
where we have neglected the $\mathcal{O}(a\tilde{\epsilon})$ terms as
$a\tilde{\epsilon}\ll a$. We will see that it is the presence of the
$\mathcal{O}(a)$ terms that induce parametric resonance within the condensate.

To solve equation (11) in this approximation, let us assume that we can
decompose the solutions as [33]

$\displaystyle\pi$
$\displaystyle=\alpha(t)\frac{e^{-i\int\omega_{1}(t^{\prime})dt^{\prime}}}{\sqrt{2\omega_{1}(t)}}+\beta(t)\frac{e^{i\int\omega_{1}(t^{\prime})dt^{\prime}}}{\sqrt{2\omega_{1}(t)}}\
,$ (12) $\displaystyle\dot{\pi}$
$\displaystyle=-i\alpha(t)\sqrt{\frac{\omega_{1}(t)}{2}}e^{-i\int\omega_{1}(t^{\prime})dt^{\prime}}+i\beta(t)\sqrt{\frac{\omega_{1}(t)}{2}}e^{i\int\omega_{1}(t^{\prime})dt^{\prime}}\
,$ (13)

where $\alpha(t)$ and $\beta(t)$ are time-dependent Bogoliubov coefficients
that satisfy the coupled differential equations

$\displaystyle\dot{\alpha}$
$\displaystyle=\frac{\dot{\omega}_{1}}{2\omega_{1}}e^{2i\int\omega_{1}(t^{\prime})dt^{\prime}}\beta(t)\qquad\dot{\beta}$
$\displaystyle=\frac{\dot{\omega}_{1}}{2\omega_{1}}e^{-2i\int\omega_{1}(t^{\prime})dt^{\prime}}\alpha(t)$
(14)

We will solve these coupled equations using the method outlined in [34].
Setting $\hat{\alpha}=e^{-i\bar{\omega}t}\alpha$ and
$\hat{\beta}=e^{i\bar{\omega}t}\beta$, where
$\bar{\omega}=\frac{1}{T}\int_{0}^{T}\omega_{1}(t)dt$, we obtain from (14),

$\displaystyle\frac{d}{dt}\begin{pmatrix}\hat{\alpha}\\\
\hat{\beta}\end{pmatrix}=\begin{pmatrix}-i\bar{\omega}&\frac{\dot{\omega}_{1}}{2\omega_{1}}e^{2i\int\delta\omega(t^{\prime})dt^{\prime}}\\\
\frac{\dot{\omega}_{1}}{2\omega_{1}}e^{-2i\int\delta\omega(t^{\prime})dt^{\prime}}&i\bar{\omega}\end{pmatrix}\begin{pmatrix}\hat{\alpha}\\\
\hat{\beta}\end{pmatrix}\ ,$ (15)

where $\delta\omega(t)=\omega_{1}(t)-\bar{\omega}$. Let
$\hat{\alpha}=\hat{\alpha}_{0}+b\hat{\alpha}_{1}+b^{2}\hat{\alpha}_{2}$ and
$\hat{\beta}=\hat{\beta}_{0}+b\hat{\beta}_{1}+b^{2}\hat{\beta}_{2}$, where
$b=2a+\epsilon\ll 1$. Note that, up to second order in $b$,
$\bar{\omega}=\omega\left(1-\frac{b^{2}}{16}\right)$.

Substitution into equation (15), solving order-by-order, and restricting
ourselves to solving the system after each period, we can define:

$\displaystyle\begin{pmatrix}\hat{\alpha}(T)\\\
\hat{\beta}(T)\end{pmatrix}=M\begin{pmatrix}\hat{\alpha}(0)\\\
\hat{\beta}(0)\end{pmatrix}=\begin{pmatrix}M_{11}&M_{12}\\\
M_{12}^{*}&M_{22}\end{pmatrix}\begin{pmatrix}\hat{\alpha}(0)\\\
\hat{\beta}(0)\end{pmatrix}\ ,$ (16)

where after a a period $T=2\pi/\Omega$, we have

$\displaystyle M_{11}$ $\displaystyle=\frac{e^{-2i\pi
q}\left(4+q^{2}\left(b^{2}\left(-\left(-8i\pi q^{3}+2i\pi q+e^{4i\pi
q}-1\right)\right)+64q^{2}-32\right)\right)}{4\left(1-4q^{2}\right)^{2}}\ ,$
(17) $\displaystyle M_{12}$
$\displaystyle=\frac{bq\left(-3ibq+4q^{2}-4\right)\sin(2\pi
q)}{4\left(4q^{4}-5q^{2}+1\right)}\ ,$ (18) $\displaystyle M_{22}$
$\displaystyle=\frac{e^{-2i\pi q}\left(-b^{2}q^{2}+e^{4i\pi
q}\left(4+q^{2}\left(b^{2}\left(-8i\pi q^{3}+2i\pi
q+1\right)+64q^{2}-32\right)\right)\right)}{4\left(1-4q^{2}\right)^{2}}\ .$
(19)

We have assumed that
$\hat{\alpha}_{1}(0)=\hat{\beta}_{1}(0)=\hat{\alpha}_{2}(0)=\hat{\beta}_{2}(0)=0$.
We see that the 1st order resonance (i.e. to 2nd order in $b$) occurs when
$q\approx\frac{1}{2}$ (see Figure 1).

After $N$ periods of the gravitational wave oscillations (or equivalently
after $N$ periods of the trap modulation), we have

$\displaystyle\begin{pmatrix}\hat{\alpha}(NT)\\\
\hat{\beta}(NT)\end{pmatrix}=\begin{pmatrix}e^{-2\pi iqN}\alpha(NT)\\\ e^{2\pi
iqN}\beta(NT)\end{pmatrix}=M^{N}\begin{pmatrix}\hat{\alpha}(0)\\\
\hat{\beta}(0)\end{pmatrix}\ .$ (20)

Diagonalizing $M$ and writing $\begin{pmatrix}\hat{\alpha}(0)\\\
\hat{\beta}(0)\end{pmatrix}=k_{1}\lambda_{1}\vec{x}_{1}+k_{2}\lambda_{2}\vec{x}_{2}$,
where $\lambda_{i},\vec{x}_{i}$ are the eigenvalues/eigenvectors of $M$ and
$k_{i}$ are coefficients related to the eigenvalues, we see

$\displaystyle\begin{pmatrix}\hat{\alpha}(NT)\\\
\hat{\beta}(NT)\end{pmatrix}=k_{1}\lambda_{1}^{N}\vec{x}_{1}+k_{2}\lambda_{2}^{N}\vec{x}_{2}\
.$ (21)

Let $\lambda_{2}$ correspond to the eigenvalue whose magnitude is less than
one around resonance. We can neglect this term because it will become
negligible after many periods. Therefore, the only relevant quantities are

$\displaystyle k_{1}$
$\displaystyle=-\frac{b\left(128\zeta^{3}-64i\zeta^{2}+3\zeta+i\right)+128\zeta^{2}-2}{4\left(1-64\zeta^{2}\right)^{3/2}}$
(22) $\displaystyle\lambda_{1}$ $\displaystyle=\pi
b^{2}\left(\frac{5\sqrt{1-64\zeta^{2}}\zeta}{512\zeta^{2}-8}+\pi\left(2\zeta^{2}-\frac{1}{32}\right)\right)-\frac{1}{4}\pi
b\sqrt{1-64\zeta^{2}}-1$ (23)

where $\zeta\equiv(q-\frac{1}{2})/b$, and

$\displaystyle\vec{x}_{1}=\begin{pmatrix}x_{0}+bx_{1}+b^{2}x_{2}\\\
1\end{pmatrix}$ (24)

with

$\displaystyle x_{0}$ $\displaystyle=\sqrt{1-64\zeta^{2}}-8i\zeta$ (25)
$\displaystyle x_{1}$
$\displaystyle=8i\zeta^{2}+\frac{\zeta(3+64\zeta(2\zeta-i))+i}{2\sqrt{1-64\zeta^{2}}}+4\zeta+\frac{3i}{16}$
(26) $\displaystyle x_{2}$
$\displaystyle=-16i\zeta^{3}+\frac{28\zeta^{2}}{3}-\frac{1}{12}i\pi^{2}\zeta+\frac{37i\zeta}{16}-\frac{3}{32}$
(27)
$\displaystyle+\frac{-219+128\zeta\left(4\zeta\left(2\pi^{2}\left(64\zeta^{2}-1\right)+16\zeta(4\zeta(-63+32\zeta(12\zeta+7i))-43i)+75\right)+29i\right)}{1536\left(1-64\zeta^{2}\right)^{3/2}}$
(28)

Let us further define
$\alpha_{2a+\tilde{\epsilon}}=\alpha_{2a}+\tilde{\epsilon}\alpha_{\tilde{\epsilon}}$
and
$\beta_{2a+\tilde{\epsilon}}=\beta_{2a}+\tilde{\epsilon}\beta_{\tilde{\epsilon}}$.
The goal is to calculate the Bogoliubov coefficients
$\alpha_{\tilde{\epsilon}}$ and $\beta_{\tilde{\epsilon}}$, which correspond
to the effect solely due to the of the gravitational wave interacting with the
trap modulation. Using the definition of the Bogoliubov coefficients, we can
write
$\hat{a}_{0}=\alpha_{2a+\tilde{\epsilon}}\hat{a}_{2a+\tilde{\epsilon}}+\beta^{*}_{2a+\tilde{\epsilon}}\hat{a}_{2a+\tilde{\epsilon}}^{\dagger}$,
where $\hat{a}_{0}$ is the the annihilation operation in the mode
decomposition with no gravitational wave or modulation and
$\hat{a}_{2a+\tilde{\epsilon}}$ is the annihilation operator in the mode
decomposition containing both a gravitational wave and magnetic field. Thus
$\hat{a}_{2a+\tilde{\epsilon}}=\alpha^{*}_{2a+\tilde{\epsilon}}\hat{a}_{0}-\beta_{2a+\tilde{\epsilon}}^{*}\hat{a}_{0}^{\dagger}$.
We can also write
$\hat{a}_{2a+\tilde{\epsilon}}=\alpha_{rel}\hat{a}_{2a}-\beta_{rel}^{*}\hat{a}_{2a}^{\dagger}$.
Therefore,

$\displaystyle\hat{a}_{2a+\tilde{\epsilon}}$
$\displaystyle=\alpha^{*}_{2a+\tilde{\epsilon}}\left(\alpha_{2a}\hat{a}_{2a}+\beta^{*}_{2a}\hat{a}_{2a}^{\dagger}\right)-\beta_{2a+\tilde{\epsilon}}^{*}\left(\alpha_{2a}^{*}\hat{a}_{2a}^{\dagger}+\beta_{2a}\hat{a}_{2a}\right)$
(29)
$\displaystyle=\left(\alpha^{*}_{2a+\tilde{\epsilon}}\alpha_{2a}-\beta_{2a+\tilde{\epsilon}}^{*}\beta_{2a}\right)\hat{a}_{2a}+\left(\alpha^{*}_{2a+\tilde{\epsilon}}\beta^{*}_{2a}-\beta_{2a+\tilde{\epsilon}}^{*}\alpha_{2a}^{*}\right)\hat{a}_{2a}^{\dagger}\
.$ (30)

With
$\alpha_{2a+\tilde{\epsilon}}=\alpha_{2a}+\tilde{\epsilon}\alpha_{\tilde{\epsilon}}$
and
$\beta_{2a+\tilde{\epsilon}}=\beta_{2a}+\tilde{\epsilon}\beta_{\tilde{\epsilon}}$,
we see

$\displaystyle\ \alpha_{rel}^{*}$
$\displaystyle=1+\tilde{\epsilon}\left(\alpha^{*}_{\tilde{\epsilon}}\alpha_{2a}-\beta_{\tilde{\epsilon}}^{*}\beta_{2a}\right)\
,$ (31) $\displaystyle\beta_{rel}^{*}$
$\displaystyle=-\tilde{\epsilon}\left(\alpha^{*}_{\tilde{\epsilon}}\beta^{*}_{2a}-\beta_{\tilde{\epsilon}}^{*}\alpha_{2a}^{*}\right)\
.$ (32)

####  2.1.1 Resonance

(a) Shaded areas show the locations of instability bands in $(a,\omega)$
plane, associated with resonances of equation (11) with $\Omega_{B}=\Omega$.
By lying within these regions for a time $t$, it is possible to improve the
sensitivity to gravitational waves. Note that maximum value of $\omega$ will
be constrained by the chemical potential of the condensate.

(b) The first resonance of our undamped system, where for illustrative
purposes, we set $N=5\times 10^{3}$ and $a=0.005$ Note that $|\beta|^{2}$ is
maximized around $q\approx\frac{1}{2}$, before there is a sharp cut-off around
$q=0.499$ and $q=0.501$. This corresponds to moving outside the instability
band.

Figure 1:

Equation (11) is a perturbed form of the Mathieu differential equation (as
$a\tilde{\epsilon}$ is a sub-dominant term), which implies that our system
includes parametric resonance effects [35, 36].

In Figure 1a, we show the locations of the first four resonances of the
system, where the shading illustrates the instability bands associated with
each individual resonance in the Mathieu equation (Figure 1a) [37]. For a
system lying within these bands (indicated by the shaded regions), evolution
is unstable and will become non-perturbative after a certain amount of time,
with the amount of time before non-perturbative effects occurs depending on
the precise location within these regions. In Figure 1b, we see that if
$q=\frac{\omega}{\Omega}$ is too far off the resonance, the instability band
is exited, which limits production of particles and nonlinearities.

By considering phonons with frequencies that that are within the instability
bands (locations of parametric resonance) for a time $t$, we can enhance the
sensitivity of the BEC to gravitational waves. In Section 2.1.2, we address
the maximum time that we can lie within the regions of instability before the
system becomes non-perturbative in $a$.

####  2.1.2 Non-linearities

For $\kappa\approx m$, similar to [28], we have the constraint of
$\dot{\pi}\ll\mu=mc_{s}^{2}$. By squaring both sides (as
$\braket{\dot{\pi}}=0$) and writing $\dot{\pi}$ as a sum of creation and
annihilation operators, we find that this is equivalent to
$\sum_{\vec{k}}|\beta_{2a,\vec{k}}|^{2}\hbar\omega_{\vec{k}}\ll
nmc^{2}_{s}L^{3}$, where $L^{3}$ is the volume of the condensate (taking the
BEC to be cubic with sides of length $L$), $n$ is the number density, $m$ is
the mass of the atoms, and $|\beta_{2a,\vec{k}}|^{2}$ is the number of phonons
present in each mode $\vec{k}$ 111Accounting for the effect of the
gravitational wave on this energy condition is negligible..

Rewriting $|\beta_{2a,\vec{k}}|^{2}$ in terms of $q=\frac{\omega}{\Omega}$ and
assuming that the BEC is constructed such that only a single phonon mode
dominates the resonance instability (i.e., the most unstable mode), we note
that at each value of $q$, there is a maximum number of trap oscillations
$N(q)$ satisfying $\frac{|\beta_{2a,q}|^{2}\hbar\omega_{q}}{\rho
L^{3}}\lessapprox 0.05$, where we relabel $|\beta_{2a,\vec{k}}|^{2}$ as
$|\beta_{2a,q}|^{2}$ to explicitly indicate the dependence on $q$. Physically,
$N(q)$ describes the maximum time that a gravitational wave could be observed
by the condensate before the condensate becomes non-linear (because, by
assumption, we take the frequency of the trap oscillation to be the same as
the period of the gravitational wave). Noting that
$|\beta_{2a,q}|^{2}=|k_{1}|^{2}|\lambda_{1}|^{2N}$, we see that

$\displaystyle N\leq\frac{\log\left[\frac{0.05mnc_{S}^{2}L^{3}}{\pi\hbar
qf_{GW}|k_{1}|^{2}}\right]}{2\log[|\lambda_{1}|]}\ .$ (33)

In Figure 2, we demonstrate the dependence of $N$ on $q$, the size of the
condensate, and the frequency of the magnetic field. We consider a cross-
section of this plot in Figure 3. Throughout the remainder of this paper, all
calculations and discussions will assume that this non-linearity condition is
saturated.

Maximum number of gravitational wave periods detectable by a BEC.

(a) $c_{s}=0.5$ cm/s, $L=100$ $\mu$m

(b) $c_{s}=1$ cm/s, $L=100$ $\mu$m

(c) $c_{s}=5$ cm/s, $L=100$ $\mu$m

(d) $c_{s}=0.5$ cm/s, $L=500$ $\mu$m

(e) $c_{s}=1$ cm/s, $L=500$ $\mu$m

(f) $c_{s}=5$ cm/s, $L=500$ $\mu$m

Figure 2: Dependence of the number of cycles $N$ needed to reach non-
linearity, on $q=\frac{f_{\rm phonon}}{f_{\rm GW}}$ and the frequency
(assuming $f_{\rm GW}=f_{B}$) for an undamped condensate. We see that, in
general, the condensate is sensitive to a greater number of oscillations when
considering a smaller frequency and larger condensates. We take the number
density to be $10^{20}$ m-3 of 39K atoms. The time to non-linearity is simply
$N/f_{\rm GW}$.

##  3 Estimating the sensitivity to gravitational waves

Figure 3: Observation time for a gravitational wave interacting with a BEC
with $q=0.501245$. Lower frequencies correspond to longer observational times.
We note that this neglects the effects of 3-body recombination, which limits
the condensate lifetime. We discuss this effect further in Section 5). For
smaller values of $q$ (i.e. closer to the resonance peak), the maximum
observational time of the condensate decreases.

As phonons are affected by the gravitational wave, the gravitational wave’s
amplitude is imprinted on the phonon’s density matrix. The question then
becomes how to extract the gravitational wave amplitude from the density
matrix. This can be done by exploiting techniques in quantum metrology, which
allows measurements to be done on quantities in quantum systems that are not
operator observables [38, 39, 40, 41]. In this Section, we will be determine
the sensitivity of an undamped condensate at zero-temperature with no extra
sources of noise in the system. In Section 4, we will consider the more
realistic case by investigating the sensitivity of a damped BEC. The influence
of noise arising in an experimental setup will be investigated in Section 5.

Given a parameter $\tilde{\epsilon}$ in a quantum system, the quantum Cramer-
Rao bound is [27]

$\displaystyle\braket{(\Delta\epsilon)^{2}}\geq\frac{1}{MH_{\epsilon}}\ ,$
(34)

where $\braket{(\Delta\epsilon)^{2}}$ is the expectation value of the
uncertainty in $\epsilon$, $M$ is the number of measurements of the system,
and $H_{\epsilon}$ is the Fisher information [27]

$\displaystyle
H_{\epsilon}=\frac{8\bigg{(}1-\sqrt{F(\rho_{\epsilon},\rho_{\epsilon+d\epsilon})}\bigg{)}}{d\epsilon^{2}}\
,$ (35)

with
$F(\rho_{\epsilon},\rho_{\epsilon+d\epsilon})=\left[\rm{Tr}\sqrt{\rho_{\epsilon}\sqrt{\rho_{\epsilon+d\epsilon}}\rho_{\epsilon}}\right]^{2}$
quantifying the overlap (fidelity) between the states $\rho_{\epsilon}$ and
$\rho_{\epsilon+d\epsilon}$ [42, 43]. We find the sensitivity to gravitational
waves is

$\displaystyle\braket{(\Delta\tilde{\epsilon})^{2}}=\frac{4}{M\left(\Re\left[\alpha_{rel}^{(1)}\right]+|\beta_{rel}|\right)^{2}\left(\cosh^{4}r_{0}+\sinh^{4}r_{0}\right)}\
.$ (36)

where $r_{0}$ is the initial squeezing of the phonons, to $\alpha_{rel}$ in
the gravitational wave amplitude, with $\alpha_{rel}$ and $\beta_{rel}$ given
by equations (31) and (32), respectively. The details of this calculation are
provided in Appendix A.

In this and all subsequent considerations, we take $M=1$. This corresponds to
a gravitational wave interacting with a $M$ BECs for $N$ periods. We also
assume that the BEC can continuously be regenerated over the course of a
year[44, 45], as suggested by [26], such that observations can be continuously
made of a gravitational wave, whose frequency remains approximately constant
over the total time of observation. In this case, we find

$\displaystyle\sqrt{\braket{(\Delta\tilde{\epsilon})^{2}}}=\sqrt{\frac{1}{N_{tot}}}\frac{2}{\left(\Re\left[\alpha_{rel}^{(1)}\right]+|\beta_{rel}|\right)\sqrt{\cosh^{4}r_{0}+\sinh^{4}r_{0}}}\
,$ (37)

where $N_{tot}$ is the total number of regenerations of the condensates. For
simplicity, we consider a year of continuous observation, so
$N_{tot}\approx\frac{\rm{1yr}}{N/f}$.

By expanding (37) around $q=\frac{1}{2}$, and taking $N$ to be large, we find
that

$\displaystyle\sqrt{\braket{(\Delta\tilde{\epsilon})^{2}}}\sim\frac{384}{\sqrt{N_{tot}}}a^{2}e^{a\left(\frac{5\pi}{2}-\pi
N\right)}\ ,$ (38)

We note that, in practice, it would be quite difficult to reconstruct the BEC
with exactly the same properties. Extra source noise would be introduced into
the system, and so such a BEC machine would be beneficial only if the noise
resulting from an inexact replication is smaller than the gain in sensitivity
resulting from multiple BEC experiments.

We also note that, in equation (37), we are calculating the sensitivity of the
scaled gravitational wave amplitude $\tilde{\epsilon}$, rather than $\epsilon$
itself. To obtain $\braket{(\Delta\epsilon)^{2}}$ from
$\braket{(\Delta\tilde{\epsilon})^{2}}$, one can either average or maximize
over the components $k_{x,y}$ of the wavevector. As both methods scale the
estimation of the gravitational wave amplitude by $\mathcal{O}(1)$, we will
not include this effect in our subsequent calculations and Figures.

As the BEC has a finite size, we note that the minimum gravitational wave
frequency that can be observed is $f\approx\frac{2c_{s}}{L}$ (where we use the
relation that resonance between the gravitational wave and phonon occurs at
$q=\frac{\omega}{\Omega}\approx\frac{1}{2}$ and $c_{s}$ is the speed of the
phonons). The maximum frequency observed is derived from the chemical
potential $\mu=mc_{s}^{2}$, which sets the upper bound on the frequency of
phonons created: $f\ll\mu$. Again using $q\approx\frac{1}{2}$, we see that the
maximum gravitational wave frequency is
$f\approx\frac{2mc_{s}^{2}}{2\pi\hbar\times 10}$, where the $10$ comes from
the inequality. We show in Figure 4 how the frequency bands of our
gravitational wave detector are dependent on the speed of sound and length of
the condensate. For lower-frequency gravitational wave detection, a larger
condensate is required while high-frequency gravitational wave detection
requires a faster speed of sound.

Figure 4: Frequency regime of a BEC gravitational wave detector for 39K atoms.
Note that the larger condensates can probe lower-frequency gravitational
waves, whereas those with faster speeds of sound can probe higher-frequency
gravitational waves.

To illustrate the sensitivity to gravitational waves, we take the number
density to be $10^{20}$ m-3 and a mass of the atoms to be $10^{-25}$ kg. In
Figures 5 and 6, we consider the sensitivity of an (undamped) BEC and
illustrate the effect of $q$, length of the condensate, and frequency of
incoming gravitational waves on strain sensitivity. We note that a BEC in the
presence of an oscillating magnetic field is most sensitive to higher
frequency gravitational waves, similar to [26] and [28]. In Section 4, we
consider a more realistic situation by modelling Beliaev damping of the
phonons within the condensate [46].

This effect of having greater sensitivity away from resonance must be
highlighted. When the system is away from
$\frac{\omega}{\Omega}=\frac{1}{2}\pm\delta$, where $\delta\ll 1$, we saw that
the sensitivity to gravitational waves is actually increased. In these off-
resonance cases, it takes longer for the system to become non-linear (as the
phonons have less energy), hence we can observe gravitational waves for a
longer time (a cross-section of this effect was illustrated in Figure 5).

(a) $L=100$ $\mu$m

(b) $L=500$ $\mu$m

Figure 5: The strain sensitivity of an undamped BEC gravitational wave when on
resonance with an oscillating magnetic field. In each case, the condensate is
observed for the maximum number of oscillations $N(q)$. We conduct the
hypothetical experiment over the course of a year. We take the number density
to be $10^{20}$ m-3 of 39K atoms).

BEC sensitivity to gravitational waves.

(a) $c_{s}=0.5$ cm/s, $L=100$ $\mu$m

(b) $c_{s}=1$ cm/s, $L=100$ $\mu$m

(c) $c_{s}=5$ cm/s, $L=100$ $\mu$m

(d) $c_{s}=0.5$ cm/s, $L=500$ $\mu$m

(e) $c_{s}=1$ cm/s, $L=500$ $\mu$m

(f) $c_{s}=5$ cm/s, $L=500$ $\mu$m

Figure 6: Contour plot of the strain sensitivity of an undamped BEC to
gravitational waves. The condensate is observed for the maximum number of
oscillations $N(q)$. We run the experiment over the course of a year and see
that undamped condensate could be sensitive to gravitational waves across the
frequency spectrum. Changing the speed of sound and length of the condensate
will affect the minimum and maximum gravitational wave frequency that can be
observed. We take the number density to be $10^{20}$ m-3 of 39K atoms).

##  4 Damping

In Section 3, we considered an undamped BEC with no additional sources of
noise and demonstrated that, if such a system could be created, then a BEC
could potentially be used to detect gravitational waves across several orders
of magnitude in frequency, depending on the speed of the phonons and the
length of the condensate. In reality, even at zero-temperature (which was
implicitly considered), the phonons within the BEC will naturally undergo
decoherence through Beliaev damping, where the damping rate is given by [47]

$\displaystyle\gamma_{B}=\frac{3}{640\pi}\frac{\hbar\omega^{5}}{mnc_{s}^{5}}\,$
(39)

for a cubic BEC with no magnetic field, where any magnetic field correction
would manifest itself as a higher-order term because $\frac{\delta B}{B}\ll
1$.

Since the density matrix is a Gaussian state, we can use a covariance matrix
to compute the Fidelity. As demonstrated in [48], the covariance matrix
$\sigma$ is damped as
$\sigma=e^{-\gamma_{B}t}\sigma_{0}+(1-e^{-\gamma_{B}t})\sigma_{\infty}$, where
$\sigma_{0}$ is the initial covariance matrix and $\sigma_{\infty}$ is the
$t\to\infty$ covariance matrix. For our purposes, we can neglect the second
term as $\gamma_{B}t\gg 1$. From [49], the quantum Fisher information depends
only the square of several combinations of elements of the covariance matrix.
Hence, the quantum Fisher information is damped as
$H_{\epsilon,\text{damped}}\sim
e^{-2\gamma_{B}t}H_{\epsilon,\text{undamped}}$. Therefore,

$\displaystyle\braket{(\Delta\tilde{\epsilon})^{2}}=\frac{1}{T_{tot}}\frac{4}{M\left(\Re\left[\alpha_{rel}^{(1)}\right]+|\beta_{rel}|\right)^{2}\left(\cosh^{4}r_{0}+\sinh^{4}r_{0}\right)}e^{2\gamma_{B}t}$
(40)

where $t=N_{max}T$ is the maximum time of observation of the condensate, $N$
is the maximum number of oscillations of the magnetic field, and
$T=\frac{2\pi}{\Omega}$ is the period of the gravitational wave. We note that
$N_{max}T<\frac{1}{\gamma_{B}}$, where $\frac{1}{\gamma_{B}}$ is approximately
the decoherence time [48].

In Figure 7, we consider how the strain sensitivity is affected when damping
is considered for various gravitational wave frequencies and values of $q$. We
see that larger condensates and faster speeds of the phonons give rise to
greater sensitivity. However, we note that the sensitivity of a BEC to
gravitational waves is optimal at lower frequencies, in contrast to the
results of [26] and [28]. In Figure 7, when fixing $q$, we see that the the
strain sensitivity becomes worse further away from the resonance peak of
$q=\frac{1}{2}$. However, for a ${}^{39}K$ BEC, the effect of damping is
negligible for most values of $q$.

Figure 7: The strain sensitivity of a damped BEC gravitational wave detector
for various values of $q$ (=ratio of phonon to GW frequency). We consider
$c_{s}=5$ cm/s and $L=500$ $\mu$m. In each case, the condensate is observed
for the maximum number of oscillations $N(q)$, before non-linearities become
important. We run the experiment over the course of a year and see that
damping as a negligible effect for a potassium-39 BEC whose number density is
$10^{20}$ m-3.

###  4.1 Squeezing

In our previous work [28], we noted that squeezing the phonon states seemed to
be a necessary feature for a BEC to detect gravitational waves. A similar
conclusion was implied by the figures in [26]: to exceed the sensitivity of
LIGO, the phonons needed to be squeezed by much more than the capabilities of
current experimental designs (7.2 dB), though it was noted that there might be
a means [50] of exceeding current limitations.

Though squeezing improves the sensitivity to gravitational waves in our
current work by a factor of

$\sqrt{\sinh^{4}r_{0}+\cosh^{4}r_{0}},$

we note that this feature is no longer an essential element of our proposal.
For $r=0.83$ (potentially achievable with current technology [51]), this
improves sensitivity by only a factor of $2.05$. Improvements in squeezing may
increase the sensitivity to gravitational waves, though we note that squeezed
states would contribute to the energy density of the condensate, and therefore
cause the condensate to become unstable quicker (Section 2.1.2. Furthermore,
the squeezing parameter would also decay, as discussed in [48].

At the present time using current techniques (which we discuss in more detail
in the next section), it is not feasible to detect gravitational waves using a
BEC, though the results of this paper indicate that this could potentially be
achieved in the future. When this occurs, then Beliaev damping and the effects
of squeezing may need to be considered in more detail. However, in the case of
observing a continuous gravitational wave by using parametric resonance,
squeezing and Beliaev damping is not the primary limiting factor as suggested
in [28]. Rather it is 3-body recombination, which we discuss in the following
section.

##  5 Experimental Implementation

###  5.1 Observable Sources

In Figure 7, we demonstrated that with a speed of sound of $c_{s}=5$ cm/s and
a cubic condensate with side lengths of $L=500$ $\mu$m, gravitational waves of
amplitude $\mathcal{O}(10^{-20})$ could be detected at frequencies around 500
Hz. In Figure 5, lower frequency gravitational waves could be detected if
their amplitude was $\mathcal{O}(10^{-21})$ and higher. However, as was noted
in [25], the amplitude of continuous gravitational waves tends to be weaker
than that of transient gravitational waves.

For this reason, it is unlikely that a BEC could detect gravitational waves
without going to larger sizes or faster speeds of sound. In Figure 8, we
illustrate the speeds of sound and condensate lengths necessary to detect
gravitational waves with a smaller amplitude, where we also account for
Beliaev damping. Currently, it is not possible to create condensates that
large, nor is it possible for the condensates to have a long enough lifetime;
as we discuss in the next section, larger speeds of sound increase the three-
body recombination rate, which in turn decreases the lifetime of the
condensate.

A Bose-Einstein condensate is an advantageous gravitational wave detector as
it is capable, in principle, of observing gravitational wave sources across
several orders of magnitude in frequency, depending on the condensate’s length
and speed of sound. At gravitational wave frequencies in the tens or hundreds
of Hertz, continuous signals from rotating neutron stars or pulsars could be
detected [23, 24]. The amplitude of such waves may be $\mathcal{O}(10^{-24})$.
For newly-created fast-spinning magnetars, the frequency of gravitational
waves could be 0.5-2 kHz, with amplitudes as large as $\mathcal{O}(10^{-21})$,
though we note that such signals will only be continuous on the order of
weeks, rather than years. [22]

Around black holes, it has been theorized that a boson cloud could form
through superradiance, and then emit continuous gravitational waves during
annhilation processes. [52] The frequency of the emitted waves will be
dependent on the mass of the bosons and distance from Earth, though could
potentially be in the kilohertz regime (smaller boson masses will have
frequencies in the tens or hundreds of Hertz). The amplitudes of such waves
could be as large as $\mathcal{O}(10^{-22})$ [52]. One candidate for boson
clouds is axions. In addition to annihilations, axions can undergo energy
level transitions, which will also emit gravitational waves. These tend to be
in the low-frequency regime ($<200$ Hz) and have amplitudes
$\mathcal{O}(10^{-24})$ and below, though one energy level transitions could
be $\mathcal{O}(10^{-22})$ (frequency of gravitational wave $\sim 40$ Hz). At
higher frequencies, there may be additional cosmological sources, such as
cosmic strings or inflationary signatures. However, we note that the signals
from such sources may be too weak to be detected by our proposed setup [53].

In Figure 8, we also illustrate the amplitude and frequency regimes of such
signals. We find that, though it may be necessary to go to larger length
scales and speeds of sound to observe most of the continuous gravitational
wave sources, it may still be possible to detect gravitational waves using
condensates of lengths $L\sim 5\times 10^{-4}$ m and speeds of sound
$c_{s}\sim 5\times 10^{-2}$ m/s.

Figure 8: Speeds of sound and length of the condensate necessary to observe
gravitational waves with a smaller amplitude. The green triangles represent
the upper bounds the strain of various pulsar sources [54] while the red stars
correspond to axion clouds based on several scenarios proposed in [52]. The
error bars for the axion clouds assume a distance of 1kPc - 100 kPc.

###  5.2 Experimental Design

We have presented the theoretical sensitivity for a BEC gravitational wave
detector in the ideal case and in the more realistic case of zero-temperature
(Beliaev) damping. We saw that Beliaev damping will decrease sensitivity to
gravitational waves at higher frequencies and smaller speeds of sound. At
lower frequencies, Beliaev damping is no longer relevant (though Landau
damping could become important at non-zero temperatures) [47]. As we will
discuss in this section, with current or reasonably projected technologies,
detection will be limited by the noise in the detector (though the precise
source of the noises will be dependent on the experimental setup).

Bose-Einstein condensates have been previously confined to cylindrical boxes
using optical traps [55, 56]. For simplicity, our theoretical analysis in the
preceding sections considered a cubic box trap but the exact geometry is not
critical to the results. A cylindrical trap can be engineered to have a phonon
resonance at any frequency available to a cubic box. In [56], such a potential
was generated with a radius of $15\pm 1$ $\mu$m and length of $63\pm 2$
$\mu$m. The “flatness” of the box corresponded to a potential $\propto
r^{13\pm 2}$. We envision a similar structure of our potential.

The simplest way to modulate the length of the trap is to modulate the power
of the trapping laser. This leverages the fact that the potential has the
shape of a high-order polynomial rather than a perfectly sharp edge. Such a
technique has been used to excite sound waves in Fermi gases in a box trap
[57].

We envision the trap to be created by suspending the optics for generating the
trap beams using a similar method as used for suspending the large mirrors in
LIGO [58]. By assuming that the noise within the BEC gravitational wave
detector follows a similar spectrum as that of LIGO, we can use LIGO’s
sensitivity curves to estimate the impact of vibrational noise on the BEC. We
also note that, similar to [56], an additional, relatively small magnetic
field gradient will need to be applied to cancel out the Earth’s gravity.

Before discussing the effects of vibrations, we discuss the dominant source of
noise: laser intensity noise. The length of a box trap, in the Thomas-Fermi
approximation, is the distance between the points at the two ends where the
trapping potential equals the chemical potential. This can be modulated by
changing the positions of the light sheets that form the hard walls at either
end of the box or by modulating the intensity of the trapping light. To
analyze the impact of light intensity noise, we consider the equation
$\mu=U_{0}e^{-2(z/w_{0})^{2}}$, where $\mu$ is the chemical potential, $U_{0}$
is the peak potential for a blue-detuned light sheet, $w_{0}$ is the Gaussian
beam waist, and $z$ is the distance from the intensity peak to the edge of the
trap defined by the equation. Since $U_{0}$ is proportional to the power $P$
in the light sheet, we note that $dz/d{U_{0}}=w_{0}^{2}/(4zU_{0})$ implies the
change in z, $\delta z$, due to a small change in power, $\delta P$, is

$\delta z=\frac{w_{0}^{2}}{4z}\frac{\delta P}{P}.$ (41)

For reasonable laser powers and focal parameters, $z\approx 2w_{0}$. For a
cubic box of length $L$, the minimum waist is roughly found by setting the
Rayleigh range equal to the length: $L=\pi w_{0}^{2}/\lambda$, which gives
$\delta z=\sqrt{L\lambda}/(8\sqrt{\pi})\cdot\delta P/P$, Assuming correlated
power fluctuations for the sheets at both ends of the trap (as from laser
power noise; fluctuations due to beam splitting to make the two sheets should
be anti-correlated and so cancel to leading order) we find finally

$\frac{\delta L}{L}=\sqrt{\frac{\lambda}{16\pi L}}\frac{\delta P}{P}.$ (42)

Using $532\;{\rm nm}$ light as in [56] and a length of $1\;{\rm mm}$ gives a
sensitivity $\delta L/L=0.003(\delta P/P)$. The current state of the art for
this wavelength could allow for $50\;{\rm W}$ beams. In an experiment lasting
three seconds, assuming the power is controlled to the shot noise limit, we
find the length noise is $\delta L/L=1.5\times 10^{-13}$. This sets a bound on
the strain sensitivity for such a technique in a single experiment.

In Figure 9, we depict the sensitivity curve of a BEC to a 1000 Hz
gravitational wave, where the observation time is taken to be the minimum of
the non-linearity time and the lifetime of the BEC (3 seconds). The minimum of
the sensitivity curve corresponds to the location where the non-linearity time
is exactly equal to the lifetime of the BEC. Moving outside the resonance
bands causes the non-linearity time to exceed the lifetime of the condensate,
which implies that there is insufficient time to use parametric resonance to
amplify the sensitivity. In this case, such a BEC would be insensitive to
gravitational waves.

Figure 9: The sensitivity of a BEC ($a=0.005$) to gravitational waves,
assuming that the BEC has side lengths of $100$ $\mu$m, the speed of sound is
$1$ cm/s, the lifetime of the BEC is limited to 3 s (without using a BEC
machine), and the incoming gravitational wave has a frequency of $1000$ Hz. We
see that the greatest sensitivity to gravitational waves is achieved slightly
off the centre of resonance and that moving outside the resonance band
destroys the sensitivity to gravitational waves.

This intrinsic BEC noise can potentially be improved by squeezing the
intensity noise of the laser, since the phase of the trapping light is not
critical. State of the art squeezing achieves $15\;{\rm dB}$ reduction in
noise [59, 60]. Supposing a substantial improvement to $30\;{\rm dB}$ is
possible in the future, the achievable strain sensitivity of $\sim 10^{-16}$
is still far behind the sensitivity of LIGO. Using an asymmetric box potential
and only looking at phonon modes along the long direction would allow for
larger $L$ and a smaller beam waist for the walls in the important direction.
A $5\;{\rm mm}$ long trap with transverse width of $100\;{\rm\mu m}$ could
improve the sensitivity by nearly an order of magnitude. We conclude that the
limit imposed by plausible near-term technologies is a strain sensitivity of
$10^{-17}$ per shot.

Mechanical vibrations of the optical setup are the other potential source of
trap length noise. To minimize the impact of vibrations, we assume the
experiment is performed in a large ultrahigh vacuum chamber with optics built
directly into a single piece of Zerodur glass [61] suspended on a vibration
control system with comparable specifications as the LIGO mirror suspension
system [58]. A phonon mode excited by gravitational waves would have different
symmetry than the phonon modes that may be excited by center of mass
oscillations of the trap. However, the frequencies of interest will dominantly
transfer as center of mass oscillations. The strong mismatch between internal
vibrational excitation resonances and the pendular modes of the optical system
as a whole mean the transfer function from vibrations transmitted through the
suspension system to length fluctuations will be much less than one. Assuming
the transfer function is 1, we find a strain sensitivity of $10^{-17}$. Given
that the transfer function will be well below this level, we find that even
with current technology vibrational noise can be suppressed below the limits
imposed by laser intensity fluctuations.

These limits are general to any technique that would use the modulation of a
BEC trap due to a gravitational wave as a signal. The uncontrolled modulations
of trap length due to laser power fluctuations or vibrations provide a noise
foreground that must be overcome for any of the proposed schemes to be
experimentally viable [26, 28].

The ultimate objective is to measure the quantum Fisher information. This can
be calculated by representing the quantum Fisher information in terms of the
covariance matrix. For a Gaussian state, the covariance matrix is
$\sigma_{mn}=\frac{1}{2}\braket{X_{m}X_{n}+X_{n}X_{m}}-\braket{X_{m}}\braket{X_{n}}$,
where $X_{2n-1}=\frac{1}{\sqrt{2}}(a_{n}+a_{n}^{\dagger})$,
$X_{2n}=\frac{1}{\sqrt{2}i}(a_{n}-a_{n}^{\dagger})$, and
$a_{n},a_{n}^{\dagger}$ are the creation and annihilation operators. Let us
also assume $\braket{X_{i}}=0$. Let $\sigma_{A}$ be the Gaussian state of the
phonons before interaction with the gravitational wave and $\sigma_{B}$ be the
Gaussian state of the phonons afterwards. Define [62]

$\displaystyle\Delta$
$\displaystyle=\det[\mathcal{\sigma}_{A}+\mathcal{\sigma}_{B}]\ ,$ (43a)
$\displaystyle\Lambda$
$\displaystyle=2^{2n}\det\left[\mathcal{\sigma}_{A}+\frac{i}{2}J\right]\det\left[\mathcal{\sigma}_{B}+\frac{i}{2}J\right]\
,$ (43b) $\displaystyle J$
$\displaystyle=\bigoplus_{k=1}^{n}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\ ,$
(43c)

where $n$ is the number of modes. The fidelity of the two single-mode states
is then given by [62]

$\displaystyle
F(\sigma_{A},\sigma_{B})=\frac{1}{\sqrt{\Delta+\Lambda}-\sqrt{\Lambda}}\ .$
(44)

and using equation (35) yields the quantum Fisher information. Thus, measuring
the quantum Fisher information reduces to a determination of the individual
elements of the covariance matrix. However, doing such measurements will only
be possible when the intrinsic noise of the BEC is sufficiently reduced.

Even if the technical noise of the trap is mitigated, there will remain a few
caveats. In Section 4, we noted that faster speeds of sound gave rise to
greater sensitivity. This was because the damping rate was $\gamma_{B}\propto
c_{s}^{-5}$. With faster speeds of sounds, a BEC will have a larger minimum
observable gravitational wave frequency ($\propto c_{s}$), though a much
larger maximum observable gravitational wave frequency ($\propto c_{s}^{2}$).
However, at larger speeds of sound, we note that the maximum time of
observation will no longer be given by the non-linearity condition of Section
2.1.2, but by three-body recombination. When three atoms collide, two are able
to form a bound, molecular state with the third escaping the system at a rate
given by $\dot{n}\propto n^{3}(t)$, where $n(t)$ is the number density and the
proportionality constant describes the rate of recombination and depends upon
the properties of the system [63, 64]. Larger speeds of sound correspond to
greater number densities, implying a greater decrease in the number atoms in
the condensate [65, 66]. We also note quantum depletion, which describes the
loss of atoms from the condensate because of interactions between the atoms
and excitations, will contribute to additional noise in the condensate, where
the number of atoms in an excited state (therefore, leaving the condensate) is
given by [67]

$\displaystyle\frac{n_{ex}}{N}=\frac{8}{3\sqrt{\pi}}(na^{3})^{1/2}\,$ (45)

where $a$ is the scattering length and is valid for a zero-temperature uniform
Bose gas.

##  6 Conclusion and Future Prospects

We have considered a Bose-Einstein condensate in a trap with oscillating
length to investigate its sensitivity towards a continuous gravitational wave
of the form $h_{+}=\epsilon\sin\Omega t$. We showed that the oscillation gives
rise to parametric resonance in the system, which can be exploited to improve
the sensitivity to gravitational waves at lower frequencies compared to higher
frequencies.

Despite the attractive aspects of this idealized setup, this proposal has
formidable challenges in its actual experimental implementation. The technical
noise present using current technology would overwhelm signals from likely
gravitational wave sources. The finite lifetime of real BECs also imposes a
difficult constraint, which we implemented as a hard limit of 3 seconds on the
length of an experiment. However, if the technical or lifetime issues can be
overcome, our results indicate that parametric resonance can make a BEC a more
promising gravitational wave detector than earlier considerations have
indicated [29, 28].

We offer three prospects for improving the sensitivity beyond the limitations
discussed on the paper. Three-body loss coefficients are relatively similar
amongst laser coolable atoms. The one atom that has produced a BEC that could
potentially have a dramatically longer lifetime is hydrogen. Since the triplet
potential of ${\rm H}_{2}$ supports no bound states, there is no three-body
loss in spin-polarized H. Hydrogen BEC experiments in magnetic traps suffer
from dipolar relaxation [68]. In principle, H polarized in the lowest-energy
Zeeman sublevel could produce BECs with very long lifetimes. This state could
be trapped in an optical potential, though using repulsive light sheets as
described in the paper would be infeasible, given the unavailability of strong
laser sources at wavelengths shorter than $121\;{\rm nm}$. Using an attractive
potential produced by a ${\rm CO}_{2}$ laser could lead to lifetimes well over
a minute.

State of the art ${\rm CO}_{2}$ lasers can produce more than $10\;{\rm kW}$ of
power. Using $25\;{\rm kW}$ of $10.6\;\mu{\rm m}$ light produced by a ${\rm
CO}_{2}$ laser would give a shot-noise limit 100 times lower than the
$50\;{\rm W}$ of $532\;{\rm nm}$ light discussed in the paper. Analyzing the
impact of laser intensity noise in an attractive, harmonic trap is beyond the
scope of this paper. However, even the repulsive optical potentials described
in this paper could potentially have far lower $\delta P/P$ by using power
build-up cavities to produce the trapping potentials.

Finally, at the price and size scale of present gravitational detectors, one
could imagine a set of machines producing many BECs for simultaneous
measurements to further improve sensitivity.

Though using a BEC to detect gravitational waves will be extremely difficult,
it should be noted that LIGO was first conceptualized in the 1980s and only
detected gravitational waves in 2015. BECs have a head start in knowing that
gravitational waves can be directly detected and some of LIGO’s technical
innovations could potentially be adapted for use by a BEC GW detector. Though
such an experiment may take years or decades to achieve, doing so will provide
a new method to detecting gravitational waves across potentially several
orders of magnitude in frequency.

## Acknowledgements

We thank Aditya Dhumuntarao, Soham Mukherjee, and Erickson Tjoa for their
useful comments and discussions. We also thank the participants of the mini-
symposium on Quantum Sensors and Their Applications in Fundamental Physics and
Astrophysics Experiments (December 13, 2019) at Washington University in St.
Louis for their helpful questions and suggestions. We thank Vladimir Dergachev
for his valuable discussion regarding LIGO’s calculations. We also thank Denis
Martynov for providing LIGO’s noise spectrum.

MR was funded by a National Science and Engineering Research Council of Canada
(NSERC) graduate scholarship. AOJ was funded by by the University of Waterloo
and the Institute for Quantum Computing. This research was supported in part
by NSERC and the Perimeter Institute for Theoretical Physics, and by AOARD
Grant FA2386-19-1-4077. Research at Perimeter Institute is supported by the
Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Economic Development & Innovation.

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##  Appendix A Sensitivity to Gravitational Waves

As mentioned in Section 3, an unknown parameter in a quantum system can be
estimated with the quantum Cramer-Rao bound, [27]

$\displaystyle\braket{(\Delta\epsilon)^{2}}\geq\frac{1}{MH_{\epsilon}}\ ,$
(46)

with $\braket{(\Delta\epsilon)^{2}}$ the expectation value of the uncertainty
in $\epsilon$, $M$ the number of measurements of the system, and
$H_{\epsilon}$ the Fisher information [27]

$\displaystyle
H_{\epsilon}=\frac{8\bigg{(}1-\sqrt{F(\rho_{\epsilon},\rho_{\epsilon+d\epsilon})}\bigg{)}}{d\epsilon^{2}}\
,$ (47)

where
$F(\rho_{\epsilon},\rho_{\epsilon+d\epsilon})=\left[\rm{Tr}\sqrt{\rho_{\epsilon}\sqrt{\rho_{\epsilon+d\epsilon}}\rho_{\epsilon}}\right]^{2}$
is the fidelity [42, 43]. Note that we can recast equations (31) and (32) as
$\alpha_{rel}=e^{-i\theta_{\alpha}}\cosh r_{rel}$ and
$\beta_{rel}=e^{-i\theta_{\beta}}\cosh r_{rel}$, where $r_{rel}$ is the
squeezing that results from the gravitational wave and
$e^{-i\theta_{\alpha}},e^{-i\theta_{\beta}}$ are phase factors. We can then
determine the $r_{rel}=\log\left[|\alpha_{rel}|+|\beta_{rel}|\right]$.

Let us suppose that before a gravitational wave interacts with the condensate,
our phonons are in a squeezed state, with squeezing parameter $r_{0}$ and
quadrature angle $\phi_{0}$, where the state of the phonons is
$\ket{\zeta_{0}}=S_{0}\ket{0}=\exp\bigg{[}\frac{1}{2}(\zeta_{0}^{*}\hat{a}^{2}-\zeta_{0}\hat{a}^{\dagger
2})\bigg{]}\ket{0}$, $S_{0}$ is the squeezing operator, and
$\zeta_{0}=r_{0}e^{i\phi_{0}}$

Let $\epsilon=\epsilon_{1}$ and $\epsilon_{2}=\epsilon+d\epsilon$. In a pure
state,
$H_{\epsilon}=\frac{8(1-|\braket{\epsilon_{1}}{\epsilon_{2}}|)}{(\epsilon_{1}-\epsilon_{2})^{2}}$.
The state after interaction with a gravitational wave is
$\ket{\epsilon_{i}}=S_{\epsilon_{i}}S_{0}\ket{0}$, where $S_{\epsilon_{i}}$
encodes the gravitational wave’s influence. Then,
$S_{rel}:=S_{\epsilon_{1}}^{\dagger}S_{\epsilon_{2}}$, we have

$\displaystyle\braket{\epsilon_{1}}{\epsilon_{2}}$
$\displaystyle=\braket{0}{S_{0}^{\dagger}S_{rel}S_{0}}{0}$ (48)
$\displaystyle=\braket{0}{S_{0}^{\dagger}\exp\bigg{[}\frac{1}{2}(\zeta_{rel}^{*}\hat{a}^{2}-\zeta_{rel}\hat{a}^{\dagger
2})\bigg{]}S_{0}}{0}$ (49)

where $\zeta_{rel}=r_{rel}e^{i\phi_{rel}}$. Noting that
$r_{rel}\propto\epsilon$, we have

$\displaystyle\braket{\epsilon_{1}}{\epsilon_{2}}\sim
1+\frac{1}{2}\braket{0}{S_{0}^{\dagger}\left(\zeta_{rel}^{*}\hat{a}^{2}-\zeta_{rel}\hat{a}^{\dagger
2}\right)S_{0}}{0}+\frac{1}{8}\braket{0}{S_{0}^{\dagger}\left(\zeta_{rel}^{*}\hat{a}^{2}-\zeta_{rel}\hat{a}^{\dagger
2}\right)^{2}S_{0}}{0}$ (50)

Thus, the quantum Fisher information is

$\displaystyle(\epsilon_{1}-\epsilon_{2})^{2}H_{\epsilon}=|\braket{0}{S_{0}^{\dagger}\left(\zeta_{rel}^{*}\hat{a}^{2}-\zeta_{rel}\hat{a}^{\dagger
2}\right)S_{0}}{0}|^{2}+\frac{1}{4}\braket{0}{S_{0}^{\dagger}\left(\zeta_{rel}^{*}\hat{a}^{2}-\zeta_{rel}\hat{a}^{\dagger
2}\right)^{2}S_{0}}{0}$ (51)

where we note that the first-order terms in $\epsilon$ vanish because of
antisymmetry. Using $S_{0}^{\dagger}aS_{0}=a\cosh
r_{0}-a^{\dagger}e^{i\phi_{0}}\sinh r_{0}$ and
$S_{0}^{\dagger}a^{\dagger}S_{0}=a^{\dagger}\cosh r_{0}-ae^{-i\phi_{0}}\sinh
r_{0}$, we find

$\displaystyle\braket{0}{a^{\dagger 2}}{0}$
$\displaystyle=-e^{-i\phi_{0}}\cosh r_{0}\sinh r_{0}$ (52)
$\displaystyle\braket{0}{a^{2}}{0}$ $\displaystyle=-e^{i\phi_{0}}\cosh
r_{0}\sinh r_{0}$ (53) $\displaystyle\braket{0}{a^{\dagger 4}}{0}$
$\displaystyle=3e^{-2i\phi_{0}}\cosh^{2}r_{0}\sinh^{2}r_{0}$ (54)
$\displaystyle\braket{0}{a^{4}}{0}$
$\displaystyle=3e^{2i\phi_{0}}\cosh^{2}r_{0}\sinh^{2}r_{0}$ (55)
$\displaystyle\braket{0}{a^{2}a^{\dagger 2}}{0}$
$\displaystyle=2\cosh^{4}r_{0}+\cosh^{2}r_{0}\sinh^{2}r_{0}$ (56)
$\displaystyle\braket{0}{a^{\dagger 2}a^{2}}{0}$
$\displaystyle=2\sinh^{4}r_{0}+\cosh^{2}r_{0}\sinh^{2}r_{0}$ (57)

Therefore,

$\displaystyle|\braket{\epsilon_{1}}{\epsilon_{2}}|\sim 1$
$\displaystyle+\frac{r_{rel}^{2}}{2}\cosh^{2}r_{0}\sinh^{2}r_{0}\left[\sin^{2}(\phi-\phi_{rel})+3\cos(2\phi_{0}-2\phi_{rel})\right]$
(58)
$\displaystyle-\frac{r_{rel}^{2}}{2}\left(\cosh^{4}r_{0}+\sinh^{4}r_{0}+\cosh^{2}r_{0}\sinh^{2}r_{0}\right)$

Therefore, after averaging over angles, we find

$\displaystyle(\epsilon_{1}-\epsilon_{2})^{2}H_{\epsilon}=\frac{1}{4}r_{rel}^{2}\left(\cosh^{4}r_{0}+\sinh^{4}r_{0}\right)$
(59)

Recalling that $r_{rel}=\log\left[|\alpha_{rel}|+|\beta_{rel}|\right]$ and
from equations (31)-(32),

$\displaystyle\ \alpha_{rel}^{*}$
$\displaystyle=1+\tilde{\epsilon}\left(\alpha^{*}_{\tilde{\epsilon}}\alpha_{2a}-\beta_{\tilde{\epsilon}}^{*}\beta_{2a}\right)\
,$ (60) $\displaystyle\beta_{rel}^{*}$
$\displaystyle=-\tilde{\epsilon}\left(\alpha^{*}_{\tilde{\epsilon}}\beta^{*}_{2a}-\beta_{\tilde{\epsilon}}^{*}\alpha_{2a}^{*}\right)\
,$ (61)

we find
$r_{rel}\sim\left(\Re\left[\alpha_{rel}^{(1)}\right]+|\beta_{rel}|\right)^{2}\epsilon^{2}$,
where $\alpha_{rel}=1+\epsilon\alpha_{rel}^{(1)}$ Therefore, with equations
(46) and (59), the sensitivity to gravitational waves is given by

$\displaystyle\braket{(\Delta\tilde{\epsilon})^{2}}=\frac{4}{M\left(\Re\left[\alpha_{rel}^{(1)}\right]+|\beta_{rel}|\right)^{2}\left(\cosh^{4}r_{0}+\sinh^{4}r_{0}\right)}\
.$ (62)