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Detecting continuous gravitational waves with superfluid
4
He
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New J. Phys.
19
(
2017
)
073023
https:
//
doi.org
/
10.1088
/
1367-2630
/
aa78cb
PAPER
Detecting continuous gravitational waves with super
fl
uid
4
He
S Singh
1
,
2
,
5
,
6
, L A De Lorenzo
3
, I Pikovski
2
,
4
and K C Schwab
3
1
Department of Physics, College of Optical Sciences and B2 Institute, University of Arizona, Tucson, AZ 85721, United States of America
2
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, United States of America
3
Applied Physics, California Institute of Technology, Pasadena, CA 91125 United States of America
4
Department of Physics, Harvard University, Cambridge, MA 02138, United States of America
5
Present address: Department of Physics, Williams College, Williamstown, MA 01267, United States of America.
6
Author to whom any correspondence should be addressed.
E-mail:
swati.singh@williams.edu
and
schwab@caltech.edu
Keywords:
gravitational waves, optomechanics, super
fl
uid helium
Abstract
Direct detection of gravitational waves is opening a new window onto our universe. Here, we study the
sensitivity to continuous-wave strain
fi
elds of a kg-scale optomechanical system formed by the
acoustic motion of super
fl
uid helium-4 parametrically coupled to a superconducting microwave
cavity. This narrowband detection scheme can operate at very high
Q
-factors, while the resonant
frequency is tunable through pressurization of the helium in the 0.1
–
1.5 kHz range. The detector can
therefore be tuned to a variety of astrophysical sources and can remain sensitive to a particular source
over a long period of time. For thermal noise limited sensitivity, we
fi
nd that strain
fi
elds on the order
of
~
-
h
10
Hz
23
are detectable. Measuring such strains is possible by implementing state of the art
microwave transducer technology. We show that the proposed system can compete with interfero-
metric detectors and potentially surpass the gravitational strain limits set by them for certain pulsar
sources within a few months of integration time.
1. Introduction
The recent detection of gravitational waves
(
GWs
)
marks the beginning of GW astronomy
[
1
,
2
]
. The
fi
rst direct
detection con
fi
rmed the existence of GWs emitted from a relativistic inspiral and merger of two large black holes
(
BHs
)
, at a distance of 400
M
parsecs
(
pc
)
[
1
]
. Indirect evidence for gravitational radiation was previously
attained by the careful observation since 1974 of the decay of the orbit of the neutron star
(
NS
)
binary system
PSR B1913
+
16 at a distance of 6.4 kpc, which agrees with the predictions from general relativity to better than
1%
[
3
]
. In this paper, we discuss the potential to use a novel super
fl
uid-based optomechanical system as a
tunable detector of narrow-band GWs, which is well suited for probing GWs from nearby pulsars As we discuss
below, in the frequency range exceeding
∼
500 Hz, this novel scheme has the potential to reach sensitivities
comparable to Advanced LIGO.
The GW detector under consideration is formed by high-
Q
acoustic modes of super
fl
uid helium parametrically
coupled to a microwave cavity mode in order to detect smal
l elastic strains. This setu
p was initially studied in
[
4
]
,and
is shown in
fi
gure
1
. The helium detector effectively acts as a Weber bar antenna
[
5
]
for GWs, but with two important
differences. Firstly, the
Q
/
T
-factor of the helium is expected to be much larger than that of metals, where
Q
is the
acoustic quality factor, and
T
is the mode temperature. Secondly, the acoustic resonance frequency can be changed by
up to 50% by pressurization of helium without affecting t
he damping rate, making the detector both narrowband and
tunable. Recent laboratory experiments
[
6
]
have realized quality factors of
= ́
Q
1.4 10
He
8
for super
fl
uid
4
He,
which appears to be limited by a combination of
3
He impurities, sample temperature
,andradiationloss.Allofthese
dissipation mechanisms can be reduced and we assume quality factors of 10
11
arepossibleinfutureexperimentswith
isotropically pure samples at lower temperatures of around 10 mK.
The power spectrum of GWs is expected to be extremely broad and is estimated to range from 10
−
16
to 10
3
Hz
[
7
–
9
]
for known sources. Ground-based optical interferometers
(
such as LIGO, Virgo, GEO, TAMA
)
allow
OPEN ACCESS
RECEIVED
1 February 2017
REVISED
21 April 2017
ACCEPTED FOR PUBLICATION
12 June 2017
PUBLISHED
21 July 2017
Original content from this
work may be used under
the terms of the
Creative
Commons Attribution 3.0
licence
.
Any further distribution of
this work must maintain
attribution to the
author
(
s
)
and the title of
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and DOI.
© 2017 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
for broad-band search for GWs in the frequency range 10 Hz
–
5 kHz. These detectors are expected to be
predominantly sensitive to the chirped, transient, GW impulse resulting from the last moments of coalescing
binaries involving compact objects
(
BHs and
/
or NSs
)
[
10
]
. Space-based interferometric detectors can be
sensitive to lower frequency GWs, as they are not limited by seismic noise
[
11
]
.
Unlikebroadband impulse sources, rapidly rotating compactobjectssuch as pulsars are expected to generate
highly coherent, continuousGWsignalsdue to the off-axis rotating mass,withfrequencies spanning from
∼
1kHz
for millisecond pulsars
(
MSPs
)
inbinaries,to 1 Hzforvery old pulsars
[
7
,
12
–
15
]
.Giventheunknownmass
distribution ofthepulsar,one can onlyestimate thestrain
fi
eldhere at earth.Several mechanisms giveupper
bounds to the strength ofGWs onearth. One suchlimitisthe
‘
spindownlimit
’
, which isgiven by theobserved
spin-downrate ofthe pulsar, and the assumption thatallofthe rotationalkinetic energywhich islost isintheform
ofGWs
[
16
]
. Anotherlimit isgiven bythe yield strength ofthe material which makes up the NS, and how much
strainthecrust can sustainbefore breaking apartdue to centripetal forces
[
17
,
18
]
. Thepresenceofstrong magnetic
fi
eldsindicate a potential mechanism for producing and sustaining such strains due to deformation ofthe NS
[
19
,
20
]
. However,without knowing thestrength and direction ofthe internal magnetic
fi
elds ina pulsar, itis
dif
fi
cultto estimate a lower limit on the size ofGWsignal.The measurementofGWs frompulsarswouldtherefore
giveuscrucial informationaboutthe stellarinterior.
Since pulsars should emit continuous and coherent GWs at speci
fi
c and known frequencies, we can use a
narrowband detector and integrate the signal for long times, averaging away the incoherent detector noise. We
show that for reasonable parameters, the super
fl
uid helium detector can approach strain sensitivities of
- ́
-
1
510 Hz
23
at around 1 kHz, depending on the size and
Q
factor of the detector. Pulsar frequencies are
observed to vary slightly due to random glitches
D
~-
--
ff
10
10
611
(
older, MSPs being more stable
)
[
21
]
,
and due to the motion of the Earth around the Sun and resulting doppler frequency shifts. The tunability of the
acoustic resonance will be essential to track these shifts during long detection integration times. Simultaneous
monitoring of the targeted pulsar electromagnetically can facilitate the required precision frequency tracking.
The frequency agility can also allow for using the same acoustic resonator to look for signals from multiple
pulsars.
Recent measurements with LIGO and Virgo have unsuccessfully searched for GW signals from 179 pulsars
and have limited the strain
fi
eld
-
h
10
25
for most pulsars after nearly a year of integration time
[
22
]
.Ina
parallel development, hundreds of new pulsars have been discovered in the last few years by analyzing gamma-
ray sources observed by the Fermi Large Area Telescope
(
Fermi-LAT
)
, some less than 0.5 kpc from earth
[
23
,
24
]
.
Together, these developments accentuate the need for new technology for GW astronomy of pulsars.
This paper is organized as follows. We start with an overview of continuous GWs from pulsars to get an
estimate for the strains produced on earth in section
2
. We then describe the super
fl
uid helium detector and
show how it functions as a detector for GWs in section
3
. In section
4
, we provide the detection system
requirements. We then compare this detector with other functional GW detectors, and show the key
fundamental differences between these detectors and our proposed detector in section
5
. Finally, we conclude
with a summary of the key features of this detector and outlook in section
6
. A review of the relevant concepts
and derivations are relegated to the appendices for the interested reader.
Figure 1.
Left
: schematic of the proposed GW sensor based on acoustic modes of super
fl
uid helium. Two cylindrical geometries
considered here are Gen1
(
radius
a
=
11 cm, length
L
=
50 cm, mass
M
=
2.7 kg
)
and Gen2
(
a
=
11 cm,
L
=
3m,
M
=
16 kg
)
.
Right
:
prototype of the detector with
a
=
1.8 cm,
L
=
4 cm,
M
=
6 g and resonant frequency 10 kHz.
2
New J. Phys.
19
(
2017
)
073023
S Singh
et al
2. Sources of continuous GWs
The generation of GWs can be studied by considering the linearized Einstein equations in the presence of matter
[
25
]
. The computations are similar to the analogous case in electromagnetism
[
26
]
, see appendix
A
for a
summary. However, in the absence of gravitational dipoles, a quadrupole moment
Q
ij
is necessary to source
GWs. The emitted power of GWs is found to be
[
27
]
=á ñ
⃛
⃛
()
P
G
c
QQ
5
,1
ij
ij
5
i.e. it depends on the third time derivative of the quadrupole moment of the system, where
ò
r
≔
QxxV
d
ij
i j
body
for a body of density
ρ
.
In the far-
fi
eld limit where size of the source
(
G
Mc
2
)
=
wavelength of the GW
(
w
c
)
=
distance to detector
(
d
)
, the gravitational metric perturbation becomes
=
()
h
G
cd
Q
2
̈
,2
ij
ij
4
where
h
is the gravitational perturbation tensor in transverse-traceless gauge. Since
~
--
G
c
10 N s kg
44442
,
one needs events with relativistic changes in quadrupole moment to have a measurable GW signal on earth. As
an estimate, if all the observed slowdown of the Crab pulsar was converted into gravitational radiation, the
power would correspond to
~ ́
P
4.5 10
3
1
W
(
10
5
times the electromagnetic radiation power from the Sun
)
[
28
]
. However, at a distance 2 kpc away from the pulsar
(
distance to earth
)
, the power
fl
ux is
--
1
0Wm
9
2
and the
metric perturbation is
~
-
h
10
24
. Even though the power
fl
ux is macroscopic and easily detectable in other
forms
(
acoustic, electromagnetic, etc
)
, the resulting strain is very small due to the remarkably high impedance of
space-time. This is at the heart of the dif
fi
culty with laboratory detection of GWs.
Estimates of gravitational radiation from pulsars is an active area of theoretical research that goes back to
early observations of pulsars
[
15
]
. The mechanism for GW generation is assumed to be an asymmetric mass
distribution. Several mechanisms are proposed for the deviation from axial symmetry in mass distribution, for
example magnetic deformations
[
18
]
, star quakes or instabilities due to gravitational or viscous effects
[
21
,
29
]
.
However, due to the unknown equation of state, there is signi
fi
cant variability in estimates of mass asymmetry
and thus GW strain from pulsars.
We now estimate the GW perturbation strain from measured spin-down rates and brie
fl
y discuss the validity
of this limit. We then present relevant numbers for a few MSPs of interest for our detector. Details about these
derivations and typical parameters for other pulsars of interest are presented in appendix
B
.
For an ellipsoidal pulsar rotating about the
z
-axis with frequency
w
p
, the two polarizations of
h
are given by
ww
=-
+
()
h
G
cd
It
4
cos 2 ,
3
zz
p
p
4
2
ww
=
́
()
h
G
cd
It
4
sin 2 ,
4
zz
p
p
4
2
where
I
zz
is the moment of inertia along the
z
-axis and
ò
characterizes the mass quadrupole ellipticity
(
=-
()
QQI
xx
yy
zz
)
. The energy
fl
ux for a continuous GW of polarization
A
from a pulsar source is given by
p
=
̇
()
( )
s
c
G
ht
16
,5
A
A
3
2
where
Î+ ́
{}
A
,
and the bar indicates time-averaging.
Typical NSs have mass 1
–
1.5
M
e
(
where
= ́
M
210
30
kg is the solar mass
)
and have a radius of around 10
km. Using these values, the moment of inertia amounts to 10
38
kg m
2
, the estimate used in previous GW
searches, see
[
22
]
. The ellipticity parameter is estimated by assuming that the observed slow-down rate
(
w
̇
p
)
of a
pulsar is entirely due to emission of GWs. This estimate is then used to compute the upper-limit estimate for
gravitation perturbation strain known as the spin down strain,
w
w
w
=-
=
̇
()
h
G
cd
I
GI
cd
4
5
2
.6
zz
p
zz
p
p
sd
4
2
32
Thus if we know the distance
(
d
)
, rotational frequency
(
w
p
)
and spin-down frequency
(
w
̇
p
)
from
astronomical observations, we can put limits on GW strain from pulsars.
While
h
sd
is a useful
fi
rst-principles upper limit, it over-estimates the strength of GWs. This has already been
con
fi
rmed by braking index measurements
[
30
,
31
]
and the negative results from recent GW detector data
[
22
]
.
However, long-lived and stable MSPs
(
w
p
<
--
210Hzs
p
14
1
)
such as the ones considered here have been
proposed as likely sources of continuous GWs
[
32
]
. The frequency stability and relatively low magnetic
fi
eld
3
New J. Phys.
19
(
2017
)
073023
S Singh
et al
indicate that unlike young pulsars like Crab and Vega, the dominant spin down mechanism in MSPs is more
likely to be gravitational radiation.
We can also set limits on GW generation mechanism by considering speci
fi
c models of the interior of NSs, as
discussed in
[
29
]
, and reviewed brie
fl
y in appendix
B
. Assuming standard nuclear matter and breaking strain for
elastic forces, the ellipticity sustained can be limited to less than
́
-
4
10
6
, irrespective of the physics leading to
deformations
[
18
,
33
]
. This can also be used to evaluate the GW strain amplitude limit
h
n
. Since the strain limits
h
sd
and
h
n
come from different physics
(
conservation of angular momentum and balancing forces in stellar
interior
)
, we use the lower of the two as the upper limit for metric strain.
Table
1
details parameters for pulsars of interest with rotational frequency higher than 500 Hz, along with
current limitations on GW strain from the LIGO
+
VIRGO collaboration. Theoretical estimates of metric strain
assuming spin-down limit, and elastic crust breakdown limit on ellipticity
(
= ́
-
410
6
)
from
[
18
]
are also
given. Table
1
also gives the strain estimate set by the helium detector outlined in
fi
gure
1
(
Gen1
)
that we will
discuss in detail in the following sections. Several of these pulsars were discovered recently by analyzing gamma-
ray sources from Fermi-LAT. The number of known fast spinning pulsars is expected to grow signi
fi
cantly as
more sources are discovered and analyzed.
Since the strain due to GWs from pulsars is expected to be very small but coherent, one needs to integrate the
signal for a long time
(
the latest result being a compilation of
∼
250 d of integration over three detectors
[
22
]
)
.
Also, in order to rule out noise we need to detect a GW signal from at least two different detectors. Strain
sensitivity also improves as
N
d
, where
N
d
is the number of detectors
[
35
]
. There has been computationally
intensive analysis of LIGO
+
VIRGO data to search for such GW signals. While being unsuccessful, they have
improved the upper bound on the emitted wave amplitudes. In the following, we outline the proposal for a
simple, low-cost, narrowband detector for these GW strains based on a super
fl
uid helium optomechanical
system. Being relatively simple and economical, super
fl
uid helium detectors can be set up in multiple locations
to improve overall detection sensitivity.
As a precursor to subsequent discussions, we present the central result of our work in
fi
gure
2
, showing the
limits set by different detectors for ten MSPs of interest from table
1
. Along with the spin-down limit and limit
set by the previous measurement
[
22
]
, we also show the limits set by two different geometries of helium detectors
that we discuss in detail in sections
3
and
4
. Here, we have assumed that the resonance frequency of the same
acoustic mode can be tuned by up to 200 Hz without changing the
Q
-factor
(
of
́
610
10
)
, thereby resonantly
targeting each pulsar with the same detector.
3. Super
fl
uid helium GW detector
The GW strain detector we propose is a resonant mass detector formed by acoustic modes of super
fl
uid helium
in a cavity parametrically coupled to a microwaves in a superconducting resonator. For the purpose of our
calculations, we will treat the super
fl
uid as an elastic medium with zero dissipation. At the temperatures we
expect to operate this detector,
<
T
10
mK, the normal
fl
uid fraction
r
n
is expected to be
r
r
<
-
10
n
0
8
, where
r
0
is the total density of the
fl
uid
[
37
]
. For temperatures below
<
T
100
mK, the dissipation of audio frequency
acoustic waves is expected and found to be dominated by a three-phonon process, falling off as
-
T
4
.
Table 1.
Table of millisecond Pulsars with frequency greater than 500 Hz:
w
p
is the rotational frequency,
wp
=
f
p
GW
is the frequency of
gravitational waves,
w
̇
p
is the measured spin down rate, and
d
is the distance to the pulsar in kilo-parsecs,
h
sd
and
h
en
are the spin-down and
elastic strain limits. These values are compared to the strain limit set by recent continuous GW surveys by LIGO
+
VIRGO
h
0
95
%
[
22
,
34
]
, and
the strain limit for three identical Gen1 helium detectors operating at
[
020
]
mode with an integration time of 250 d
(
except for J1748
−
2446ad, where mode
[
201
]
was used
)
. Pulsars indicated by superscript
*
were discovered by the Fermi gamma ray telescope.
Pulsar
wp
2
p
f
GW
wp
̇
2
p
dh
sd
h
n
h
0
95
%
h
He,1
95%
(
Hz
)(
Hz
)(
Hz s
−
1
)(
kpc
)
́
-
10
25
́
-
10
25
́
-
10
25
́
-
10
25
J0034
−
0534
532.71
1065.43
- ́
-
1.4 10
15
1
́
-
1
.3 10
2
48
0.49
́
-
1
.1 10
1
J1301
+
0833
*
542.38
1084.76
- ́
-
3.2 10
15
0.7
́
-
2.8 10
2
71
1.1
́
-
1
.1 10
1
J1747
−
4036
*
609.76
1219.51
- ́
-
4.9 10
15
3.4
́
-
6.7 10
3
19
No data
́
-
9.2 10
2
J1748
−
2446O
596.44
1192.87
- ́
-
9.3 10
15
5.9
́
-
5
.4 10
3
10
2.6
́
-
9.6 10
2
J1748
−
2446P
578.50
1157
- ́
-
8.9 10
14
5.9
́
-
1
.7 10
2
9.6
1.6
́
-
1
.0 10
1
J1748
−
2446ad
716.36
1432.7
- ́
-
1.7 10
14
5.9
́
-
6.7 10
3
15
1.8
́
-
3
.3 10
1
J1810
+
1744
*
601.41
1202.82
- ́
-
1.6 10
15
2.5
́
-
5
.3 10
3
2
4
0
.4
9
́
-
9.4 10
2
J1843
−
1113
541.81
1083.62
- ́
-
2.9 10
15
2.0
́
-
9.4 10
3
25
0
.46
́
-
1
.1 10
1
J1902
−
5105
*
574.71
1149.43
- ́
-
3.0 10
15
1.2
́
-
1
.5 10
2
47
No data
́
-
1
.0 10
1
J1939
+
2134
641.93
1283.86
- ́
-
4.3 10
14
1
.5
́
-
4.4 10
2
46
0.48
́
-
8.6 10
2
J1959
+
2048
622.12
1244.24
- ́
-
6.2 10
15
1
.5
́
-
1
.7 10
2
44
0
.7
4
́
-
9.0 10
2
4
New J. Phys.
19
(
2017
)
073023
S Singh
et al
An elastic body
(
with dimensions
l
GW
)
in a gravitational
fi
eld will undergo deformation due to changes in space-
time as a GW passes by. The equation of motion for the displacement
fi
eld
(
)
rt
u
,
of an elastic body is given by
[
38
]
rm lm
r
¶
¶
-- + =
()()
()
t
u
uuhx
.
1
2
̈
,7
2
2
L
2
L
L
where
ρ
is the density,
l
m
,
L
L
are the Lamé coef
fi
cients for the elastic body and
hx
̈
is the effective amplitude of
the wave for a particular orientation of the detector that exerts an effective tidal force on the detector.
This acoustic deformation can be broken into its eigenmodes
x
=
å
()
()()
tt
u
rwr
,
nn
n
. For this analysis, we
assume our acoustic antenna is in a single eigenmode of frequency
w
m
, thus dropping index
n
. We have used the
notation where
()
w
r
is a dimensionless spatial mode function with unit amplitude, and the actual amplitude of
the displacement
fi
eld is in
x
()
t
.
Rigid boundary walls and zero viscosity enables us to describe the acoustic modes accurately via a wave
equation as opposed to Navier
–
Stokes equations typically used for
fl
uid
fl
ow. The spatial modes are obtained by
solving the acoustic equations of motion
[
39
]
, as outlined in appendix
C
. These acoustic modes of helium in a
superconducting cavity were experimentally studied by some of the authors in
[
4
]
. We found them to be well-
modeled by this theory, and have
Q
-factors exceeding 10
8
at 45 mK.
For the purposes of this paper, we will simply add the
fi
nite linear dissipation to the acoustic resonance,
parameterized as a
fi
nite
Q
. For a damped acoustic resonator, equation
(
7
)
can be simpli
fi
ed to show that the
displacement
fi
eld
x
()
t
satis
fi
es the equation of motion
[
40
,
41
]
å
mx
w
xwx
++=
⎛
⎝
⎜
⎞
⎠
⎟
̇
()
Q
hq
̈
1
4
̈
,8
m
m
ij
ij
ij
He
2
where
Q
He
is the
Q
-factor and
μ
is the reduced mass for the particular eigenmode,
ò
m
r
=
V
w
d
2
, and
q
ij
is the
dynamic part of the quadrupole moment,
ò
rd
=+-
⎜⎟
⎛
⎝
⎞
⎠
·()
qwxxw
V
wr
2
3
d.
9
ij
ij
i j
ij
In their analysis of various antenna geometries for GW detection, Hirakawa and co workers introduced two
quantities to compare GW antennas spanning different size and symmetry groups
[
41
]
. These are the effective
area
(
A
G
)
characterizing the GW-active part of the vibrational mode, and the directivity function
(
d
A
)
, which
characterizes the directional and polarization dependence of the antenna. They are de
fi
ned as
å
m
=
()
A
M
q
2
10
G
ij
2
and
qf
=
å
å
()
()
()
()
d
qe
q
k
,
5
4
,11
A
ij
ij
A
ij
2
2
where
M
is the total mass of the antenna and
e
ij
A
is the unit vector for incoming GW signal polarization
A
(
Î+ ́
{}
A
,
)
in arbitrary direction
qfy
(
)
k
,,
. The Euler angles
qfy
(
)
,,
transform from the pulsar
Figure 2.
Strain sensitivity of various detectors for 10 MSPs of interest for the helium detector versus measurement time for two
helium detectors with same sensitivity operating simultaneously, at a bath temperature of 5 mK,
=
[
][]
lmn
020
, and
Q
-factor of 6
×
10
10
for both Gen1
(
mass
=
2.66 kg, in blue
)
and Gen2
(
mass
=
15.9 kg, in red
)
detectors. We also show the current limits on strain set
by LIGO
+
VIRGO collaboration
[
22
,
34
]
and calculated limit of Advanced LIGO operating at design sensitivity for 365 d
[
36
]
. As seen
in the
fi
gure, the current limit on pulsar GW strains can be surpassed within a few days of integration time for Gen1, and in under a few
days for Gen2. Also shown is the spin-down limit for these pulsars.
5
New J. Phys.
19
(
2017
)
073023
S Singh
et al
coordinate system to the detector co-ordinate system and are discussed in appendix
A
, along with the explicit
form of
e
ij
A
. An important distinction between the proposed detector and other GW sensors, particularly the
interferometric ones, is that the orientation of the detector
(
parameterized by
ψ
)
can be adjusted to optimize the
directivity function for the astrophysical source in consideration due to its small size. This acts as another tunable
parameter that can give signi
fi
cant enhancement in sensitivity, as shown in
fi
gure
3
for mode
[]
020
.
In terms of the above expressions, the mean squared force from a continuous GW source of polarization
A
is
p
mw
q f
mw
==
()
()
()
f
G
c
MAd
s
MAdht
2
5
,
1
40
,12
G
G
G
AA
G
G
A
A
2
3
24
2
where
w
w
=
2
Gp
is the GW frequency. Here we have assumed a
δ
-function GW spectrum.
As an example, we choose a cylindrical cavity of radius
=
a
10.8 cm
, length
L
=
50 cm
(
from now onwards
referred to as Gen1 or with subscript
(
)
He, 1
)
. We focus on the acoustic mode
=
()
f
1071 Hz
0,2,0
, which has an
effective mass
m
=
M
0.625
, and a large GR-active area of
p
=
A
a
0.629
G
2
due to its quadrupolar shape, shown
in
fi
gure
C1
. Another geometry considered in this work is a cylindrical cavity of the same radius, but length
L
=
3m
(
from now onwards referred to as Gen2 or with subscript
(
)
He, 2
)
. Since the resonance frequency of the
[]
020
mode is independent of length, it remains unchanged. However, increasing the mass gives us a larger
effective mass for the same area. Figure
3
shows the various directivity functions for this acoustic mode that
capture the angular dependence of the sensitivity of the detector. Appendix
C
discusses several characteristics of
the
fi
rst few modes of the cylindrical cavity that have a non-zero quadrupolar tensor in more detail.
4. Noise mechanisms and minimum detectable strain
The system we are proposing and have been exploring in the laboratory
[
4
,
6
]
is a parametric transducer
[
42
]
and
essentially similar to other optomechanical systems
[
43
]
: the acoustic motion of the super
fl
uid and resulting
perturbation of the dielectric constant modulates the frequency of a high-
Q
superconducting microwave
resonator. The details of the coupled acoustic and microwave system, sources of dissipation
(
phonon scattering,
effect of isotopic impurities, radiation loss,
)
requirements on thermal stability, etc are the subject of another
manuscript
[
6
]
. Here we take a few central results of this analysis.
The noise sources relevant to this system are the Brownian motion of the
fl
uid driven by thermal
/
dissipative
forces, the additive noise of the ampli
fi
er which is used to detect the microwave
fi
eld, the added noise of the
stimulating microwave
fi
eld
(
phase noise
)
, and possible back-action forces due to
fl
uctuations of the
fi
eld inside
the microwave cavity
(
due to phase noise and quantum noise
)
. The effect of vortices in super
fl
uid helium due to
Figure 3.
The directivity patterns for acoustic mode
=
[
][]
lmn
020
of the cylindrical cavity. The
+ ́
,
polarization and total directivity
functions are given for two different polarizations of the detector
(
Euler angle
y
p
=
0, 2
)
. The orientation of the detector can be
adjusted to optimize the directivity function for the astrophysical source in consideration.
6
New J. Phys.
19
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2017
)
073023
S Singh
et al
earth
’
s rotation on the
Q
-factor is unclear. However, using an annular cylinder or an equatorial mount allows for
long integration times without the possibly detrimental effects due to vortices.
Active and passive vibration isolation schemes deployed in interferometric sensors can be easily transplanted
to our proposed detector. Seismic isolation requirements for this detector are less stringent than LIGO primarily
due to the high frequency of operation
(
∼
1 kHz
)
. Furthermore, due to the mismatch between the speed of sound
in helium and niobium, there is natural acoustic isolation from the container. We will assume for the purpose of
this discussion that the challenging job of seismically isolating the super
fl
uid cell from external vibrations has
been accomplished, as has been done for other GW detectors, including LIGO
[
44
]
, cryogenic test masses for
KAGRA
[
45
]
, and various resonant detectors
[
46
]
. Due to the high frequency and narrow bandwidth of the
astrophysical source of interest, the strain noise due to Newtonian gravity
fl
uctuations are expected not to be
relevant for this detector
[
44
]
.
For a suf
fi
ciently intense microwave pump, with suf
fi
ciently low phase noise, the thermal Brownian motion
of the helium will dominate the noise. Assuming the device is pumped on the red sideband,
w
ww
=-
pp
c
m
, and
that the system is the side-band resolved limit,
w
k
>
m
c
, the upconversion rate of microwave photons is given
by:
k
G
=D
(·)
pgn
4
p
c
opt
SQL
0
2
, where
w
pp
,
w
c
, and
w
m
are the pump, cavity, and acoustic mode frequency
respectively,
kw
=
Q
cc
Nb
is the cavity damping rate,
D
p
SQL
is the amplitude of the zero-point
fl
uctuation of
the pressure of the acoustic
fi
eld,
n
p
is the amplitude of the pump inside the cavity measured in quanta, and
g
0
is
the coupling between the acoustic and microwave
fi
eld. For the geometry we consider here, Gen1:
l
=
0.5 m,
a
=
0.108 m,
w
p
= ́
2
1071 H
z
m
,
w
p
= ́
21.6GHz
c
, and
p
=- ́ ́
-
g
27.510
0
1
1
Hz.
To achieve a readout with noise temperature of 1 mK, which means that the added noise of the ampli
fi
er is
equal to the thermal noise amplitude when the helium is thermalized at 1 mK, requires
= ́
n
610
p
9
microwave pump photons and a phase noise of
−
145
dB
c
/
Hz. To begin to dampen and cool the acoustic
resonance with cavity backaction force, would require
=
n
10
p
12
, and a phase noise of
−
145
dB
c
/
Hz. Microwave
sources have been realized using sapphire resonators with phase noise of
−
160
dB
c
/
Hz at 1 kHz
[
47
,
48
]
.To
incorporate a similar low noise oscillator in our detector, one has to implement a microwave readout using
interferometric frequency discriminator, as used in
[
49
]
.Together with a tunable superconducting cavity, it is
possible to realize a source with suf
fi
cient low noise to broaden and cool this mode with backaction, as
demonstrated earlier in resonant bar antennas
[
50
]
.
Due to the very low dielectric constant of helium
(
=
1.05
He
)
, the bare optomechanical coupling constant is
small compared to typical micro-scale optomechanical systems:
w
=D ¶ ¶D
·
g
p
p
c
0
SQL
p
=- ́ ́
-
27.510
1
1
Hz: this is the frequency shift of the Nb cavity,
w
c
, due to the
zero-point
fl
uctuations of the acoustic
fi
eld of the helium,
D
p
SQL
. However, the relevant quantity is
cooperativity,
g
=G
C
opt
He
, which compares the rate of signal photon up-conversion,
G
opt
, to the loss rate of
acoustic quanta to the thermal bath,
g
w
=
Q
He
He He
. With quantum limited microwave detection
(
now
possible with a number of ampli
fi
ers
)
, detection at the SQL is achieved when
C
=
1, and is the onset of signi
fi
cant
backaction effects such as optomechanical damping and cooling. The key point is that for this system we expect
to be able to realize very large
n
p
. This is due to the very high
Q
possible in Nb,
(
~
Q
10
Nb
11
is now routine for
accelerator cavities
[
51
,
52
]
, even when driven to very high internal
fi
elds of 10
7
Vm
−
1
corresponding to
=
n
10
p
23
,
)
and dielectric losses and resulting heating at microwave frequency in liquid helium are expected to
be negligible up to very high pump powers. Assuming the dielectric loss angle in helium is less than 10
−
10
, our
estimates suggest that
=
n
10
p
16
should be achievable before dissipative effects lead to signi
fi
cant heating of
helium at 5 mK, far beyond the internal pump intensity used with micro-optomechanical systems and far above
the onset of backaction effects,
C
=
1 for
= ́
n
810
p
11
. As a result, we are optimistic that SQL limited detection
and signi
fi
cant backaction cooling and linewidth broadening are possible. As seen in other resonant GW
antennas
[
50
,
53
]
, one has to be careful about amplitude noise of the pump, since it starts deteriorating the
acoustic
Q
-factor at large powers due to backaction forces.
Since the frequency and phase of the pulsar
’
s GW signal should be known through observations of the
electromagnetic signal, single quadrature back-action evading, quantum non-demolition measurement
techniques could be implemented
[
54
]
. This has the advantage of avoiding the back-action forces from the cavity
fi
eld
fl
uctuations and can lower the phase noise requirements of the microwave pump. Similar noise evasion
techniques have already been studied in resonant GW antennas
[
55
]
.
In the following discussion we assume that noise at the detection frequency is dominated by the thermal
noise of the acoustic mode. As highlighted in the previous paragraphs, this assumption is valid for a suf
fi
ciently
intense microwave pump, with suf
fi
ciently low phase noise. This involves implementing state of the art
microwave transducer technology in our system. Detailed quantitative analysis of the contribution of various
types of transducer noise to strain sensitivity will be the subject of future work. We also assume acoustic
Q
-
factors of around 10
10
in these calculations. While other low loss materials such as sapphire, quartz and silicon
have demonstrated acoustic
Q
-factors higher than the one we demonstrated for super
fl
uid helium
(
́
1
.4 10
8
)
,
it appears possible for helium to reach signi
fi
cantly higher
Q
-factors than those measured so far with technical
7
New J. Phys.
19
(
2017
)
073023
S Singh
et al