Ultra-high Q Acoustic Resonance in Superfluid
4
He
L. A. De Lorenzo
1
and K. C. Schwab
1
1
Applied Physics, California Institute of Technology, Pasadena, CA 91125 USA
∗
(Dated: July 28, 2016)
We report the measurement of the acoustic quality factor of a gram-scale, kilo-hertz frequency
superfluid resonator, detected through the parametric coupling to a superconducting niobium mi-
crowave cavity. For temperature between 400mK and 50mK, we observe a
T
−
4
temperature de-
pendence of the quality factor, consistent with a 3-phonon dissipation mechanism. We observe Q
factors up to 1
.
4
·
10
8
, consistent with the dissipation due to dilute
3
He impurities, and expect that
significant further improvements are possible. These experiments are relevant to exploring quantum
behavior and decoherence of massive macroscopic objects, the laboratory detection of continuous
wave gravitational waves from pulsars, and the probing of possible limits to physical length scales.
INTRODUCTION
Recently there has been increasing interest in super-
fluid optomechanical systems[1–3]. Superfluid
4
He offers
extremely low losses in both the mechanical and elec-
tromagnetic domains (spanning microwave to optical fre-
quencies). Furthermore superfluid
4
He is a quantum con-
densate which can be both cooled deeply below its tran-
sition temperature
T
λ
,
T/T
λ
≈
10
−
2
and isotopically pu-
rified to extreme purity, realizing very small normal-state
fractions of
ρ
n
/ρ
0
<
10
−
8
, where
ρ
n
is the density of the
normal fluid, and
ρ
0
is the total density of the fluid. To-
gether with an ultra-sensitive detection scheme, this sys-
tem may allow the uniquely quantum behavior which has
been observed recently in micromechanical oscillators[4–
6] to be revealed in a far larger, gram scale systems. Fur-
thermore, this system may find utility in diverse applica-
tions such as the search for fundamental limits to physical
length scales[7], the detection of continuous gravitational
waves[8], the understanding of quantum decoherence of
massive systems and the emergence of the classical world
from the underlying quantum nature[9–14].
A key parameter in this systems is the loss rate of
the mechanical element.
Here we study the dissipa-
tion rate of acoustic modes in
4
He coupled to a very
low loss electromagnetic resonator, and report loss rates
which are an order of magnitude smaller than our initial
measurements[1], and a factor of more than 10
3
times
less than recent observations in other superfluid optome-
chanical experiments[2, 3].
Since the 1950’s, acoustic loss in
4
He has been well
studied both experimentally and theoretically[15]. In the
low temperature regime where losses are dominated by
the 3-phonon scattering process, the acoustic loss of first
sound in helium-4 is given by[16]
α
=
π
2
60
(
G
+ 1)
2
ρ
4
~
3
c
6
4
(
k
B
T
)
4
ω
He
(arctan (2
ω
He
τ
)
−
arctan (∆
Eτ
))
(1)
where
G
= (
ρ/c
)
∂c/∂ρ
= 2
.
84 is the Gr ̈uneisen’s
parameter,[17]
ρ
= 145 kg/m
3
and
c
4
= 238 m/s are
the helium density and speed of sound[17],
~
is the
Planck constant,
k
B
is the Boltzmann constant,
ω
He
is the frequency of the acoustic wave,
T
is the tem-
perature,
τ
= 1
/
(0
.
9
·
10
7
T
5
) is the thermal phonon
lifetime[18], ∆
E
= 3
γ
ρ
2
ω
He
is the energy discrepancy be-
tween the initial and final states in the 3PP,
ρ
= 3
k
B
T/c
is the average thermal momentum, and
γ
≈ −
10
48
(s/kg
·
m)
2
is a constant which characterizes the weak non-
linearity of the dispersion relation for low momentum
phonons[15, 19] The loss expected form the 3PP is plot-
ted as in Fig. (2). Below
≈
300 mK where
ωτ >>
1, the
first arctan function reaches its maximum value of
π/
2
and the loss follows a
T
4
law, in good agreement with our
data. The second arctan function is
≈
0 down to 100 mK
but rises to
≈−
π/
2 by 40 mK, increasing the attenuation
a)
b)
0.05-1.6 K
4.2 K
300 K
T
MC
PZT
0
500
1000
1500
2000
2500
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Time (s)
Voltage (V)
12201 Hz at 25 kohms with 400 mVpp Drive
OS1
OS2
Lock-In
T
N
= 6 K
4
He
8118 Hz
8678 Hz
10138 Hz
2
1
-1
-2
-3
3
0
Max: 3.5
Min: -3.5
FIG. 1. a) finite element simulations of the pressure profiles
of the three highest Q acoustic eigenmodes, with mode fre-
quencies shown above each figure. Note that in each of these
high Q modes, the radial node is located at the position of
the helium fill line. b) Measurement setup: Microwave oscil-
lator OS2 is used to pump the niobium resonator on its red
sideband while audio oscillator OS1 drives the piezo at a he-
lium acoustic eigenfrequency. Parametric coupling between
the acoustic and microwave modes results in an upconverted
peak at the microwave cavity resonance which is mixed down
to an acoustic frequency and measured on a lock-in amplifier.
arXiv:1607.07902v1 [quant-ph] 26 Jul 2016
2
by a factor of 2. Eqn. (1) predicts a minimum point at
≈
450 mK, where
Q
begins to rise with increasing
T
; our
data shows a relatively sharp minimum at 600mK. Fur-
thermore, above
≈
600 mK, the roton population is no
longer negligible and phonon-roton scattering becomes
important, which is not included in Eqn. (1).
We form a parametric system coupling low frequency
acoustic vibrations in a superfluid filled cavity, approxi-
mately 4 cm in length and 3.6 cm in diameter, to high fre-
quency microwave modes of the surrounding cylindrical
niobium resonator[1]. The density modulation produced
by the acoustic modes produces a proportional modu-
lation of permitivitty which couples to the microwave
modes. We employ the TE
011
mode of the microwave
cavity, which is typically the highest Q mode in these
systems and has a frequency
ω
C
/
2
π
= 10
.
6 GHz when
the resonator is filled with
4
He. Input and output cou-
pling to the microwave mode is achieved via two loops of
wire recessed into the cavity lid. The intrinsic loss rate
of the TE
011
mode was measured to be
κ
int
= (2
π
)
·
31
Hz; for the most recent runs of our experiment we have
overcoupled the cavity such that
κ
in
=
κ
out
= 2
π
·
230
Hz . We apply a red detuned microwave pump tone at
ω
P
=
ω
C
−
ω
He
while driving the acoustic mode at
ω
He
with a piezo transducer attached to the niobium cavity,
to produce an upconverted sideband at the microwave
cavity resonance[20]. We detect acoustic modes in the su-
perfluid at frequencies within 1% of their expected values
for a right cylindrical acoustic resonator. Quality factors
are determined by recording the free decay of the acoustic
oscillations.
In this paper we compare our recent measurements
to the initial measurements we reported in Ref.[1]. To
improve the mechanical quality factor of the superfluid
acoustic modes, we have focused on two known sources of
loss: the temperature dependent 3PP described by Eq.
(1) and the clamping losses from attaching the cell to the
dilution refrigerator (DR). Experimental improvements
were identical for both final runs shown in Fig. (2), with
the sole exception being the correction of a thermal short
in the fridge which improved the fridge cooling power at
each cool stage for the final run (triangles).
A significant difficulty with very low temperature
4
He
experiments is effectively thermalizing the sample to mil-
likelvin temperatures. Since the speed of sound in
4
He,
c
4
, is
≈
1
/
10 the speed of sound in most metals,
4
He has
comparatively an extremely high phonon heat capacity
at low temperatures. Furthermore,
4
He thermalizes only
through tranmission of phonons between the fluid and
the walls of the container (Nb in this case.) However,
due to the severe acoustic impedance mismatch, most of
this acoustic energy is reflected resulting in a high bound-
ary resistance, Kapitza resistance, and is given approx-
imately by:
R
k
= 15
~
3
ρ
Nb
c
3
Nb
/
2
π
2
k
4
B
T
3
ρ
4
c
4
A
, where
ρ
Nb
and
c
Nb
refer to the density and speed of sound in
niobium and
A
is the surface area of contact[21]. Since
a)
b)
Time (s)
0
4000
8000
12000
Lock-In Amplitude (mV)
0
1
2
3
4
5
6
8112 Hz
T = 44 mK
Q = 1.35·10
8
Raw data
Fit
Temperature (mK)
10
20
30
50
100
200
300
500
1000
Q
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
3PP
He-3
Purified He-3
8119 Hz
8678 Hz
10138 Hz
FIG. 2. a) is the free decay of the helium acoustic resonance
at 8112 Hz, exhibiting a quality factor of 1
.
35
·
10
8
. b) shows
quality factor versus fridge temperature for the three highest
Q superfluid acoustic modes. The circles and triangles repre-
sent data from our two most recent experimental runs, with
improvements specified in the text. The red line shows the
dissipation expected from the 3PP which scales as
T
4
at low
temperatures (Eq. (1)). The blue line shows the dissipation
expected from
3
He impurities for concentrations
n
3
<
10
−
8
when the mean free path of the
3
He atom is less than the
container dimensions and the acoustic dissipation is concen-
tration independent, while the gray line show the expected
dissipation for
n
3
= 2
·
10
−
10
when the mean free path ex-
ceeds the container size by a factor of 100; both cases assume
a mode frequency of 8112 Hz.
the heat capacity of helium,
C
, depends on temperature
as
∝
T
3
for temperature far below
T
λ
, we expect the
thermal time constant
τ
=
R
k
C
≈
10 seconds and to be
independent of temperature below
∼
200 mK (when the
R
k
is larger than the thermal resistance of the copper
wire which suspends the Nb cell from the mixing cham-
ber plate.) However, the very high heat capacity and
3
the convective counter-flows in the fluid lead to a large
thermal conductance through the capillaries which fill the
cell and connect the helium sample to higher temperature
reservoirs. Together with the high boundary resistance,
this leads to substantial difficulty to cool the sample to
the base temperature of the dilution refrigerator. For in-
stance, it would take a heat leak of
̇
Q
= 25
nW
(0
.
1
nW
)
into the helium resonator to hold the superfluid sample
at 40mK (10mK).
In previous runs of the experiment, we observed the he-
lium temperature to be significantly above the fridge base
temperature, indicating a failure to sufficiently thermal-
ize the helium in the fill-line at various refrigerator fridge
stages (1K pot, still, cold plate, mixing chamber.) In the
data reported here, we have lowered
R
k
at three tempera-
ture stages by adding silver sinter heat exchangers to the
fill line. Because the acoustic impedance mismatch be-
tween helium and a metal is high, large surface areas are
required to achieve small values of
R
k
; we have added 6
.
6
m
2
of surface area to the base plate, and an additional 3
.
3
m
2
at 100 mK and 975 mK. By thermally anchoring the
helium in the fill line at each stage, heat leaks thru the
helium from higher stages of the fridge are reduced. We
have also increased the thermal resistance of the fill line
between stages of the DR by using capillaries of smaller
diameter (150
−
200
μ
m), and longer length (1 m).
However, even with these improvements, we find that
at low temperatures, our helium sample has much longer
thermalization times than expected. For instance, in our
most recent experimental run, we find
τ
≈
6 hours when
heating the fridge from 40 to 50 mK, and we note that
even at the lowest temperatures, the superfluid
Q
con-
tinued to improve by about 1% per day, suggesting a
long term thermal relaxation. We have not identified the
reason for the long thermal relaxation time.
Our final improvement to reduce the helium sample
temperature (triangles only in Fig. (2)) came from in-
creasing the cooling power of our dilution refrigerator by
correcting a previously undetected thermal short between
1.6 K and 975 mK. The result of these changes was an
improvement at the mixing chamber to temperatures be-
low 20 mK and an improvement in quality factor of the
8111 Hz mode from 14
·
10
6
to 135
·
10
6
. If the acoustic Q
is limited by the three phonon process in each case, the
helium temperature has decreased from 82mK to 44 mK.
For mechanical resonators, a significant loss channel is
often clamping loss thru the suspension system. In our
most recent experiments, we have decreased the suspen-
sion loss by replacing the rigid copper block mounted to
cell’s midpoint with a 0.13 cm diameter, 6.7 cm length
copper wire. The thermal resistance through this wire
at 40mK is expected to be
≈
10
4
K/W
, a factor of 100X
less than the Kapitza resistance between the helium and
the niobium walls. We have also replaced the lid of the
cell, moving the helium fill line and microwave couplers
to the location of the first radial acoustic node, thereby
reducing acoustic loss from acoustic radiation into the fill
line.
Finally, we note that acoustic loss is expected to arise
from the dilute
3
He impurity, which behaves as a classical
viscous gas in the superfluid[22]. Following this assump-
tion we calculate the acoustic attenuation from
3
He (at
natural concentration of
≈
1 in 10
6
helium atoms) to
be[23]:
α
3
He
≈
(
7
3
√
k
B
m
∗
3
π
1
σ
)
(
1
ρ
4
c
3
4
)
(
√
Tω
2
)
(2)
where
σ
=
πd
2
= 6
·
10
−
20
m
2
is the He-3 scattering cross
section and
m
∗
3
= 2
.
34
m
3
is the effective mass of a
3
He
atom at zero concentration[24]. The expected loss from
3
He is plotted as a blue line in Fig. (2). The losses due
to
3
He are expected to be independent of concentration
until the mean free path of the
3
He atom exceeds the con-
tainer dimensions; in this case, for concentrations above
x
=
n
3
/n
4
>
10
−
8
, where
n
3
and
n
4
are the number den-
sity of
3
He and
4
He atoms. The loss rate from
3
He can
be diminished by decreasing the
3
He concentration with
isotopically purified samples of helium[23]. In our most
recent experimental run, the quality factor of the 8.119
kHz mode has become comparable to the
3
He impurity
limit at low temperatures. In an attempt to improve the
Q further, we warmed the system to above 4 K, flushed
the cell and refilled it with a sample of helium with only
2
·
10
−
10 3
He concentration[25], but did not note an im-
provement in quality factor of the 8 kHz mode. We did
not measure the isotopic purity of the helium removed
from the cell following the experiment and are unable to
confirm that we adequately removed the
3
He impurity
from the setup.
The low temperature, the highest acoustic Q data
points were taken with an incident microwave power at
the niobium cavity of 0
.
4 pW, corresponding to 3
·
10
4
pump photons in the cavity. We are able to apply powers
up to 4 nW or 3
·
10
8
pump photons in the cavity while
maintaining a superfluid acoustic
Q
above 10
8
. Higher
powers begin to heat the helium and degrade the me-
chanical
Q
. For instance at 40 (400) nW, the helium
temperature extracted from the
Q
is 61 (122) mK. This
level of heating is much higher than expected given the
low dielectric loss tangent of helium
tan
(
δ
)
<
10
−
10
[26]
and of our niobium cell 1
/Q
= 5
·
10
−
8
. A possible expla-
nation is the normal metal remaining in our microwave
circuits: the coupling loops embedded in the Nb lid are
BeCu and the SMA caps are Cu; in future runs of this
experiment these will be replaced with superconducting
materials.
Reaching and acoustic
Q >
10
10
will require lower-
ing the helium temperature from our current 44 mK to
<
15 mK. To reduce the heat leak from the fill-line to
the cell we will install a superfluid leak tight cryogenic
valve located on the mixing chamber[27]. With the valve
4
in place, the fill line from the mixing chamber to room
temperature can be evacuated to vacuum and heat leaks
resulting from helium film flow up the capillaries will be
eliminated. We can also replace the copper wire used for
suspension with a high purity (5N) annealed silver wire,
which due to higher thermal conductance will allow for
an even smaller diameter wire to be used, minimizing
further the mechanical contact to the cell.
Reaching such high quality factors is likely to require
further limiting clamping losses. In our current setup,
there are four mechanical connections to the cell: the
suspension and cooling wire, the fill line, and two mi-
crowave connections. A simple improvement in suspen-
sion loss can be accomplished by utilizing a higher Q
material (such as silver) for the suspension wire,[28],
and to engineer the vibrational modes of the suspension
such that there are no modes at the helium acoustic fre-
quency. It is possible to eliminate one microwave connec-
tion by operating the niobium cavity in reflection or to
eliminate both microwave connections by using antenna
coupling[29]. The fill line could be removed by welding
the cavity lid in place and pre-filling the cell to a pressure
of 2
.
3
·
10
7
Pa (230 bar) at 77 K or 9
·
10
7
Pa (900 bar)
at 300 K. Alternatively, the fill line and suspension wire
can be combined to a single connection.
The viscous loss expected from the
3
He impurity can
be lowered by using isotopically purified samples of he-
lium. To reach
Q >
10
10
in our system requires a
3
He
concentration of
x
= 2
·
10
−
10
, (see Fig. (2)) which we
have already purchased.[25] Samples with
3
He concentra-
tions as low as
x
= 10
−
12
are available[30].
Finally we note that prospects for measurements of the
thermal motion of the acoustic modes appears well within
reach. Assuming a microwave cavity with
Q
= 10
9
, neg-
ligible heating of the helium from dielectric loss, and an
acoustic quality factor limited by the three phonon pro-
cess (Eqn. (1)), it will be possible to measure the thermal
motion of the mode at 14 (8) mK with a precision of 10%,
where
Q
= 10
10
(
Q
= 10
11
) with a source of phase noise
−
143 (
−
136) dBc/Hz at an 8 kHz offset from carrier.
This level of phase noise is possible with demonstrated
microwave sources[31]
In conclusion, we have realized a superfluid
4
He acous-
tic resonator with exceedingly high quality factor,
Q
=
1
.
4
·
10
8
, which for the lowest dissipation mode appears
to be limited by the dissipation due to dilute
3
He im-
purities. Only three other materials (silicon, sapphire,
and quartz)[32–34] have ever shown lower acoustic losses,
with loss rates approximately 10x lower than what we
demonstrate here. Given the unquie properties of su-
perfluid
4
He it appears possible for
4
He to achieve sig-
nificantly lower loss rates. Further development of this
unique system will find applications which span from
the detection of continuous gravitational waves from
sources such as nearby pulsars[8], quantum decoherence
of macroscopic systems[9–14], to studies which search
for limits to physical length-scales at extremely small
distances[7].
We acknowledge funding provided by the Institute for
Quantum Information and Matter, an NSF Physics Fron-
tiers Center (NSF IQIM-1125565) with support of the
Gordon and Betty Moore Foundation (GBMF-1250) NSF
DMR-1052647, and DARPA-QUANTUM HR0011-10-1-
0066. L.D. acknowledges support from the NSF GRFP
under Grant No. DGE-1144469.
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∗
schwab@caltech.edu
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