Ultrasound Attenuation of Superfluid 3He in Aerogel
H.C. Choi, N. Masuhara, B.H. Moon, P. Bhupathi, M.W. Meisel, and Y. Lee∗
Microkelvin Laboratory, Department of Physics, University of Florida, Gainesville, FL 32611-8440, USA
N. Mulders
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
S. Higashitani, M. Miura, and K. Nagai
Faculty of IAS, Hiroshima University, Kagamiyama 1-7-1, Higashi-Hiroshima 739-8521, Japan
(Dated: October 27, 2018)
We have performed longitudinal ultrasound (9.5 MHz) attenuation measurements in the B-phase
of superfluid 3He in 98% porosity aerogel down to the zero temperature limit for a wide range of
pressures at zero magnetic field. The absolute attenuation was determined by direct transmission
of sound pulses. Compared to the bulk fluid, our results revealed a drastically different behavior in
attenuation, which is consistent with theoretical accounts with gapless excitations and a collision
drag effect.
Liquid 3He has attracted intense interest for many
decades in the field of low temperature physics [1]. In
its normal state, liquid 3He has served as a paradigm
for a Fermi liquid whose nature transcends 3He physics.
The superfluid phases of 3He exhibit exotic and intrigu-
ing features associated with the broken symmetries in the
condensate, having an unconventional structure of the or-
der parameter with spin triplet p-wave pairing. Liquid
3He is arguably the most well-understood system mainly
because of its extreme intrinsic pureness at low temper-
atures. Therefore, it has provided important insights
in understanding other unconventional superconductors
such as the high temperature superconductors, the heavy
fermion superconductors, and in particular the more re-
cently discovered Sr2RuO4, which is also thought to have
the p-wave symmetry [2]. However, the same virtue has
hampered the effort in pursuing answers to an important
overarching question: how does the nature of a quantum
condensate (spin triplet p-wave superfluid in this case)
respond to increasing impurity or disorder?
Observation of superfluid transitions in liquid 3He im-
pregnated in high porosity aerogel in 1995 [3, 4] opened
a novel path to introducing static disorder in liquid 3He.
Aerogel possesses a unique structure, whose topology is
at the antipode of widely studied porous media such as
Vycor glass and metallic sinters. Due to its open struc-
ture, there are no well-defined pores in aerogel and conse-
quently, the liquid is in the proximity to the bulk. Ninety
eight percent porosity aerogel, which has been used in
most of the studies including this work, offers a corre-
lated network of strand-like aggregates of SiO2 molecules
whose structure can be characterized by the geometrical
mean free path (ℓ ≃ 100 - 200 nm), the diameter of strand
(r ≈ 3 nm), and the average inter-strand distance (d ≃ 25
- 40 nm). The coherence length of pure superfluid 3He,
ξ0, which varies from 20 nm (34 bar) to 80 nm (0 bar),
is at least an order of magnitude larger than the strand
diameter but is comparable to ℓ and d. As a result, the
scattering off the aerogel strand would have a significant
influence on the superfluid. It is now well established
that the superfluid transition temperature is significantly
depressed from that of the bulk, and the effect of pair-
breaking is progressively magnified at lower pressures,
leading to the possibility of a quantum phase transition
at Pc ≈ 6 bars [5]. To date, three distinct superfluid
phases have been experimentally identified, namely the
A-like, B-like, and A1-like phases [4, 6, 7, 8, 9]. The B-like
phase and the A1-like phase in aerogel show striking sim-
ilarity to their counterparts in the bulk superfluid [9, 10].
Detailed NMR studies [7, 8, 10] suggest that the aerogel
B-phase has the same order parameter structure as the
bulk B-phase. The aerogel A1-phase only appears in the
presence of magnetic field as is the case in the bulk [9].
However, the aerogel A-phase exhibits quite a different
behavior from the bulk A-phase (e.g. in NMR frequency
shift and superfluid density), although the overwhelming
experimental evidence suggests that it is an equal spin
pairing state. Various interpretations or novel proposi-
tions on the possible order parameter structure have been
suggested for this phase [11, 12, 13].
Nuclear magnetic resonance and ultrasound spec-
troscopy have been used in concert to investigate the mi-
croscopic structure of the superfluid phases [1, 14]. These
two experimental methods encompass complementary in-
formation on the orbital (ultrasound) and spin (NMR)
structure of the Cooper pairs. Rich spectra of order
parameter collective modes in bulk superfluids, which
are the fingerprints of specific broken symmetries in the
system, have been mapped by ultrasound spectroscopic
techniques [14]. In 2000, Nomura et al. [15] performed ul-
trasound attenuation measurements on 98% aerogel us-
ing a 16.5 MHz cw acoustic impedance technique. Their
work was limited to a single pressure at 16 bars and down
to 0.6 mK. Although their technique was not adequate in
determining absolute attenuation, they managed to ex-
tract the absolute sound attenuation after making auxil-
http://arxiv.org/abs/0704.0419v1
Time (µs)
FIG. 1: Acoustic response from the receiver vs. time at 34
bars for select temperatures ranging from 0.3 mK to 2.5 mK.
The aerogel superfluid transition is marked by a small arrow.
iary assumptions. A Bayreuth group [16] performed ab-
solute sound attenuation measurements in aerogel (97%
porosity) using a direct sound transmission technique at
10 MHz. They experienced poor transducer response,
and observed self-heating and no depression in the aero-
gel superfluid transition. We conducted high frequency
sound transmission experiments in 98% porosity aero-
gel, covering the whole phase diagram of the superfluid
phases in aerogel, from 8 to 34 bars and from the transi-
tion temperatures to as low as 200 µK.
In this experiment, two matched LiNbO3 longitudi-
nal sound transducers with the fundamental resonance at
9.5 MHz were used as a transmitter and a receiver. The
6.3 mm diameter transducers were separated by a Ma-
cor spacer maintaining a 3.05 (± 0.02) mm sound path
between the transducers where the aerogel sample was
grown in situ. This scheme ensures the best contact be-
tween the transducer surface and the aerogel, which is
crucial for clean sound transmission at the boundaries.
A 1 µs pulse was generated by the transmitter and de-
tected by the receiver. Temperature was determined by
a melting pressure thermometer (MPT) for T ≥ 1 mK
and a Pt NMR thermometer for T ≤ 1 mK which was
calibrated against the MPT. No non-linear response or
self-heating was observed at the excitation level used in
this work. All the data presented here, except for 8 bars,
were taken while warming with a typical warming rate
of 3 µK/min. A detailed description on the experimen-
tal cell and experimental techniques can be found else-
where [17, 18].
The temporal responses of the receiver taken at 34 bars
are shown in Fig. 1 for select temperatures ranging from
0.3 to 2.5 mK. The primary response, which starts to rise
around 8 µs, shows a rather broad response due to ring-
ing of the high Q transducer (Q ∼ 103). The step-like
structure of the receiver signal is caused by the slight mis-
match in the spectra of the transducers [18]. Below the
aerogel superfluid transition (marked around 2.1 mK by
an arrow in Fig. 1) the primary response starts to grow
and the trailing echoes emerge from the background, as
the sound attenuation decreases in the superfluid. No
change in the receiver signal was observed at the bulk
superfluid transition. The multiple echoes follow a bona
fide exponential decay in time. Absolute sound attenua-
tion was obtained in the following manner [19]. First, the
relative attenuation at each temperature was calculated
using the area under the primary response curve by inte-
grating the signal from the rising edge to a fixed point in
time (23 µs point). The absolute attenuation at 0.4 mK
and 29 bars, obtained using the primary signal and the
echoes, was used as a reference point in converting the
relative attenuation into the absolute attenuation. Due
to a drastic mismatch in the acoustic impedance at the
the transducer-aerogel/3He boundary, the signal absorb-
tion at the surface of transducers was ignored [19]. The
possible background contributions to attenuation from
the quasi-particle scattering off the cavity wall [20] and
the non-parallel alignment of the two transducers are es-
timated to be negligible.
The absolute attenuations on warming for several pres-
sures are plotted as a function of temperature in Fig. 2(a).
The superfluid transition is marked by the smooth drop
in attenuation. Our aerogel superfluid transition tem-
peratures are in excellent agreement with the previously
reported values for all pressures [5, 21]. At 9.5 MHz in
the bulk B-phase, a strong attenuation peak appears right
below the superfluid transition. This peak is the result of
the combined contributions from pair-breaking and cou-
pling to the order parameter collective modes. Above the
polycritical pressure, the B to A transition on warming
is registered as a sharp step in attenuation. In aero-
gel, none of these features exist. However, we did ob-
serve a sharp step in attenuation on cooling for P > 14
bars, which implies the existence of the supercooled A-
phase [19]. We were able to identify a rather smooth B
to A transition on warming for 29 and 34 bars within ≈
150 µK below the superfluid transition. This observation
is consistent with the previous results obtained using a
transverse acoustic impedance technique [13]. Therefore,
most of the attenuation data presented here are in the
aerogel B-phase. In the bulk B-phase with a clean gap,
the attenuation follows α ∝ e−∆(T )/kBT below the atten-
uation peak, practically reaching zero attenuation below
T/Tc ≈ 0.6, due to thermally activated quasi-particles,
where ∆(T ) is the temperature dependent gap and kB
is the Boltzmann constant. In contrast, the attenuation
in aerogel decreases rather slowly with temperature and
remains high even at T/Tc ≈ 0.2. Furthermore, a pe-
culiar shoulder feature appears at T/Tc ≈ 0.6 for higher
pressures. This feature weakens gradually and eventually
disappears at lower pressures, Fig. 2(a).
Sound propagation for higher harmonics up to 96 MHz
was measured for several temperatures and pressures,
but no evidence of sound propagation was found above
30 MHz even at 0.3 mK, where the lowest attenuation
is expected. Below about 10 mK, the scattering process
is dominated by the temperature independent impurity
scattering off the aerogel, and at 9.5 MHz, ωτi ∼ 0.1 for
all pressures where τi = ℓ/vf (see below for ℓ). There-
fore, the sound mode should remain in the hydrodynamic
limit. This claim is bolstered by the observation of the
strong frequency dependence in attenuation and the ab-
sence of a temperature dependence in the normal fluid
attenuation [15]. The coupling between the normal com-
ponent of the superfluid 3He and the mass of the elas-
tic aerogel modifies the conventional two-fluid hydrody-
namic equations [22, 23]. This consideration leads to two
(slow and fast) longitudinal sound modes with different
sound speeds, cs = ca
ρsρa/ρ, and cf = c1
1+ρaρs/ρnρ
1+ρa/ρn
Here, cf(s) represents the speed of the fast (slow) mode,
ρn(s) is the normal fluid (superfluid) density (ρ = ρn+ρs),
ρa is the aerogel density, c1 is the speed of hydrodynamic
sound in 3He, and finally ca is the sound speed of the bare
aerogel. From the time of flight measurements, we found
the sound speed in aerogel consistently lower (by ≈ 20%)
than c1 for all pressures studied and in good agreement
with the values obtained using the expression above [24].
Detailed analysis of sound velocity for various pressures
will be presented in a separate publication.
Low mass density and the compliant nature of aero-
gel necessitate the consideration of effective momentum
transfer upon quasi-particle scattering off the aerogel,
which generates dragged motion of aerogel. Ichikawa et
al. [25] incorporated the collision drag effect in calculat-
ing the dispersion relation in the normal fluid. Their
model offered a successful explanation for the experimen-
tal results of the Northwestern group [15]. Recently, Hi-
gashitani et al. [26, 27] extended this model to study the
longitudinal sound (fast mode) propagation in superfluid
3He/aerogel within the framework of the two-fluid model.
The drag effect can be described phenomenologically by
a frictional force, ~Fd =
(~vn −~va), introducing an addi-
tional relaxation time τf , where ~vn(a) is the normal fluid
component (aerogel) velocity. This effect is of particu-
lar importance when ωτi < 1, and the total attenuation
(Eq. (130) of ref. [27]) is
ω2/2cf
1 + ρaρs/ρnρ
ρ2aτf/ρρn
1 + ρa/ρn
4η/3ρc21
1 + ρaρs/ρnρ
), (1)
where η is the shear viscosity of liquid 3He. The first
term (αf ) arises from the frictional damping caused by
the aerogel motion relative to the normal fluid compo-
nent, and the second term (αv) from the conventional
hydrodynamic sound damping associated with the viscos-
ity. This expression allows us to extract ℓ in this system
from our absolute attenuation at the transition temper-
ature, αc. The inset of Fig. 3 shows our results of αc for
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0
 10 bar
 14 bar
 20 bar
 25 bar
 33 bar
 34 bar
T / T
 10 bar
 12 bar
 14 bar
 21 bar
 25 bar
 34 bar
T (mK)
FIG. 2: (a) Absolute attenuation for various pressures vs.
temperature (color in on-line version). Thin solid lines are
the results of a quadratic fit to the low temperature part
(T/Tc <∼ 0.4) of the data at each pressure. (b) Normalized
sound attenuation vs. normalized temperature. The results
of theoretical calculation (solid lines, color in on-line version)
are plotted along with the experimental results at 34 bars for
comparison.
various pressures. The solid lines are the result of calcu-
lation using Eq. (1) for three different mean free paths,
ℓ = 100, 120, and 140 nm. As can be seen, ℓ = 120 nm
produces an excellent fit to our data for the whole pres-
sure range, which is in good agreement with the val-
ues obtained from the thermal conductivity (90 nm) [28]
and spin diffusion (130 nm) [29] measurements. With
the knowledge of the mean free path, one can calculate
the full temperature dependence of sound attenuation in
the superfluid phase. The results of the calculation (in
the unitary limit) following the prescription described in
ref. [27] are displayed in Fig. 2(b) along with the experi-
mental results at 34 bars. The calculation reproduces all
the important features observed in our measurements. In
particular, the conspicuous shoulder structure appearing
near T/Tc ≈ 0.6 at 33 bars softens at lower pressures and
is completely absorbed in an almost linear temperature
dependence below 20 bars. This behavior is the charac-
teristic of αf [27]. A fast decrease in ρn right below Tc
produces the bump in αf , and αf → 0 as T → 0. On
the other hand, αv decreases monotonically and reaches
a finite value due to non-zero ρn and the impurity states
0 5 10 15 20 25 30 35
0 10 20 30
P (bar)
 100 nm
 120 nm
 140 nm
P (bar)
FIG. 3: Normalized zero temperature attenuation vs. pres-
sure. The dashed line is a guide for eye. Inset: Pressure
dependence of sound attenuation at Tc. The solid lines (color
on-line) are the results of theoretical fit for ℓ = 100, 120, and
140 nm (see text).
induced inside the gap as T → 0. The quantitative agree-
ment between the theory and experiment, however, is not
yet satisfactory. The calculation utilizes the isotropic ho-
mogeneous scattering model (IHSM) [30], which tends to
overestimate ∆(T ) and ρs compared to the experimen-
tally determined values [3, 23]. As shown in ref. [31],
the inhomogeneity gives rise to the reduction of the aver-
age value of the order parameter and consequently yields
larger η and ρn, which in turn increases α0 but decreases
the frictional contribution. It is also expected that the
non s-wave scattering components make non-trivial con-
tributions to the viscous and frictional relaxation times
in a direction that improves the quantitative agreement.
Theoretical calculations based on the IHSM [27, 32]
predict that the impurity states would completely fill the
gap, leading to a gapless superfluid when τiTc < 1 for the
B-phase in the unitary limit. We estimate 0.3 < τiTc < 1
for 10 < P < 34 bars with ℓ = 120 nm. The normalized
zero temperature attenuation (α0/αc) obtained by ex-
trapolating the low temperature part of the attenuation
(solid lines in Fig. 2(a)) is plotted in Fig. 3, where α0/αc
increases as the sample pressure is reduced and seems
to approach unity near Pc ≈ 6 bars. Since the viscosity
ratio is directly related to the density of states at zero
energy through η(0)/η(Tc) = n(0)
z, z = {2,4} for the
{Born, unitary} limit where n(0) is the normalized den-
sity of states at zero energy [27], the finite α0/αc is strong
evidence of a finite n(0). The gapless behavior has been
experimentally suggested by recent thermal conductivity
(for P ≤ 10 bars) [28] and heat capacity (for 11 ≤ P ≤ 29
bars) [33] measurements. The pressure dependence of
α0/αc is in qualitative agreement with the combined re-
sults of Fisher et al. and Choi et al. Although all of
these experimental techniques (including ours) are lim-
ited to probe the impurity states near the Fermi level,
the behavior is consistent with the theoretical predictions
with gapless excitations. Unlike the thermodynamic and
transport measurements, the high frequency ultrasound
measurement has a potential to unveil a larger portion
of the impurity states profile from the frequency depen-
dence.
We acknowledge support from an Alfred P. Sloan Re-
search Fellowship (YL), NSF grants DMR-0239483 (YL),
DMR-0305371 (MWM), and a Grant-in-Aid for Scientific
Research on Priority Areas (No. 17071009) from MEXT
of Japan (SH and KN). We would like to thank J.-H.
Park for his technical assistance, and Jim Sauls, Peter
Wölfle, and Bill Halperin for useful discussions.
∗ yoonslee@phys.ufl.edu
[1] D. Vollhardt and P. Wölfle, The Superfluid Phases of
Helium Three, (Taylor and Francis, London, 1990).
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[18] H.C. Choi, Ph.D. thesis, University of Florida (2007).
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[24] For example, c = 350 (± 10) m/s at 34 bars from our
measurement, and cf = 370 m/s.
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[29] J.A. Sauls et al., Phys. Rev. B 72, 024507 (2005).
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mailto:yoonslee@phys.ufl.edu

  We have performed longitudinal ultrasound (9.5 MHz) attenuation measurements
in the B-phase of superfluid $^3$He in 98% porosity aerogel down to the zero
temperature limit for a wide range of pressures at zero magnetic field. The
absolute attenuation was determined by direct transmission of sound pulses.
Compared to the bulk fluid, our results revealed a drastically different
behavior in attenuation, which is consistent with theoretical accounts with
gapless excitations and a collision drag effect.

Ultrasound Attenuation of Superfluid 3He in Aerogel
H.C. Choi, N. Masuhara, B.H. Moon, P. Bhupathi, M.W. Meisel, and Y. Lee∗
Microkelvin Laboratory, Department of Physics, University of Florida, Gainesville, FL 32611-8440, USA
N. Mulders
Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA
S. Higashitani, M. Miura, and K. Nagai
Faculty of IAS, Hiroshima University, Kagamiyama 1-7-1, Higashi-Hiroshima 739-8521, Japan
(Dated: October 27, 2018)
We have performed longitudinal ultrasound (9.5 MHz) attenuation measurements in the B-phase
of superfluid 3He in 98% porosity aerogel down to the zero temperature limit for a wide range of
pressures at zero magnetic field. The absolute attenuation was determined by direct transmission
of sound pulses. Compared to the bulk fluid, our results revealed a drastically different behavior in
attenuation, which is consistent with theoretical accounts with gapless excitations and a collision
drag effect.
Liquid 3He has attracted intense interest for many
decades in the field of low temperature physics [1]. In
its normal state, liquid 3He has served as a paradigm
for a Fermi liquid whose nature transcends 3He physics.
The superfluid phases of 3He exhibit exotic and intrigu-
ing features associated with the broken symmetries in the
condensate, having an unconventional structure of the or-
der parameter with spin triplet p-wave pairing. Liquid
3He is arguably the most well-understood system mainly
because of its extreme intrinsic pureness at low temper-
atures. Therefore, it has provided important insights
in understanding other unconventional superconductors
such as the high temperature superconductors, the heavy
fermion superconductors, and in particular the more re-
cently discovered Sr2RuO4, which is also thought to have
the p-wave symmetry [2]. However, the same virtue has
hampered the effort in pursuing answers to an important
overarching question: how does the nature of a quantum
condensate (spin triplet p-wave superfluid in this case)
respond to increasing impurity or disorder?
Observation of superfluid transitions in liquid 3He im-
pregnated in high porosity aerogel in 1995 [3, 4] opened
a novel path to introducing static disorder in liquid 3He.
Aerogel possesses a unique structure, whose topology is
at the antipode of widely studied porous media such as
Vycor glass and metallic sinters. Due to its open struc-
ture, there are no well-defined pores in aerogel and conse-
quently, the liquid is in the proximity to the bulk. Ninety
eight percent porosity aerogel, which has been used in
most of the studies including this work, offers a corre-
lated network of strand-like aggregates of SiO2 molecules
whose structure can be characterized by the geometrical
mean free path (ℓ ≃ 100 - 200 nm), the diameter of strand
(r ≈ 3 nm), and the average inter-strand distance (d ≃ 25
- 40 nm). The coherence length of pure superfluid 3He,
ξ0, which varies from 20 nm (34 bar) to 80 nm (0 bar),
is at least an order of magnitude larger than the strand
diameter but is comparable to ℓ and d. As a result, the
scattering off the aerogel strand would have a significant
influence on the superfluid. It is now well established
that the superfluid transition temperature is significantly
depressed from that of the bulk, and the effect of pair-
breaking is progressively magnified at lower pressures,
leading to the possibility of a quantum phase transition
at Pc ≈ 6 bars [5]. To date, three distinct superfluid
phases have been experimentally identified, namely the
A-like, B-like, and A1-like phases [4, 6, 7, 8, 9]. The B-like
phase and the A1-like phase in aerogel show striking sim-
ilarity to their counterparts in the bulk superfluid [9, 10].
Detailed NMR studies [7, 8, 10] suggest that the aerogel
B-phase has the same order parameter structure as the
bulk B-phase. The aerogel A1-phase only appears in the
presence of magnetic field as is the case in the bulk [9].
However, the aerogel A-phase exhibits quite a different
behavior from the bulk A-phase (e.g. in NMR frequency
shift and superfluid density), although the overwhelming
experimental evidence suggests that it is an equal spin
pairing state. Various interpretations or novel proposi-
tions on the possible order parameter structure have been
suggested for this phase [11, 12, 13].
Nuclear magnetic resonance and ultrasound spec-
troscopy have been used in concert to investigate the mi-
croscopic structure of the superfluid phases [1, 14]. These
two experimental methods encompass complementary in-
formation on the orbital (ultrasound) and spin (NMR)
structure of the Cooper pairs. Rich spectra of order
parameter collective modes in bulk superfluids, which
are the fingerprints of specific broken symmetries in the
system, have been mapped by ultrasound spectroscopic
techniques [14]. In 2000, Nomura et al. [15] performed ul-
trasound attenuation measurements on 98% aerogel us-
ing a 16.5 MHz cw acoustic impedance technique. Their
work was limited to a single pressure at 16 bars and down
to 0.6 mK. Although their technique was not adequate in
determining absolute attenuation, they managed to ex-
tract the absolute sound attenuation after making auxil-
http://arxiv.org/abs/0704.0419v1
Time (µs)
FIG. 1: Acoustic response from the receiver vs. time at 34
bars for select temperatures ranging from 0.3 mK to 2.5 mK.
The aerogel superfluid transition is marked by a small arrow.
iary assumptions. A Bayreuth group [16] performed ab-
solute sound attenuation measurements in aerogel (97%
porosity) using a direct sound transmission technique at
10 MHz. They experienced poor transducer response,
and observed self-heating and no depression in the aero-
gel superfluid transition. We conducted high frequency
sound transmission experiments in 98% porosity aero-
gel, covering the whole phase diagram of the superfluid
phases in aerogel, from 8 to 34 bars and from the transi-
tion temperatures to as low as 200 µK.
In this experiment, two matched LiNbO3 longitudi-
nal sound transducers with the fundamental resonance at
9.5 MHz were used as a transmitter and a receiver. The
6.3 mm diameter transducers were separated by a Ma-
cor spacer maintaining a 3.05 (± 0.02) mm sound path
between the transducers where the aerogel sample was
grown in situ. This scheme ensures the best contact be-
tween the transducer surface and the aerogel, which is
crucial for clean sound transmission at the boundaries.
A 1 µs pulse was generated by the transmitter and de-
tected by the receiver. Temperature was determined by
a melting pressure thermometer (MPT) for T ≥ 1 mK
and a Pt NMR thermometer for T ≤ 1 mK which was
calibrated against the MPT. No non-linear response or
self-heating was observed at the excitation level used in
this work. All the data presented here, except for 8 bars,
were taken while warming with a typical warming rate
of 3 µK/min. A detailed description on the experimen-
tal cell and experimental techniques can be found else-
where [17, 18].
The temporal responses of the receiver taken at 34 bars
are shown in Fig. 1 for select temperatures ranging from
0.3 to 2.5 mK. The primary response, which starts to rise
around 8 µs, shows a rather broad response due to ring-
ing of the high Q transducer (Q ∼ 103). The step-like
structure of the receiver signal is caused by the slight mis-
match in the spectra of the transducers [18]. Below the
aerogel superfluid transition (marked around 2.1 mK by
an arrow in Fig. 1) the primary response starts to grow
and the trailing echoes emerge from the background, as
the sound attenuation decreases in the superfluid. No
change in the receiver signal was observed at the bulk
superfluid transition. The multiple echoes follow a bona
fide exponential decay in time. Absolute sound attenua-
tion was obtained in the following manner [19]. First, the
relative attenuation at each temperature was calculated
using the area under the primary response curve by inte-
grating the signal from the rising edge to a fixed point in
time (23 µs point). The absolute attenuation at 0.4 mK
and 29 bars, obtained using the primary signal and the
echoes, was used as a reference point in converting the
relative attenuation into the absolute attenuation. Due
to a drastic mismatch in the acoustic impedance at the
the transducer-aerogel/3He boundary, the signal absorb-
tion at the surface of transducers was ignored [19]. The
possible background contributions to attenuation from
the quasi-particle scattering off the cavity wall [20] and
the non-parallel alignment of the two transducers are es-
timated to be negligible.
The absolute attenuations on warming for several pres-
sures are plotted as a function of temperature in Fig. 2(a).
The superfluid transition is marked by the smooth drop
in attenuation. Our aerogel superfluid transition tem-
peratures are in excellent agreement with the previously
reported values for all pressures [5, 21]. At 9.5 MHz in
the bulk B-phase, a strong attenuation peak appears right
below the superfluid transition. This peak is the result of
the combined contributions from pair-breaking and cou-
pling to the order parameter collective modes. Above the
polycritical pressure, the B to A transition on warming
is registered as a sharp step in attenuation. In aero-
gel, none of these features exist. However, we did ob-
serve a sharp step in attenuation on cooling for P > 14
bars, which implies the existence of the supercooled A-
phase [19]. We were able to identify a rather smooth B
to A transition on warming for 29 and 34 bars within ≈
150 µK below the superfluid transition. This observation
is consistent with the previous results obtained using a
transverse acoustic impedance technique [13]. Therefore,
most of the attenuation data presented here are in the
aerogel B-phase. In the bulk B-phase with a clean gap,
the attenuation follows α ∝ e−∆(T )/kBT below the atten-
uation peak, practically reaching zero attenuation below
T/Tc ≈ 0.6, due to thermally activated quasi-particles,
where ∆(T ) is the temperature dependent gap and kB
is the Boltzmann constant. In contrast, the attenuation
in aerogel decreases rather slowly with temperature and
remains high even at T/Tc ≈ 0.2. Furthermore, a pe-
culiar shoulder feature appears at T/Tc ≈ 0.6 for higher
pressures. This feature weakens gradually and eventually
disappears at lower pressures, Fig. 2(a).
Sound propagation for higher harmonics up to 96 MHz
was measured for several temperatures and pressures,
but no evidence of sound propagation was found above
30 MHz even at 0.3 mK, where the lowest attenuation
is expected. Below about 10 mK, the scattering process
is dominated by the temperature independent impurity
scattering off the aerogel, and at 9.5 MHz, ωτi ∼ 0.1 for
all pressures where τi = ℓ/vf (see below for ℓ). There-
fore, the sound mode should remain in the hydrodynamic
limit. This claim is bolstered by the observation of the
strong frequency dependence in attenuation and the ab-
sence of a temperature dependence in the normal fluid
attenuation [15]. The coupling between the normal com-
ponent of the superfluid 3He and the mass of the elas-
tic aerogel modifies the conventional two-fluid hydrody-
namic equations [22, 23]. This consideration leads to two
(slow and fast) longitudinal sound modes with different
sound speeds, cs = ca
ρsρa/ρ, and cf = c1
1+ρaρs/ρnρ
1+ρa/ρn
Here, cf(s) represents the speed of the fast (slow) mode,
ρn(s) is the normal fluid (superfluid) density (ρ = ρn+ρs),
ρa is the aerogel density, c1 is the speed of hydrodynamic
sound in 3He, and finally ca is the sound speed of the bare
aerogel. From the time of flight measurements, we found
the sound speed in aerogel consistently lower (by ≈ 20%)
than c1 for all pressures studied and in good agreement
with the values obtained using the expression above [24].
Detailed analysis of sound velocity for various pressures
will be presented in a separate publication.
Low mass density and the compliant nature of aero-
gel necessitate the consideration of effective momentum
transfer upon quasi-particle scattering off the aerogel,
which generates dragged motion of aerogel. Ichikawa et
al. [25] incorporated the collision drag effect in calculat-
ing the dispersion relation in the normal fluid. Their
model offered a successful explanation for the experimen-
tal results of the Northwestern group [15]. Recently, Hi-
gashitani et al. [26, 27] extended this model to study the
longitudinal sound (fast mode) propagation in superfluid
3He/aerogel within the framework of the two-fluid model.
The drag effect can be described phenomenologically by
a frictional force, ~Fd =
(~vn −~va), introducing an addi-
tional relaxation time τf , where ~vn(a) is the normal fluid
component (aerogel) velocity. This effect is of particu-
lar importance when ωτi < 1, and the total attenuation
(Eq. (130) of ref. [27]) is
ω2/2cf
1 + ρaρs/ρnρ
ρ2aτf/ρρn
1 + ρa/ρn
4η/3ρc21
1 + ρaρs/ρnρ
), (1)
where η is the shear viscosity of liquid 3He. The first
term (αf ) arises from the frictional damping caused by
the aerogel motion relative to the normal fluid compo-
nent, and the second term (αv) from the conventional
hydrodynamic sound damping associated with the viscos-
ity. This expression allows us to extract ℓ in this system
from our absolute attenuation at the transition temper-
ature, αc. The inset of Fig. 3 shows our results of αc for
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0
 10 bar
 14 bar
 20 bar
 25 bar
 33 bar
 34 bar
T / T
 10 bar
 12 bar
 14 bar
 21 bar
 25 bar
 34 bar
T (mK)
FIG. 2: (a) Absolute attenuation for various pressures vs.
temperature (color in on-line version). Thin solid lines are
the results of a quadratic fit to the low temperature part
(T/Tc <∼ 0.4) of the data at each pressure. (b) Normalized
sound attenuation vs. normalized temperature. The results
of theoretical calculation (solid lines, color in on-line version)
are plotted along with the experimental results at 34 bars for
comparison.
various pressures. The solid lines are the result of calcu-
lation using Eq. (1) for three different mean free paths,
ℓ = 100, 120, and 140 nm. As can be seen, ℓ = 120 nm
produces an excellent fit to our data for the whole pres-
sure range, which is in good agreement with the val-
ues obtained from the thermal conductivity (90 nm) [28]
and spin diffusion (130 nm) [29] measurements. With
the knowledge of the mean free path, one can calculate
the full temperature dependence of sound attenuation in
the superfluid phase. The results of the calculation (in
the unitary limit) following the prescription described in
ref. [27] are displayed in Fig. 2(b) along with the experi-
mental results at 34 bars. The calculation reproduces all
the important features observed in our measurements. In
particular, the conspicuous shoulder structure appearing
near T/Tc ≈ 0.6 at 33 bars softens at lower pressures and
is completely absorbed in an almost linear temperature
dependence below 20 bars. This behavior is the charac-
teristic of αf [27]. A fast decrease in ρn right below Tc
produces the bump in αf , and αf → 0 as T → 0. On
the other hand, αv decreases monotonically and reaches
a finite value due to non-zero ρn and the impurity states
0 5 10 15 20 25 30 35
0 10 20 30
P (bar)
 100 nm
 120 nm
 140 nm
P (bar)
FIG. 3: Normalized zero temperature attenuation vs. pres-
sure. The dashed line is a guide for eye. Inset: Pressure
dependence of sound attenuation at Tc. The solid lines (color
on-line) are the results of theoretical fit for ℓ = 100, 120, and
140 nm (see text).
induced inside the gap as T → 0. The quantitative agree-
ment between the theory and experiment, however, is not
yet satisfactory. The calculation utilizes the isotropic ho-
mogeneous scattering model (IHSM) [30], which tends to
overestimate ∆(T ) and ρs compared to the experimen-
tally determined values [3, 23]. As shown in ref. [31],
the inhomogeneity gives rise to the reduction of the aver-
age value of the order parameter and consequently yields
larger η and ρn, which in turn increases α0 but decreases
the frictional contribution. It is also expected that the
non s-wave scattering components make non-trivial con-
tributions to the viscous and frictional relaxation times
in a direction that improves the quantitative agreement.
Theoretical calculations based on the IHSM [27, 32]
predict that the impurity states would completely fill the
gap, leading to a gapless superfluid when τiTc < 1 for the
B-phase in the unitary limit. We estimate 0.3 < τiTc < 1
for 10 < P < 34 bars with ℓ = 120 nm. The normalized
zero temperature attenuation (α0/αc) obtained by ex-
trapolating the low temperature part of the attenuation
(solid lines in Fig. 2(a)) is plotted in Fig. 3, where α0/αc
increases as the sample pressure is reduced and seems
to approach unity near Pc ≈ 6 bars. Since the viscosity
ratio is directly related to the density of states at zero
energy through η(0)/η(Tc) = n(0)
z, z = {2,4} for the
{Born, unitary} limit where n(0) is the normalized den-
sity of states at zero energy [27], the finite α0/αc is strong
evidence of a finite n(0). The gapless behavior has been
experimentally suggested by recent thermal conductivity
(for P ≤ 10 bars) [28] and heat capacity (for 11 ≤ P ≤ 29
bars) [33] measurements. The pressure dependence of
α0/αc is in qualitative agreement with the combined re-
sults of Fisher et al. and Choi et al. Although all of
these experimental techniques (including ours) are lim-
ited to probe the impurity states near the Fermi level,
the behavior is consistent with the theoretical predictions
with gapless excitations. Unlike the thermodynamic and
transport measurements, the high frequency ultrasound
measurement has a potential to unveil a larger portion
of the impurity states profile from the frequency depen-
dence.
We acknowledge support from an Alfred P. Sloan Re-
search Fellowship (YL), NSF grants DMR-0239483 (YL),
DMR-0305371 (MWM), and a Grant-in-Aid for Scientific
Research on Priority Areas (No. 17071009) from MEXT
of Japan (SH and KN). We would like to thank J.-H.
Park for his technical assistance, and Jim Sauls, Peter
Wölfle, and Bill Halperin for useful discussions.
∗ yoonslee@phys.ufl.edu
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mailto:yoonslee@phys.ufl.edu
