Current–phase relationship measurements in the flow of ... - Physics

Physica B 280 (2000) 130}131
Current}phase relationship measurements in the #ow
of super#uid He through a single ori"ce
Olivier Avenel*, Yury Mukharsky, ED ric Varoquaux
CEA-DRECAM, Service de Physique de l+E! tat Condense& , Centre d+E! tudes de Saclay, 91191 Gif-sur-Yvette Cedex, France
CNRS-Laboratoire de Physique des Solides, BaL t. 510, Universite& Paris-Sud, 91405 Orsay, France
Abstract
We report accurate measurements of the current-phase relationship in the #ow of super#uid He through a single
ori"ce. While ideal sinusoidal 2π-periodic Josephson behavior prevails close to ¹
, increasingly strong π-periodic
admixture is usually observed at lower temperatures. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Current}phase relationship; He super#uid; Josephson e!ect
The miniature Helmholtz resonator [1] that we used
earlier to demonstrate Josephson e!ects in super-
#uid He [2] and to measure the Earth's rotation in
super#uid He [3] is now being used to make accurate
measurements of the mass current J(θ) of super#uid
He-B through a single ori"ce as a function of the phase
di!erence θ across it. In these new experiments the ori"ce
is a 0.182.6 μm slit in a 0.1 μm thick silicon nitride
membrane and the parallel path is a 5.9 cm rotation
pick-up loop.
The slope of the current}phase relationship is obtained
by recording the small amplitude free oscillation resonance
frequency ω,
dJ/dθ"ω/ω
!1. (1)
ω
, the Helmholtz frequency of the parallel path alone, is
measured as the large signal resonance frequency.
This device is a very sensitive rotation sensor [4], and
coupling to the Earth's rotation is used to vary the phase
bias θ across the ori"ce. In the laboratory frame, the
rotation of the Earth induces a circulation κ
"2 Ω
) A
in the pick-up loop of oriented area A. The corresponding
phase di!erence along the loop is θ
"2πκ
/κ
, κ
being the quantum of circulation. The equilibrium phase
* Corresponding author.
E-mail address: avenel@drecam.saclay.cea.fr (O. Avenel)
di!erence θ across the ori"ce satis"es the load line
equation
J(θ)"!(θ!θ
), (2)
which expresses the fact that the current through the
ori"ce balances the current through the parallel path
when the membrane is at rest. The curves J(θ) are obtained
by integration of Eqs. (1) and (2) using resonance
frequency measurements as a function of cryostat
orientation.
Di!erent shapes are usually observed at the same temperature
in each cool-down through the transition temperature
¹
, as shown in Fig. 1a.
But J(θ) is usually robustly "xed in each run and
the general trend is to go from 2π periodicity to π
periodicity when the temperature is lowered, Fig. 1b,
although on one occasion no π periodicity was observed
at the lowest ¹. The curves in Fig. 1b are probably not
hysteretic, dissipative phase slips cannot be resolved but,
due to rotational noise, the small signal resonance is
di$cult to track when J(θ) is nearly parallel to the load
line.
Hysteretic behavior prevails at higher pressure, as represented
in Fig. 1c. At low temperatures up to three
well-de"ned resonances can be tracked at some cryostat
orientations, and phase slips take place. Switching from
one branch to another is done by applying pulses to the
membrane.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 5 1 7 - 3

O. Avenel et al. / Physica B 280 (2000) 130}131 131
existence of π/2 states in such an array when J(θ) has
π periodicity and is hysteretic [6].
We conjecture that the di!erent J(θ) obtained in successive
cool-downs come from di!erent textural states,
established when cooling through ¹
. Indeed, two di!erent
current-phase relationships have been predicted in
He-B for parallel and anti-parallel n vectors on each
side of the ori"ce [7]. No calculation has been done for
arbitrary orientations which may well be met in the
present resonator with submillimeter size cylindrical volumes
on each side. Our earlier measurements [2] did not
reveal departures from 2π periodicity, at least down to
0.7¹
. The #exible membrane used then was coated with
Al through which the magnetic "eld of the displacement
sensor could leak [1]. The present coating with Pb is less
permeable to "eld. Also, the supporting material of the
ori"ce has been changed from ferromagnetic Ni to inert
Si
N
. No orientation of n is really favored in the present
set-up, which probably explains the di!erent determinations
of J(θ) seen in Fig. 1.
Note added in proof
A recent calculation indeed predicts di!erent current}
phase relationships for di!erent orientations of the n vector
on each side of a single pinhole [8]. See also Refs.
[9,10].
References
Fig. 1. Current}phase relationships. (a) 0.2 bar, 4 cool-downs at
&0.845¹
. (b) 0.2 bar, at 0.889, 0.837, 0.677 and 0.469¹
;
vertical scale multiplied by 2 for the two dotted lines at low ¹.
(c) 10 bars, same run at 0.968, 0.945, 0.905 and 0.756¹
. The
straight lines are the load lines, Eq. (2), shifted horizontally by
reorienting the cryostat.
A positive slope of J(θ) around π can explain the
observation of π states in an array of holes [5], even
in non-hysteretic regimes. We would then predict the
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(Eds.), Proceedings of the ICEC11, Berlin, Germany, Butterworths,
1986, p. 587.
[2] O. Avenel, E. Varoquaux, Jpn. J. Appl. Phys. (Suppl.) 26 (3)
(1987) 1798.
[3] O. Avenel, E. Varoquaux, Czech. J. Phys. 46 (S6) (1996)
3319.
[4] Y. Mukharsky, O. Avenel, E. Varoquaux, Physica B, these
proceedings.
[5] S. Backhaus et al., Nature 392 (1998) 687.
[6] O. Avenel et al., Nature 397 (1999) 484 and 662.
[7] E.V. Thuneberg, Europhys. Lett. 7 (1988) 441.
[8] S.-K. Yip, Phys. Rev. Lett. 83 (1999) 3864.
[9] J.K. Viljas, E.V. Thuneberg, Phys. Rev. Lett. 83 (1999)
3868.
[10] A. Marchenkov et al., Phys. Rev. Lett. 83 (1999) 3860.