Phase slippage in superfluid 3He-B

where Jc is the critical mass flow through the
orifice and f2~ is an odd function bounded by 1 ~Permanent address: Physics Department, University of Florida, Gainesville, FL 32611, USA. 2 Present address: Low Temperature Laboratory, Helsinki University of Technology, SF 02150 Espoo, Finland.
and of period 2-rr. In the ideal case treated by Josephson [2] of an infinitely thin tunnel barrier separating two bulk superconductors, J ~ is a sine function. Equations (1) and (2) are known to hold quite generally in superconductors. Their validity for superfluids has been established with the help of a low frequency hydromechanical resonator fitted with a submicronic aperture [3, 4]. This aperture acts as a weak link connecting the inner c h a m b e r of the resonator to an outside superfluid bath. Phase slippage takes place in this weak link, changing the phase difference between the inside and the outside by multiples of 2v. Similar experiments supporting in essence the same findings are now being repeated in other laboratories [5, 6]. In superfluid 4He, the c u r r e n t - p h a s e relation is highly degenerate and consists of straight slanting segments. Phase slips are hysteretic and cause an energy loss proportional to the critical mass flow [3] and the quantum of hydrodynamical circulation, AE = K4Jc . (3)
(4)
In the next scclion, we tr~ to go beyond this rather formal and restricted interpretation and probe its physical content in a little more depth, Wc shall reach the conclusion that morc detailed experimental investigations of the current-phasL relation in the anisotropic superfluid arc needed both to ascertain the limits of cq. (4) and to check the much more sophisticated theoretical results that have appeared recently in the literature [11-19 I. The last section of this paper is devotcd t~ brief descriptions of other cxperiments, besides the m e a s u r e m e n t of J(6,#), which can bc performed with the two-holef improvements to the cxpcrimcntal set-up which we have achieved to carry oul these cxperimcms,
Although 2~r phase slips are quite accurately resolved and eq. (3) verified, the detailed mechanism of the slips, which almost certainly involves the motion of quantized vortex filaments across the orifice, is a matter of speculation [7].
0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
310
F. Varoquau.~ ¢'t al.
l'haxe ,Ylipl~a:,,c in ~'ulwr/hdd 'lh'-H
In superfluid : H e , the coherence length ~, is of the order of 600 A at T - 0 and low pressure. In contrast with the case of 4He, such a length is no hmgcr very small compared to the smallest dimensions of the micro-orificc. A nearly smusoidal c u r r e n t - p h a s e relation is observed ncar 7 ' where ,~.(T) becomes comparable to the microorifice size [4]. Hysteresis sets in as the temperaturc is lowered and more complicated patterns of behavior develop [8]. Examples of such behaviors arc given in the next section. In somc cases, thc observed c u r r e n t - p h a s e relation is well accounted for by a model associating an ideal Joscphson junction to a series inductance. Such a model, frst proposed for superconducting microbridges by Deavcr and Pierce [9, 10], inw)lvcs a single p a r a m e t e r e~ to dcscribe the non-ideal relation between J and 8¢: J--J sin g , ~,# g+c~sin g.
at -~'
(1)
where p~ is the chemical potential (of an atom in the case of 4He, and of a pair of atoms in that of
3He). Along with the recognition of the importance of the phase came the expectation that the mass current through a small orifice would be related to the phase difference along the orifice by some form of D C Josephson relation, S = Scf2~(aq~ ) , (2)
Physica B 178 (1992) 309-317 North-Holland
PHYSICA
,',
Phase slippage in superfluid 3He-B
E. V a r o q u a u x ~', O. A v e n e l b, G . I h a s b'l a n d R. S a l m e l i n b'2
2. A simple phase slip model
The parametrized form of the c u r r e n t - p h a s e relation given by (4) and shown in fig. 1 may be viewed [4, 8] as nothing but a convenient interpolation between a highly degenerate, hysteretic situation similar to the behavior in 4He, in the limit ~--+ :c and the ideal Josephson case, in the limit a--~0. In this restricted situation, it provides a smooth description of the cross-ovcr between the pure quantum case ( ~ x - 0 ) and a near-classical mechanical limit (c~---, :~).
We review some applications of the hydrodynamic Josephson effects. The relationship between the quantum mechanical phase difference along a micro-orificeand the flow through it is discussed in terms of a simple mode[ which accounts for the observations performed in 3He-B above 0.7T. Possible uses of the superfluid hydromechanical resonator as a sensitive and stable absolute gyromeler are described.
~'Ltlboratoire de Physique des Solides, UniversitO Paris-Sud, 91405 Orsay, France "Service de Physique de l'Etat Condense!, Centre d'Etudes NuclOaires de Saclay, 91191 GiJ~ France
1. Josephson-type phenomena in superfluids
This p a p e r gives a brief review of pending problems which can be addressed with the help of hydromechanical devices in which quantum mechanical phase slippage occurs. As pointed out by Anderson in 1965 [1], the phase of the wave function of a superfluid is a macroscopic variable whose rate of change is governed by the A C Josephson relation [2],