AVENEL 0. CEA Centre D'etudes De Saclay, 91 - Gif-Sur-Yvette (FR)
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COMMISSARIAT A L'ENERGIE ATOMIQUE CENTRE D'ETUDES DE SACLAY CEA-CONF-10564 MIST ocrvicc des Bases de Données Spécialisées F91191 GIF SUR YVETTE CEDEX PHASE SLIPPAGE IN SUPERFLUID 3HE-B AVENEL 0. CEA Centre d'Etudes de Saclay, 91 - Gif-sur-Yvette (FR). Direction des Sciences de la Matière IHAS G. CCA Centre d'Etudes de Saclay, 91 - Gif-sur-Yvette (FR). Direction des Sciences de la Matière/ Florida Univ, Gainesville, FL (US). Dept. of Physics SALMELIN R. CEA Centre d'Etudes de Saclay, 91 - Gif-sur-Yvette (FR). Direction des Sciences de la Matière; Helsinki Univ. of Technology, Espoo (FI) VAROQUAUX E. Paris-11 Univ., 91 - Orsay (FR) 3 Communication présentée à : Symposium on Superfluid He in Rotation Espoc o tt-r(FI \) 10-H Jun 1991 PHASE SLIPPAGE in SUPERFLUID 3HE-B E. Varoquaux Laboratoire de Physique des Solides, Université Paris-Sud, 91405 — Orsay — France 0. Avenel G. Ihas f R. Salmelinj: Service de Physique de l'Etat Condensé Centre d'Etudes Nucléaires de Saclay, 91191 - Gif - France f Permanent Address: Physics Department, University cf Florida, Gainesville - FL 32611 - USA J Present address: Low Temperature Laboratory, Helsinki University of Technology - SF - 02150 Espoo - Finland Abstract We review some applications of the hydrodynamic Josephson effects. The relationship between the quantum mechanical phase difference along a micro-orifice and the flow through it is discussed in terms of a simple model which accounts for the observations performed in 3He-B above 0.7 Tc. Possible uses of the superfluid hydromechanics! resonator as a sensitive and stable absolute gyrometer are described. F. Kôrber Symposium on Superfluid 3He in Rotation, I3SPOO, Espagne SPI-C/91-052 June 10-14, 1991 (to be published to Physica B) I. Josephson-type phenomena in superfluids This paper 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 wavefunction of a superfluid is a macroscopic variable wnose rate of change is governed by the ac Josephson relation [2]: where ft 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 dc Josephson relation: 3 = Jc /(^) , (2) where Jc is the critical mass flow through the orifice and /« is an odd function bounded by 1 and of period 2r. In the ideal case treated by Josephson [2] of an infinitely thin tunnel barrier separating two bulk superconductors, /« is a sine function. Equations (1) and (2) are known to hold quite generally in super- conductors. Their validity for superfluids has been established with the help of a low frequency hydromechanical resonator outfitted with a submicronic aperture [3][4]. This aperture acts as a weak link connecting the inner chamber 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 2r. Similar experiments supporting in essence the same findings are now being repeated in other laboratories [5][6l. In superfluid! 4He, the current-phase relation is highly degenerate and consists in 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: A£ = «4 Jc . (3) Although 2x 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]. In superfluid 3He, the coherence length £o is of the order of 600 A at 7=0 and low pressure. In contrast with the case of *He, such a length is no longer very small compared to the smallest dimensions of the micro-orifice. A nearly sinusoïdal current-phase relation is observed near Tc where £o(T) becomes comparable to the micro-orifice size [4]. Hysteresis sets in as the temperature is lowered and more complicated patterns of behavior develop [8]. Examples of such behaviors are given in the next section. In some cases, the observed current-phase relation is well accounted for by a model associating an ideal Josephson junction to a series inductance. Such a model, first proposed for superconducting microbridges by Deaver and Pierce [9-10], involves a single parameter a to describe the non-ideal relation between J and 6<p: J = Jc sin £ , (4) 6<p = £ + a sin £ . The parametrized form of the current-phase relation given by (4) and shown in fig.l may be viewed [4, 8] as nothing but a convenient interpolation between a highly degenerate, hysteretic situation similar to the behavior in *He, in the limit a —» CD, and the ideal Josephson case, in the limit a —» 0. In this restricted acceptation, it provides a smooth description of the cross-over between the pure quantum case (o=0) and a near-classical mechanical limit (a —» <u). In the next section, we try to go beyond this rather formal and restricted interpration and probe its physical content in a little more depth. We shall reach the conclusion that more detailed experimental investigations of the current-phase relation in the anisotropic superfluid are needed both to ascertain the limits of eq.(4) and to check the much more sophisticated theoretical results that have appeared recently in the literature [11-19]. The last section of this paper is devoted to brief descriptions of other experiments, besides the measurement of J(ti<p), which can be performed with the two-hole hydromechanical resonator and of improvements to the experimental set-up which we have achieved to carry out these experiments. II. A simple phase slip model A number of calculations of the current-phase relation for superfluid flow in a micro-orifice have now been performed, starting with the work of Monien and Tewordt [12]. These calculations fall into two broad categories, those aiming at an analytical description of the phase slippage phenomenon [13-16] and those providing realistic numerical simulations of experimental current versus phase curves [11, 12, 17-19]. Let us use as a starting point for a very much simplified approach the Gross-Pitaevskii-Ginzburg-Landau formalism for an s-wave superfluid as applied to the 2-dimensional flow constriction depicted in Fig.2. We assume that the presence of the orifice constricting the flow somehow depletes the wavefunction amplitude / in a significant way. That is, a potential energy barrier builds up over a localized region of the z-axis, of length Lb, from which the Cooper pairs are effectively expelled. On each side of this depletion region, the bulk wavefunction corresponding to a local -srfluid velocity vs prevails: tfbuik = /o exp{-i£t/K + ik.r] , with k = 2/n3i/s/K = Vp. Inside the barrier, as the amplitude of the wavefunction is small, the cubic term in the Ginzburg-Landau equation can be dropped and replaced by the potential barrier of height U whose existence we have assumed. Thus, at this crude level of approximation, the wavefunction is governed inside the region of length l\, by a simple Schrôdinger equation, in which U > E. The solution for the one-dimensional case is well-known and consists in a sum of two exponentially damped plane waves travelling in opposite directions: fa) = /+ exp{-iârt + xftb} + f_ exp{-iat - x/£b} - The length £b, equal to ft/ [2flfs( #-&&;)]% is characteristic of the depleted region and is of the order of the coherence length £o or larger. Coefficients / , and /_ are determined by matching the wavefunction to the bulk plane wave solutions, eq.(5), at each boundary with a phase difference 8<pb. The usual textbook solution for the transmission of a particle through a barrier is recovered: The corresponding mass current through the weak link of area sw is: J = Iks* {i/flip - ip Eq.(5) for the current exhibits a sinusoïdal dependence in terms of the phase difference across the barrier 6y>b and an exponential decay in terms of the length of the barrier Zb provided Zb>£b- Similar results have been obtained by Rainer and Lee [14] using a quasiclassical description of the scattering of quasiparticles which has a much firmer theoretical basis and wider scope than the above rudimentary approach. A parallel can also be drawn with a result obtained by Kurkijàrvi [15] for a point-like orifice, also in the framework of quasiclassical theory. In this last case, the current- phase relation in the p-wave superfluid is identical to that of the a-wave superfluid, showing that the structure of the order parameter disappears when the dimensions of the orifice are small with respect to the coherence length. The results of refs.[14] and [15] can be used to express the barrier parameters £b and Zb in the idealized cases of a tunnelling process through a 3He-*He film [14] or a point-like orifice [15]. Several authors [11-13, 15] have obtained a skewed current-phase relation, which is a characteristic of the experimental data [4, 8]. At least part of this effect arises from a very simple cause. The full phase difference across the weak link, o<p, includes, besides the phase difference across the barrier <Vt>, the phase change S<pb in the two regions on each side of it where i/s, and hence 7y?, are sizable but behave strictly in accord with classical ideal fluid dynamics.