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 J = Jc sin £ , (4) 6 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 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 The total phase difference Sip is the sum of the phase drop through the 'hydraulic' region fyto and through the barrier 6 (8) Combining eqs.(7) and (8) yields eq.(4) with £ = 6ipb and with the following expression of the non-ideality parameter: <9> Both a and Jc can be obtained from the experimental staircase patterns, examples of which are given in figs. 3, 4 and 5. The fits to the experimental data are performed by modelling numerically the operation of the two-hole hydromechnical resonator [81. This modelization involves the ratio 1 of the currents in the long parallel channel, (Kpss\/2inzl{)S<(>, and in the weak link in the limit of vanishing current, Jc6 The fitting parameters a and Ï and the derived quantities Jc and lhSw/l*Sh obtained in the B-phase at zero pressure for various temperatures are summarized in Table 1. The superfluid densities have been taken from the work of Archie et al [20]. T/7c a 1 Ww*h Ps/P Je [10-n g/s] 0.69 11 6t 0.87 0.43 163 0.77 6 6 0.78 0.30 66 0.83 1 10.5 0.83 0.20 7.3 0.89 0.4 30 1.36 0.12 1.1 Table 1. f This value of 1 is assumed and does not result from a fit. The conclusions that can be drawn from Figs. 3-5 and Table 1 are the following. Firstly, the fits of the model to the 3He-B data at temperatures above 0.7 Tc are probably as good as can be with the presently achieved quality of the resonator data. They are likely to be insensitive to the detailed shape of the current-phase relation, as long as the transition from the hysteretic to the dispersive behavior is correctly described. Below 0.7 TÇ, more complicated staircase patterns develop which cannot be described with the help of eq.(4). These patterns are not even reproducible during a given run, indicating that textural effects are probably coming into play. Secondly, we note in Table 1 the gratifying result that Zh is of the order of Zw (assuming si ~ sw) as we anticipated, except close to Tc where the coherence length becomes of the order of the orifice size and brings in corrections to the simple picture. Also, the experimental uncertainty on a becomes large close to Tc> and this leads to large error bars on l^/lw. We also note in Table 1 the large change of /c with temperature which varies much more rapidly than ps: the barrier height, £b> is also greatly affected by temperature and reflects the fact that the coherence length £(7) is larger than the micro-orifice width at ~ 0.9 Tc and smaller at lower temperatures. We conclude from this analysis that, at least when £(T) is small with respect to lw, the flow at the mouths of the orifice is 'classical', i.e. is governed by the laws ideal fluid dynamics. Since the orifice is short, the 'classical' behavior prevails on most of the length of the streamlines and the Josephson-type behavior is fairly localized in space and resembles the point-like, or tunnel junction, ideal behavior. Thus, at present, our experimental observations in 3He-B — and to a lesser extent, in 3He-A [8] — at T > 0.7 TC and P ~ 0 bar, for our orifice, are compatible with the simplest possible model for the behavior of the weak link. This model only calls for an ideal Josephson junction in series with a hydraulic inductance [9]. This conclusion does not rule out that non-trivial effects can take place in the weak link that reflect peculiarities of the multicomponent order parameter behavior in constricted flow [11]. Further progress on the experimental side is needed to check the refined predictions of the numerical analysis of réf.[11]. This requires, among other things, a direct measurement of J(6 III. Further experimental challenges The closed superfluid path threading the two orifices of the resonator defines a loop in which hydrodynamic circulation, = 3j2'irSd, i.e. a random circulation proportional to ji/d . Thus, this experiment calls for an annular loop of small cross-section, both to magnify the effect and to disfavor the formation of vortices. In our present set-up, the parallel channel is at least an order of magnitude too large and the cooling rate is too slow by several orders of magnitude to really obtain quenched superfluid states. In most instances in 4He and 3He-B, the cell was indeed found in an inital state of low trapped circulation, but not always. In some cases, large circulations were found to be trapped in the initially unperturbed superfluid. Also, severe overdriving of the cell is known [4, 8] to produce persistent bias currents equivalent t'a a phase difference across the micro-orifice of any value between 0 and 2r In relation with the problem of the existence of rémanent vorticity, and also with Zurek's conjecture, we note that this bias phase difference is botih difficult to create, and, once created, is remarkably stable. The A-phase shows a prefered bias of TT and poses a different problem. These observations call for further studies of the trapping mechanism for vortices and rémanent circulation. It remains to be seen whether the situation in superfluids differs from what it has been found to be in superconductors by Cabrera et al. [29] who have never observed a fractional quantization state in a narrow superconducting annulus. The basic practical problem which needs to be overcome to meet these new experimental challenges is to still further reduce the influence of external vibration on the cell. Let us recall that the operation of the apparatus depends on the measurement of the position of an aluminized Kapton membrane [4, 8] detected via a (magnetic) flux transformer by a SQUID. This membrane is directly coupled to the superfluid flow channels so that its motion both detects and can change the flow. Therefore, random motions of this membrane due to vibrations both affect the superfluid flow and the resolution of the measurements. Random motions resulting in a rotation of the loop threading the two channels also induce a noise on the measurements. To minimize both these effects, the apparatus has recently been mounted on a 15 metric ton concrete table suspended on four air legs [30]. This table is in turn mounted on a 45 ton slab isolated from the laboratory building by polyurethane foam. The air supension has a critically damped double resonance response which is nearly flat up to about 1.2 Hz and provides a low-pass cut-off at higher frequencies for both horizontal and vertical perturbations. The foam suspension has a cut-off frequency of 15 Hz. The Dewar is placed in such a way that the location of the cell nearly coïncides with the center of gravity of the mount. There are no vibrations detectable on this table from 1 Hz to 250 Hz greater than the noise level (0.5 ng) of the accelerometer [31] used for the diagnosis. Although all plumbing and electrical connections are brought to the cryostat through multiple, massive concrete blocks and lead anchors, with suitable flexible couplings in between, sizable acceleration levels could be detected on the cryostat head. These are interpreted to be associated with 8 the metal parts of the cryostat acting as sound transducers to noises in the laboratory. The worst vibrations at ~ 50 and 100 Hz would correspond to motions of the cell membrane of ~ 6 A if coupled directly to it. To alleviate direct sound transmission, a sound proof enclosure is bang constructed for the cryostat head. This enclosure will also provide additional electrical shielding, although in the present arrangement electrical noise is not a problem. The best quality data has so far only been obtained at "off hours" when the influence of the building noise is minimal. Analysis of the resonator noise shows that the mean perturbations are smaller than 2 parts in a thousand (~ 0.2 À in amplitude) and can be decomposed between a "common mode" and a "differential" component. Most of the noise (~ 90 %) is in the former channel and can be digitally filtered out with no loss of accuracy, leaving an overall signal-to-noise ratio of a few parts in ten thousands. Further improvement will be possible by data averaging, opening the way for very detailed studies of the behavior of superfluid weak links. IV. Acknowledgements This work has benefited from discussions with a large number of people. We express particular gratitude to Wayne Saslow and William Zimmermann for very useful suggestions. References P.W. Anderson, Rev. Mod. Phys. 38, 298 (1966). B.D. Josepson, Phys. Lett. 1, 251 (1962). 3] O. Avenel, E. Varoquaux, Phys. Rev. Lett. 55, 2704 (1985). 4] O. Avenel, E. Varoquaux, Jpn. J. Appl. Phys. 26-3, 1798 (1987), Phys. Rev. Lett. 60, 416 (1988). [5] W. Zimmermann (private communication) has reported the observation of a staircase pattern at the Washington APS meeting in April 1987, Bull. Am. Phys. Soc. 32, 1104 (1987). [6] J. Davis, A. Amar, R. Packard, Poster at QFS89, Gainesville, PL (1989), A. Amar, J.C. Davis, R.E. Packard, R.L. Lozes, Physiaca B 165&166, 753 (1990). [7] E. Varoquaux, W. Zimmermann, Jr., O. Avenel, in Proc. NATO Workshop on Excitations in 2D and 3D Quantum Liquids — Exeter (GB) Aug. 1990, éd. A.F.G. Wyatt, to be published by Plenum Press. [8] O. Avenel, E. Varoquaux, in Quantum Fluids and Solids - 1989, eds. G. Ihas, Y. Takano, AIP, New-York, 1989, p.3. [9] B.S. Deaver, J.M. Pierce, Phys. Lett. 38A, 81 (1972). Wayne Saslow has called to our attention the fact that Deaver and Pierce's model is in effect more appropriate to a hydrodynamic microbridge than to an electrodynamic one where rigid banks boundary conditions are known to apply reasonable well. 10] K.K. Likharev, Rev. Mod. Phys. 51, 101 (1979). 11] P.I. Soininen, N.B. Kopnin, M.M. Salomaa, Europhys. Lett. 14, 49 (1991) and recent preprint. [12] H. Monien, L. Tewordt, J. Low Temp. Phys.62, 277 (1986), Can. J. Phys. 65, 1388 (1987). 13 J.R. Hook, Jpn. J. Appl. Phys. 26-3, 159 (1987). 14' D. Rainer, P. Lee, Phys. Rev. 35B, 318 (1987). 15 J. Kurkijarvi, Phys. Rev. B 38, 11184 (1988). X E.V. Thuneberg, J. Kurkijarvi, J.A. Sauls, Physica B 165&166, 755 (1990). 17 E.V. Thuneberg, Europhys. Lett. 7, 441 (1988). 18 N.B. Kopnin, M.M. Salomaa, Phys. Rev. B41, 2601 (1990). 19 N.B. Kopnin, M.M. Salomaa, Physica B 165&166, 629 (1990). 20 C.N. Archie, T.A. Alvesalo, J.D. Reppy, R.C. Richardson, Phys. Rev. Lett. 43, 139 (1979). [21] M. Cerdonio, S. Vitale, Phys. Rev. B29, 481 (1984) and references therein. [22] M. Bonaldi, M. Cerdonio, P. Falferi, A. Goller, M. Mazzer, A. Miotello, G.A. Prodi, F. Sorge, R. Tommasini, L. Vanzo, S. Vitale, S. Zerbini, in Proc. 4th Marcel Grossmann Meeting on General Relativity, ed. R. Ruffini, Elsevier, Amsterdam, 1986, p.1309. 23] Physics Today, May 1984, p.20 and references therein. 24] W.W. Chow, J. Gea-Banacloche, L.M. Pedrotti, V.E. Sanders, W. Schleich, M.O. Scully, Rev. Mod. Phys. 57, 61 (1985). [25] V.B. Braginsky, A.G. Polnarev, K.S. Thorne, Phys. Rev. Lett. 53, 863 (1984). [26] W.F. Vinen, Proc. Roy. Soc. 243, 400 (1958),and in Liquid Helium, Proc. Int. School of Physics Enrico Fermi, Course XXI (Academic Press, New York, 1963) p.336. D.D. Awschalom, K.W. Schwarz, Phys. Rev. Lett. 52, 49 (1984). V.Zurek, Nature 317, 505 (1985). B. Cabrera, C.E. Cunningham, D. Saroff, Phys. Rev. Lett. 62, 2040 10 (1989). [30] Air mounts réf. AL133-15XX, Barry Controls, D-6096, Raunheim Germany. [31] Model QA-900, Sundstrand Data Control, Redmont, WA 98052 - USA. 11 Figure captions Fig. 1 Slanted sine current-phase relation for a = 0 (pure sine), 1 (limit of hysteretic behavior) and a = 3, as given by eq.(4) and normalized to slope 1 at the origin. The vertical straight lines with arrows show hysteretic phase slips for positive-going and negative-going excursions of the current. The horizontal lines underline the periodicity by 2ir of the current-phase relation. Fig. 2 At the top, two-dimensional sketch of the orifice, which has the shape of a short, narrow rectangular slit, showing with dashed lines the central region where the superfluid density is depleted. At the bottom, the equivalent one- dimensional potential barrier whose height yields the correct depletion of ps and whose spatial extent is i fa- Fig. 3 Staircase patterns in 3He-B at zero pressure and a temperature of 0.77 Tc. The operating frequency of the hydromechanical resonator is 4.43 Hz. The top and bottom traces are unaltered data for a bias phase difference across the weak link of 0 and ir respectively, as determined from the position and the length of the first plateaus. The raw data is shifted by ± 1 amplitude unit, while digitally filtered interwoven patterns start from the origin. The units in the horizontal and vertical axes are uncalibrated, but consistent between Figs. 3, 4 and 5. Fig. 4 3 Staircase patterns in He—B at zero pressure and 0.83 Tc for two different operating frequencies, 3.60 Hz (top) and 3.30 Hz (bottom), and two values of the bias phase (0 and w). The raw data displays short scale fluctuations which are mainly due to the individual phase slips. The smooth curves, are staircases simulated numerically using the current-phase relation given by eq.(4). Parameters a and 1 are varied to adjust the computed pattern to the data. Their values are given in Table 1. Fig-5 Same as in Fig. 4 for a temperature of 0.89 Tc. Note the magnification of the horizontal and vertical scales whose units are consistent with those of Figs. 3 and 4. Fig. 6 Fluid in a thin torus undergoing the superfluid phase transition at various locations. According to this scenario, the superfluid pools have uncorrelated quantum mechanical phases and a typical size 12 T7 2TL/ -Ti I e c u E P, LU D D Q. Z UJ D_ D 4 6 8 DRIVE LEVEL DRIVE LEVEL LU D D I- i—« J OL LU CL DRIVE LEVEL