Superfluid helium interferometry:

an introduction

Eric Varoquaux

CNRS–Universite Paris-Sud, Laboratoire de Physique des Solides,Batiment 510, F-91405 Orsay Cedex, France

andCommissariat a l’Energie Atomique, Service de Physique de l’Etat Condense,

Batiment 772, Centre de Saclay, F-91191 Gif-sur-Yvette Cedex, France

Abstract

These lecture Notes describe, at an introductory level, how superfluids can be usedto measure absolute rotations. To make them self-contained to some degree, I firstintroduce briefly the two-fluid model for superfluid helium and the concept of su-perfluid order parameter. These ideas, which were put forward for the superfluidheliums, are now widely used, in particular for the BEC gases which are the maintopic of this Volume. They are presented in the somewhat different perspective ofhelium physics.

A second part will deal with the Josephson effects, the real engine behind super-fluid interferometry. Theses effects were predicted in the early sixties for supercon-ductors and were promptly observed in the laboratory [1]). It was quickly realisedthat they would also exist in superfluids [2] but the search took longer and conclu-sive experiments were performed in the eighties only in the B-phase of superfluid3He [3]. How these experiments are done, and how they can be used to measurethe rotation of the Earth by superfluid interferometry is surveyed in the last twoSections.

1 Helium as a quantum liquid

1.1 Bose-Einstein condensation

Helium remains liquid at absolute zero under moderate pressure. This anomalyis due to the fact that it is a rare gas element with weak long-distance inter-atomic forces and with a light atomic mass. The hard core of the potential(the size of the atomic cloud, 2.6 A for 4He) causes the atoms to bump into one

Preprint submitted to Elsevier Preprint 21 December 2000

another, ‘caging’ them into cells of size approximately equal to the interatomicspacing a (3.6 A for 4He at zero pressure).

"!#

"!# q"!# q

-2.6 A

QQQ

QQQ

3.6 A

Fig. 1.

This caging of the atoms is a consequence of strongcorrelations between neighbours. It also causes eachof them to have a large zero point energy. An esti-mate of this residual kinetic energy is given by

K.E. =~

2

2m

(2π

a

)2

= 17 K.

This large kinetic energy balances out most of theattractive potential energy. It pushes the atoms

away from one another and is responsible for the low density of helium (as faras condensed matter goes). Low density helium does not solidify; crystallineorder does not set in at very low temperature [4].

This poses the problem of how to fulfil Nernst’theorem which states that en-tropy vanishes at absolute zero. The solution was proposed in 1938 indepen-dently by F. London and L. Tisza as explained in London’s book [4]: orderingtakes place in momentum space.

Liquids that live at T = 0 because of quantum mechanical effects are calledquantum fluids. Quantum fluids, 4He, 3He, but also the inside of neutron stars,differ from cold atomic gases in a number of ways. They are thermodynam-ically stable, provided the proper pressure and temperature environment ismaintained. They do not reside in traps and can be fairly extended. Moreimportantly, their number densities is much higher, of the order of 1022 percm3 for helium, 1038 per cm3 for neutron stars.

For a free ideal gas, the BEC temperature is given by

kB Tc =2π ~2

m

(N

V

1

ζ(3/2)

)3/2

, ζ(3/2) = 2.612 .

With V/N = (1/3.6)3 A−3 and the atomic mass of 4He, Tc = 3.13 K. Thisvalue is surprisingly close to the temperature for the onset of superfluidity in4He, Tλ = 2.17 K. One would indeed think that the free gas formula wouldbe a poor approximation for a dense medium. In particular, the bare atomicmass m should be heavily renormalised by the strong local correlations which“dress” the atom. The above numerical agreement is not a sure proof by itselfthat superfluid 4He undergoes a Bose-Einstein condensation at Tλ.

The existence of a Bose-Einstein condensate has been confirmed directly bythe observation of a characteristic hump in the neutron diffraction spectrum[5]. The dependence of the BEC temperature on density has been measured

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recently by making helium less dense and more gas-like by diluting it in anextremely fine porous–medium matrix and measuring the onset of superfluiditywith the help of a torsional oscillator [6]. Both types of experiments are difficultand conceptual difficulties persist. But it remains that superfluidity possessesthe hallmarks of BEC and it is widely admitted that the lack of viscosityexhibited by superfluid flow is associated with the existence of a condensate[7].

1.2 The two-fluid model

At the BEC transition, superfluidity sets in. The question arises of how todescribe such a state. Landau proposed in 1941 [8] that the fluid behaves asif made up of a superfluid fraction which moves without friction with velocityvs and carries no entropy, and a ‘normal’ fluid which carries entropy andinteracts with the surrounding walls of the cell. The normal fraction moveswith velocity vn and consists of the elementary excitations of the system whichare thermally excited at non-zero temperatures. Phonons, that is sound wavesin the liquid, are part of this elementary excitation gas and do contribute tothe entropy (and the specific heat) of the liquid as they do in a solid. However,they do not account for all of it and Landau postulated that a second typeof elementary excitations must exist in liquid 4He that he called ‘rotons’. Theenergy spectrum of the combined excitations, phonons plus rotons, has theshape shown in Fig. 2.

The energy spectrum of elementary exci-

Fig. 2. Energy spectrum

tations in 4He has been measured by neu-tron scattering over the years to a veryhigh accuracy [9]. A most remarkable fea-ture is the sharpness of the neutron scat-tering peak: as envisioned by Landau, theelementary excitations are very long livedand well-defined. In a normal liquid, suchas 3He at 0.3 K, phonons and rotons alsoexist but the energy spectrum is very broad:excitations can be created with a wide rangeof energies and momenta and the fluid in-teracts readily with any external pertur-bations, in particular with walls and mov-ing objects. This causes viscous damping. The sharpness of the excitationenergy spectrum prevents superfluid 4He to couple to the sample holder andaccounts for the lack of viscosity in superfluid flow. There is no final state inthe fluid which a low-energy excitation can be scattered into, at least for flowvelocities appreciably smaller than the limit set by the dashed line in Fig.2.

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Dissipation cannot occur. It should be appreciated that Landau’s criterion forsuperfluidity makes no reference to the existence of a condensate; the linkageis indirect as will appear below.

A two-fluid hydrodynamics emerges from this model, a flow of viscous fluidwith density ρn at velocity vn, and an inter-penetrating flow of non-viscousfluid with density ρs = ρ − ρn at velocity vs, ρ being the total density of thefluid. At T = 0, there are no thermally excited elementary excitations, ρn = 0and ρs is equal to the total density of the liquid. The flow of vs is that ofan inviscid fluid and is irrotational: ∇×vs = 0. The two-fluid hydrodynam-ics equations can be written down with the help of the conservation laws formass, momentum, energy, ... and using Galilean invariance [8]; vs obeys theEuler equation as does the flow velocity of an ideal inviscid fluid. The stan-dard derivation can be found, for example, in Landau and Lifshitz’s course onhydrodynamics [10].

In the two-fluid model, the superfluid state is described with the help of twoadditional variables, ρs and vs, which we shall find again in the superfluidorder parameter below.

1.3 The superfluid order parameter

For a nearly-ideal BEC gas with a number density of atoms n, the order pa-rameter can be written as a complex number

√n exp (iϕ) as comes out in a

natural way from the microscopic description of a near-ideal Bose gas [11]. Thespirit of the definition of the order parameter for a dense, strongly interacting,non-uniform system, as given by Penrose and Onsager [12], is to look at thelarge-scale correlations in the superfluid. In a usual fluid, the correlations de-crease rapidly as r-r′ increases. A superfluid can sustain a persistent current:large scale correlations should be strong so that, when a particle is deflectedat r by an obstacle and kicked out of the condensate, a sister particle is imme-diately relocated in the condensate at r′ with no loss of order in momentumspace. Such correlations are described by the density matrix,

ρ(r, r′) =∑n

〈φ|ψ†(r)|φn〉〈φn|ψ(r′)|φ〉 ,

ψ†(r) and ψ(r′) being the boson creation and annihilation field operators,|φn〉 a complete set of eigenstates of the system and |φ〉 the state in whichthe average is expressed, which we shall take as the ground state |φ0〉. Amongthe intermediate states |φn〉, those of special relevance to the kicking-out andrelocation process discussed above are those which connect the ground statewith N bosons to the ground state with N − 1 bosons. So we give special

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attention to the following matrix element:

Φ(r) = 〈φ0(N − 1)|ψ(r)|φ0(N)〉 . (1)

The (condensate) ground state is occupied by a macroscopic number of parti-cles N0 . N , so that Φ∗(r)Φ(r) = N0/V ; Φ(r) is large in absolute value andindependent of r (to the extent that n0 = N0/V is constant in space). Thedensity matrix can be split into two parts as follows [13]:

ρ(r, r′) = Φ∗(r)Φ(r′) +∑

other matrix elements .

The summation over all the remaining contributions is of order n0 becauseit spreads over many excited (non k=0) states. These excited states are notmacroscopically populated and only have short range coherence between them:their sum decays as r-r′ becomes large. The term Φ∗(r)Φ(r′) in the densitymatrix is equal to n0 and remains constant as r-r′ becomes large; it describesthe long-range correlations in the condensate. Its existence is the reason forwhich Penrose and Onsager chose to describe superfluid order by the followingquantity

Φ(r) =√n0(r) eiϕ(r) .

The difference with the near-ideal BEC case mentioned above is that n0 maybe appreciably smaller than n. The incoherent terms in the density matrixcontribute to the total density, ρ(r, r) = n0 +

∑k6=0 nk: the condensate is

depleted from all the particles which have non-zero momentum because ofinter-particle collisions and which populate the states k with distribution nk.While this depletion is a small effect in low density atomic gases, it is large inliquid helium. Penrose and Onsager have noticed that this depletion is easilycalculated for a gas of hard sphere bosons such as the one pictured in Fig. 1.1;they have found that, due to hard sphere repulsion, only about 10 % of thehelium atoms can be in the condensate.

The strong depletion of the condensate raises the following question: how isit that the condensate fraction is only 10 % while the superfluid fraction inthe two-fluid model is 100 % at T = 0? Simply because these are not thesame quantities. The superfluid density stands in fact for the inertia of thesuperfluid fraction, as measured with a torsional oscillator for instance. Thisquantity is different from the density of bosons in the macroscopically occupiedquantum state seen as a hump in the neutron diffraction spectrum. When thesuperfluid is set into motion, the condensate enforces long-range order anddrags the excited states along through the short-range correlations; there isentrainment of all the atoms in the fluid by the condensate. Microscopic theoryis needed to describe this process in detail, as done for instance by Ceperley[14] using path integral Monte Carlo methods.

5

But, even though the above question can be settled, we still have the problemthat the Gross-Pitaevskii (GP) equation, which applies so wonderfully wellto the BEC gases [11], has the ambiguity in helium of being derived for the

order parameter√n0 eiϕ and of being applied to

√ρs eiϕ in order to agree

with the two-fluid hydrodynamics, as mentioned below. This unsatisfactorystate of affairs cannot be resolved. The GP equation, in spite of its many veryuseful features when applied to helium as reviewed for instance by Berloff andRoberts [15], is simply not a good microscopic description of a dense superfluid.It takes no account of finite-range hard-core interactions and disregards thefull hierarchy of N-body collisions. It provides a phenomenological descriptionof superfluid helium with a remote connection only to first-principles theory.Following this phenomenological bend, we shall consider below the quantity

Φ(r) =√ρs(r) eiϕ(r) (2)

and treat it as an order parameter with the properties of a ‘macroscopic’wavefunction. The particle current density is therefore given by

j = (~/2i) [Φ∗(r)∇Φ(r)− Φ(r)∇Φ∗(r)] = (~/2i)ρs(r)∇ϕ(r) . (3)

From Eq.(3), the superfluid velocity appears in a natural way as the gradientof the macroscopic phase ϕ

vs =~

m∇ϕ . (4)

It can be checked that, when the order parameter (2) is plugged into the GPequation, the equations of motion of the superfluid component in the two-fluidmodel are recovered (see, for instance, [11] or [15]). In addition, the charac-teristic length in the GP equation governs the healing length over which theorder parameter grows from zero, right at a solid wall for instance, to its valuein the bulk of the superfluid. It is, however, a phenomenological parameterwhose value in 4He, 2.5 A, can be derived from a number of experiments [16].The healing length is a temperature dependent quantity and diverges at Tλ.

The situation with 3He, which is a liquid of fermions, is slightly different.Superfluidity in such a case is of the BCS type (Bardeen–Cooper–Schrieffer),bosons appearing as a result of Cooper pair formation. The hard core repulsionin helium is strong and prevent pairing to form in an s-state for which theprobability of overlap of the two atoms in the pair is too large. Pairing occurs ina p-state, with orbital quantum number L = 1. The antisymmetry of the pairwavefunction then requires S = 1. The order parameter reflects the internalstructure of the pair and is a 3×3 matrix [17] to account for the three azimuthalspin and orbit states. However, for the B-phase, we can disregard this internalstructure because it averages out when there is no strong perturbations ofthe superfluid (no strong heat current, flow, applied magnetic field, ...). The

6

B-phase is said to be ‘pseudo-isotropic’ and its hydrodynamics, in this simplesituation, reduces to that of superfluid 4He. The healing length is given by theBCS theory as ξ = ~ vF/∆, vF being the Fermi energy and ∆ the superfluidgap above the Fermi sea. It is much larger than in 4He: at T and P = 0,ξ = 650 A.

2 The Josephson effects

Josephson predicted the effects which bear his name in 1962 for supercon-ductors within the framework of the BCS theory. As discussed below, thereare two distinct phenomena, ac and dc, which take place between two piecesof superconducting material which are brought into weak interaction withone another, for instance by allowing the quantum tunnelling of Cooper pairsthrough a thin oxide barrier. The existence of these effects was promptly con-firmed experimentally [1]. The search for analogous effects was then startedin superfluids [2], but it took until the late-eighties to finally observe themin liquid helium [3]. Interference effects which have a strong kinship with theJosephson effects have been seen in the BEC gases [18].

2.1 N and ϕ are canonically conjugate

Let us follow Anderson’s simple approach [1] and construct a wave functionwith a given number N of particles by applying boson creation operators tothe vacuum state |0〉:

|N〉 = ψ†1(r1)ψ†2(r2)...ψ†N(rN)|0〉 .

Multiply the field operators by an identical phase factor:

|N〉 = eiϕψ†1(r1)eiϕψ†2(r2)...eiϕψ†N(rN)|0〉 = eiNϕ |N〉 ,

which implies that −i∂|N〉/∂ϕ = N |N〉. This simple argument can be ex-tended to coherent superpositions of states with different numbers of particles[19]. Thus, it is established, at least for that class of states, that particle num-ber and phase are canonically conjugate variables with the consequence thatthey obey the following uncertainty relation:

∆N ∆ϕ ∼ 1 .

For systems with fixed number of particles, the phase is undetermined. Forsystems with a condensate where particles can be kicked in and out of thecondensate, particle number can fluctuate locally and the phase becomes a

7

meaningful quantity. In fact, if we assume that the particle number fluctu-ations are of order

√N , then those of the phase are of order 1/

√N . For N

large, both the phase and the particle number are well-defined quantities, theirrelative fluctuations being of order 1/

√N .

2.2 The ac-Josephson equation

We now turn to the time evolution of ϕ and N which can be expressed quitegenerally by:

i~N = [H, N ] = i∂H∂ϕ

,

i~ϕ= [H, ϕ ] = −i∂H∂N

,

H being the Hamiltonian of the system.

These equations hold for the operators N and ϕ but these quantities can alsobe treated to a very good approximation as c-numbers because, as discussedabove, their relative quantum uncertainties are very small. Let us then takean average over a volume of superfluid which is small compared to the sizeof the sample but still contains a large number of atoms. This coarse-grainedaveraging procedure is well suited to a dense system such as helium (densecompared to the cold atomic gases). The rate of change of the phase is thusgiven, writing 〈H〉 as E, the energy, by

~∂ϕ

∂t= −∂E

∂N= −µ , (5)

µ being the chemical potential. This relation is the ac Josephson relation.For superconductors (for which it was initially derived), µ = 2eV , 2e beingthe electrical charge of the Cooper pair, V the applied electric potential. Forsuperfluids, µ = vaP , va being the atomic volume for 4He, twice the atomicvolume for 3He. The contribution of the entropy to the chemical potential,ST , is negligible at low temperature.

The ac Josephson relation applies more readily to phase and pressure differ-ences. In particular, when applied to the gradient of the phase, it can be cast,using Eq.(4), into the Euler equation:

∂v

∂t+∇(vaP +

1

2mav

2) = 0 ,

ma being the atomic mass of the effective boson.

8

The second Heisenberg equation of motion, that for N , expresses particlenumber conservation:

~∂N

∂t=∂E

∂ϕ. (6)

What we have just done is simply to reproduce the equations for the motionof the superfluid component in the two-fluid hydrodynamics from the factthat N and ϕ are canonically conjugate. However, as stressed by Anderson[2], the range of validity of Eq.(5) is quite wide and it will apply even whenhydrodynamics is expected to break down as for tunnelling supercurrents. Inthe same kind of situations, to which we turn below, the internal energy Edepends in a non-trivial way on ϕ, as may be expected from Eq.(6).

2.3 The Josephson supercurrent

When applied between two regions of the super-

Fig. 3. Weak link

fluid, Eqs.(5) and (6) describe the supercurrentflowing from one region to the other. This sit-uation becomes especially interesting when thetwo regions, the two superfluid baths, are wellseparated and only weakly coupled to one an-other so that a well defined phase difference be-tween them δϕ can be sustained.

For superconductors, this was the case studiedby Josephson, the weak link being provided by athin layer of insulating oxide through which theCooper pairs could tunnel quantum-mechanically.For superfluids, the only practical weak link sofar is a microscopic aperture. As is well knownin superconductivity, weak links, or micro-bridges,lead to the same kind of effects as tunnel junc-tions [20].

Let us consider superflow through such a micro-aperture, as pictured in Fig. 3.For simplicity, we restrict the problem to one dimension along z . We describethe barrier (the weak link) as a square potential wall of height U over lengthl. We want to compute the areal current density through the barrier and weshall take care of the finite lateral extent of the weak link at the end of thecalculation.

In the bulk of the fluid, the wave function corresponding to a state with energyE is taken as a plane wave with identical amplitudes A = ρ1/2

s on both sides

9

of the barrier, but with phases which differ by δϕ. Here, we make explicit useof the definition (2) of the order parameter.

Inside the barrier, A is severely depressed: this is a consequence of the as-sumption that the barrier is a weak link between the two baths. Hence, theinteractions within the fluid can be neglected (for instance, the V0(Φ∗Φ)Φ termin the GP equation becomes small). The equation of motion then reduces to:

i~∂Φ

∂t= − ~

2

2ma

∇2Φ + UΦ , U > E ,

and also has a plane wave solution exp−i(Et/~−kz). The momentum takestwo values corresponding to the two possible directions of propagation:

k± = ±(i/~)√

2ma(U − E) .

Let b = ~/√

2ma(U − E) : the barrier height is characterised by a penetrationlength. The wave function inside the barrier can be found by standard methods[21] to be:

Φ(z) =A

sinh(l/b)

sinh

(z

b

)eiδϕ − sinh

(z − lb

).

The modulus of Φ midway in the barrier is such that:

Φ∗(l/2) Φ(l/2) =A2

2 cosh2(l/2b)[1 + cos δϕ] . (7)

Knowing the wavefunction, we can compute the current density, Eq.(3), in astraightforward manner. The total current through a micro-aperture of effec-tive cross section s is found to be:

J = j s =~s

2i

A2

sinh2 (l/b)

[sinh

(z

b

)e−iδϕ − sinh

(z − lb

)]×[

(eiδϕ/b)cosh(z

b

)− 1

bcosh

(z − lb

)]− complex conjugate

= Jc sin (δϕ) , with Jc =A2 s

b sinh(l/b). (8)

Eq.(8) describes the dc Josephson effect. Although this equation has beenobtained here in a simplified manner, it is nearly identical to the result ofmuch more involved theories [22,23].

The supercurrent J is periodic by 2π in δϕ as it can be expected to be sincechanging the phase by 2π on one side of the barrier must leave the overall

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physical situation unchanged. It vanishes for δϕ = ±π not because the velocity,∝ δϕ, goes to zero but because ρs inside the barrier, which is proportional toA2 sin(δϕ)/δϕ, does. The modulus of the wave function at the midpoint inthe barrier, Eq.(7), vanishes: superfluidity is actually destroyed at that point,which is why the supercurrent goes to zero and the phase can slip by 2π.

The critical current Jc decreases exponentially with l/b. The assumption ofweak coupling implies that l/b is large. Weak coupling is achieved in practicefor a micro-aperture by making its diameter (or the shorter lateral size if it isnot round) of the order of a few healing lengths. The length along the streamin the aperture ∼ l must also be of the same order so that δϕ can remain ofthe order of π.

If the coupling is not weak, a more elaborate calculation is necessary: the sinefunction is replaced by a more general periodic function f2π(δϕ), the current-phase relation of a non-ideal weak link. Often, this relation is not even single-valued and, when the phase is varied, the current jumps discontinuously fromone determination to another: the weak link is then said to be hysteretic.This behaviour is due to the nucleation of vortices and is accompanied bydissipation while the ideal Josephson case where f2π(δϕ) is a sine function isdissipation-less [20].

Since the healing length is very small for 4He, it cannot be expected to exhibita near-ideal Josephson effect in the micro-apertures that can be manufacturedat present, except very close to the λ point where the experiment has beenconducted recently [24]. The experiments which have first shown the existenceof this effect have been carried out in 3He [3].

3 The superfluid Helmholtz resonator

We now turn to a brief description of the experimental confirmation of theexistence of a near-ideal dc Josephson effect in superfluids.

To carry out the experiment, one must first have a superfluid sample, that is,one must cool 4He below Tλ = 2.17 K, or 3He to less than 10−3 K at zero pres-sure. This is done in a nuclear demagnetisation cryostat which can maintainsub-milliKelvin temperatures for several weeks in a row. The experimental cellitself is immersed in the main superfluid bath. It consists of a small chamberwith two openings to the main bath, as depicted in Fig. 4. One opening is themicro-aperture in which all the action takes place because the flow velocity ishighest there, the other is a longer and wider channel in which the superflowis subcritical and which provides a separate path to equilibrate temperature,pressure and phase differences in a controlled way. This parallel channel also

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provides a closed path in the superfluid as seen in Fig. 4 whose importancewill appear later.

One side wall of the chamber is a

Fig. 4. Two–aperture Helmholtz resonator

flexible membrane: it is made of Kap-ton, a polyimide material which doesnot become stiff at low temperature.This membrane is used to drive theflow of superfluid through both open-ings: it is coated with a thin alu-minium film and can be driven elec-trostatically. Since the aluminiumcoating is superconducting, its mo-tion can be tracked by placing a su-perconducting coil close to it and making a very sensitive electrodynamic mi-crophone, capable of reading out position changes with a resolution of 7×10−5

nm/√

Hz. It is important to realise that this device, with the restoring forceprovided by the flexible membrane and the inertia of the accelerated fluid inthe two vents, behaves as a resonator, a flexible wall Helmholtz resonator.

The operation of such resonators has

Fig. 5. Total current vs phase

been described in a number of publi-cations, for instance [25,26]. Since thecurrent-phase relation J(δϕ) is non-linear (as well as periodic and possiblymulti-valued), the resonator response tothe drive is also non-linear. Its motioncan be understood with the help of thediagram in Fig. 5 which shows the res-onator characteristics where J(δϕ) istaken as a sine function. The volumeswept by the membrane in its motioncorresponds to the combined flow throughthe two channels. Upon increasing thedrive level, hence δϕ = va~

∫Pdt (be-

cause of the ac Josephson relation), theresonance amplitude increases in a stepwise manner: the current climbs an ad-ditional step for each increase of the (ac) peak excursion of δϕ by 2π. For asmoothly varying J(δϕ), the steps are rounded; in the hysteretic case, theybecome steep and kinky. This response pattern to drive level is exactly simi-lar to the staircase pattern of rf-SQUIDs. Two-aperture Helmholtz resonatorsare the superfluid analogues of electrodynamic SQUIDs which have becomewidely used, in particular to measure very small magnetic fields.

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The experiments have been conducted both in 4He and in 3He with micro-apertures in the shape of rectangular slits 0.3 × 5 µm2 micro-machined in a0.2 µm thick nickel foil. In 4He, the size of the aperture is large compared tothe healing length (except very close to the λ-point) and the effects which areobserved are due to the nucleation of vortices. These effects are only remotelyconnected to the Josephson effects but they do involve the ac Josephson re-lation. The current-phase relation in this case is highly non-ideal: it consistsof straight segments extending from −Jc to Jc, parallel to one another andshifted in phase by 2π.

In superfluid 3He in the B–phase at

Fig. 6. Staircase patterns in 3He-B

zero pressure, the current– phase re-lation was observed in 1987 [3] to bevery nearly a sine function close tothe superfluid transition temperatureTc as deduced from the shape of thestaircase patterns. A set of such stair-case patterns is shown in Fig. 6. Theresonance peak amplitude is plottedin arbitrary (instrumental) units interms of the drive level, also in arbi-trary units. The four curves for T/Tc

= 0.69, 0.77, 0.83 and 0.89 from top tobottom all start from zero with no ap-plied drive; they are shifted along theordinates for clarity. At T = 0.89Tc,the modulation in the response is quitesmooth and regular; it is weak be-cause the critical current is small closeto Tc. As the temperature is reduced, the critical current, marked by the ap-pearance of the first step when the drive level is increased, grows and J(δϕ) isseen to gradually become less ideal, and even hysteretic below ∼ 0.8Tc. Theseobservations, which were confirmed in 1997 by a different method at Berkeley[27], show the existence of an ideal dc Josephson effect in superfluids.

4 Rotation sensors

Ever since the Foucault pendulum experiment in 1851, physicists have beenfascinated by measurements of the rotation of the Earth, although just gazingat the stars gives enough evidence of it (at least, since Galileo). Such aninterest arises because these measurements illustrate the basic foundations ofmechanics, and of optics and light waves with the Sagnac effect [28], and,more recently, of the properties of systems at the crossing between matter and

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waves, electrons [29], neutrons [30], BEC atoms [31] and superfluid helium[32]. Pushing to extremely high sensitivity, they can even provide a test ofEinstein’s General Relativity [33].

Let us consider a loop (e.g. a hollow torus) filled with liquid helium thatwe cool through the transition temperature. When superfluid ordering takesplace, it does so in the inertial frame of reference, exactly in the same way asthe Foucault pendulum starts oscillating in that same frame when the pieceof string which holds it back is burnt. This means that liquid helium actuallysets into motion with respect to the Earth-bound laboratory (provided thetorus is properly oriented with respect to the axis of rotation of the Earth).This effect, predicted by Landau, has been observed by Hess and Fairbankand is one of the hallmarks of superfluidity [7]. In most experiments however,it is blurred by stray vorticity [32].

In the experiments described here, the spurious effect of vorticity is subtractedby orienting the loop which picks up the rotation so that it cuts either no fluxfrom Ω⊕ (the rotation axis of the Earth lies in the plane of the loop), ormaximum flux. A schematic view of the experiment is given in Fig. 7. Thewhole cold part of the apparatus is rotated about its (vertical) axis. Theexperimental results are then checked to be independent of the arrangementof the stray vorticity in the cell by changing this arrangement (shaking thecryostat!) and repeating the measurements. The liquid motion is detected byusing a two-aperture Helmholtz resonator of the kind described in the previousSection.

If a rotation-induced supercurrent is present in the micro-aperture, to whichcorresponds a phase difference δϕ⊕, the operating point of the resonator isshifted (refer to Fig. 5) and the small-signal resonance frequency changes. Thismethod applies to the case of 3He which is more favourable [34] because thenon-linear device is fully non-dissipative in the ideal Josephson regime. Theintrinsic noise is expected be of the order of ~. A residual rotation noise of7 10−8 (rad/s)/

√Hz, or 10−3×Ω⊕/

√Hz, obtained in preliminary measure-

ments has been reported at the 22nd International Low Temperature PhysicsConference in Finland in 1999 [35]. This figure is undoubtly not the ultimatenoise level of the superfluid 3He rotation-sensor. The sensitivity can be in-creased by enlarging the pick-up loop, which at present has an area of 6 cm2.The long-term stability of superfluid gyrometers is intrinsically very good —quantum-mechanical motion is drift-free — and the practical limitations arebeing currently explored.

But, as an inquisitive reader will notice, the device which has just been de-scribed seems to be very little quantum-mechanical. All what appears to beneeded is a viscousless fluid with a well-defined critical velocity. But this isprecisely where the quantum mechanics lies, in the existence of a condensate

14

Fig. 7. Resonator with rotation pick-up loop

which exhibits these very features.

To see in more details how this comes about, let us consider the velocitycirculation along a closed contour lying in the fluid along the pick-up loop. Inthe inertial frame (that which is fixed with respect to the distant stars), thisquantity is defined as:

κ =∮

loopv · dl .

In the laboratory frame, which is rotating with respect to the inertial framewith velocity Ω, the expression of the fluid velocity at point r becomes v’=v + Ω× r. The circulation becomes, along the same contour as above,

κ′ = κ+∮

loopΩ× r · dl .

By rearranging the triple product and introducing the vector area S spannedby the (oriented) contour, we find κ′ = κ+2 Ω ·S . The circulation changes by2 Ω ·S because of the rotation. The corresponding change of the phase aroundthe superfluid loop is

δϕ = (ma/~)2 Ω · S . (9)

Let us now take a different point of view and derive the phase shift due torotation in another way by looking at the propagation of the phase informationaround the loop. Let us consider for simplicity a circular loop, of radius R.This loop can be the superfluid loop as we discuss at present, or the loopformed by the arms of a ring-laser.

The time taken for a perturbation in the phase to travel around the loop isτ = 2πR/v, v being the velocity of propagation. Possible choices for thisvelocity could be the Fermi velocity in 3He if quasi-particles are considered, orthe sound velocity if the change in the phase corresponds to a local variation

15

of the fluid velocity, or the velocity of light if electromagnetic disturbances orlight waves are involved. The exact choice will turn out to be irrelevant andwe do not specify v any further at this point.

If, during that time τ , the loop rotates at angular ve-

Fig. 8.

locity Ω, the weak link, referring to Fig. 8, moves by

δ` = τ (Ω×R) = 2πΩR2

v=

2ΩS

v.

In the case of a ring laser, the beam-splitting prismmoves by the same quantity, which induces a phase-shift

δϕ = k δ` = (2k/v) Ω · S , (10)

k being the laser light wave number. The comparison between (9) and (10)tells us that the superfluid helium condensate behaves as a ‘giant matter wave’obeying the de Broglie relation, ~k = mav.

One could argue that laser physics is different from superfluid physics andthat this comparison is purely formal. We know on other grounds that thisis not the case, for example through the analogy with superconductivity forwhich the relationship between the BCS theory and electromagnetism is wellspelled out. The recent work with BEC gases also illustrates very clearly thewave-like properties of atomic condensates. With superfluid helium, we areled to mix this standpoint with fluid mechanics and we find a rather extremecase of giant matter waves.

We can go one step further in the discussion of the quantum character ofthese superfluid helium interferometry experiments and mention a final point:where does the interference set in? The detection of phase shifts along theloop makes use of the dc Josephson effect, which results from the beats of thewave-functions from both sides of the weak link as described by Eq.(7). Thisinterference effect is a full-fledged quantum-mechanical effect and representsa key feature of the 3He gyrometer [35]. For the 4He case, [32], the quantumfeatures are buried deeper and lie in the nucleation of quantised vortices whichthen go on behaving as purely classical objects. Thus, not only do these experi-ments provide a laboratory demonstration of Einstein’s Equivalence Principle,which is the actual basis of the Sagnac effect, and of de Broglie’s duality ofwaves and particles in condensed matter, but they also provide an illustrationof how closely the borderline between quantum and classical physics can beapproached.

To summarise:

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• Helium can remain liquid at absolute zero because of quantum effects: or-dering takes place in momentum space as a Bose-Einstein condensation.• The condensate formation is described by a superfluid order parameter

which can be chosen on phenomenological grounds to reproduce Landau’stwo-fluid model: Φ =

√ρs exp(iϕ).

• Josephson effects take place between two weakly coupled baths of superfluid.They offer a concrete example of quantum effects in hydrodynamics andprovide a sensitive ‘interferometric’ mean of measuring small changes of ϕ.• Superfluid interferometry can be used to measure rotations with high reso-

lution in a drift-free manner. The present sensitivity is 7 10−8(rad/s)/√

Hz .• Phase coherence in a superfluid arises from the existence of a single-particle

quantum state with a macroscopic occupation number. In the language ofhydrodynamics, for distances larger than the healing length, it boils down tothe existence of a velocity potential and to the quantisation of circulation.

Acknowledgements

The author wishes to acknowledge useful discussions on these problems withO. Avenel, P. Hakonen and Yu. Mukharsky as well as their comments, andthose of R. Packard, on the manuscript.

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