## Main

The dynamics of flow between two weakly coupled macroscopic quantum reservoirs can be highly counterintuitive. In both superconductors and superfluids, currents will oscillate through a constriction (weak link) between two reservoirs in response to a static driving force, which, in a classical system, would simply yield flow in one direction. In superconductors, such junctions have given rise to a range of technologies. Although promising analogous devices3,4,5 based on weak links have been demonstrated in superfluid 3He, practical development will be hampered by the difficulty of working at the very low temperatures (T<10−3 K) required. Quantum oscillations were observed in superfluid 4He at a temperature 2,000 times higher. To understand the fundamental nature of these oscillations, and to make progress towards device development, it is necessary to know the relationship between current and phase difference across the junction, Is(ϕ). The measurement of Is(ϕ) reported here reveals a transition between two distinct quantum regimes and opens the door to the development of superfluid 4He interference devices analogous to the d.c.-SQUID, which will be highly sensitive to rotation.

When superfluid 4He, well below its transition temperature Tλ=2.17 K, is forced through a constriction, it will accelerate until it reaches a critical velocity, vc, at which a quantized vortex is nucleated. This is shown schematically in Fig. 1. The vortex moves across the path of the fluid, decreasing the quantum phase difference between the reservoirs by 2π and decreasing the fluid velocity6 by a quantized amount Δvs. This phase-slip process repeats, such that the flow through the constriction follows a sawtooth waveform. The critical velocity decreases towards zero as T is increased towards Tλ, but Δvs is mostly independent of T. When vcvs, the flow actually reverses direction whenever a phase slip occurs. If this situation were to continue as TTλ and vc drops below Δvs/2, on phase slipping the flow would end up with a velocity greater than vc in the opposite direction. This could not be an energy conserving process. At about the same temperature that this would occur, the healing length of the superfluid, ξ4=0.34(1−T/Tλ)−0.67 nm, becomes comparable to the diameter of the constriction, d. Superfluidity is then suppressed in the confined geometry of the constriction, which now acts as a barrier between the two reservoirs of superfluid, analogous to a Dayem bridge in superconductors7. In this limit, where the wavefunctions on either side of the barrier partially overlap, the dynamics of flow through the aperture are expected to be described by the Josephson effect equations, which predict sinusoidal, rather than sawtooth, oscillations. Flow features of a hydrodynamic resonator were found to be consistent with such sinusoidal behaviour8. The system can be brought from one limit to the other by varying the strength of the coupling through the aperture. This coupling strength depends on the ratio ξ4/d and we control ξ4 by varying T.

A schematic of our experimental cell, described in more detail elsewhere9, is shown in Fig. 2a. A cylindrical inner reservoir of diameter 8 mm and height 0.6 mm is bounded on the top by an 8-μm-thick flexible Kapton diaphragm on which a 400-nm-thick layer of superconducting lead has been evaporated. An array of 4,225 apertures spaced on a 3 μm square lattice in a 50-nm-thick silicon nitride membrane is mounted in a rigid aluminium plate forming the walls and bottom of the inner reservoir. Flow measurements both above and below Tλ indicate that the apertures are d=38±9 nm in diameter. Pressures can be induced across the array by application of an electrostatic force between the diaphragm and a nearby electrode, thereby pulling up on the diaphragm. Above the electrode is a superconducting coil (not shown) in which a persistent electrical current flows, producing a magnetic field that is modified by the superconducting plane of the diaphragm. Motion of the diaphragm, indicating fluid flow through the array, induces changes in the persistent current flowing in the coil, which are detected with a SQUID. The output of the SQUID is proportional to the displacement of the diaphragm x(t). We can resolve a displacement of 2×10−15 m in 1 s.

The inner reservoir sits in a sealed can filled at room temperature with 4He through a capillary, which, to decouple the can and inner reservoir from environmental fluctuations, is then blocked close to the can with a cryogenic valve. The can is immersed in a pumped bath of liquid helium that is temperature stabilized to within ±50 nK using a distributed resistive heater and high-resolution thermometer10 in a feedback loop.

Superfluid 4He with superfluid density ρs is described by a macroscopic quantum wavefunction . In unrestricted space, the flow velocity is proportional to the gradient of the phase: . Here ħ is Plank’s constant h divided by 2π and m4 is the 4He atomic mass. The superfluid current Is through our array is a function of the phase difference Δφ between the two reservoirs. In general this phase difference evolves according to the Josephson–Anderson phase-evolution equation, dΔφ/dt=−Δμ/ħ, where Δμ=m4P/ρsΔT) is the chemical potential difference across the array. Here ρ is the fluid total mass density, s is the entropy per unit mass, and ΔP and ΔT are the pressure and temperature differences across the array. If Isφ) is 2π periodic, a constant Δμ gives rise to oscillations at the Josephson frequency, fJμ/h. This can occur in either the strong-coupling phase-slip regime or the weak-coupling Josephson regime. Our goals here are to determine the detailed time evolution of these oscillations and the underlying current–phase relation Isφ) as we change the coupling strength by varying the temperature. Hereafter we use ϕ to denote Δφ.

Figure 2b and c shows sections of two flow transients excited by a step in the pressure ΔP across the array. The curves show the displacement x(t) of the diaphragm as fluid is driven though the apertures under the influence of a time-dependent chemical potential gradient. The slope of these curves, dx/dt, is proportional to the total mass current through the aperture array, I(t)=ρ Adx/dt, where A is the diaphragm area. For both flow transients, the pressure step is such that a conventional fluid would be driven in the positive direction (positive slope). In Fig. 2b, where TλT=7.4 mK, ξ4/d=0.4 and the coupling is relatively strong. The regularly spaced slope discontinuities in the first half of the plot are the signatures of phase slips that occur whenever the continuously accelerating flow reaches a maximum current Ic. By contrast in Fig. 2c, where TλT=0.8 mK, ξ4/d=1.8 and the system is in the Josephson weak-link regime. The sharp phase-slip discontinuities have been smoothed out into sinusoidal Josephson oscillations. The Δμ induced by the initial pressure step relaxes to equilibrium throughout each transient. When Δμ reaches zero, the Josephson frequency oscillations cease and lower-frequency resonant ‘Helmholtz’ or ‘pendulum-mode’ oscillations begin, with current amplitude less than Ic. These are the larger displacement oscillations in the second halves of Fig. 2b and c.

The method we used to determine Is(ϕ) is conceptually similar to that used in ref. 11 for 3He. The superfluid current as a function of time Is(t) is determined from transient data such as that shown in Fig. 2, with a small correction owing to a small normal flow component In(t). The phase difference across the aperture array is determined by integrating the phase-evolution equation: , where the phase offset is determined by the fact that ϕ=0 when Is=0. Elimination between Is(t) and ϕ(t) of the common variable of time then yields the current–phase relation Is(ϕ).

Integration of the phase-evolution equation requires knowledge of both ΔP(t) and ΔT(t). ΔP(t) is directly determined by the diaphragm displacement: ΔP=k x/A, where k is a measured spring constant. An absolute calibration of ΔP is provided by the Josephson frequency relation for ΔT=0: ΔP=ρ h fJ/m4 (with k and A this, in turn, provides the calibration for x). A temperature difference ΔT(t) is created whenever superfluid flows into or out of the inner cell (the thermo-mechanical effect) and is calculated using the measured current and a simple heat-flow equation9.

The current–phase functions for several temperatures are shown in Fig. 3. A smooth transformation occurs from the low-temperature strong-coupling regime where Is(ϕ) is linear with limiting values, into the weak-coupling regime, within a few millikelvin of Tλ, where Is(ϕ) morphs into a sine function. For TλT>5 mK, Is(ϕ) is mostly linear and the system is in the phase-slip regime. Under the influence of a constant Δμ, ϕ will increase linearly until it reaches a critical value ϕc (the maximum value of ϕ for each plot) then slip back discontinuously by 2π. Is drops from Is(ϕc) to Is(ϕc−2π). Because ϕc is less than 2π at these temperatures, ϕ and Is reverse direction when a phase slip occurs. As the temperature is increased, going from top to bottom in Fig. 3, ϕc decreases. Around the temperature at which ϕc reaches π, Is(ϕ) morphs into a sinusoid, which is the signature of an ideal Josephson weak link. Each of the curves in Fig. 3 is obtained by averaging the Is(ϕ) data from between 5 and 70 transients such as those in Fig. 2.

An intriguing question that remains is why, in the temperature regimes investigated here, the array seems to act like a single aperture or a single weak link. The amplitude of the oscillations in the phase-slip regime, but close to the transition to weak-coupling behaviour, indicates that all of the apertures are acting together2. One simple argument for phase coherence across the array is that phase gradients parallel to the wall containing the apertures correspond to lateral currents, which are not energetically favourable. It has been suggested that whereas thermal fluctuations can be strong in a single aperture, they may be suppressed in an array12. It is not at all clear in the strong-coupling limit how the apertures interact and give rise to the synchronous generation of phase slips. We are working on extending these measurements to lower temperatures, and there is preliminary evidence that the array becomes less synchronous as T drops.

We find that the measured Is(ϕ) is well described by an empirical model consisting of a purely linear kinetic inductance in series with an ideal (purely sinusoidal) weak link. For the latter, Is(θ1)=Icsin(θ1). For the linear inductance, Is(θ2)=ħ θ2/m4L. Here θ1 is the phase across the ideal weak link, θ2 is the phase across the linear inductance L and ϕ=θ1+θ2. The model can be characterized in terms of Ic and the ratio of two inductances, α=L/LJ. Here LJ is the kinetic inductance of the ideal weak link evaluated at θ1=0, LJ=(ħ/m4)${\left(\text{d}{I}_{\text{s}}/\text{d}{\theta }_{1}\right)}_{{\theta }_{1}=0}^{-1}$=ħ/m4Ic. The overall current–phase relation can be written parametrically: Is=Icsin(θ1), ϕ=θ1+αsin(θ1). An analogous model has been applied to superconducting Josephson junctions13. It has been found to be inapplicable to 3He weak links14. The model parameters can be determined from the measured Is(ϕ) in a very simple way: Ic is the maximum of Is(ϕ), which occurs at ϕ=ϕm, and α=ϕm−(π/2) is the deviation of the peak position from π/2. In the limit α→0, the linear inductance is negligible and Is(ϕ)=Icsin(ϕ). In the limit α1, Is(ϕ) is linear except near ϕm. For α≥1, there exists a critical phase ϕc at which dIs/dϕ=−∞. The model is multiple valued in this case, and a phase slip occurs when the system falls off the cliff at ϕc onto an adjacent branch of Is(ϕ). When α1, and the size of the phase slip is ΔIs=2πIc/α. The transition between the discontinuous phase-slip regime and the continuous weak-coupling regime occurs when α=1, ϕm=(π/2)+1, ϕc=π and ΔIs=0.

The measured parameters Ic and α (independent of any model) are plotted versus temperature in Fig. 4a and b. The model prediction for Is(ϕ), using only the measured Ic and α values, is plotted in Fig. 4c for four different temperatures, along with the actual measured Is(ϕ). The agreement is striking, and shows that, within the temperature range we have investigated, the entire Is(ϕ) can be accurately reproduced by this model at a given temperature from the measured Ic and α alone. Although the model is empirical, it lends insight into how the evolution of Is(ϕ) can be viewed as the transition from a multiple-valued hysteretic function to one that is single valued.

The experiment described here reveals the evolution of the function Is(ϕ) characterizing the union of two superfluid 4He reservoirs. This evolution shows a transition between two important and distinct quantum phenomena: phase slips, associated with the generation of singly quantized vortices, and the Josephson effect, associated with the weak coupling of two quantum systems through a potential barrier. We find that a simple two-parameter model accurately describes the entire temperature regime under study. The sin(ϕ) behaviour revealed at the higher temperatures will lead to the development of a superfluid 4He interferometer, an analogue of the superconducting d.c.-SQUID. Such a device, operating near 2 K, a regime accessible by mechanical cryo-coolers, will lead to practical devices useful in inertial navigation, geodesy and basic physics.