Abstract
The rich dynamics of flow between two weakly coupled macroscopic quantum reservoirs has led to a range of important technologies. Practical development has so far been limited to superconducting systems, for which the basic building block is the socalled superconducting Josephson weak link^{1}. With the observation of quantum oscillations^{2} in superfluid ^{4}He near 2 K, we can now envision analogous practical superfluid helium devices. The characteristic function that determines the dynamics of such systems is the current–phase relation I_{s}(ϕ), which gives the relationship between the superfluid current I_{s} flowing through a weak link and the quantum phase difference ϕ across it. Here we report the measurement of the current–phase relation of a superfluid ^{4}He weak link formed by an array of nanoapertures separating two reservoirs of superfluid ^{4}He. As we vary the coupling strength between the two reservoirs, we observe a transition from a strongly coupled regime in which I_{s}(ϕ) is linear and flow is limited by 2π phase slips, to a weakcoupling regime where I_{s}(ϕ) becomes the sinusoidal signature of a Josephson weak link.
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 devices^{3,4,5} based on weak links have been demonstrated in superfluid ^{3}He, 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 ^{4}He 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, I_{s}(ϕ). The measurement of I_{s}(ϕ) reported here reveals a transition between two distinct quantum regimes and opens the door to the development of superfluid ^{4}He interference devices analogous to the d.c.SQUID, which will be highly sensitive to rotation.
When superfluid ^{4}He, well below its transition temperature T_{λ}=2.17 K, is forced through a constriction, it will accelerate until it reaches a critical velocity, v_{c}, 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 velocity^{6} by a quantized amount Δv_{s}. This phaseslip process repeats, such that the flow through the constriction follows a sawtooth waveform. The critical velocity decreases towards zero as T is increased towards T_{λ}, but Δv_{s} is mostly independent of T. When v_{c}<Δv_{s}, the flow actually reverses direction whenever a phase slip occurs. If this situation were to continue as T→T_{λ} and v_{c} drops below Δv_{s}/2, on phase slipping the flow would end up with a velocity greater than v_{c} 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, ξ_{4}=0.34(1−T/T_{λ})^{−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 superconductors^{7}. 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 behaviour^{8}. 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 ξ_{4}/d and we control ξ_{4} by varying T.
A schematic of our experimental cell, described in more detail elsewhere^{9}, 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 8μmthick flexible Kapton diaphragm on which a 400nmthick layer of superconducting lead has been evaporated. An array of 4,225 apertures spaced on a 3 μm square lattice in a 50nmthick 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 T_{λ} 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 ^{4}He 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 highresolution thermometer^{10} in a feedback loop.
Superfluid ^{4}He with superfluid density ρ_{s} 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 m_{4} is the ^{4}He atomic mass. The superfluid current I_{s} 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 phaseevolution equation, dΔφ/dt=−Δμ/ħ, where Δμ=m_{4}(ΔP/ρ−sΔT) 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 I_{s}(Δφ) is 2π periodic, a constant Δμ gives rise to oscillations at the Josephson frequency, f_{J}=Δμ/h. This can occur in either the strongcoupling phaseslip regime or the weakcoupling Josephson regime. Our goals here are to determine the detailed time evolution of these oscillations and the underlying current–phase relation I_{s}(Δφ) 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 timedependent 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 T_{λ}−T=7.4 mK, ξ_{4}/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 I_{c}. By contrast in Fig. 2c, where T_{λ}−T=0.8 mK, ξ_{4}/d=1.8 and the system is in the Josephson weaklink regime. The sharp phaseslip 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 lowerfrequency resonant ‘Helmholtz’ or ‘pendulummode’ oscillations begin, with current amplitude less than I_{c}. These are the larger displacement oscillations in the second halves of Fig. 2b and c.
The method we used to determine I_{s}(ϕ) is conceptually similar to that used in ref. 11 for ^{3}He. The superfluid current as a function of time I_{s}(t) is determined from transient data such as that shown in Fig. 2, with a small correction owing to a small normal flow component I_{n}(t). The phase difference across the aperture array is determined by integrating the phaseevolution equation: , where the phase offset is determined by the fact that ϕ=0 when I_{s}=0. Elimination between I_{s}(t) and ϕ(t) of the common variable of time then yields the current–phase relation I_{s}(ϕ).
Integration of the phaseevolution 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 f_{J}/m_{4} (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 thermomechanical effect) and is calculated using the measured current and a simple heatflow equation^{9}.
The current–phase functions for several temperatures are shown in Fig. 3. A smooth transformation occurs from the lowtemperature strongcoupling regime where I_{s}(ϕ) is linear with limiting values, into the weakcoupling regime, within a few millikelvin of T_{λ}, where I_{s}(ϕ) morphs into a sine function. For T_{λ}−T>5 mK, I_{s}(ϕ) is mostly linear and the system is in the phaseslip regime. Under the influence of a constant Δμ, ϕ will increase linearly until it reaches a critical value ϕ_{c} (the maximum value of ϕ for each plot) then slip back discontinuously by 2π. I_{s} drops from I_{s}(ϕ_{c}) to I_{s}(ϕ_{c}−2π). Because ϕ_{c} is less than 2π at these temperatures, ϕ and I_{s} reverse direction when a phase slip occurs. As the temperature is increased, going from top to bottom in Fig. 3, ϕ_{c} decreases. Around the temperature at which ϕ_{c} reaches π, I_{s}(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 I_{s}(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 phaseslip regime, but close to the transition to weakcoupling behaviour, indicates that all of the apertures are acting together^{2}. 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 array^{12}. It is not at all clear in the strongcoupling 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 I_{s}(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, I_{s}(θ_{1})=I_{c}sin(θ_{1}). For the linear inductance, I_{s}(θ_{2})=ħ θ_{2}/m_{4}L_{ℓ}. Here θ_{1} is the phase across the ideal weak link, θ_{2} is the phase across the linear inductance L_{ℓ} and ϕ=θ_{1}+θ_{2}. The model can be characterized in terms of I_{c} and the ratio of two inductances, α=L_{ℓ}/L_{J}. Here L_{J} is the kinetic inductance of the ideal weak link evaluated at θ_{1}=0, L_{J}=(ħ/m_{4}){(\text{d}{I}_{\text{s}}/\text{d}{\theta}_{1})}_{{\theta}_{1}=0}^{1}=ħ/m_{4}I_{c}. The overall current–phase relation can be written parametrically: I_{s}=I_{c}sin(θ_{1}), ϕ=θ_{1}+αsin(θ_{1}). An analogous model has been applied to superconducting Josephson junctions^{13}. It has been found to be inapplicable to ^{3}He weak links^{14}. The model parameters can be determined from the measured I_{s}(ϕ) in a very simple way: I_{c} is the maximum of I_{s}(ϕ), which occurs at ϕ=ϕ_{m}, and α=ϕ_{m}−(π/2) is the deviation of the peak position from π/2. In the limit α→0, the linear inductance is negligible and I_{s}(ϕ)=I_{c}sin(ϕ). In the limit α≫1, I_{s}(ϕ) is linear except near ϕ_{m}. For α≥1, there exists a critical phase ϕ_{c} at which dI_{s}/dϕ=−∞. The model is multiple valued in this case, and a phase slip occurs when the system falls off the cliff at ϕ_{c} onto an adjacent branch of I_{s}(ϕ). When α≫1, and the size of the phase slip is ΔI_{s}=2πI_{c}/α. The transition between the discontinuous phaseslip regime and the continuous weakcoupling regime occurs when α=1, ϕ_{m}=(π/2)+1, ϕ_{c}=π and ΔI_{s}=0.
The measured parameters I_{c} and α (independent of any model) are plotted versus temperature in Fig. 4a and b. The model prediction for I_{s}(ϕ), using only the measured I_{c} and α values, is plotted in Fig. 4c for four different temperatures, along with the actual measured I_{s}(ϕ). The agreement is striking, and shows that, within the temperature range we have investigated, the entire I_{s}(ϕ) can be accurately reproduced by this model at a given temperature from the measured I_{c} and α alone. Although the model is empirical, it lends insight into how the evolution of I_{s}(ϕ) can be viewed as the transition from a multiplevalued hysteretic function to one that is single valued.
The experiment described here reveals the evolution of the function I_{s}(ϕ) characterizing the union of two superfluid ^{4}He 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 twoparameter 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 ^{4}He interferometer, an analogue of the superconducting d.c.SQUID. Such a device, operating near 2 K, a regime accessible by mechanical cryocoolers, will lead to practical devices useful in inertial navigation, geodesy and basic physics.
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Acknowledgements
T. Haard made important contributions to the construction and design of the apparatus. M. Abreau also helped with the construction of the apparatus. The aperture arrays were fabricated by A. Loshak. We thank D. H. Lee and H. Fu for helpful discussions. This work was supported in part by the NSF (grant DMR 0244882) and NASA.
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Hoskinson, E., Sato, Y., Hahn, I. et al. Transition from phase slips to the Josephson effect in a superfluid ^{4}He weak link. Nature Phys 2, 23–26 (2006). https://doi.org/10.1038/nphys190
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DOI: https://doi.org/10.1038/nphys190
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