Interferometric Measurement of the Current-Phase Relationship of a Superfluid Weak Link

Weak connections between superconductors or superfluids can differ from classical links due to quantum coherence, which allows flow without resistance. Transport properties through such weak links can be described with a single function, the current-phase relationship, which serves as the quantum analog of the current-voltage relationship. Here, we present a technique for inteferometrically measuring the current-phase relationship of superfluid weak links. We interferometrically measure the phase gradient around a ring-shaped superfluid Bose-Einstein condensate containing a rotating weak link, allowing us to identify the current flowing around the ring. While our Bose-Einstein condensate weak link operates in the hydrodynamic regime, this technique can be extended to all types of weak links (including tunnel junctions) in any phase-coherent quantum gas. Moreover, it can also measure the current-phase relationships of excitations. Such measurements may open new avenues of research in quantum transport.

Popular Summary

When certain gases of atoms are cooled to near absolute zero, they form a unique state of matter known as a Bose-Einstein condensate, where the atomic particles behave like a single wave. Bose-Einstein condensates exhibit superfluidity, the ability to flow without resistance. If a flow exists in a Bose-Einstein condensate, inserting a barrier to the flow—for example, by obstructing the flow or constricting the channel in which the atoms flow—will often not destroy the superfluidity. Such a barrier is called a “weak link.” Different types of weak links have different properties, just as different components in ordinary electrical circuits have different properties. Ordinary electrical components are characterized by their current-voltage relation; an analogous characterization of weak links is their current-phase relation, which reveals how much current flows through the weak link in response to a difference in the phase of the wave on either side of the link. We demonstrate a way to measure the current-phase relationship of weak links in Bose-Einstein condensates using interferometry.

We insert a weak link into a ring-shaped Bose-Einstein condensate consisting of approximately $8\times 10^5$ ${}^{23}\mathrm{Na}$ atoms held by an optical dipole trap and rotate the weak link to generate flow. Because of the wavelike nature of Bose-Einstein condensates, we can interfere this ring-shaped Bose-Einstein condensate with a second, unperturbed Bose-Einstein condensate. From the resulting spiral patterns due to constructive and destructive interference, we determine the current flowing around the ring and the phase difference of the superfluid across the weak link. For our particular weak link, we find that the resulting current-phase relationship is roughly linear.

This new method paves the way for understanding potential devices with weak links in Bose-Einstein condensates, such as gyroscopes. Our method can be extended to weak links in any phase-coherent quantum gas, creating new research opportunities in superfluid transport, perhaps offering the possibility of exploring unique quantum states like those that similar measurements with superconducting weak links are currently exploring.

Figure 1

(a) In situ image of the ring and disk BECs with dimensions shown. (b) Example interferogram after 15 ms TOF (left) when there is no current in the ring, including traces of the azimuthal interference fringes to guide the eye (right). (c) Interferograms for various winding numbers, where the arrow indicates the direction of flow. (d) Traces of the interference fringes to guide the eye and count the number of spiral arms. The extracted winding number is shown below the traces.

Figure 2

(a) Schematic of the atoms in the trap with a weak link applied. The coordinate system used throughout is shown; $\theta =0$ corresponds to the $\widehat{x}$ axis. (b) A close-up of the weak-link region. When the weak link is rotated at $\mathrm{\Omega }$, atoms flow through the weak link (solid) and around the ring (dashed) as shown by the stream lines. Larger velocities along the stream lines correspond to darker lines. (c) The resulting density $n(\theta )$, velocity $v(\theta )$, and phase $\varphi (\theta )$ as a function of angle, with the phase drop $\gamma$ across the weak link shown. (d) Method of extracting the phase from an interferogram (left). First, we trace the interference fringes around the ring (center), and then we fit the discontinuity across the region where the barrier was (right).

Figure 3

Plot of the normalized current around the bulk of the ring, $I_{\text{bulk}}/I_0=\alpha /2\pi$, versus the rotation rate $\mathrm{\Omega }$ of the weak link for four different weak-link potential strengths $U$: (a) $0.45{\mu }_0$, (b) $0.6{\mu }_0$, (c) $0.7{\mu }_0$, (d) $0.8{\mu }_0$. The solid lines are the prediction of our model (see text). The dashed, vertical lines show the predicted transitions between the different winding number branches. The thin, gray, diagonal lines represent the case where all the atoms move around the bulk of the ring with the weak link, i.e., $I_{\text{bulk}}=n_{1\mathrm{D}}R\mathrm{\Omega }$.

Figure 4

(a) Derivative of the initial bulk current $d{I}_{\text{bulk}}/d\mathrm{\Omega }$ versus $U$, normalized to the expected value in the limit where $U/{\mu }_0\ge 1$, $n_{1\mathrm{D}}R$. The solid line shows the prediction of the local-density approximation model. (b)–(d) Extracted current-phase relationships from the data in Fig. 3, for three different weak-link potential strengths $U$: (b) $0.45{\mu }_0$, (c) $0.6{\mu }_0$, (d) $0.7{\mu }_0$. $\gamma$ is the phase across the weak link and $I_{\mathrm{WL}}$ is the current through it, normalized to $I_0=n_{1\mathrm{D}}R{\mathrm{\Omega }}_0\approx 5\times 10^5\mathrm{atoms}/\mathrm{s}$. The solid curves represent the prediction of our theoretical model. The dashed lines merely guide the eye by connecting the multiple branches of the current-phase relationship.