Observation of Shapiro steps in an ultracold atomic Josephson junction

TL;DR

This work demonstrates Shapiro steps in an ultracold atomic Josephson junction by driving a movable barrier through a Bose-Einstein condensate and modulating its position. The steps appear as quantized chemical-potential differences $\Delta\mu = n h f_m$, corroborated by both experiment and classical-field simulations, with step widths obeying a Bessel-function dependence on the modulation amplitude. Microscopic dynamics reveal that dissipation is carried by phonons and solitonic excitations (including vortex rings) emitted from the barrier, providing a direct view of the microscopic mechanisms behind Shapiro transport in a neutral superfluid. The results establish a platform for metrological applications, including measurements of the equation of state $\mu(n)$ and the inverse thermodynamic density of states, and open avenues for advanced atomtronic circuits and quantum simulations of Josephson physics.

Abstract

The current-voltage characteristic of a driven superconducting Josephson junction displays discrete steps. This phenomenon, called the Shapiro steps, forms today's voltage standard! Here, we report the observation of Shapiro steps in a driven Josephson junction in a gas of ultracold atoms. We demonstrate that the steps exhibit universal features, and provide insight into the microscopic dissipative dynamics that we directly observe in the experiment. We find that the steps are directly connected to phonon emission and nucleation of solitonic excitations, whose dynamics we follow in space and time. The experimental results are underpinned by extensive numerical simulations based on classical-field dynamics and may enable metrological and fundamental advances.

Figures (17)

• Figure 1: Shapiro steps in an ultracold atomic Josephson junction. (A) The system consists of a cylindrically shaped superfluid which is split by a movable optical barrier creating a weak link. The barrier as well as the two confining end-caps are realized with tightly focused laser beams. In the measurement protocol, the barrier is moved linearly (dc particle current) through the superfluid with an additional harmonic modulation (ac particle current), which is sketched in the figure. (B) At the end of the protocol, a real space absorption image of the atomic cloud is taken, using matter wave imaging Asteria_2021supp. From the atom number imbalance between the two superfluids, their chemical potential difference is derived. The pictures are taken with a modulation current $I_\mathrm{m} = 0.8 \, I_\mathrm{c}$. The position of the barrier is marked as a dashed orange line. (C) Measured Shapiro steps for $f_\mathrm{m} = 90Hz$, and different modulation amplitudes $I_\mathrm{m}$. The horizontal lines indicate the ideal Shapiro step height, given by Eq. \ref{['eq:deltamu']}. Every data point is the average of about $25$ measurement runs, the vertical bars indicate the error of the mean. (D) Comparison between experiment and classical-field simulation at $T=35nK$supp. (E) Simulations of the system at different $I_\mathrm{m}$supp.
• Figure 1: Calibration of the barrier height. A barrier block of $5 \times 8$ spots is projected into the center of the BEC. The atom number in the block area is measured and plotted against the barrier intensity. The solid line is a linear fit, according to $n=\mu/g$. The intersection with the abscissa defines the light intensity, for which $V_0=\mu$
• Figure 2: Characteristics of the Shapiro steps. (A) Dependence of the step height on the modulation frequency $f_\mathrm{m}$, where the step height is determined using sigmoid fits (continuous lines). (B) Measurements of the step height (dots) are compared to the numerical simulations (triangles) and the prediction $\Delta \mu \, / \, h = f_\mathrm{m}$ (continuous line). (C and D) Measurements (dots) and simulation (triangles) of zeroth (C) and first (D) step widths ($I_0$ and $I_1$) for varying $I_\mathrm{m}/I_c$. The solid blue lines are a fit to the experimental data, following Eq. \ref{['eq:besselfunc']}, where both curves are fitted simultaneously with the same fit parameter $\alpha_\mathrm{fit}$, yielding $\alpha_\mathrm{fit}=2.15\pm0.08$. The dashed orange lines indicate the theoretical prediction, following Eq. \ref{['eq:besselfunc']}, see text.
• Figure 2: RCSJ circuit. RCSJ model circuit to describe a Josephson junction.
• Figure 3: Microscopic system dynamics. Time evolution of the atomic density in the experiment (A, B) and in the simulation (C). (A) Constant barrier velocity. We show the time evolution for three different currents, indicated in the $I-\Delta \mu$ plot on the left. (B) Constant plus periodically modulated barrier velocity (Shapiro protocol), where $I_\mathrm{m} = 0.8I_\mathrm{c}$ and $f_\mathrm{m} = 90Hz$. (C) Classical-field simulations of the driven junction, with $I_\mathrm{m} = 0.8I_\mathrm{c}$ and $f_\mathrm{m} = 90Hz$. Each horizontal line in the images corresponds to a transversely integrated absorption image, which are stacked together. The color code indicates the relative change of the line density with respect to a reference image without barrier.