Shapiro steps in strongly-interacting Fermi gases

Abstract

We report the observation of Shapiro steps in a periodically driven Josephson junction between strongly-interacting Fermi superfluids of ultracold atoms. We observe quantized plateaus in the current-potential characteristics, the height and width of which mirror the external drive frequency and the junction nonlinear response. Direct measurements of the current-phase relationship showcase how Shapiro steps arise from the synchronization between the relative phase of the two reservoirs and the external drive. Such mechanism is further supported by the detection of periodic phase-slippage processes, in the form of vortex-antivortex pairs. Our results are corroborated by a circuital model and numerical simulations, overall providing a clear understanding of Shapiro dynamics in atomic Fermi superfluids. Our work demonstrates phase-coherent and synchronization effects in driven strongly-interacting superfluids, opening prospects for studying emergent non-equilibrium dynamics in quantum many-body systems under external drives.

Figures (13)

• Figure 1: Current injection in a homogeneous atomic JJ. (A) (i) In situ image of the JJ at unitarity, averaged over $15$ repetitions. (ii) 1D density profile of the junction integrated along the y direction. (B) 1D density profiles as a function of time for (i) $v_{DC} = 1.5mm/s$ and $x_{AC} = 0$, (ii) $v_{DC} = 0.4mm/s$ and $x_{AC} = 2\mu m$, modulating with a frequency of $175Hz$. Each row corresponds to the integrated average density profile over 5 experimental realizations. (C) Sketch of the phase particle dynamics in the washboard potential for the conditions of (B), with the particle transparency indicating the corresponding time. (i) The voltage state of the JJ corresponds to the particle rolling down the potential, causing a voltage increase. (ii) Introducing an alternating current modulates the tilt of the potential over time. For overdamped JJ, the phase particle velocity locks to the external drive, $\langle \dot \phi \rangle_t = n\omega$, yielding to a synchronized motion that crosses $n$ potential minima during each modulation cycle. The insets show the current for the two illustrated cases.
• Figure 2: Shapiro steps in strongly-interacting Fermi superfluids. (A- B) Current-voltage characteristic for $I_{AC}/I_c=0$ (light blue), $I_{AC}/I_c= 1.6(1)$ and $\omega=2\pi \times 210 \,$Hz (dark blue) (A), and for $I_{AC}/I_c= 2.4(1)$ and $\omega=2\pi \times 175 \,$Hz (B). Dashed light blue lines represent the fit with the stationary solution of the undriven overdamped RCSJ model; dark blue ones show the results for the driven scenario, where shades account for the fitting error on $I_c$SM. Inset: phenomenological fit with $2$ independent sigmoid functions SM. (C) Shapiro step height characterization. Colors label the 1st (red), 2nd (green), 3rd (blue), and 4th (purple) step. Dotted lines represent $\Delta \mu = n \hbar \omega$. Data points represent an average over at least $3$ measurements at the same $\omega$ and different $I_{AC}$. (D) Step half-width characterization: symbols report the measured step width for $175 \,$Hz, and variable $I_{AC}$. Error bars represent the error of the centroid of the multiple sigmoid fit. Solid lines correspond to the numerical solutions of the current-driven overdamped RCSJ model. $G$ results from the fit of the DC curve SM. The gray shaded area marks the inaccessible parameter region.
• Figure 3: Phase dynamics and dynamical locking in a driven atomic JJ. (A) Current-voltage characteristic for a BEC junction under DC (light green symbols), and DC+AC drive at $\omega = 2 \pi \times 175\,$Hz and $I_{AC}/I_{C} = 1.4(1)$ (dark green symbols). (B) Current-phase relation under same experimental conditions as A, vertically shifted by $0.3\, \pi$ for the DC+AC case. Dashed lines represent the fit (light green) with the analytical and numerical solutions (dark green) of the overdamped RCSJ model. The shaded area accounts for the $I_c$ fitting error. (C) Time evolution of the interference pattern between the reservoirs for $I_{AC}/I_{c} = 1.4(1)$ and $I_{DC}/I_c = 0.14$ (i), $I_{DC}/I_c = 0.83$ (ii). Each row reports the integrated fringe profile, averaged over 5 repetitions. Red arrows signal the occurrence of phase slips; the yellow dotted line marks the barrier position. (D) Measured phase time-evolution from a sinusoidal fit of the fringe patterns. Solid lines in panels (A-B-D) represent the numerical simulation results (see main text), with $I_{AC} = 0$ (orange) and $I_{AC}/I_c = 1.4$ (dark purple). (E) Density profile in time-of-flight at $t/T= 1.03$ (i), $1.38$ (ii), $1.73$ (iii), $2.08$ (iv), $2.43$ (v). Vortices are marked in yellow dashed circles.
• Figure 4: Vortex-antivortex pairs as phase-slips proxis in a unitary JJ. Number of emitted vortex-antivortex pairs as a function of the bias current $I_{DC}/I_c$ for $I_{AC}/I_c = 0$ (light blue), and $I_{AC}/I_c = 2.8(2)$ (dark blue). We measure the number of vortices after three periods of driving with a frequency $\omega=2\pi \times 175$ Hz. The background color highlights the regions of the $0$-th, $1$-st, and $2$-nd Shapiro steps, with a color gradient reflecting the measured $\Delta \mu$ under the same driving condition. Error bars represent the standard deviation of the mean over 3 to 5 realizations.
• Figure S.1: Density time evolution (A) Time evolution of the relative integrated density along the $y$ direction, $n_{1D}$, with respect to the average value at each time $\langle n_{1D} \rangle$ for $v_{DC} = 1.1\mu m/ms$, $\omega = \mathrm{2\pi\times}175Hz$ and $x_{AC} = 2.0\mu m$. Density modulations arising from the instantaneous chemical potential difference at the barrier position $\Delta \mu_b (t)$ travel inside the two reservoirs. The horizontal black dashed line indicates the time at which the measurement is performed. White points show the extracted position of the barrier as a function of time. (B) Trajectories of the barrier for $\omega = \mathrm{2\pi\times}175Hz$, $x_{AC} = 2.0\mu m$ and $v_{DC} = 0\mu m/ms$ (dark blue points), $v_{DC} = 0.4\mu m/ms$ (medium blue points) and $v_{DC} = 1.1\mu m/ms$ (light blue points). The black solid lines markes the edges of the junction. The inset shows the barrier position with respect to the trajectories with the same parameters as extracted by the fits, but $x_{AC} = 0$. Curves corresponding to different velocities are shifted for better visualization. (C) Fourier transform of the quantity in the inset of B. Dashed lines are guides to the eye.