Full Shapiro spectroscopy of current-phase relationships

Abstract

Extracting the current-phase relationship (CPR) of a single superconducting junction is challenging in practice and traditionally involves embedding the junction in a larger superconducting circuit containing SQUIDs and/or resonators. Applying ac driving to the junction has proven to be a viable and less invasive way to extract information about the few lowest harmonics of the CPR, by locating the integer and fractional Shapiro steps in the IV-curve of the driven junction. Here, we present an alternative driving-based method that allows to extract the full harmonic content of a CPR in a non-invasive way, by fitting the measured critical currents of the driven junction as a function of driving power. We test our method, both using numerical simulations and in experiments, and we show that it works very accurately, also in the presence of noise.

Figures (3)

• Figure 1: (a) RCSJ model describing an ac-driven superconducting element with an unknown CPR, characterized by a resistance $R$ and capacitance $C$. (b) Numerical calculation of the average differential resistance $d\langle V \rangle / RdI_\text{dc}$ in the overdamped limit, as a function of $I_\text{ac}$ and $I_\text{dc}$, for a CPR with four harmonics. Blue dots show the boundaries of the zero-voltage region extracted from the numerical data. (c) The reconstructed CPR (dashed orange) after fitting the blue dots to Eqs. (\ref{['boundary_plus']},\ref{['boundary_minus']}) and the true CPR used to generate (b) (dark blue). (d) Illustration of the analytic method to extract a CPR with only two harmonics. The blue and red curves show $I_c^+(I_\text{ac})$ and $-I_c^-(I_\text{ac})$, respectively, and the dashed lines indicate the values of $I_c^\pm$ that need to be read off.
• Figure 2: Simulations of Shapiro spectroscopy in the presence of noise, where we used the same CPR as in Fig. \ref{['fig:IV_boundary']}(b). For each $I_\text{ac}$ in the calculated differential resistance in (a,b) a random offset was added to $I_\text{dc}$, drawn from a zero-mean normal distribution with (a) $\sigma = 0.1\,I_0$ and (b) $\sigma = 0.2\,I_0$. (c) the corresponding reconstructed CPRs for $\sigma = 0.1\,I_0$ (dashed orange) and $\sigma = 0.2\,I_0$ (dashed red) compared to the true CPR (solid blue). (d) Error plot as a function of the noise intensity for $3$ restarts with $600$ measurements per noise point.
• Figure 3: (a) SEM image of the planar Ge SQUID used for the experiment. Inset shows the material stack used to realize the junctions. (b) Differential resistance measurement of the balanced SQUID at a constant drive power ($P_{\rm rf} = 15\,\mathrm{dBm}$) in the slow driving regime ($f=230\,\mathrm{MHz}$) as a function of $\varphi_\text{ext}$ and $I_\text{dc}$. The inset shows the same measurement without drive, taken over a slightly large range $\varphi_\text{ext} \in [0.37,0.63]$; blue and green lines indicate the fluxes at which we perform the fast-drive spectroscopy. (c,d) Shapiro spectroscopy with fast driving ($f=2.3\;\mathrm{GHz}$) at $\varphi_\text{ext} = 0.5$ and $\varphi_\text{ext} = 0.38$, respectively. The cyan dash-dotted curves show the critical currents as given by Eqs. (\ref{['boundary_plus']},\ref{['boundary_minus']}) using the fit parameters.