Revisiting the adiabatic limit in ballistic multiterminal Josephson junctions

Abstract

Motivated by recent experiments on multiterminal Josephson junctions (MJJs) that probe different ranges of the size and bias voltage parameters, we explore the regime of increasing bias voltage in large-scale devices, where the electrochemical potential becomes comparable to the 1D energy level spacing. We find that the relative number of quantum-correlated pairs formed by colliding Floquet--Kulik quartet levels is equal to the inverse of the number of channels. This observation motivates a model for the intermediate regime in which the ballistic central two-dimensional normal metal is treated as a continuum under the adiabatic approximation, while Andreev modes propagate in a background of voltage- and flux-tunable nonequilibrium electronic populations. The model predicts characteristic voltage scales that govern the mesoscopic oscillations of the critical current, and these scales are at the crossroads of interpreting experiments in all sectors of the MJJs: quartets, topology, and Floquet theory. Our model is specifically inspired by the recent Harvard and Penn State group experiments.

Figures (9)

• Figure 1: The two-terminal JJ: The horizontal Andreev tubes that carry the pairs between the left and right superconducting leads (a); the 2D JJ embedded in a superconducting loop (b); and the voltage-biased two-terminal JJ (c).
• Figure 2: The four-terminal JJ: The device (a) and the circuit containing a loop (b).
• Figure 3: The four-terminal JJ evaporated on top of a ballistic 2D-metal of dimension $L\times L$. The distance between the superconducting contacts $S_k$ and $S_l$ is denoted by $R_{k,l}$. The diameter of each contact is denoted by $d$. The spectrum is continuous if $L$ is sent to infinity.
• Figure 4: The interference $I_{q,c}\left(\Phi,\delta\mu_N\right)$ in the critical current between the quartets emitted by the contacts $S_R$ and $S_B$, in the presence of the nonequilibrium quasiparticle populations in the central-$N$. The magnetic fluxes are $\frac{\Phi}{\Phi_0}=0$ (magenta) and $\frac{\Phi}{\Phi_0}=\frac{1}{2}$ (green). Panel a shows the same data as Fig. \ref{['fig:interference3']}, where the critical currents are identical for $\frac{\Phi}{\Phi_0}=0$ and for $\frac{\Phi}{\Phi_0}=\frac{1}{2}$. Panels c and d show the interference $I'_{q,c}\left(\Phi,\delta\mu_N\right)$ between all types of quartets emitted by the contacts $S_R$ and $S_B$, see Eq. (\ref{['eq:Iqc-4']}). Panels b and d include the electroflux effect with the transformation $\delta \mu_N\rightarrow \delta \mu_N'\pm s_0$, see Eqs. (\ref{['eq:s0-1']})-(\ref{['eq:s0-2']}). Going from panels a and b to panels c and d, the electroflux effect qualitatively shifts in opposite directions the $\frac{\Phi}{\Phi_0}=0$ and the $\frac{\Phi}{\Phi_0}=\frac{1}{2}$ data, which produces the bias-voltage sensitive noninversion-inversion cross-overs.
• Figure 5: The first model for the interference $I_{q,3T,c}^{(1)}$, see Eq. (\ref{['eq:Iqc-1']}). We evaluate the three-terminal quartet critical current for short-junctions, in the absence of nonequilibrium populations. The quartet critical current is plotted as a function of the reduced magnetic flux $\frac{\Phi}{\Phi_0}$. The values of the ratios $\frac{I_{q,3T,c,2}^{(odd)}}{I_{q,3T,c,1}^{(odd)}}$ between the corresponding critical currents of both three-terminal quartet junctions are shown on the figure. In the considered short-junction limit, those parameters are independent on the value of the bias voltage. This figure corresponds to the limiting case of a $4e$ charge-Superconducting Quantum Interference Device (SQUID) with Josephson couplings that are symmetric or nonsymmetric.