Fig. 2: Space dependence of rectification amplitudes with a vortex.

a Schematic illustration of the Josephson junction with real-space coordinates. The analysis is performed for a vortex configuration having winding number \({V}_{0}^{{{{\rm{L}}}}}=+l\) with l = 1, 2, 3. SC-L, N, and SC-R mean the Left-side superconductor (SC), the Normal metal, and the Right-side SC. \({N}_{x}^{{{{\rm{L}}}}}\), \({N}_{x}^{{{{\rm{R}}}}}\), \({N}_{x}^{{{{\rm{N}}}}}\), N_y, and a denote the number of sites in the left and right-side SCs, and in the normal metal along the x-direction, and number of sites along the y-direction, and lattice constant. Evolution of the rectification amplitude η with regard to the vortex core coordinates for different winding numbers: (b, c) \({V}_{0}^{{{{\rm{L}}}}}=1\), (f, g) \({V}_{0}^{{{{\rm{L}}}}}=2\), and (j, k) \({V}_{0}^{{{{\rm{L}}}}}=3\). Spatially resolved harmonics of the supercurrent: I₁ (red-solid line), I₂ (green-dotted), and J₁ (blue-dashed) indicate the odd-parity first harmonic, the odd-parity second harmonic, and the even-parity first harmonic amplitude, respectively. Longitudinal scan: I₁, I₂, and J₁ at a given \({y}_{0}^{{{{\rm{L}}}}}\) as a function of \({x}_{0}^{{{{\rm{L}}}}}\) for (d) \({V}_{0}^{{{{\rm{L}}}}}=1\), (h) \({V}_{0}^{{{{\rm{L}}}}}=2\), and (l) \({V}_{0}^{{{{\rm{L}}}}}=3\). Lateral scan: I₁, I₂, and J₁ at a given \({x}_{0}^{{{{\rm{L}}}}}\) as a function of \({y}_{0}^{{{{\rm{L}}}}}\) for (e) \({V}_{0}^{{{{\rm{L}}}}}=1\), (i) \({V}_{0}^{{{{\rm{L}}}}}=2\), and (m) \({V}_{0}^{{{{\rm{L}}}}}=3\). The gray dotted lines refer to the position of the maximal rectification and are a guide to indicate the values I₁, I₂, and J₁. We set \({y}_{0}^{{{{\rm{L}}}}}\) as (b, d) 0.5a and (f, h, j, l) 5.5a, and \({x}_{0}^{{{{\rm{L}}}}}\) as (c, e, g, i)  − 23a and (k, m)  − 6a. The aspect ratio is (b–i) α = 3/2 and (j–m) α = 1. The maximal rectification η occurs for vortex core positions corresponding to a supercurrent with I₁, I₂, and J₁ components that are comparable in size. The sign change of η is related to the vanishing of J₁ and to the zeros of I₁ when the amplitude is comparable to J₁. Multiple sign reversals of η are observed for \({V}_{0}^{{{{\rm{L}}}}}=3\). Parameters: ∣Δ₀∣ = 0.02t (superconducting energy gap amplitude), t_int = 0.90 (transparency at the interface), \({N}_{x}^{{{{\rm{N}}}}}=10\), N_y = 30, and \({z}_{0}^{{{{\rm{L}}}}}=10a\) (vortex size).