Fig. 3: Evaluation of the first and second harmonics and the rectifications as a function of the space with a vortex.

Amplitude of the first harmonics of the Josephson current and the rectification versus the vortex core positions \({{{{\boldsymbol{r}}}}}_{0}^{{{{\rm{L}}}}}=({x}_{0}^{{{{\rm{L}}}}},{y}_{0}^{{{{\rm{L}}}}})\) for different vortex winding \({V}_{0}^{{{{\rm{L}}}}}\): (a–c) \({V}_{0}^{{{{\rm{L}}}}}=1\), (d–f) \({V}_{0}^{{{{\rm{L}}}}}=2\), (g–i) \({V}_{0}^{{{{\rm{L}}}}}=3\). \({\bar{I}}_{1}\), \({\bar{J}}_{1}\), and η indicate the odd- and even-parity first harmonics of the supercurrent evaluated by the direct Cooper pairs tunneling and the amplitude of the rectification. The color bars indicate the amplitude of (a, d, g) \({\bar{I}}_{1}({{{{\boldsymbol{r}}}}}_{0}^{{{{\rm{L}}}}})\), (b,e,h) \({\bar{J}}_{1}({{{{\boldsymbol{r}}}}}_{0}^{{{{\rm{L}}}}})\), and (c, f, i) the rectification amplitude η. In (a, b, d, e, g, h), gray-dotted and dashed lines stand for the vortex core positions, \({y}_{0}^{{{{\rm{L}}}}}\) and \({x}_{0}^{{{{\rm{L}}}}}\), evaluated in Fig. 2. The aspect ratio is (a–f) α = 3/2 and (g–i) α = 1. In (c, f, i), the rectification amplitude is evaluated by scanning all the positions of the vortex cores by performing the computation of the supercurrent for the weak link, assuming the following 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).