deposited on top of a substrate (Fig. \[schema\_system\]). The wire is assumed to be infinite along the $x$-axis, and a Zeeman field $\vec{h}=(h_x,\,0,\,h_z)$ is applied within the plane $x-z$, thereby making an arbitrary angle $\theta$ with the wire.

![We consider an infinite superconducting wire along the $x$-axis. A magnetic field is applied within the plane $x-z$ corresponding to the surface of the substrate, leading to a Zeeman coupling where $\vec{h}=(h_x,\,0,\,h_z)$. Inversion symmetry is broken along the direction of the unit vector $\vec{n}$ normal to the substrate surface, thus inducing Rashba spin-orbit interaction in the wire. \[schema\_system\]](Fig_1.png)

The second quantized Hamiltonian of the 1D wire is expressed as: $$\mathcal{H} = \mathcal{H}_{0} + \mathcal{H}_{\vec{h}} + \mathcal{H}_\text{SO} + \mathcal{H}_\text{BCS}\; ,
\label{Hamiltonian1}$$ where

$$\begin{aligned}
\mathcal{H}_{0} &= \displaystyle \sum_{\rho} \int \mathrm{d}x \, \psi^{\dagger}_{\rho}\,\left(\frac{-\hbar^2}{2\,m}\,\partial^2_x - \mu \right)\,\psi_{\rho}\; , \\
\mathcal{H}_{\vec{h}} &= \displaystyle \sum_{\rho,\, \rho^{\prime}} \int \mathrm{d}x \, \psi^{\dagger}_{\rho}\,\left[\vec{h}\cdot\vec{\sigma}\right]_{\rho\,\rho^{\prime}}\,\psi_{\rho^{\prime}}\; ,\\
\mathcal{H}_\text{SO} &= 
\alpha\,\frac{\hbar}{\text{i}}\,\displaystyle \sum_{\rho,\, \rho^{\prime}} \int \mathrm{d}x \, \psi^{\dagger}_{\rho}\,\left[\sigma_z\right]_{\rho\,\rho^{\prime}}\,\partial_x\,\psi_{\rho^{\prime}}\; ,\\
\mathcal{H}_\text{BCS} &= \displaystyle \int \mathrm{d}x \left(\Delta(x)\,\psi^{\dagger}_{\uparrow}\,\psi^{\dagger}_{\downarrow} + \Delta^{\star}(x)\,\psi_{\downarrow}\,\psi_{\uparrow}\right)\;.\end{aligned}$$

\[H\]

The annihilation electronic field operator at position $x$ is written $\psi_\rho=\psi_\rho(x)$, where $\rho$ is the spin index. The summation is made over the spin $\rho$, and the Pauli matrices $\sigma_i$ ($i = x, \, y,\, z$) have the 