Field-Free Transverse Aharonov--Bohm Phase Gate for an Orbital $l$-Qubit
The Aharonov--Bohm (AB) effect is usually read out through phase differences associated with spatially distinct electron paths. We show that confined orbital modes provide a same-path alternative: a core-confined magnetic flux writes opposite propagation phases on the co-propagating modes $|\pm l\rangle$ of a straight annular electron guide while the transported electron-wave support remains field free. In a spin-resolved Dirac treatment, the phase is carried by the overlap of the field-free vector potential $A_\phi$ with the mode's azimuthal conserved-current texture. The spin-dependent radial-gradient current becomes a boundary term that cancels when the complete finite-wall evanescent tail is retained, leaving the spin-independent orbital phase $\Delta\phi_{ln}\propto l\Phi L_{\rm int}\langle\rho^{-2}\rangle_{ln}/v_z$. The matched $|\pm l\rangle$ modes therefore realize a same-path $R_z(2\delta_l)$ gate, with differential internal-mode readout and common-mode phase rejection. For $a=75\,\mathrm{nm}$, $R=95\,\mathrm{nm}$, $L_{\rm int}=1\,\mathrm{mm}$, $E_z=10\,\mathrm{meV}$, and $|l|=10$, the gate angle is $2.315\,\mathrm{rad/G}$ and $R_z(\pi)$ occurs at $1.357\,\mathrm{G}$. Finite-barrier, mode-spacing, disorder-mismatch, and readout-visibility checks quantify the main implementation constraints. More broadly, the result connects a mode-resolved AB energy shift to a measurable propagation operation and shows how the spatially distributed conserved current of a Dirac wave can become an operational quantum-control resource.