Field-Free Transverse Aharonov--Bohm Phase Gate for an Orbital $l$-Qubit

Abstract

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 $|\pm l⟩$ 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 $A_{\varphi }$ 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 $\Delta {\varphi }_{ln}\propto l\Phi L_{\mathrm{int}}⟨{\rho }^{-2}{⟩}_{ln}/v_z$. The matched $|\pm l⟩$ modes therefore realize a same-path $R_z(2{\delta }_l)$ gate, with differential internal-mode readout and common-mode phase rejection. For $a=75\mathrm{nm}$, $R=95\mathrm{nm}$, $L_{\mathrm{int}}=1\mathrm{mm}$, $E_z=10\mathrm{meV}$, and $|l|=10$, the gate angle is $2.315rad/G$ 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.

10 pages, 2 figures, 1 table. Revised version with updated device-scale parameters, numerical benchmarks, and minor improvements to the presentation. The main theoretical results are unchanged