Electron Wave-Spin Qubit

TL;DR

The paper reframes the electron as a wave--entity described by the Dirac equation in a cylindrical cavity, where spin qubits are represented by spatial current-density configurations rather than point-like spin vectors. It derives spin-resolved Dirac solutions with a toroidal ground-state current, and defines a qubit state $\psi=\cos(\Theta/2)\,\psi_{\uparrow}+\sin(\Theta/2)\,e^{i\Phi}\,\psi_{\downarrow}$ whose phase $\Phi$ fixes the real-space current orientation, linking quantum information to measurable current patterns. While both pictures yield the same total magnetic moment $|\bm\mu|=\mu_B$ in uniform fields, the wave--entity framework predicts a poloidal magnetization with genus-1 topology versus the wave-particle axial dipole, leading to distinct couplings with structured fields and AB-like energy responses (discussed as a companion AB-test in related work). This deterministic, spatially grounded ontology offers new perspectives for spin-qubit design, coherence robustness, and foundational questions about quantum-classical descriptions.

Abstract

As a continuation of our earlier investigations into electron wave--spin~\citep{GaoJOPCO22,EntropyEvaSpin2024}, we analyze the electron spin and its qubit in a cavity by treating the electron as a physical wave obeying the Dirac equation. In this view, a qubit is a current--density configuration whose orientation is fixed by the relative phase, rather than a particle carrying simultaneous ``up'' and ``down'' spin states with assigned probabilities. The resulting magnetic--moment density, derived from the current, displays a richer vector distribution and topology than the fixed axial dipole weighted by probability density in the conventional wave--particle model. Both frameworks yield the same total moment of one Bohr magneton and are indistinguishable in uniform external fields, yet their ontological differences predict distinct couplings to structured fields and spin--spin interactions. These contrasts motivate further exploration of dynamical consequences within the wave--entity framework, including Aharonov--Bohm--like responses that provide testable alternatives to conventional wave--particle duality.

Figures (3)

• Figure 1: (a) Conventional particle--spin model, depicting the electron as a rotating corpuscular ball. (b) Toroidal contour of the current density plotted at two--thirds of its peak value. The electron is confined within a cylindrical cavity of radius $R = 8~\text{nm}$, height $d = 4~\text{nm}$, and potential energy $U = 10~\text{meV}$. The spin--up and spin--down components are degenerate in the ground state $(n l m = 101)$, with eigenenergy $\mathcal{E}_{101} - m_{e}c^{2} = 8.06~\text{meV}$.
• Figure 2: (a) Bloch--sphere representation of a qubit as an abstract state vector. (b) Three--dimensional contour plots of the current density for the wave--spin qubit, drawn at two--thirds of the peak value, reveal toroidal topologies circulating around $\phi=\Phi = 0$ (red, $x$--axis) and $\phi=\Phi = \tfrac{\pi}{2}$ (green, $y$--axis). These configurations highlight the phase--dependent character of the wave--spin qubit. Parameters: cylindrical cavity with $R = 8~\text{nm}$, $d = 4~\text{nm}$, $U = 10~\text{meV}$. The spin--up and spin--down components are degenerate in the ground state $(n l m = 101)$, with $\mathcal{E}_{101} - m_{e}c^2 = 8.06~\text{meV}$.
• Figure 3: Contour plots of magnetic--moment density magnitudes for the ground state $(101)$ in a cylindrical cavity ($R=8~\mathrm{nm}$, $d=4~\mathrm{nm}$, $U=10~\mathrm{meV}$). (a) Wave--particle picture: the moment density mirrors the probability distribution, resulting in a genus--0, sphere--like topology. (b) Wave--entity picture: the current density generates a genus--1 toroidal topology, with the corresponding poloidal vector pattern confined to the meridional plane, as described in Eq. \ref{['eq:M_we']}. Contours are drawn at two--thirds of the maximum.