Antisymmetric SU(2) adjoint generator as the universal origin of geometric phases in quantum two-level systems and classical polarization

TL;DR

The paper addresses the unification of geometric phases across classical polarization and quantum two-level dynamics by identifying the antisymmetric part of the adjoint SU(2) generator as the universal geometric-content kernel. It shows that for ideal retarders the antisymmetric block $A=\sin\delta\,[\mathbf{n}]_\times$ encodes the instantaneous angular velocity on the Poincaré sphere, and that the Pancharatnam–Berry phase is fully determined by the tangential component of this velocity via $\gamma_{geom}=-\tfrac{1}{2}\Omega_{solid}$. The same antisymmetric structure governs the adjoint action on the Bloch sphere for qubits, yielding a geometric phase that is independent of adiabaticity or cyclicity and can be accessed operationally through Mueller-matrix measurements or quantum process tomography (QPT) by extracting $A_{\mathcal{E}}$. The framework provides concrete classical and quantum examples, outlines practical extraction methods, and discusses extensions to nonunitary dynamics and higher-dimensional systems, offering a unified, searchable language for designing and diagnosing geometric-phase effects in optics and quantum information.

Abstract

We establish that the antisymmetric part of the adjoint SU(2) generator serves as the universal algebraic origin of geometric phases across classical polarization optics and quantum two-level systems. For classical ideal retarders, we demonstrate that the antisymmetric block of the Mueller matrix exclusively determines the geometric phase, encoding the angular-velocity pseudovector governing Stokes-vector evolution on the Poincaré sphere. Remarkably, the symmetric component is geometrically neutral. Quantum mechanically, the identical antisymmetric generator emerges in the adjoint SU(2) action on the Bloch sphere and fully governs the geometric phase of pure qubit states, independent of adiabaticity or cyclicity. This unified framework provides direct operational access to geometric phases via measured Mueller matrices or quantum process tomography, enabling precise control and diagnosis of geometric effects in both classical and quantum domains.

Figures (2)

• Figure 1: Representative closed trajectory on the Poincaré/Bloch sphere in $(S_1,S_2,S_3)$ coordinates. The closed curve and its shaded interior represent the solid angle enclosed by the Stokes/Bloch vector. The instantaneous tangential component of the angular velocity, $\boldsymbol{\Omega}_\perp$, lies in the tangent plane to the sphere and is generated exclusively by the antisymmetric adjoint operator $A = \sin\delta\,[\mathbf{n}]_\times$ for classical pure retarders and by $A_U = \sin\delta\,[\mathbf{n}]_\times$ for quantum two-level unitaries. This common antisymmetric generator fully determines the geometric phase. The actual velocity of the Stokes/Bloch vector along the trajectory is $\dot{\mathbf{s}} = \boldsymbol{\Omega}_\perp \times \mathbf{s}$.
• Figure 2: Local geometry of the Stokes/Bloch vector and the antisymmetric adjoint generator at a point $\mathbf{s}$ on the Poincaré/Bloch sphere. For visual clarity the point is shown at the north pole of the sphere, but the construction is local and applies to any point. The unit vector $\mathbf{s}$ is normal to the sphere and to the tangent plane $T_{\mathbf{s}}$ at that point. The instantaneous angular velocity $\boldsymbol{\Omega}$ is decomposed into a component $\boldsymbol{\Omega}_\parallel$ parallel to $\mathbf{s}$ and a component $\boldsymbol{\Omega}_\perp$ that lies in the tangent plane. The velocity of the Stokes/Bloch vector is $\dot{\mathbf{s}}=\boldsymbol{\Omega}_\perp\times\mathbf{s}$; it is therefore tangent to the sphere and orthogonal to $\boldsymbol{\Omega}_\perp$ within the tangent plane. Reversing $\boldsymbol{\Omega}_\perp$ reverses the circulation of the trajectory and the sign of the geometric phase.