Antisymmetric Mueller Generator Uniquely Determines Geometric Phase in Classical Polarization and Quantum Two-Level Systems

Geometric phases, fundamental to both classical optics and quantum mechanics, arise from the path a system takes through its possible states, rather than the forces acting upon it. José J. Gil from Universidad de Zaragoza and colleagues now demonstrate a surprising connection between these phases in seemingly disparate systems, revealing a universal origin rooted in the antisymmetric part of the Mueller matrix. The research establishes that this component, present in any ideal retarder, uniquely determines the geometric phase observed in classical polarization optics and, crucially, also governs the geometric phase of a qubit, irrespective of how quickly or cyclically the system evolves. This discovery identifies a single algebraic structure underlying geometric phases in both classical and quantum realms, offering a new operational method for isolating these contributions from experimental measurements of optical properties or quantum process tomography.

The antisymmetric component of the Stokes-vector motion on the Poincaré sphere fully determines the associated geometric phase, while the symmetric block is geometrically neutral and does not contribute to the phase. Researchers further demonstrate that the same antisymmetric generator arises in the adjoint action of any SU(2) unitary operator, fully determining the geometric phase of a qubit independently of adiabaticity or cyclicity. This identifies a single algebraic structure underlying geometric phases in both classical and quantum two-level systems, and provides an operational criterion for isolating geometric-phase contributions directly from measured Mueller matrices.

Retardance Kernel Governs Classical and Quantum Phases

This work demonstrates that the antisymmetric part of the Mueller matrix of an ideal retarder, A(t) = sin δ(t)[n(t)]×, serves as a universal geometric kernel governing both classical Pancharatnam-Berry (PB) phases and quantum geometric phases. Only retarders with A ≠ 0 can produce geometric phases; those with A = 0 are geometrically neutral. The same kernel governs Bloch-sphere trajectories and geometric phases of qubits, independently of dynamical conditions. The method involves representing a pure retarder with the Jones operator U = exp −i 2 δ n·σ, where δ is the retardance and n is the unit vector defining the eigenpolarizations on the Poincaré sphere.

The adjoint action of U generates the rotation R = cos δ I + (1 −cos δ) nnT + sin δ [n]×. The antisymmetric part of this rotation, A = 1 2(R −RT ), is proportional to the cross-product operator about the retarder axis n, specifically A = sin δ [n]×. This antisymmetric generator determines the instantaneous angular velocity, Ω, of the Stokes/Bloch vector s via the relation ̇s = Ω× s, where Ω ↔ A. The geometric phase is calculated from the tangential component of the angular velocity, Ω⊥= Ω−(Ω· s) s, using the infinitesimal geometric phase formula dγgeom = −1 2 Ω⊥dt. This demonstrates that the geometric phase depends solely on the tangential motion of the Stokes/Bloch vector on the unit sphere.

The solid angle enclosed by a closed curve on the sphere is calculated as Ωsolid = I (1 −cos θ) dφ, directly relating the enclosed solid angle to the geometric phase. For an incident Stokes vector s0 = (0, 1, 0), the trajectory s(α) = R(α) s0 traces a great circle on the Poincaré sphere, enclosing a solid angle of Ωsolid = 2π as α varies from 0 to π, resulting in a geometric phase of γgeom = −π. Similarly, for a spin-1/2 particle in a conical magnetic field, the Bloch vector traces a cone enclosing a solid angle of Ωsolid = 2π(1 −cos θ), yielding a geometric phase of γgeom = −π(1 −cos θ). This unified algebraic viewpoint provides a practical criterion for predicting or engineering geometric phases from experimentally determined polarization transformations or reconstructed SU(2) adjoint maps.

Mueller Matrix Defines Geometric Phase Equivalence

Scientists have discovered a fundamental link between geometric phases in classical polarization optics and quantum two-level systems, revealing a shared algebraic structure governing these phenomena. The research demonstrates that the antisymmetric part of the Mueller matrix uniquely determines the geometric phase observed in classical optics. Importantly, the symmetric part of the Mueller matrix contributes nothing to the phase. The team further established that this same antisymmetric generator arises in the adjoint action of any SU(2) unitary operator, fully determining the geometric phase of a qubit, irrespective of whether the evolution is adiabatic or cyclic.

This finding establishes a single algebraic structure underlying geometric phases in both classical and quantum systems. Measurements confirm that the infinitesimal geometric phase, dγgeom, is completely determined by the antisymmetric generator, and is proportional to the tangential component of the angular velocity. Experiments with a quarter-wave plate, varying its azimuth from 0 to π, yielded a geometric phase of −π, corresponding to a solid angle of 2π. Conversely, half-wave plates, with an antisymmetric generator of zero, produced no geometric phase. A similar quantum example, involving a spin-1/2 particle in a magnetic field, demonstrated that the geometric phase of −1/2 Ωsolid arises entirely from the antisymmetric adjoint generator. The research confirms that the antisymmetric part of the Mueller matrix, A = sin δ[n]×, is the universal geometric kernel governing polarization optics and qubit behavior. This unified algebraic viewpoint provides a practical criterion for predicting or engineering geometric phases from experimentally determined polarization transformations or reconstructed SU(2) adjoint maps.

Geometric Phases Mirror Classical Polarization Optics

Researchers have established a fundamental link between geometric phases observed in classical polarization optics and those governing the behaviour of qubits, the fundamental units of quantum information. The work demonstrates that the antisymmetric component of the Mueller matrix uniquely determines the geometric phase experienced by light, mirroring a similar role in defining the geometric phase of a qubit. This antisymmetric component effectively encodes the angular velocity driving the evolution of polarization states, fully determining the associated geometric phase in both classical and quantum systems. This discovery reveals a shared algebraic structure underlying geometric phases in seemingly disparate areas of physics, providing a method for isolating these contributions directly from measured optical properties or from the reconstructed behaviour of qubits. Specifically, the research provides an operational criterion for predicting or engineering geometric phases from experimentally determined polarization transformations or from the behaviour of qubits undergoing specific processes. While the current work focuses on two-level quantum systems and classical polarization, the authors acknowledge that the same algebraic framework could potentially extend to more complex scenarios, including partially polarized light and higher-dimensional quantum systems, representing a promising avenue for future investigation.