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arXiv:2407.19887

Geometric phase evolution

Nathan A. Hagen, Luis A. Garza-Soto

Published Jul 29, 2024• 0 citations
arXivHTMLPDF

Abstract

Geometric phase has historically been defined using closed cycles of polarization states, often derived using differential geometry on the Poincare sphere. Using the recently-developed wave model of geometric phase, we show that it is better to define geometric phase more generally, allowing every polarized wave to have a well-defined value at any point in its path. Using several example systems, we show how this approach provides more insight into the wave's behavior. Moreover, by tracking the continuous evolution of geometric phase as a wave propagates through an optical system, we encounter a natural explanation of why the conventional Poincare sphere solid angle method uses geodesic paths rather than physical paths of the polarization state.

DOI
10.48550/arxiv.2407.19887

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Geometric phase has historically been defined using closed cycles of polarization states, often derived using differential geometry on the Poincare sphere. Using the recently-developed wave model of geometric phase, we show that it is better to define geometric phase more generally, allowing every polarized wave to have a well-defined value at any point in its path. Using several example systems, we show how this approach provides more insight into the wave's behavior. Moreover, by tracking the continuous evolution of geometric phase as a wave propagates through an optical system, we encounter a natural explanation of why the conventional Poincare sphere solid angle method uses geodesic paths rather than physical paths of the polarization state.
We use the recently developed wave model of geometric phase to track the continuous evolution of geometric phase as a wave propagates through optical elements and throughout an optical system. By working directly with the wave properties, we encounter a natural explanation of why the conventional Poincaré sphere solid angle method must use geodesic paths rather than the physical paths of the polarization state-the "geodesic rule"-and show that the existing rules for the solid angle algorithm are incomplete. Finally, we use the physical model to clarify the differences between the Pancharatnam connection and the geometric phase of a wave.
Since Pancharatnam's 1956 discovery of optical geometric phase and Berry's 1984 discovery of geometric phase in quantum systems, researchers analyzing geometric phase have focused almost exclusively on algebraic approaches using the Jones calculus, or on spherical trigonometry approaches using the Poincaré sphere. The abstracted mathematics of the former and the abstracted geometry of the latter obscure the physical mechanism that generates geometric phase. We show that optical geometric phase derives entirely from the superposition of waves and the resulting shift in the location of the wave maximum. This wave-based model provides a way to visualize how geometric phase arises from relationships between waves, and from the transformations induced by optical elements. We also derive the relationship between the geometric phase of a wave by itself and the phase exhibited by an interferogram, and provide the conditions under which the two match one another.
When the control field of a qubit, a polarization state, or a spin-12\tfrac1221​ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle Ω[C]Ω[C]Ω[C] intrinsically, and displacing the degeneracy by $ε\uhat$ closes the path with enclosed solid angle $Ω(ε\uhat)=Ω[C]+2α+O(ε)$, where ααα is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane (α=0α=0α=0)---supplied by the curvature at the degeneracy. Berry's πππ invariant under reversal of the displacement and the values ±π/2\pmπ/2±π/2 under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.
The wave function of a quantum system may acquire an additional phase factor (the geometric phase), when its parameters undergo a cyclic change. An optical manifestation of the geometric phase is the change in the phase of a light wave produced when its state of polarization is taken around a circuit on the Poincare sphere. This phase change can be verified by interferometric measurements. We describe an interferometer in which the two beams traverse the same path in opposite directions, so that their optical paths are always equal, and the phase difference between them is varied by operating on the geometric phase. Measurements made with this interferometer, at a light level at which the probability of more than one photon being present simultaneously in the interferometer was negligibly small, confirmed that the interference effects observed due to the geometric phase persist down to the single photon level.
With twisted stacks of N polarizers P or retarders R, the polarization of a light beam can be cycled around the Poincaré sphere on N similar arcs of great P or small R circles. We calculate the phase changes around these cycles (geometric for P; geometric + dynamical for R). In the continuum limit N → ∞ of a smoothly twisting medium, a P stack forces the light to follow its changing polarization, and the phase is the solid angle of the associated loop on the sphere; for an R stack, on the other hand, it is only in the adiabatic limit of slow twist (where the dynamical phase is large) that the geometric phase corresponds to that of the loop specified by the changing eigenpolarization of the medium. The predicted phase shifts are observed as fringe shifts in an interferometer for N=2, 3 and 4.
We describe a recently developed generalization of the Poincaré sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group SU(3) and its generators in the defining representation, and uses geometrical objects and operations in an eight-dimensional real Euclidean space. This construction is then used to develop a generalization of the well known Pancharatnam geometric phase formula, for evolution of a three-level system along a geodesic triangle in state space.