Math 2.0

Optics, electromagnetic theory Definition

Pancharatnam–Berry phase

Builds on Polarization of lightJones calculusPoincaré sphereHolonomy

Definition

Describe a state of full polarization by a unit Jones vector u∈ℂ2u \in \C^2, where uu and eiαue^{i\alpha} u give the same state. Let ⟨u,v⟩=u1‾v1+u2‾v2\ip{u}{v} = \overline{u_1} v_1 + \overline{u_2} v_2. Let u ⁣:[0,1]→ℂ2u \colon [0, 1] \to \C^2 be a smooth curve of unit vectors such that u(1)=eiγu(0)u(1) = e^{i\gamma} u(0) for some real γ\gamma, so that the curve of states is closed. Assume that Im ⟨u(t),u′(t)⟩=0\mathrm{Im}\, \ip{u(t)}{u'(t)} = 0 for all tt, that is, each state is in phase with its neighbours in the sense of Pancharatnam. The number γ\gamma modulo 2π2\pi is the Pancharatnam–Berry phase of the closed curve of states.

Remarks

The phase depends only on the closed curve CC traced on the Poincaré sphere. If CC bounds a region of solid angle Ω\Omega, then γ≡±Ω/2(mod2π)\gamma \equiv \pm \Omega/2 \pmod{2\pi}, where the sign depends on the orientation of CC and on the conventions. Geometrically, γ\gamma is the holonomy of the natural connection on the bundle of unit Jones vectors over the sphere.

Example: take the state along a loop that goes from horizontal to 45∘45^\circ linear along the equator, then to a circular pole, then back to horizontal. This geodesic triangle encloses one eighth of the sphere, so Ω=π/2\Omega = \pi/2 and ∣γ∣=π/4\abs{\gamma} = \pi/4.

Non-example: a curve that goes from horizontal to 45∘45^\circ along the equator and comes back the same way encloses no area. Its Pancharatnam–Berry phase is 00.

For a discrete cycle of states u1,u2,u3u_1, u_2, u_3, no two orthogonal, the phase equals arg⁡(⟨u1,u2⟩⟨u2,u3⟩⟨u3,u1⟩)\arg\big(\ip{u_1}{u_2} \ip{u_2}{u_3} \ip{u_3}{u_1}\big) up to the same sign convention.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Holonomy, concept, no evidence levelHolonomyconceptJones calculus, concept, no evidence levelJones calculusconceptPolarization of light, concept, no evidence levelPolarization of lightconceptRiemann sphere, concept, no evidence levelRiemann sphereconceptSphere, concept, no evidence levelSphereconceptStokes parameters, concept, no evidence levelStokes parametersconceptPoincaré sphere, object, no evidence levelPoincaré sphereobjectPancharatnam–Berry phase, definition, no evidence levelPancharatnam–Berry ph…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Holonomy concept
  2. Jones calculus concept
  3. Polarization of light concept
  4. Riemann sphere concept
  5. Sphere concept
  6. Stokes parameters concept
  7. Poincaré sphere object
  8. Pancharatnam–Berry phase definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Geometric_phase
About this page
TagD2HBEQ484B
URIm2://D2HBEQ484B
Kinddefinition
WritingW0 stub
MSC78A10, 81Q70
DefinesPancharatnam–Berry phase
UsesPolarization of light, Jones calculus, Poincaré sphere, Holonomy
Changed3 November 2026

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