In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian.[1] The phenomenon was independently discovered by S. Pancharatnam (1956),[2] in classical optics and by H. C. Longuet-Higgins (1958)[3] in molecular physics; it was generalized by Michael Berry in (1984).[4]
It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase.
It can be seen in the conical intersection of potential energy surfaces[3][5] and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385–386 of the textbook by Bunker and Jensen.[6] In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be simply connected, or there will be nonzero holonomy.
Waves are characterized by amplitude and phase, and may vary as a function of those parameters. The geometric phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could involve rotations but also translations of particles, which are apparently undone at the end. One might expect that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursions correspond to a loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the geometric phase, and its occurrence typically indicates that the system's parameter dependence is singular (its state is undefined) for some combination of parameters.
To measure the geometric phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the geometric phase. This mechanics analogue of the geometric phase is known as the Hannay angle.
^Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623. S2CID 121930907.
^S. Pancharatnam (1956). "Generalized Theory of Interference, and Its Applications. Part I. Coherent Pencils". Proc. Indian Acad. Sci. A. 44 (5): 247–262. doi:10.1007/BF03046050. S2CID 118184376.
^ abH. C. Longuet Higgins; U. Öpik; M. H. L. Pryce; R. A. Sack (1958). "Studies of the Jahn-Teller effect .II. The dynamical problem". Proc. R. Soc. A. 244 (1236): 1–16. Bibcode:1958RSPSA.244....1L. doi:10.1098/rspa.1958.0022. S2CID 97141844.See page 12
^M. V. Berry (1984). "Quantal Phase Factors Accompanying Adiabatic Changes". Proceedings of the Royal Society A. 392 (1802): 45–57. Bibcode:1984RSPSA.392...45B. doi:10.1098/rspa.1984.0023. S2CID 46623507.
^G. Herzberg; H. C. Longuet-Higgins (1963). "Intersection of potential energy surfaces in polyatomic molecules". Discuss. Faraday Soc. 35: 77–82. doi:10.1039/DF9633500077.
^Molecular Symmetry and Spectroscopy,
2nd ed. Philip R. Bunker and Per Jensen, NRC Research Press, Ottawa (1998) [1]
ISBN 9780660196282
In classical and quantum mechanics, geometricphase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic...
Geometricphase analysis is a method of digital signal processing used to determine crystallographic quantities such as d-spacing or strain from high-resolution...
astronomy, the geometric albedo of a celestial body is the ratio of its actual brightness as seen from the light source (i.e. at zero phase angle) to that...
associated with the Berry phase or geometricphase. The concept was first introduced by S. Pancharatnam as geometricphase and later elaborately explained...
Sub-Mycenaean period and into the Geometric where cult activity is once more reasonably certain. Furtwängler proposes three phases of building at the sanctuary...
be used to induce a geometricphase on one of the computational states of the qubits. This leads to an entangling conditional phase shift of the relevant...
mathematical lens, but it has the drawbacks of the hologram (diffraction). Geometricphase lenses, also known as polarization-directed flat lenses are made by...
angle by only half a round. This is an early example of the geometricphase. This geometricphase of π {\displaystyle \pi } is observable in the difference...
discovered the properties of what is now known as the geometricphase (sometimes known as the Pancharatnam phase) for polarized beams passing through crystals...
Geometric art is a phase of Greek art, characterized largely by geometric motifs in vase painting, that flourished towards the end of the Greek Dark Ages...
[when?] the geometricphase concept has been adopted for OAM generation. The geometricphase is modulated to coincide with the spatial phase dependence...
coupling at conical intersections is responsible for the existence of Geometricphase, which was discovered by Longuet-Higgins in this context. Degenerate...
for geometricphase analysis. It allows to generate 2D maps of various crystallographic quantities using high-resolution images. Since the geometric phase...
described by robust ground state degeneracy and quantized non-Abelian geometricphases of degenerate ground states. Microscopically, topological orders correspond...
In a roof prism without a phase-correcting coating, s-polarized and p-polarized light each acquire a different geometricphase as they pass through the...
the Hannay angle is a mechanics analogue of the whirling geometricphase (or Berry phase). It was named after John Hannay of the University of Bristol...
Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger (2003). The GeometricPhase in Quantum Systems. Springer. pp. §12.5, p. 292 ff. doi:10.1007/978-3-662-10333-3...
dynamical phase θ m ( t ) = − 1 ℏ ∫ 0 t E m ( t ′ ) d t ′ {\displaystyle \theta _{m}(t)=-{\frac {1}{\hbar }}\int _{0}^{t}E_{m}(t')dt'} and geometricphase γ m...
Geometric dimensioning and tolerancing (GD&T) is a system for defining and communicating engineering tolerances via a symbolic language on engineering...
In a roof prism without a phase-correcting coating, s-polarized and p-polarized light each acquire a different geometricphase as they pass through the...
dictionary. In geometric optics, stigmatism refers to the image-formation property of an optical system which focuses a single point source in one phase optics...
composition. In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions...
In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. Each set of initial conditions...
the classical phase space. This led him to discover the phase-space star-product of a pair of functions. The modern theory of geometric quantization was...
Global Information