aqora

Geometric phases without geometry

Amar C. Vutha, David P. DeMille

Geometric phases arise in a number of physical situations and often lead to systematic shifts in frequencies or phases measured in precision experiments. We describe, by working through some simple examples, a method to calculate geometric phases that relies only on standard quantum mechanical calculations of energy level shifts. This connection between geometric phases and off-resonant energy shifts simplifies calculation of the effect in certain situations. More generally, it also provides a way to understand and calculate geometric phases without recourse to topology.Geometric phases arise in a number of physical situations and often lead to systematic shifts in frequencies or phases measured in precision experiments. We describe, by working through some simple examples, a method to calculate geometric phases that relies only on standard quantum mechanical calculations of energy level shifts. This connection between geometric phases and off-resonant energy shifts simplifies calculation of the effect in certain situations. More generally, it also provides a way to understand and calculate geometric phases without recourse to topology.

arXiv:0907.5116Jul 29, 20094 citations

Geometric Phases

Péter Lévay

The differential geometric aspects of Geometric Phases are reviewed.

arXiv:math-ph/0509064Sep 28, 20050 citations

The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics

Arno Böhm, Alí Mostafazadeh, Hiroyasu Koizumi +2 more

Jun 12, 2003306 citations

Wave description of geometric phase

Luis A. Garza-Soto, Nathan A. Hagen, Dorilian Lopez-Mago +1 more

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.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.

arXiv:2211.06900Journal of the Optical Society of America AJan 4, 202322 citations

Geometric-phase-propagator approach to time-dependent quantum systems

F. Κ. Diakonos, P. A. Kalozoumis, A. I. Karanikas +2 more

A field-theoretical approach to the scattering off an oscillating quantum system is developed. As a key ingredient it employs the adiabatic eigenstate basis and consists of a perturbative scheme for the calculation of the geometric phases influencing the transmission through the time-dependent potential landscape. The main advantage is the identification of basic diagrams which allow for an immediate interpretation of the underlying elementary physical processes contributing to the scattering and transmission behavior. We apply our method to the simple, but prototypical, problem of transmission through an one-dimensional oscillating $\ensuremath{\delta}$ potential and demonstrate how it enables a deeper understanding of the relevant physical processes.A field-theoretical approach to the scattering off an oscillating quantum system is developed. As a key ingredient it employs the adiabatic eigenstate basis and consists of a perturbative scheme for the calculation of the geometric phases influencing the transmission through the time-dependent potential landscape. The main advantage is the identification of basic diagrams which allow for an immediate interpretation of the underlying elementary physical processes contributing to the scattering and transmission behavior. We apply our method to the simple, but prototypical, problem of transmission through an one-dimensional oscillating $\ensuremath{\delta}$ potential and demonstrate how it enables a deeper understanding of the relevant physical processes.

Physical Review AJun 18, 20126 citations

The Electromagnetic Duality Formulation of Geometric Phases

Yuchao Zhang, Kang Li B

International Journal of Theoretical PhysicsNov 9, 20141 citation

A Simple Model for Understanding and Calculating Geometric Phase

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A Simple Model for Understanding and Calculating Geometric Phase

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