Explore the fascinating world of geometric phases in this comprehensive 90-minute lecture by Michael Berry at the International Centre for Theoretical Sciences. Delve into the history and mathematics behind geometric phases, from their discovery in crystal optics in 1830 to modern applications in condensed matter physics and quantum computation. Learn about parallel transport, anholonomy, and the distinction between dynamical and geometric phases. Examine practical demonstrations, including polarization rotation in optical fibers and spinning neutrons. Investigate advanced topics such as adiabatic approximations, parameter dynamics, and geometric magnetism. Gain insights into the fundamental role of separation in scientific practice and trace the evolution of geometric phase concepts through a detailed timeline. Conclude with a hands-on demonstration of conoscopy and access to further resources for continued exploration of this influential field in physics.
Geometric Phases and the Separation of the World by Michael Berry
International Centre for Theoretical Sciences via YouTube
Overview
Syllabus
Prehistory: The first geometric phase, discovered in the optics of crystals in 1830
"The radiant Stranger", TrinityCollege Dublin, 24 May 2018
Phase: Describes the stages of any cyclic process
Underlying mathematics Gauss approximate 1800: parallel transport in the presence of curvature
Foucault pendulum
Underlying parallel transport: Anholonomy
The geometric phase
Dynamical phase and geometric phase
Measure the geometric phase by interference
Polarisation rotation in a coiled optical fiber
Polarisation rotation of spinning neutrons
Why is the phase geometric?
Exchange sign is a topological phase Pi for Pauli
A new calculation, with Pragya Shukla: probability distribution of curvature C for random parameter-dependent states
Numerical simulation, 10000 sample hamiltonians
Real symmetric matrix, eg. time-reversal symmetry
Beyond adiabatic i driven parameters Rt
Represent solution by unit spin expectation vector
The series eventually diverges, because higher terms involve higher derivatives
Divergence is inevitable, in order to accommodate transitions - exponentially weak, i.e beyond all orders epsilon power n
Optimal truncation: smoothest birth of the transition
Where is the phase?
Beyond adiabatic 2: dynamics of parameters Rt
Pi is a special case of the phase, amounting to a reversal
Geometric magnetism from the polarisation-rotation phase of light
Separation is fundamental to the practice of science
Geometric phase timeline
1983. Simon: connection with fiber bundles, Chern class
Eponymous nomenclature
Back to the beginning: easy way to see hamilton's cone and its geometric phase: do-it-yourself cononscopy
Fringes are contours of cone separation
Some references, all downloadable from http://michaelberryphisics.wordpress.com
Taught by
International Centre for Theoretical Sciences
