On the correspondence between classical geometric phase of gyro-motion and quantum Berry phase

Abstract

We show that the geometric phase of the gyro-motion of a classical charged particle in a uniform time-dependent magnetic field described by Newton's equation can be derived from a coherent Berry phase for the coherent states of the Schrödinger equation or the Dirac equation. This correspondence is established by constructing coherent states for a particle using the energy eigenstates on the Landau levels and proving that the coherent states can maintain their status of coherent states during the slow varying of the magnetic field. It is discovered that orbital Berry phases of the eigenstates interfere coherently to produce an observable effect (which we termed "coherent Berry phase"), which is exactly the geometric phase of the classical gyro-motion. This technique works for particles with and without spin. For particles with spin, on each of the eigenstates that makes up the coherent states, the Berry phase consists of two parts that can be identified as those due to the orbital and the spin motion. It is the orbital Berry phases that interfere coherently to produce a coherent Berry phase corresponding to the classical geometric phase of the gyro-motion.The spin Berry phases of the eigenstates, on the other hand, remain to be quantum phase factors for the coherent states and have no classical counterpart.

Geometric phase in Brillouin flows

“…Second, a geometric phase which increases with the surface of the closed circuit C in the control parameter space [29][30][31] . Geometric phase effects are well documented in charged particle physics 32,33 . They have, for instance, been studied for the cyclotron motion 34,35 , wave propagation and Faraday effects 36 and the bounce motion of trapped particles 37 .…”

Unified perspective on single cyclotron electron with radiation-reaction from classical to quantum

“…[38,39]. Given a specified magnetic field A = − 1 2 By, 1 2 Bx, 0 in Landau gauge, a magnetic coherent state (MCS) can be generated by coherent superposition of the Landau levels ψ n,m [40][41][42][43],…”