Classical geometric phase of gyro-motion is a coherent quantum Berry Phase

Authors: H. Zhu, H. Qin

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 is a coherent quantum Berry phase for the coherent states of the Schrodinger equation or the Dirac equation. This equivalence 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 such that a coherent Berry phase for the coherent states can be naturally defined, 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, only result in an incoherent Berry phase for the coherent states, which remains to be a quantum phase factor for the coherent states and has no classical counterpart.

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