It is shown that the non-adiabatic Hannay's angle of an integrable non-degenerate classical hamiltonian dynamical system may be related to the Aharonov-Anandan phase it develops when it is looked mathematically as a quantum dynamical system.
It is known that the dynamics and geometric phase of a quantum system could be mapped mathematically into a classical system of coupled oscillators without loss of physics. In this work, we show that this quantum-classical mapping can also be used to simulate non-Abelian dynamics and phase of the quantum system with energy degeneracy by interpreting the quantum degeneracy as the resonance between the eigenfrequencies of the classical oscillators. The quantum evolution and Wilczek-Zee phase in the degenerate subspace can also be described by the classical non-Abelian evolution and a non-Abelian geometric angle in the resonant subspace. By studying the adiabatic evolution in this classical non-Abelian case, our results can also be used to generalize the concept of averaging principle and Hannay's angle into the classical system with resonant eigenfrequencies. The quantum-classical mapping of a tripod-scheme Hamiltonian with two degenerate dark states is employed to illustrate the classical non-Abelian dynamics and geometric angles.
The ``geometrical phase factor,'' found by Berry in his study of the quantum adiabatic theorem, has a classical analogue discovered recently by Hannay and Berry himself. This classical analogue consists of an extra angle swept by the system in its classical evolution during an adiabatic excursion. As in Berry's phase, Hannay's angle reflects a nontrivial holonomy of the system and it has a purely geometric character. In this paper we calculate Hannay's angles for the classical Grassmannian system associated to a quantum spin in a magnetic field. We also show their close relation to the corresponding quantum-mechanical phase of Berry for the same system.
We study the concepts of adiabatic driving and geometric phases of classical integrable systems under the Koopman-von Neumann formalism. In close relation to what happens to a quantum state, a classical Koopman-von Neumann eigenstate will acquire a geometric phase factor $exp\left\{ iΦ\right\} $ after a closed variation of the parameters λ in its associated Hamiltonian. The explicit form of Φ is then derived for integrable systems, and its relation with the Hannay angles is shown. Additionally, we use quantum formulas to write a classical adiabatic gauge potential that generates adiabatic unitary flow between classical eigenstates, and we explicitly show the relationship between the potential and the classical geometric phase.