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Physical Review A

Non-Abelian effects in a quadrupole system rotating around two axes

Josef W. Zwanziger, M. Koenig, Alexander Pines

Published Sep 1, 1990• 48 citations
Publisher

Abstract

The $^{35}\mathrm{Cl}$ nuclear quadrupole resonance spectrum of a single crystal of sodium chlorate [R. Tycko, Phys. Rev. Lett. 58, 2281 (1987)] rotating about two axes reflects a non-Abelian gauge potential. This gauge potential is an example of Wilczek and Zee's [Phys. Rev. Lett. 52, 2111 (1984)] generalization of Berry's phase to the adiabatic transport of degenerate states.

DOI
10.1103/physreva.42.3107

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The 35Cl^{35}\mathrm{Cl}35Cl nuclear quadrupole resonance spectrum of a single crystal of sodium chlorate [R. Tycko, Phys. Rev. Lett. 58, 2281 (1987)] rotating about two axes reflects a non-Abelian gauge potential. This gauge potential is an example of Wilczek and Zee's [Phys. Rev. Lett. 52, 2111 (1984)] generalization of Berry's phase to the adiabatic transport of degenerate states.
A high-order adiabatic-approximation method is proposed to study the time evolution of degenerate quantum-mechanical systems with slowly changing Hamiltonians. We not only discuss the quantum adiabatic theorem, induced gauge structure, and non-Abelian Berry's phase factors in the first-order approximation, but we also give nonadiabatic corrections for problems in the second-order approximation. With the nuclear quadrupole resonance as an explicit example, we analyze the above-mentioned questions in detail and point out the observable effects of non-Abelian Berry's phase factors for nonadiabatic transitions between two instantaneous states in an improvement of Tycko's experiment.
A geometrical characteristic of the mixing effect induced by the non-Abelian Berry's phase appearing in a nuclear quadrupole system (I = 3/2) rotating about two axes is examined by using the rotating frame formulation. We show that the geometry-dependent parts of the mixing effect appear in the magnitude of the probability amplitude of the spin states and the rate of non-trivial mixing among these states after one period of motion of the system. These values depend only on the geometrical parameters which determine the trajectory traced by the principal-axis defined as the direction of the symmetry axis of the quadrupolar interaction.
For a molecular system with an odd number of electrons, the electronic Hamiltonian possesses a twofold Kramers degeneracy for all nuclear configurations. It is shown that this leads to a non-Abelian gauge-field term in the effective nuclear Hamiltonian, with permissible gauge transformations belonging to SU(2). This should lead to observable effects analogous to the Berry phase for the nondegenerate case. In particular, an atom in a slowly rotating electric field exhibits changes in its component of angular momentum along the field because of the non-Abelian phase factor.