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.
The
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.
The full effect of adiabatically varying Hamiltonians on systems prepared in eigenstates of the stationary Hamiltonian often includes geometric Berry's phases. For degenerate systems these effects may result in transitions between distinct degenerate eigenstates that can be described in terms of a non-Abelian gauge potential. I demonstrate the explicit dynamic origin of such transitions between degenerate angular-momentum sublevels in an atom that is subject to collinear electric and magnetic fields that rotate adiabatically. The origin of the effects becomes clear when eigenstates of the total Hamiltonian, which includes the rotating fields, are calculated. The total Hamiltonian removes the degeneracy, and the eigenstates are in some cases linear combinations of the angular-momentum sublevels. Thus a system prepared in a specific sublevel may not remain in that sublevel, and the transition probability is exactly that given by an analysis of the geometric phase.
For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.
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.
Nuclear-magnetic-resonance gyroscopes that detect rotation as a shift in the precession frequency of nuclear spins have attracted a lot of attention. Under a feedback-generated drive, the precession frequency in the steady state is supposed to be dependent only on the angular momentum and an applied magnetic field. However, nuclei with spins larger than 1/2 experience electric quadrupole interaction with electric-field gradients at the cell walls. This quadrupole interaction shifts the precession frequencies of the nuclear spins, which brings inaccuracy to the rotation measurement as the quadrupole interaction constant
Cq is difficult to precisely measure. In this paper, the effects of quadrupole interaction on nuclear-magnetic-resonance gyroscopes are theoretically studied. We find that, when the constant
Cq is small compared to the characteristic decay rate of the system, which is the case in most gyroscopes for practical use, as the strength of the feedback-driving field increases, the quadrupole shift monotonically decreases, regardless of the system parameters in including
Cq. Furthermore, the larger driving amplitude can reduce the quadrupole shift's fluctuation caused by fluctuations of the system parameters, and, thus, it can reduce systematic errors resulted from the quadrupole interaction. In the large-
Cq regime, more than one precession frequency exists, and the nuclear spins may precess with a single frequency or multifrequencies depending on initial conditions. In this regime with large driving amplitudes, the nuclear spin can restore the single-frequency precession. These results are obtained by solving an effective master equation of the nuclear spins in a rotating frame from which both the steady-state solutions and the dynamics of the system are shown.