Abstract
Adiabatic perturbation is shown to be singular from the exact solution of a spin-1/2 particle in a uniformly rotating magnetic field. Due to a non-adiabatic effect, its quantum trajectory on a Bloch sphere is a cycloid traced by a circle rolling along an adiabatic path. As the magnetic field rotates more and more slowly, the time-energy uncertainty, proportional to the length of the quantum trajectory, calculated by the exact solution is entirely different from the one obtained by the adiabatic path traced by the instantaneous eigenstate. However, the non-adiabatic Aharonov- Anandan geometric phase, measured by the area enclosed by the exact path, approaches smoothly the adiabatic Berry phase, proportional to the area enclosed by the adiabatic path. The singular limit of the time-energy uncertainty and the regular limit of the geometric phase are associated with the arc length and arc area of the cycloid on a Bloch sphere, respectively. Prolate and curtate cycloids are also traced by different initial states outside and inside of the rolling circle, respectively. The axis trajectory of the rolling circle, parallel to the adiabatic path, is shown to be an example of transitionless driving. The non-adiabatic resonance is visualized by the number of cycloid arcs.
Highlights
The rotation of the magnetic field is slowed down, the non-adiabatic Aharonov-Anandan (AA) phase[13], the area enclosed by the quantum trajectory goes to the adiabatic Berry phase, the area enclosed by the adiabatic path
Quantum dynamics is governed by the time-dependent Schrödinger equation i d |ψ〉 = H (t)|ψ〉, dt where the time-dependent Hamiltonian is given by the Zeeman interaction
The question we address now is how the quantum evolution follows the adiabatic path as the rotation of the magnetic field becomes slower
Summary
The rotation of the magnetic field is slowed down, the non-adiabatic Aharonov-Anandan (AA) phase[13], the area enclosed by the quantum trajectory goes to the adiabatic Berry phase, the area enclosed by the adiabatic path. The time-energy uncertainty, the length of the quantum trajectory, does not converge to the minimum time-energy uncertainty of the adiabatic path. This singular feature of the adiabatic approximation is explained by the arc length and arc area of the cycloid. The non-adiabatic resonance condition is visualized by the number of perfect arcs of the cycloid. Our results could be tested with a single qubit, a neutron, or light polarization, and could have important implications for the application of the adiabatic perturbation, for example, adiabatic quantum computing and adiabatic quantum dynamics
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