Oh Sangchul, Hu Xuedong, Nori Franco, Kais Sabre
Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Qatar Foundation, PO Box 5825, Doha, Qatar.
Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260-1500, USA.
Sci Rep. 2016 Feb 26;6:20824. doi: 10.1038/srep20824.
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.
从自旋1/2粒子在均匀旋转磁场中的精确解可以看出，绝热微扰是奇异的。由于非绝热效应，其在布洛赫球上的量子轨迹是一个由沿着绝热路径滚动的圆所描绘出的摆线。随着磁场旋转得越来越慢，由精确解计算出的与量子轨迹长度成正比的时间 - 能量不确定性，与由瞬时本征态所描绘的绝热路径得到的结果完全不同。然而，由精确路径所围成的面积测量得到的非绝热阿哈罗诺夫 - 阿南丹几何相位，会平滑地趋近于与绝热路径所围成面积成正比的绝热贝里相位。时间 - 能量不确定性的奇异极限和几何相位的正则极限分别与布洛赫球上摆线的弧长和弧面积相关。长幅摆线和短幅摆线分别由滚动圆外部和内部的不同初始状态所描绘。滚动圆的轴轨迹平行于绝热路径，它被证明是无跃迁驱动的一个例子。非绝热共振通过摆线弧的数量直观地展现出来。
