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理论物理通讯：英文版

Evolution of Arbitrary States under Fock-Darwin Hamiltonian and a Time-Dependent Electric Field

徐晓飞, 杨涛, 翟智远, 潘孝胤

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

路径积分的方法被采用在在系统的飞机受到一个时间依赖者电场的 FockDarwin (FD ) Hamiltonian 下面计算一个控告的粒子的 eigenstates 的时间进化。准确分析表情为 eigenstates 的进化被建立。这结果然后提供一个一般答案给时间依赖者 Schr ? dinger 方程。

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We develop a complete analytical description of the time evolution of squeezed states of a charged particle under the Fock-Darwin (FD) Hamiltonian and a time-dependent electric field. This result generalizes a relation obtained by Infeld and Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski for states of the one-dimensional harmonic oscillator. We relate the evolution of a state-vector subjected to squeezing to that of state which is not subjected to squeezing and for which the time evolution under the simple harmonic oscillator dynamics is known (e.g., an eigenstate of the Hamiltonian). A corresponding relation is also established for the Wigner functions of the states, in view of their utility in the analysis of cold-ion experiments. In Appendix A, we compute the response functions of the FD Hamiltonian to an external electric field, using the same techniques as in the main text.
The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, Rényi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.