非退化系统的Morris-Shore变换
Morris-Shore transformation for non-degenerate systems
摘要：
Morris-Shore（MS）变换是将多态量子系统的动力学分解为一组两态系统和非耦合单态的有力工具。它假设两组状态，其中第一组中的任何状态可以耦合到第二组中的任何状态，但是每组内的状态彼此之间不耦合。另一个重要的条件是每个集合中状态的简并性，尽管来自不同集合的状态之间的所有耦合可以通过相同的失谐而从共振失谐。简并条件限制了MS变换在各种物理上感兴趣的情况下的应用，例如，在存在电场和/或磁场或光位移的情况下，这提升了每个状态集合中的简并，例如，当这些集合包括具有非零角动量的能级的磁子能级时。本文将MS变换推广到两个集合中的状态都是非简并的情况。为此，我们开发了另一种方法的推导莫里斯-肖尔变换，它可以应用于非简并的状态集。我们提出了一个广义本征值的方法，通过它，在小失谐的限制，从简并，我们能够产生一个有效的哈密顿量，是动态等效的非简并哈密顿量。通过两步相似变换，有效哈密顿量可以映射到Morris-Shore基。在推导了一般框架之后，我们展示了这种技术在流行的Lambda三态系统，以及四态三脚架，双Lambda和钻石系统中的应用。在所有这些系统中，我们的形式主义允许我们将它们的量子动力学简化为更简单的两态系统，即使存在各种失谐，例如由频率漂移的外部场产生的失谐。
英文摘要：
The Morris-Shore (MS) transformation is a powerful tool for decomposition of the dynamics of multistate quantum systems to a set of two-state systems and uncoupled single states. It assumes two sets of states wherein any state in the first set can be coupled to any state in the second set but the states within each set are not coupled between themselves. Another important condition is the degeneracy of the states in each set, although all couplings between the states from different sets can be detuned from resonance by the same detuning. The degeneracy condition limits the application of the MS transformation in various physically interesting situations, e.g. in the presence of electric and/or magnetic fields or light shifts, which lift the degeneracy in each set of states, e.g. when these sets comprise the magnetic sublevels of levels with nonzero angular momentum. This paper extends the MS transformation to such situations, in which the states in each of the two sets are nondegenerate. To this end, we develop an alternative way for the derivation of Morris-Shore transformation, which can be applied to non-degenerate sets of states. We present a generalized eigenvalue approach, by which, in the limit of small detunings from degeneracy, we are able to generate an effective Hamiltonian that is dynamically equivalent to the non-degenerate Hamiltonian. The effective Hamiltonian can be mapped to the Morris-Shore basis with a two-step similarity transformation. After the derivation of the general framework, we demonstrate the application of this technique to the popular Lambda three-state system, and the four-state tripod, double-Lambda and diamond systems. In all of these systems, our formalism allows us to reduce their quantum dynamics to simpler two-state systems even in the presence of various detunings, e.g. generated by external fields of frequency drifts.
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