arXiv 1302.07542013-02-12quant-ph

任意N能级量子系统的态与经典耦合振子位置之间的精确映射

An exact mapping between the states of arbitrary N-level quantum systems and the positions of classical coupled oscillators

Thomas E. Skinner

备注：: 6 pages, 1 figure; added illustrative examples, discussed and clarified implications, corrected typo in figure
发表信息：: Phys. Rev. A 88, 012110 (2013)
DOI：: 10.1103/PhysRevA.88.012110

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摘要：

代表任意N能级量子系统（包括耗散系统）的态的动力学可以精确地由经典耦合振子的动力学建模。在量子态和振子的位置之间存在直接的一一对应。因此，系统可观测量的量子相干性、期望值和测量概率可以从相应的经典态实现。N能级系统的时间演化表示为超空间中实态矢量的旋转，如以前已知的密度矩阵态，但在这里推广到薛定谔态。然后，将n维中的单个转子直接映射到一个物理维度中的n个振荡器。与密度矩阵形式的N^2不同，对于薛定谔态，表示N能级系统所需的振子数量与N成线性比例。虽然二能级量子动力学与实数旋转的等价性（SU（2），SO（3）同态）是众所周知的，但物理空间不能推广到任意的N能级系统，用一个物理维度上的耦合谐振子系统来表示量子动力学对任意的N能级系统都是通用的。经典耦合常数的值很容易从系统哈密顿量中获得，从而可以构建经典力学系统，从而可以提供对抽象量子系统动力学的直观了解，以及表征量子和经典力学之间界面的度量。

英文摘要：

The dynamics of states representing arbitrary N-level quantum systems, including dissipative systems, can be modelled exactly by the dynamics of classical coupled oscillators. There is a direct one-to-one correspondence between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. The time evolution of an N-level system is represented as the rotation of a real state vector in hyperspace, as previously known for density matrix states but generalized here to Schrodinger states. A single rotor in n dimensions is then mapped directly to n oscillators in one physical dimension. The number of oscillators needed to represent N-level systems scales linearly with N for Schrodinger states, in contrast to N^2 for the density matrix formalism. Although the well-known equivalence (SU(2), SO(3) homomorphism) of 2-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary N-level systems, representing quantum dynamics by a system of coupled harmonic oscillators in one physical dimension is general for any N. Values for the classical coupling constants are readily obtained from the system Hamiltonian, allowing construction of classical mechanical systems that can provide visual insight into the dynamics of abstract quantum systems as well as a metric for characterizing the interface between quantum and classical mechanics.

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