量子映射的粒子轨迹
Particle Trajectories for Quantum Maps
AI总结：
本文研究量子化环面上半经典粒子在Kraus算子间接测量下的轨迹，证明其总变差收敛于经典轨迹，并给出数值模拟及均匀缺陷测度刻画。
摘要：
我们研究了在量子化环面的设定下，半经典量子粒子在Kraus算子的重复间接测量下的轨迹。在两次测量之间，系统通过哈密顿传播子或元辛算子进行演化。我们证明了在这两种情况下，量子轨迹在总变差意义上收敛到其对应的经典轨迹，该经典轨迹由半经典缺陷测度的传播定义。这种收敛在经典系统的埃伦费斯特时间内成立，当系统混沌程度较低时，该时间更长。此外，我们给出了这些效应的数值模拟。在证明这一结果的过程中，我们提供了一类我们称之为均匀缺陷测度的半经典缺陷测度的刻画。我们还证明了环面上流复合函数的导数估计。
英文摘要：
We study the trajectories of a semiclassical quantum particle under repeated indirect measurement by Kraus operators, in the setting of the quantized torus. In between measurements, the system evolves via either Hamiltonian propagators or metaplectic operators. We show in both cases the convergence in total variation of the quantum trajectory to its corresponding classical trajectory, as defined by propagation of a semiclassical defect measure. This convergence holds up to the Ehrenfest time of the classical system, which is larger when the system is less chaotic. In addition, we present numerical simulations of these effects. In proving this result, we provide a characterization of a type of semi-classical defect measure we call uniform defect measures. We also prove derivative estimates of a function composed with a flow on the torus.
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