利用希尔伯特变换和经典链模拟量子漫步
Using Hilbert transform and classical chains to simulate quantum walks
摘要：
我们提出了一种模拟策略，它使用一个经典的线性耦合弹簧链装置来模拟量子动力学，特别是量子行走。通过这种策略，我们从经典演化中获得了量子波函数。特别地，这个目标是通过链上粒子的经典动量和它们的希尔伯特变换来实现的，从这些动量和希尔伯特变换的动量对关联函数中，我们分别构造出波函数的实部和虚部。利用这种波函数，我们证明了经典链的能量和热扩散密度可以与波函数的模平方有关。这种关系表明了“声子无规行走”的概念，从而为理解弹道热输运提供了一个新的视角。本文的结果可能对费曼提出的用经典器件模拟量子物理的设想给出一个明确的答案。
英文摘要：
We propose a simulation strategy which uses a classical device of linearly coupled chain of springs to simulate quantum dynamics, in particular the quantum walks. Through this strategy, we obtain the quantum wave function from classical evolution. Specially, this goal is achieved with the classical momenta of the particles on the chain and their Hilbert transform, from which we construct the many-body momentum and Hilbert transformed momentum pair correlation functions yielding the real and imaginary parts of the wave function, respectively. With such wave function, we show that the classical chain's energy and heat spreading densities can be related to the wave function's modulus square. This relation indicates a concept of "phonon random walks", and thus it provides a new perspective to understand ballistic heat transport. The results here may give a definite answer to Feynman's idea of using a classical device to simulate quantum physics.
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