Evanescence in Coined Quantum Walks

Details Evanescence in Coined Quantum Walks Evanescence in Coined Quantum Walks

2005-06-06

arxiv.org/abs/quant-ph/0506048v4

J. Phys. A: Math. Gen. 38 (2005) 8641-8665

Physics

Quantum Physics

32 pages AMS LaTeX, 5 figures in .eps format. Rewritten in response to referee comments, including some additional references.

Scientific paper

10.1088/0305-4470/38/40/011

In this paper we complete the analysis begun by two of the authors in a previous work on the discrete quantum walk on the line [J. Phys. A 36:8775-8795 (2003) quant-ph/0303105 ]. We obtain uniformly convergent asymptotics for the "exponential decay'' regions at the leading edges of the main peaks in the Schr{\"o}dinger (or wave-mechanics) picture. This calculation required us to generalise the method of stationary phase and we describe this extension in some detail, including self-contained proofs of all the technical lemmas required. We also rigorously establish the exact Feynman equivalence between the path-integral and wave-mechanics representations for this system using some techniques from the theory of special functions. Taken together with the previous work, we can now prove every theorem by both routes.

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