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Journal of the Physical Society of Japan

Ballistic Quantum Walk of a Phase-Modulated Wave Packet

Takasi Endo, Ken'ichiro Katsuki, Kouichi Toyoshima, Yutaka Hirayoshi

Published Feb 24, 2012• 1 citation
Publisher

Abstract

A continuous-time quantum walk of a phase-modulated wave packet is considered in a one-dimensional equispacing lattice {| k >| k ∈ Z }. The interaction \(\hbar c/\Delta x\) between adjacent sites drives the packet unidirectionally, where Δ x denotes the lattice spacing. If the initial state is spread over N consecutive sites and phase-modulated by exp (i k θ), the packet travels ballistically with a velocity v = (1 - N -1 ) c cos θ.

DOI
10.1143/jpsj.81.034002

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A continuous-time quantum walk of a phase-modulated wave packet is considered in a one-dimensional equispacing lattice {| k >| k ∈ Z }. The interaction (\hbar c/\Delta x) between adjacent sites drives the packet unidirectionally, where Δ x denotes the lattice spacing. If the initial state is spread over N consecutive sites and phase-modulated by exp (i k θ), the packet travels ballistically with a velocity v = (1 - N -1 ) c cos θ.
A continuous-time quantum walk in a one-dimensional discrete system is considered. There are interactions ℏv/Δx\hbar v/\Delta xℏv/Δx between adjacent sites, where Δ x denotes the lattice spacing. If the initial state is spread over N sites, the packet travels ballistically with velocity v for N ≫1.
The wave function temporal evolution on the one-dimensional (1D) lattice is considered in the tight-binding approximation.The lattice consists of N equal sites and one impurity site (donor).The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor.The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2.The wave packet width grows with time ∼ t 1/3 and its amplitude decreases ∼ t -1/3 .The wave packet reflects multiply from the lattice ends.Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.
We investigate the impact of decoherence and static disorder on the dynamics of quantum particles moving in a periodic lattice. Our experiment relies on the photonic implementation of a one-dimensional quantum walk. The pure quantum evolution is characterized by a ballistic spread of a photon's wave packet along 28 steps. By applying controlled time-dependent operations we simulate three different environmental influences on the system, resulting in a fast ballistic spread, a diffusive classical walk, and the first Anderson localization in a discrete quantum walk architecture.
The wave function temporal evolution on the one-dimensional(1D) lattice is considered in the tight-binding approximation. The lattice consists of N equal sites and one impurity site(donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time ～ t1/3and its amplitude decreases ～ t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.
We investigate the quantum transport of delocalized states in continuous-time quantum walks (CTQWs) on a one-dimensional lattice containing a single defect. The defect is modeled by assigning complex-valued hopping amplitudes to the edges that connect the site corresponding to the mean position of the delocalized initial state to its nearest neighbors. We find that this single defective site is sufficient to enhance the ballistic spreading of an initially Gaussian wave packet. Extending these results, we implement a time-dependent alternation protocol between two distinct defect configurations, each individually yielding poor propagation of the state. The combination of these two unfavorable configurations improves the transport efficiency of the quantum walker, revealing a manifestation of Parrondo's paradox in CTQWs with delocalized initial states. This study provides insights into the role of complex-phase defects and time-dependent protocols in CTQWs, demonstrating that the interplay between quantum interference and graph engineering can effectively enhance quantum transport in discrete lattices.