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arXiv:1510.08905 · Physical Review A

Revivals in quantum walks with a quasiperiodically-time-dependent coin

Christopher Cedzich, Reinhard F. Werner

Published Mar 21, 2016• 24 citations
arXivHTMLPDF

Abstract

We provide an explanation of recent experimental results of Xue et al. [P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, Phys. Rev. Lett. 114, 140502 (2015)], where full revivals in a time-dependent quantum walk model with a periodically changing coin are found. Using methods originally developed for ``electric'' walks with a space-dependent, rather than a time-dependent, coin, we provide a full explanation of the observations of Xue et al. We extend the analysis from periodic time dependence to quasiperiodic behavior with periods incommensurate to the step size. Spectral analysis, one of the principal tools for the study of electric walks, fails for time-dependent systems, but we find qualitative propagation behavior of the time-dependent system in close analogy to the electric case.

DOI
10.1103/physreva.93.032329

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Acta Physica Sinica•Jan 1, 2016•0 citations
We provide an explanation of recent experimental results of Xue et al. [P. Xue, R. Zhang, H. Qin, X. Zhan, Z. H. Bian, J. Li, and B. C. Sanders, Phys. Rev. Lett. 114, 140502 (2015)], where full revivals in a time-dependent quantum walk model with a periodically changing coin are found. Using methods originally developed for ``electric'' walks with a space-dependent, rather than a time-dependent, coin, we provide a full explanation of the observations of Xue et al. We extend the analysis from periodic time dependence to quasiperiodic behavior with periods incommensurate to the step size. Spectral analysis, one of the principal tools for the study of electric walks, fails for time-dependent systems, but we find qualitative propagation behavior of the time-dependent system in close analogy to the electric case.
We introduce quantum walks with a time-dependent coin, and show how they include, as a particular case, the generalized quantum walk recently studied by Wojcik et al. [Phys. Rev. Lett. 93, 180601 (2004)] which exhibits interesting dynamical localization and quasiperiodic dynamics. Our proposal allows for a much easier implementation of this particularly rich dynamics than the original one. Moreover, it allows for an additional control on the walk, which can be used to compensate for phases appearing due to external interactions. To illustrate its feasibility, we discuss an example using an optical cavity. We also derive an approximated solution in the continuous limit (long-wavelength approximation) which provides physical insight about the process.
We study the dynamics of a generalization of a quantum coin walk on the line, which is a natural model for a diffusion modified by quantum or interference effects. In particular, our results provide surprisingly simple explanations for recurrence phenomena observed by Bouwmeester et al. [Phys. Rev. A 61, 13410 (1999)]] in their optical Galton board experiment, and a description of a stroboscopic quantum walk given by Buerschaper and Burnett [quant-ph/0406039] through numerical simulations. We also provide heuristic explanations for the behavior of our model which show, in particular, that its dynamics can be viewed as a discrete version of Bloch oscillations.
We demonstrate a quantum walk with time-dependent coin bias. With this technique we realize an experimental single-photon one-dimensional quantum walk with a linearly ramped time-dependent coin flip operation and thereby demonstrate two periodic revivals of the walker distribution. In our beam-displacer interferometer, the walk corresponds to movement between discretely separated transverse modes of the field serving as lattice sites, and the time-dependent coin flip is effected by implementing a different angle between the optical axis of half-wave plate and the light propagation at each step. Each of the quantum-walk steps required to realize a revival comprises two sequential orthogonal coin-flip operators, with one coin having constant bias and the other coin having a time-dependent ramped coin bias, followed by a conditional translation of the walker.
We study the dynamics of a generalization of quantum coin walk on the line which is a natural model for a diffusion modified by quantum or interference effects. In particular, our results provide surprisingly simple explanations to phenomena observed by Bouwmeester et al. (Phys. Rev. A, 61, 13410 (1999)) in their optical Galton board experiment, and a description of a stroboscopic quantum walks given by Buershaper and Burnett (quant-ph/0406039) through numerical simulations. We also provide heuristic explanations for the behavior of our model which show, in particular, that its dynamics can be viewed as a discrete version of Bloch oscillations.
Quantum walk research has mainly focused on evolutions due to repeated applications of time-independent unitary coin operators. However, the idea of controlling the single particle evolution using time-dependent unitary coins has still been a subject of multiple studies as it not only hosts interesting possibilities for quantum information processing but also opens a much richer array of phenomena including static and dynamic localizations. So far, such studies have been performed only for single quantum walkers. In case of multi-walker systems, time-dependent coins may generate measurable phenomena not described by the single-particle model, due to entanglement and interaction among the walkers. In this context, we present here a thorough numerical study of an one dimensional system of two quantum walkers exhibiting rich collective dynamics controlled by simple time-dependent unitary coins proposed in [Phys. Rev. A \textbf{80}, 042332(2009)] and [Phys. Rev. A \textbf{73},062304(2006)]. We study how the interplay of coin time-dependence, simple interaction schemes, entanglement and the relative phase between the coin states of the particles influences the evolution of the quantum walk. The results show that the system offers a rich variety of collective dynamical behavior while being controlled by time dependent coins. In particular, we find and characterize fascinating two-body localization phenomena with tunable quasiperiodic dynamics of correlations and entanglements which are quantities of quantum origin.
Quantum walk (QW), the quantum mechanical counterpart of classical random walk, has recently been studied in various fields. The evolution of the discrete time quantum walk can be described as follows: the walker changes its spin state by the coin operator C, then takes one step left or right according to its spin state. For homogeneous quantum walk, the coin operator is independent of time and the standard deviation of the position grows linearly in time. It is quadratically faster than that in the classical random walk. In this work, we numerically study the dynamical behaviors of spreading in a one-dimensional discrete time quasiperiodic quantum walk (DTQQW). The DTQQW is that the coin operator is dependent on time and takes two different coins C() and C() arranged in generalized Fibonacci (GF) sequences. The GF sequences are constructed from A by the recursion relation: AAmBn, BA, for m, n are positive integers. They can be classified into two classes according to the wandering exponent . For 0, they belong to the first class, and for 0, they belong to the second class. For one dimensional system, the behaviors of two classes of GF systems are different either for the electronic spectrum of an electron in quasiperiodic potentials or for the quantum phase transitions of the quasiperiodic spin chains. In this paper, we discuss the cases of two different C operators (C();C()) arranged in GF sequences and find that the spreading behaviors are superdiffusion (the standard deviation of the position ~t; 0:5 1) for the two classes of GF DTQQW. For the second class of GF DTQQW, the exponent values are larger than those of the first class of GF DTQQW in the case of two identical C operators. By exploring the probability distribution in the real space, we find that for the first class of GF DTQQW, the probability distributions are almost the same for different initial states and are similar to the classical Gaussian distribution. For the probability distributions of the second class of GF DTQQW, there are two peaks at the two edges and the height of the two peaks can be different for different initial states. They are similar to the ballistic distribution of the homogeneous quantum walk. Therefore, we conclude that for the first class of GF DTQQW, the spreading behaviors are close to those of the classical random walk ( = 0:5) while for the second class of GF DTQQW, they are close to those of the homogeneous quantum walk ( = 1). This result is quite different from the characteristics of the quantum phase transitions in two classes of GF quasiperiodic quantum spin chains.