Quantum walks are known to have nontrivial interaction with absorbing boundaries. In particular, Ambainis et. al. [2] showed that in the (Z, C 1 , H) quantum walk (one-dimensional Hadamard walk) an absorbing boundary partially reflects information. These authors also conjectured that the left absorption probabilities P (1) n (1, 0) related to the finite absorbing Hadamard walks (Z, C 1 , H, {0, n}) satisfy a linear fractional recurrence in n (here P n (1, 0) is the probability that a Hadamard walk particle initialized in |1 |R is eventually absorbed at |0 and not at |n ). This result, as well as a third order linear recurrence in initial position m of P (m) n (1, 0), was later proved by Bach and Borisov [3] using techniques from complex analysis. In this paper we extend these results to general two state quantum walks and three-state Grover walks, while providing a partial calculation for absorption in d-dimensional Grover walks by a d − 1-dimensional wall. In the one-dimensional cases, we prove partial reflection of information, a linear fractional recurrence in lattice size, and a linear recurrence in initial position. arXiv:1905.04239v1 [quant-ph]
Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Q n where n is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for t = O(n), periodic modal mixing for t = O(n 2 ), and stability for t = O(n 3 ). At the end of this paper, we demonstrate the existence of periodic modal mixing in other sufficiently regular quantum systems. arXiv:1909.12680v1 [quant-ph]
In this paper, we study Grover walks on a line with one and two absorbing boundaries. In particular, we present some results for the absorbing probabilities both in a semi-finite and finite line. Analytical expressions for these absorbing probabilities are presented by using the combinatorial approach. These results are perfectly matched with numerical simulations. We show that the behavior of Grover walks on a line with absorbing boundaries is strikingly different from that of classical walks and that of Hadamard walks.
Absorbing boundaries introduce non-unitary dynamics in quantum mechanical systems. In particular, it has been shown that absorbing boundaries partially reflect quantum information in a variety of one-dimensional quantum walks. The methods used to compute absorption probabilities for quantum walks can be extended to calculate absorption probabilities for general finite and discrete quantum mechanical systems. These methods can also be used to compute moments of the associated absorption time distributions. We conclude the paper by discussing a modification of the quantum Zeno effect in these absorption systems.
This paper introduces a waveform design method using Multi-Tone Feedback Frequency Modulation (MT-FFM), a generalization of the single oscillator feedback FM method developed by Tomisawa [1]. The MT-FFM utilizes a collection of K harmonically related oscillators each governed by a design parameter z k which are utilized as a discrete set of parameters that may be modified to generate a richer set of modulation functions than in the single oscillator case. The resulting modulation function is represented using a form of Kapteyn series composed of Generalized Bessel Functions. This paper describes the structure of the MT-FFM waveform, derives the Kapteyn series representation of the waveform's modulation function, and demonstrates the design method with a waveform design example.
The generalized Bessel function (GBF) is a multi-dimensional extension of the standard Bessel function. Two dimensional GBFs have been studied extensively in the literature and have found application in laser physics, crystallography, and electromagnetics. However, a more rigorous treatment of higher-dimensional GBFs is lacking. The GBF exhibits a rich array mathematical structure in regards to its partial differential equation representation, its asymptotic characterization, and its level sets. In this talk/paper, we explore these properties and connect spectral and ambiguity function optimization of a multi-tone SFM signal to finding the location of the roots of these generalized Bessel functions.
The generalized Bessel function (GBF) is a multi-dimensional extension of the standard Bessel function. Two dimensional GBFs have been studied extensively in the literature and have found application in laser physics, crystallography, and electromagnetics. However, a more rigorous treatment of higher-dimensional GBFs is lacking. The GBF exhibits a rich array mathematical structure in regards to its partial differential equation representation, its asymptotic characterization, and its level sets. In this talk/paper, we explore these properties and connect spectral and ambiguity function optimization of a multi-tone SFM signal to finding the location of the roots of these generalized Bessel functions.
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