Resonant-tunneling in discrete-time quantum walk

Matsue, Kaname; Matsuoka, Leo; Ogurisu, Osamu; Segawa, Etsuo

doi:10.1007/s40509-017-0151-9

Cited by 14 publications

(16 citation statements)

References 15 publications

(27 reference statements)

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“…Then the unitarity of S 2 (ξ) implies that the perfect transmitting happens iff 1 + ∆e −2iξ = 0. This condition agrees with the result on [13].…”

Section: Then the Eignequation (Esupporting

confidence: 92%

“…The Fourier transform of Σ gives an explicit formula of the S-matrix determined by the matrix C x . Moreover, our arguments are deeply related with the resonant-tunneling of QWs (see [13]). Precisely, a generalized eigenfunction of U will be constructed in ℓ ∞ (Z; C 2 ) (see [6]), and the S-matrix Σ(θ) appears in this eigenfunction.…”

Section: Introductionmentioning

confidence: 82%

“…Let us see a meaning of the result from the view point of a quantum walk's dynamics; resonant-tunneling in discrete-time quantum walk by [13]. We set the following initial state of the quantum walk by…”

Section: 39)mentioning

confidence: 99%

See 2 more Smart Citations

Explicit expression of scattering operator of some quantum walks on impurities

Komatsu¹,

Konno²,

Morioka³

et al. 2019

Preprint

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In this paper, we consider the scattering theory for a one-dimensional quantum walk with impurities which make reflections and transmissions. We focus on an explicit expression of the scattering operator. Our construction of the formula is based on the counting paths of quantum walkers. The Fourier transform of the scattering operator gives an explicit formula of the scattering matrix which is deeply related with the resonant-tunneling for quantum walks.

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“…Then the unitarity of S 2 (ξ) implies that the perfect transmitting happens iff 1 + ∆e −2iξ = 0. This condition agrees with the result on [13].…”

Section: Then the Eignequation (Esupporting

confidence: 92%

Section: Introductionmentioning

confidence: 82%

See 1 more Smart Citation

Explicit expression of scattering operator of some quantum walks on impurities

Komatsu¹,

Konno²,

Morioka³

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this paper, we tend to extend the model from that in [19]: (1) we generalize the connected finite graph, the internal graph; (2) we increase the number of tails, that is, the number of directions for observing the behaviour of scattering on the internal; (3) we observe the distribution of penetration into the internal, that is, a kind of conditional probability on the internal. In the next section, using a simple example, the internal graph is a 3-cycle with two tails, we demonstrate and illustrate what we intend.…”

Section: Introductionmentioning

confidence: 99%

A dynamical system induced by quantum walk

Higuchi¹,

Segawa²

2019

J. Phys. A: Math. Theor.

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We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an ℓ ∞ -infinite external source from one of the tails as the initial state. We show that for any connected internal graph, a stationary state exists, moreover a perfect transmission to the opposite tail always occurs in the long time limit. We also show that the lower bound of the norm of the stationary measure restricted to the internal graph is proportion to the number of edges of this graph. Furthermore when we add more tails (e.g., r-tails) to the internal graph, then we find that from the temporal and spatial global view point, the scattering to each tail in the long time limit coincides with the local one-step scattering manner of the Grover walk at a vertex whose degree is (r + 1). *

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“…Roughly speaking, the generalized eigenfunction is not in H but in ℓ ∞ (Z; C 2 ). This is a generalization of tunneling solutions of DTQWs given in [24]. The S-matrix will be represented by the distorted Fourier transformation and the generalized eigenfunction.…”

Section: 2mentioning

confidence: 99%

Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

Morioka

2019

Rev. Math. Phys.

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We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is construction of generalized eigenfunctions of the time evolution operator. Roughly speaking, the generalized eigenfunctions are not square summable but belong to ℓ ∞ -space on Z. Moreover, we derive a characterization of the set of generalized eigenfunctions in view of the time-harmonic scattering theory. Thus we show that the S-matrix associated with the quantum walk appears in the singularity expansion of generalized eigenfunctions.

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Resonant-tunneling in discrete-time quantum walk

Cited by 14 publications

References 15 publications

Explicit expression of scattering operator of some quantum walks on impurities

Explicit expression of scattering operator of some quantum walks on impurities

A dynamical system induced by quantum walk

Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

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