Two-dimensional quantum walk under artificial magnetic field

Yalçınkaya, I.; Gedik, Zafer

doi:10.1103/physreva.92.042324

Cited by 32 publications

(33 citation statements)

References 37 publications

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“…Previous work on DTQWs coupled to electric or magnetic fields [26,27,33] have shown that walks with field values which are rational multiples of 2π ('rational fields') follow very peculiar dynamics. Fig.…”

Section: Simulations Outside the Continuous Limitmentioning

confidence: 99%

Quantum walks and discrete gauge theories

2016

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A particular example is produced to prove that quantum walks can be used to simulate fullfledged discrete gauge theories. A new family of 2D walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local U (1) gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called E × B drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.

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Section: Simulations Outside the Continuous Limitmentioning

confidence: 99%

Quantum walks and discrete gauge theories

2016

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“…For more general space dependent coins there would also be x-dependent coefficients. The first form in (37) is often used in the (theoretical or experimental) construction of walks, and is called a shift-coin decomposition. It is a product of two kinds of operations: On the one hand, the "coin" operations are just the localized unitary operators.…”

Section: Walks Shifts and Coinsmentioning

confidence: 99%

“…where P is a constant projection, which in the second equality has been taken as |1 1|. This special case suffices, because we allow products, and by conjugation with a constant coin we can include also shifts with P = |φ φ|, or like the first factor in (37). A walk will be called decomposable, if it can be written as a finite product…”

Section: Walks Shifts and Coinsmentioning

confidence: 99%

“…Of course, our scheme is gauge invariant in the sense that, up to a gauge transformation, the result depends only on the field, and not on the vector potentials. While there are several examples in the literature [13,16,20,35,37] which describe an external electromagnetic field in accordance with our general scheme, the general case has not been formulated and studied at its natural level of generality. Hence we give a unifying background to the works mentioned, and also help to correct some faulty approaches [no cite].…”

Section: Introductionmentioning

confidence: 99%

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Quantum walks in external gauge fields

Cedzich

Geib

Werner

et al. 2019

Journal of Mathematical Physics

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Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as "minimal coupling", and consists of replacing the momentum operators in the Hamiltonian by modified ones with an added vector potential. In lattice systems it is not so clear how to do this, because there is no continuous translation symmetry, and hence there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degree of freedom, for walks that allow jumps to a finite neighborhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.

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“…Recently, DTQWs on square lattices under artificial magnetic fields have been considered [48]. An artificial or synthetic magnetic field can be simulated as follows [49]: instead of using charged particles in an actual magnetic field, one typically uses neutral particles upon which the effects of a fictitious magnetic field are imposed, e.g.…”

Section: Introductionmentioning

confidence: 99%

Continuous-time quantum walks on planar lattices and the role of the magnetic field

2020

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We address the dynamics of continuous-time quantum walk (CTQW) on planar 2D lattice graphs, i.e. those forming a regular tessellation of the Euclidean plane (triangular, square, and honeycomb lattice graphs). We first consider the free particle: on square and triangular lattice graphs we observe the well-known ballistic behavior, whereas on the honeycomb lattice graph we obtain a sub-ballistic one, although still faster than the classical diffusive one. We impute this difference to the different amount of coherence generated by the evolution and, in turn, to the fact that, in 2D, the square and the triangular lattices are Bravais lattices, whereas the honeycomb one is non-Bravais. From the physical point of view, this means that CTQWs are not universally characterized by the ballistic spreading. We then address the dynamics in the presence of a perpendicular uniform magnetic field and study the effects of the field by two approaches: (i) introducing the Peierls phase-factors, according to which the tunneling matrix element of the free particle becomes complex, or (ii) spatially discretizing the Hamiltonian of a spinless charged particle in the presence of a magnetic field. Either way, the dynamics of an initially localized walker is characterized by a lower spread compared to the free particle case, the larger is the field the more localized stays the walker. Remarkably, upon analyzing the dynamics by spatial discretization of the Hamiltonian (vector potential in the symmetric gauge), we obtain that the variance of the space coordinate is characterized by pseudooscillations, a reminiscence of the harmonic oscillator behind the Hamiltonian in the continuum, whose energy levels are the well-known Landau levels.

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Two-dimensional quantum walk under artificial magnetic field

Cited by 32 publications

References 37 publications

Quantum walks and discrete gauge theories

Quantum walks and discrete gauge theories

Quantum walks in external gauge fields

Continuous-time quantum walks on planar lattices and the role of the magnetic field

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