Two-dimensional Bloch oscillations: a Lie-algebraic approach

Mossmann, S.; Schulze, Andreas; Witthaut, Dirk; Korsch, Hans Jürgen

doi:10.1088/0305-4470/38/15/010

Cited by 8 publications

(15 citation statements)

References 22 publications

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“…The aim of the present study is to investigate the transport and localization properties for a bi-chromatic driving extending previous studies by Liu and Zhu [6], Suqing et al [7] and Yashima et al [8]. In addition, we will briefly consider the even less explored realm of polychromatic driving We base our analysis on a Lie-algebraic approach introduced by one of the authors [9] (see also [10] for an extension to two space dimensions). Following Ref.…”

Section: Introductionmentioning

confidence: 91%

Quantum transport and localization in biased periodic structures under bi- and polychromatic driving

Klumpp¹,

Witthaut²,

Korsch³

2007

J. Phys. A: Math. Theor.

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We consider the dynamics of a quantum particle in a one-dimensional periodic potential (lattice) under the action of a static and time-periodic field. The analysis is based on a nearest-neighbor tightbinding model which allows a convenient closed form description of the transport properties in terms of generalized Bessel functions. The case of bichromatic driving is analyzed in detail and the intricate transport and localization phenomena depending on the communicability of the two excitation frequencies and the Bloch frequency are discussed. The case of polychromatic driving is also discussed, in particular for flipped static fields, i.e. rectangular pulses, which can support an almost dispersionless transport with a velocity independent of the field amplitude.

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Section: Introductionmentioning

confidence: 91%

Quantum transport and localization in biased periodic structures under bi- and polychromatic driving

Klumpp¹,

Witthaut²,

Korsch³

2007

J. Phys. A: Math. Theor.

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“…Note that, for general k y , there is a transverse displacement not only from the Berry curvature but also from the group velocity [36,48]. Reversing the force, such that the set of paths run in the opposite direction, over k = (0, k y ) → (−K x , k y ), the displacement becomes…”

Section: A Relation To the Chern Numbermentioning

confidence: 99%

“…One important consequence of dimensionality is that the real-space Bloch oscillations in 2D become Lissajouslike [34][35][36]. For separable potentials, 1D Bloch oscillations along the x and y axes are simply superposed.…”

Section: D Bloch Oscillationsmentioning

confidence: 99%

“…In principle, these effects may be measured directly from the real-space trajectory of such a wave packet, as has been previously proposed [32,33]. However, in two dimensions, Bloch oscillations are complicated [34][35][36], and it can be difficult to disentangle the effects of Berry curvature from the usual effects arising from the group velocity. We propose how this may be overcome using a "time-reversal" protocol.…”

Section: Introductionmentioning

confidence: 99%

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Mapping the Berry curvature from semiclassical dynamics in optical lattices

Price¹,

Cooper²

2012

Phys. Rev. A

196

231

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We propose a general method by which experiments on ultracold gases can be used to determine the topological properties of the energy bands of optical lattices, as represented by the map of the Berry curvature across the Brillouin zone. The Berry curvature modifies the semiclassical dynamics and hence the trajectory of a wave packet undergoing Bloch oscillations. However, in two dimensions these trajectories may be complicated Lissajous-like figures, making it difficult to extract the effects of Berry curvature in general. We propose how this can be done using a "time-reversal" protocol. This compares the velocity of a wave packet under positive and negative external force, and allows a clean measurement of the Berry curvature over the Brillouin zone. We discuss how this protocol may be implemented and explore the semiclassical dynamics for three specific systems: the asymmetric hexagonal lattice, and two "optical flux" lattices in which the Chern number is nonzero. Finally, we discuss general experimental considerations for observing Berry curvature effects in ultracold gases.

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“…As the problem of discrete diffraction in waveguide arrays with curved axis or transverselly-imposed index gradients is analogous to the problem of one-dimensional or two-dimensional Bloch oscillations of electrons in periodic potentials with an applied electric field or of cold atoms in optical lattices, some results are already available in the literature. In particular, in recent works [28][29][30] an algebraic approach has been developed, capable of providing rather general results for wave packet center of mass evolution and wave packet spreading in certain lattice models. In this approach, after the introduction of a dynamical Lie algebra, an explicit form of the evolution operator is first derived, and then the expectation values of operators are calculated in the Heisenberg picture.…”

Section: Beam Propagation In Curved Waveguide Arraysmentioning

confidence: 99%

Discrete diffraction and shape-invariant beams in optical waveguide arrays

Longhi

2009

Phys. Rev. A

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General properties of linear propagation of discretized light in homogeneous and curved waveguide arrays are comprehensively investigated and compared to those of paraxial diffraction in continuous media. In particular, general laws describing beam spreading, beam decay and discrete far-field patterns in homogeneous arrays are derived using the method of moments and the steepest descend method. In curved arrays, the method of moments is extended to describe evolution of global beam parameters. A family of beams which propagate in curved arrays maintaining their functional shape -referred to as discrete Bessel beams-is also introduced. Propagation of discrete Bessel beams in waveguide arrays is simply described by the evolution of a complex q parameter similar to the complex q parameter used for Gaussian beams in continuous lensguide media. A few applications of the q parameter formalism are discussed, including beam collimation and polygonal optical Bloch oscillations.

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Two-dimensional Bloch oscillations: a Lie-algebraic approach

Cited by 8 publications

References 22 publications

Quantum transport and localization in biased periodic structures under bi- and polychromatic driving

Quantum transport and localization in biased periodic structures under bi- and polychromatic driving

Mapping the Berry curvature from semiclassical dynamics in optical lattices

Discrete diffraction and shape-invariant beams in optical waveguide arrays

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