On Tight‐Binding Approximations in Optical Lattices

Ablowitz, Mark J.; Curtis, Christopher W.; Zhu, Yi

doi:10.1111/j.1467-9590.2012.00558.x

Cited by 30 publications

(82 citation statements)

References 22 publications

(62 reference statements)

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“…In the deep lattice limit the scalar field φ(r, z) is well-approximated by coupled evanescent modes centered at the lattice sites. Rigorous studies of these types of approximations were carried out in [25].…”

Section: Derivation Of Tight-binding Approximationmentioning

confidence: 99%

Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons

Ablowitz

Cole

2017

Phys. Rev. A

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A systematic approach for deriving tight-binding approximations in general longitudinally driven lattices is presented. As prototypes, honeycomb and staggered square lattices are considered. Timereversal symmetry is broken by varying/rotating the waveguides, longitudinally, along the direction of propagation. Different sublattice rotation and structure are allowed. Linear Floquet bands are constructed for intricate sublattice rotation patterns such as counter rotation, phase offset rotation, as well as different lattice sizes and frequencies. An asymptotic analysis of the edge modes, valid in a rapid-spiraling regime, reveals linear and nonlinear envelopes which are governed by linear and nonlinear Schrödinger equations, respectively. Nonlinear states, referred to as topologically protected edge solitons are unidirectional edge modes. Direct numerical simulations for both the linear and nonlinear edge states agree with the asymptotic theory. Topologically protected modes are found; they possess unidirectionality and do not scatter at lattice defect boundaries.

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Section: Derivation Of Tight-binding Approximationmentioning

confidence: 99%

Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons

Ablowitz

Cole

2017

Phys. Rev. A

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“…We then define φ 1,v = φ Again, please see Equations (11,12), and (13) for reference. The term c (tb) 0 is given by (14) andẼ is defined in (10).…”

Section: Discrete and Dirac Systems For Imaginary-mass Potentialsmentioning

confidence: 99%

“…A plot of this function is in Figure 3 to the left by v 12 = v 1 + v 2 , and use the fact that 2r 1 + d = v 12 , then we get thatW…”

Section: Discrete and Dirac Systems For Imaginary-mass Potentialsmentioning

confidence: 99%

“…In the tight-binding limit, see Refs. [11] and [12] for more details, the nonlinearities, after transformation are approximated by To leading order then, we have the nonlinear equation for the slowly evolving envelopes…”

Section: Dirac Systems For Real-mass Potentialsmentioning

confidence: 99%

“…In response to this ever growing body of physical literature, mathematical and theoretical studies of these effects have appeared in Refs. [11][12][13][14][15][16][17] and [18,19].…”

Section: Introductionmentioning

confidence: 99%

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Dynamics in ‐Symmetric Honeycomb Lattices with Nonlinearity

Curtis

Zhu

2015

Stud Appl Math

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We examine the impact of small parity-time (PT ) symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of PT perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the PT perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing PT perturbations offer potentially novel means for the control of light in honeycomb lattices.

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Honeycomb Schrödinger Operators in the Strong Binding Regime

Fefferman

Lee-Thorp

Weinstein

2017

Comm Pure Appl Math

107

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In this article, we study the Schrödinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues. We consider this Schrödinger operator in the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model (Wallace, 1947 [56]). Furthermore, we establish as corollaries, in the regime of strong binding, results on (a) the existence of spectral gaps for honeycomb potentials that break PT symmetry and (b) the existence of topologically protected edge states-states that propagate parallel to and are localized transverse to a line defect or "edge"-for a large class of rational edges, and that are robust to a class of large transverse-localized perturbations of the edge. We believe that the ideas of this article may be applicable in other settings for which a tight-binding model emerges in an extreme parameter limit.We begin with a review of Floquet-Bloch theory. See, for example, [11,37,38,49] and [3,23,24,34,53]. Fourier Analysis on L 2 .R=ƒ/Let fv 1 ; v 2 g be a linearly independent set in R 2 , and introduce the following:Lattice: ƒ D Zv 1˚Z v 2 D fm 1 v 1 C m 2 v 2 W m 1 ; m 2 2 ZgI Dual lattice: ƒ D ZK 1˚Z K 2 D fm E K

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On Tight‐Binding Approximations in Optical Lattices

Cited by 30 publications

References 22 publications

Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons

Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons

Dynamics in ‐Symmetric Honeycomb Lattices with Nonlinearity

Honeycomb Schrödinger Operators in the Strong Binding Regime

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