An exactly soluble model of a shallow double well

We present a framework for the construction of solvable models of optical settings with genuinely twodimensional landscapes of refractive index. Solutions of the associated non-separable Maxwell equations in paraxial approximation are found using the time-dependent supersymmetry. We discuss peculiar theoretical aspects of the construction. In particular, we focus on the existence of localized solutions specific for the new systems. Sufficient conditions for their existence are discussed. Localized solutions vanishing for large | x|, which we call light dots, as well as the guided modes that vanish exponentially outside the wave guides, are constructed. We consider different definitions of the parity operator and analyze general properties of the PTsymmetric systems, e.g. presence of localized states or existence of symmetry operators. Despite the models with parity-time symmetry are of the main concern, the proposed framework can serve for construction of non-PT -symmetric systems as well. We explicitly illustrate the general results on a number of physically interesting examples, e.g. wave guides with periodic fluctuation of refractive index or with a localized defect, curved wave guides, two coupled wave guides or a uniform refractive index system with a localized defect. arXiv:1808.08225v2 [math-ph]

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“…It corresponds to two parallel wave guides that were discussed in [73]. We can find two guided modes of S 2 .…”

Section: Coupled Wave Guidesmentioning

confidence: 80%

“…Figure 2: Optical wave guide with a localized defect. Plots of the real (a) and imaginary (b) parts of V 1 , see (73), when L 1 = √ 1 + z 2 . In (c) the imaginary part of V 1 when L 1 = 1, see (19) and (70).…”

Section: Optical Wave Guide With a Localized Defectmentioning

confidence: 99%

Photonic systems with two-dimensional landscapes of complex refractive index via time-dependent supersymmetry

Contreras-Astorga

Jakubský

2019

Phys. Rev. A

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“…The potential confinement experienced by electrons in the double quantum dot (DQD) device is well described or approximated by the two-dimensional potential function owing to the high degree of confinement in ẑ direction 25 . However, the availability of exactly solvable double-well potential solutions is limited to onedimensional solutions [37][38][39][40][41][42] and has not yet been developed for two-dimensional solutions. Therefore, in this study, we will adopt the solution of the one-dimensional double-well potential Schrödinger equation presented by Caticha 37 to investigate the exchange coupling in ST 0 qubit architecture.…”

Section: A Conventional Methods Of Describing Electron Statesmentioning

confidence: 99%

On the validity of microscopic calculations of double-quantum-dot spin qubits based on Fock-Darwin states

Xuan

Wang

2018

Sci. China Phys. Mech. Astron.

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We consider two typical approximations that are used in the microscopic calculations of doublequantum dot spin qubits, namely, the Heitler-London (HL) and the Hund-Mulliken (HM) approximations, which use linear combinations of Fock-Darwin states to approximate the two-electron states under the double-well confinement potential. We compared these results to a case in which the solution to a one-dimensional Schrödinger equation was exactly known and found that typical microscopic calculations based on Fock-Darwin states substantially underestimate the value of the exchange interaction, which is the key parameter that controls the quantum dot spin qubits. This underestimation originates from the lack of tunneling of Fock-Darwin states, which is accurate only in the case with a single potential well. Our results suggest that the accuracies of the current two-dimensional molecular-orbit-theoretical calculations based on Fock-Darwin states should be revisited since underestimation could only deteriorate in dimensions that are higher than one. arXiv:1801.00902v1 [quant-ph]

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“…As a second illustrative example, let us consider a potential V (x) that describes a binary lattice and assume that in the interval (lattice period) −a/2 ≤ x < a/2 the potential V (x) is approximated by the reflectionless double-well potential [101,102] V (x) V a (x) = 2(σ 2 −1)…”

Section: B the Binary (Double-well) Potentialmentioning

confidence: 99%

Non-Hermitian skin effect beyond the tight-binding models

Longhi

2021

Preprint

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The energy bands of non-Hermitian systems exhibit nontrivial topological features that arise from the complex nature of the energy spectrum. Under periodic boundary conditions (PBC), the energy spectrum describes rather generally closed loops in complex plane, characterized by integer nonzero winding numbers. Such nontrivial winding provides the topological signature of the non-Hermitian skin effect (NHSE), i.e. the macroscopic condensation of bulk states at the lattice edges under open boundary conditions (OBC). In spite of the great relevance of band winding in the non-Hermitian topological band theory and the related NHSE, most of current results rely on tight-binding models of non-Hermitian systems, while exact Bloch wave function analysis of the NHSE and related topological band theory is still lacking. While tight-binding models can correctly describe narrow-band electronic states with a relatively weak degree non-Hermiticity, they are not suited to describe high-energy wide-band electronic states and/or regimes corresponding to strong non-Hermiticity. Here we consider the single-particle continuous Schrödinger equation in a periodic potential, in which non-Hermiticity is introduced by an imaginary vector potential in the equation, and show that the NHSE is ubiquitous under OBC and characterized by a non-vanishing integer winding number, even thought the energy spectrum under PBC always comprises an open curve, corresponding to high-energy electronic states. We also show that the interior of the PBC energy spectrum corresponds to the complex eigenenergies sustaining localized (edge) states under semiinfinite boundary conditions.

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An exactly soluble model of a shallow double well

Cited by 5 publications

References 30 publications

Photonic systems with two-dimensional landscapes of complex refractive index via time-dependent supersymmetry

Photonic systems with two-dimensional landscapes of complex refractive index via time-dependent supersymmetry

On the validity of microscopic calculations of double-quantum-dot spin qubits based on Fock-Darwin states

Non-Hermitian skin effect beyond the tight-binding models

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