Spatiotemporal deformations of reflectionless potentials

Horsley, S. A. R.; Longhi, Stefano

doi:10.1103/physreva.96.023841

Cited by 25 publications

(27 citation statements)

References 41 publications

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“…Such a result has a deep physical consequence and indicates that discrete wave dynamics is distinct in different inertial reference frames. For example, recent works have predicted that reflectionless potentials of the continuous Schrödinger equation, such as Kramers-Kronig potentials p-1 [34,35], become reflective on a lattice when they are at rest [36], while potentials that are reflective at rest can become fully transparent when they move on the lattice at sufficiently high speeds [37]. This is of course in contrast with the continuous limit of the Schrödinger equation because wave scattering from a potential barrier or well is invariant for a Galilean boost: a potential barrier can not become reflectionless by just observing the scattering process in a moving reference frame!…”

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confidence: 99%

Bound states of moving potential wells in discrete wave mechanics

Longhi¹

2017

EPL

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-Discrete wave mechanics describes the evolution of classical or matter waves on a lattice, which is governed by a discretized version of the Schrödinger equation. While for a vanishing lattice spacing wave evolution of the continuous Schrödinger equation is retrieved, spatial discretization and lattice effects can deeply modify wave dynamics. Here we discuss implications of breakdown of exact Galilean invariance of the discrete Schrödinger equation on the bound states sustained by a smooth potential well which is uniformly moving on the lattice with a drift velocity v. While in the continuous limit the number of bound states does not depend on the drift velocity v, as one expects from the covariance of ordinary Schrödinger equation for a Galilean boost, lattice effects can lead to a larger number of bound states for the moving potential well as compared to the potential well at rest. Moreover, for a moving potential bound states on a lattice become rather generally quasi-bound (resonance) states.Introduction. -One of the cornerstones of nonrelativistic quantum mechanics is the Schrödinger equation, which describes the temporal evolution of a particle wave function based on its initial state. Traditionally, in wave mechanics space and time are considered continuous. However, on several occasions authors have debated about the nature of space-time manifold and the possibility of considering variant forms of the Schrödinger equation in which the wave function is defined on discrete lattice sites of space, time, or space-time, instead of on the space-time continuum [1][2][3][4][5][6][7][8][9][10][11][12]. Fundamental limits to a minimum measurable length were suggested in the early days of quantum physics, notably by Heisenberg, and appear, for example, in modern theories of loop quantum gravity, where spacetime looks granular [13]. A phenomenological approach to account for a minimum length scale is to consider an extension of the uncertainty principle by deforming the canonical commutation relations of position and momentum operators [14], leading to an extended form of the Schödinger equation [15,16]. Other simple models consider the non-relativistic Schrödinger equation defined on a discrete lattice [7][8][9][10][11][12][17][18][19][20][21][22], leading to so-called discrete wave mechanics [7] or discrete quantum mechanics [20]. While earlier models of discrete wave mechanics [7][8][9][10][11][12] did not find great relevance as foundational theories, in

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confidence: 99%

Bound states of moving potential wells in discrete wave mechanics

Longhi¹

2017

EPL

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“…Kramers-Kronig potentials are a rather broad class of unidirectionally or bidi-rectionally reflectionless complex potentials, introduced by Horsley and coworkers in a recent work [63], in which the real and imaginary parts of the potentials are related one another by spatial Kramers-Kronig relations. Such potentials show rather interesting properties, such as unidirectional or bidirectional transparency, invisibility, perfect absorption, and robustness to spatio-temporal deformations, which have been investigated in several recent works [54,63,[65][66][67][68][69][70][71][72][73][74]. The main result of our study is that all classes of reflectionless potentials mentioned above become reflective when probed by an accelerated wave-packet.…”

Section: Introductionmentioning

confidence: 77%

Scattering of accelerated wave packets

2018

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Wave-packet scattering from a stationary potential is significantly modified when the wave packet is subject to an external time-dependent force during the interaction. In the semiclassical limit, wave-packet motion is simply described by Newtonian equations, and the external force can, for example, cancel the potential force, making a potential barrier transparent. Here we consider wave-packet scattering from reflectionless potentials, where in general the potential becomes reflective when probed by an accelerated wave packet. In the particular case of the recently introduced class of complex Kramers-Kronig potentials we show that a broad class of timedependent forces can be applied without inducing any scattering, while there is a breakdown of the reflectionless property when there is a broadband distribution of initial particle momentum, involving both positive and negative components.

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“…From the appearance of the permittivity profile only, we can explain how the rays are laterally shifted, but not why there is also no reflection from such a medium. However this is a general feature of the graded-index media; it is not easy to see why certain profiles with arbitrarily large contrasts are reflectionless, even in one dimension (such as the spatial Kramers-Kronig media [1,2] or the Pöschl-Teller media [5,6]). …”

Section: The Beam Shiftermentioning

confidence: 99%

“…Instead, mathematical techniques have been used to make progress, particularly with a view to designing nonscattering media. In one dimension, media whose graded-index susceptibility satisfies the spatial Kramers-Kronig relations are unidirectionally reflectionless for all angles of incidence [1,2] and remain reflectionless in two dimensions when their profiles are rescaled and translated along a second spatial coordinate. The analogous problem in higher dimensions is much harder to solve.…”

Section: Introductionmentioning

confidence: 99%

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Designing scattering-free isotropic index profiles using phase-amplitude equations

2018

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The Helmholtz equation can be written as coupled equations for the amplitude and phase. By considering spatial phase distributions corresponding to reflectionless wave propagation in the plane and solving for the amplitude in terms of this phase, we designed two-dimensional graded-index media which do not scatter light. We give two illustrative examples, the first of which is a periodic grating for which diffraction is completely suppressed at a single frequency at normal incidence to the periodicity. The second example is a medium which behaves as a "beam shifter" at a single frequency; acting to laterally shift a plane wave, or sufficiently wide beam, without reflection.

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Spatiotemporal deformations of reflectionless potentials

Cited by 25 publications

References 41 publications

Bound states of moving potential wells in discrete wave mechanics

Bound states of moving potential wells in discrete wave mechanics

Scattering of accelerated wave packets

Designing scattering-free isotropic index profiles using phase-amplitude equations

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