Absence of Floquet scattering in oscillating non-Hermitian potential wells

Abstract: Scattering of a quantum particle from an oscillating barrier or well does not generally conserve the particle energy owing to energy exchange with the photon field, and an incoming particle-free state is scattered into a set of outgoing (transmitted and reflected) free states according to Floquet scattering theory. Here we introduce two families of oscillating non-Hermitian potential wells in which Floquet scattering is fully suppressed for any energy of the incident particle. The scattering-free oscillating potentials are synthesized by application of the Darboux transformation to the time-dependent Schr\"{o}dinger equation. For one of the two families of scattering-free potentials, the oscillating potential turns out to be fully invisible.

• The paper demonstrates that applying the Darboux transformation to the time-dependent Schrödinger equation creates potential wells that completely suppress Floquet scattering.
• Numerical pseudospectral simulations confirm that these non-Hermitian potentials prevent energy channel mixing unlike conventional Hermitian systems.
• The study suggests that uni-directional invisibility in oscillating potentials could inspire novel designs in quantum transport and synthetic optical structures.

Absence of Floquet Scattering in Oscillating Non-Hermitian Potential Wells

Introduction

This paper addresses the scattering phenomena in oscillating non-Hermitian potential wells, specifically focusing on Floquet scattering theory. The study introduces two families of oscillating non-Hermitian potential wells where Floquet scattering is completely suppressed. This suppression occurs due to the application of the Darboux transformation to the time-dependent Schrödinger equation. The paper highlights an intriguing feature: one of the families of potential wells exhibits complete invisibility in the context of oscillating potentials.

Floquet Scattering Overview

Floquet scattering involves a quantum particle interacting with oscillating barriers wherein energy is not conserved due to interaction with an external photon field. This generally results in the scattered state comprising multiple energy channels, each with a probability amplitude derived from the initial energy. Conventionally, an incident free state can lead to reflected and transmitted states at different energy levels. The paper revisits this concept under the influence of non-Hermitian potentials showing no Floquet scattering, which could redefine existing understandings in quantum physics.

Methodology: Darboux Transformation

The core method utilized is the Darboux transformation, which is instrumental in constructing these non-standard potential wells. A first-order Darboux transformation operators $\hat{\mathcal{D}}$, defined in terms of a solution to a primary time-dependent Schrödinger equation, reveals new potentials that are solutions themselves. By choosing specific forms for this solution, two families of potentials that suppress Floquet states emerge.

1. Complex Exponential Potential: This form uses solutions based upon hyperbolic functions encapsulating parameters $\alpha$, $\beta$, and $\mu$, forming potentials where time variations manage all scattering channels to nullify deviations from initial energy.
2. Trigonometric Potential: Implementing sinusoidal and cosine dependencies characterized by parameter choices aligns the potential in ways enforcing invisibility. Here, the potentials ensure no reflected or altered energy channels appear, maintaining a unitary transition.

Practical Implications and Numerical Confirmation

The paper emphasizes that despite solving for non-Hermitian potentials, they exhibit uni-directional invisibility, crucially distinct from PT-symmetric scenarios where unidirectionality is typically a feature. Through numerical simulations using pseudospectral techniques, the wave packet propagation verifies theoretical predictions demonstrating the absence of Floquet scattering empirically.

• Complex and Hermitian potentials were compared, showing that while Hermitian potentials cause considerable Floquet states and scattering, the derived non-Hermitian ones presented no such complexity.
• An important experimental implication exists about quantum systems' studies and diffraction scenarios, potentially paving the path to further realizations that incorporate gain/loss profiles like synthetic optical structures.

Conclusion

The study opens new avenues in the investigation of quantum scattering, especially the non-Hermitian systems largely unexplored in oscillating potentials. Future efforts directed towards synthesizing such potentials could lead to innovative applications in mesoscopic quantum transport and optics. Unveiling static and oscillating potential interactions without scattering beckons further foundational scrutiny possibly extending towards synthesis strategies in Hermitian realms as well.