Local symmetries in one-dimensional quantum scattering

Abstract

We introduce the concept of parity symmetry in restricted spatial domains -- local parity -- and explore its impact on the stationary transport properties of generic, one-dimensional aperiodic potentials of compact support. It is shown that, in each domain of local parity symmetry of the potential, there exists an invariant quantity in the form of a non-local current, in addition to the globally invariant probability current. For symmetrically incoming states, both invariant currents vanish if weak commutation of the total local parity operator with the Hamiltonian is established, leading to local parity eigenstates. For asymmetrically incoming states which resonate within locally symmetric potential units, the complete local parity symmetry of the probability density is shown to be necessary and sufficient for the occurrence of perfect transmission. We connect the presence of local parity symmetries on different spatial scales to the occurrence of multiple perfectly transmitting resonances and propose a construction scheme for the design of resonant transparent aperiodic potentials. Our findings are illustrated through application to the analytically tractable case of piecewise constant potentials.

10 pages, 7 figures

References in corpus (7)

• Symmetry implies independence
• Perfect transmission and highly asymmetric light localization in photonic multilayers
• One-Dimensional Kronig-Penney Model with Positional Disorder: Theory versus Experiment
• Local Symmetries and Order-Disorder Transitions in Small Macroscopic Wigner Islands
• Anomalous localisation in the aperiodic Kronig-Penney model
• Magnetoconductance switching in an array of oval quantum dots
• Transmission in waveguides with compositional and structural disorder: experimental effects of disorder cross-correlations

Cited by in corpus (17)

• Compact localized states and flat bands from local symmetry partitioning
• Local symmetries and perfect transmission in aperiodic photonic multilayers
• Invariants of broken discrete symmetries
• Latent symmetry induced degeneracies
• Quantum network transfer and storage with compact localized states induced by local symmetries
• Invariant currents and scattering off locally symmetric potential landscapes
• $\mathcal{P}\mathcal{T}$ symmetry breaking in waveguide arrays with competing loss/gain pairs
• Local symmetry dynamics in one-dimensional aperiodic lattices
• Nonlocal discrete continuity and invariant currents in locally symmetric effective Schrödinger arrays
• Systematic pathway to $\mathcal{P}\mathcal{T}$ symmetry breaking in scattering systems
• Non-Local Currents and the Structure of Eigenstates in Planar Discrete Systems with Local Symmetries
• Dynamics of Local Symmetry Correlators for Interacting Many-Particle Systems
• Transfer efficiency enhancement and eigenstate properties in locally symmetric disordered finite chains
• Super-Lagrangian and variational principle for generalized continuity equations
• Robustness of perfect transmission resonances to asymmetric perturbation
• Duality of bounded and scattering wave systems with local symmetries
• Non-local divergence-free currents for the account of symmetries in two-dimensional wave scattering