Approximate quantum cloaking and almost trapped states

TL;DR

This paper introduces approximate quantum cloaks derived from transformation optics that can hide potentials from matter waves and support almost trapped states, with applications in ion trapping and quantum system simulation.

Contribution

It presents a novel method to create approximate quantum cloaks for matter waves using isotropic transformation optics, enabling undetectability and trapping of quantum states.

Findings

Approximate cloaks make potentials undetectable to matter waves at most energies.

Cloaks support almost trapped states with negligible external wave influence.

Potential applications include ion traps and quantum system simulations.

Abstract

We describe families of potentials which act as approximate cloaks for matter waves, i.e., for solutions of the time-independent Schr\"odinger equation at energy $E$, with applications to the design of ion traps. These are derived from perfect cloaks for the conductivity and Helmholtz equations, by a procedure we refer to as isotropic transformation optics. If $W$ is a potential which is surrounded by a sequence $\{V_n^E\}_{n=1}^\infty$ of approximate cloaks, then for generic $E$, asymptotically in $n$ (i) $W$ is both undetectable and unaltered by matter waves originating externally to the cloak; and (ii) the combined potential $W+V_n^E$ does not perturb waves outside the cloak. On the other hand, for $E$ near a discrete set of energies, cloaking {\it per se} fails and the approximate cloaks support wave functions concentrated, or {\it almost trapped}, inside the cloaked region and negligible outside. Applications include ion traps, almost invisible to matter waves or customizable to support almost trapped states of arbitrary multiplicity. Possible uses include simulation of abstract quantum systems, magnetically tunable quantum beam switches, and illusions of singular magnetic fields.