Topological Considerations

D. J. Thouless

Sami M. Al-Jaber

Steven R. White

Mauro Antezza

We investigate both entanglement and energy transport properties for two-level systems in the vicinity of a photonic topological insulator (PTI) interface, which supports a nonreciprocal (unidirectional), scattering-immune and topologically-protected surface plasmon polariton in the bandgap of the bulk material. Moreover, we demonstrate that despite the presence of considerable imperfections at the interface of the PTI, the efficiency of the SPP-assisted energy transport is almost unaffected by discontinuities. We also show that the SPP properties allow energy transport over considerably much larger distances then in the reciprocal case, and we point out a particularly simple way to tune the transport. Finally, we analyze the specific case of a two-emitter-chain and unveil the origin of the efficiency amplification. The efficiency amplification and the practical advantages highlighted in this work might be particularly useful in the development of new devices intended to manage energy at the atomic scale, e.g. in quantum technologies.

Peter Hertel

Sohang Gandhi

It is well known that the standard canonical uncertainty relation does not apply to the angular variable ? and its conjugate LZ. That is, the relation ? ø ? L Z > h/2 is false. The break down of the result has to do with difference in topology between the line and the circle. It is thus desirable to generalize the standard uncertainty relation topologically and find satisfactory results for the non-Euclidean spaces. This problem is intimately related to the issue of finding a consistent definition for quantum mechanics on "curved spaces". Just as the Heisenberg uncertainty relation was pivotal in understanding the basic structure of standard quantum mechanics, a solution to this problem should shine some light onto the proper conduct of quantum mechanics on general topological spaces. In this study we explore in detail how the standard uncertainty relation may breakdown. We also address the importance of topological considerations in quantum mechanics in general - we shall show how a change in topological character can change the nature of the quantum mechanics for a system and how the consideration of the topology of a system can greatly organize the solution of a problem and in some cases even be necessary for a. full understanding of the problem. We then discuss the derivation of satisfactory uncertainty relations for the compact, homogeneous spaces of the circle, the n-torus and the n-sphere. Finally, we draw out any implications to the issue of properly defining quantum mechanics on the non- Euclidean spaces.