Controlling light with non-Hermitian Schrödinger dynamics

从数据传输的医疗应用到量子计算的新技术，许多现代技术都依赖于光波导中的光控制。因此，控制机制的任何改进都会产生很大的连锁反应。直观地说，为了最大限度地提高任何光学设备的效率，人们会寻求最小化吸收和损耗，几十年来（如果不是几个世纪的话），这一直是控制方案设计的指导原则。直到最近，人们才开始探索利用工程损耗来主动控制动态的想法，并带来了大量的新应用。支撑这些想法的数学是从基础量子物理学中借来的，从一个被称为非厄米量子理论和pt对称量子理论的领域。虽然在波导中实现由非厄米哈密顿量产生的量子动力学在过去十年中取得了重大成功，但许多重要方面仍未得到探索。特别是在没有损失的情况下非常重要的显式时间相关方案的应用，由于其非平凡的基础数学，很少被研究。为了将应用推进到下一个水平，必须理解时间相关的非厄米量子系统的数学基础，并使物理应用的实践者能够使用。这项拨款的主要目标是对非厄米哈密顿人描述的系统进行分类，然后利用其丰富的数学性质，这些系统具有明确的时间相关参数。这是一项具有挑战性的任务，因为量子绝热定理，作为大多数厄米系统的基础，在非厄米系统中失效了。最近，我们利用动力系统的语言和几何学和群论的工具，在具体的模型系统上取得了实质性的进展。我们将在此基础上实现为下一代非厄米波导应用提供数学基础的愿景。

Much of modern technology, ranging from medical applications over data transmission to novel technologies for quantum computing, rely on the control of light in optical waveguides. Thus, any improvement in the control mechanisms has a large ripple on effect. Intuitively, to maximise efficiency of any optical device, one would seek to minimise absorption and losses, and for decades (if not centuries) this has been a guiding principle in the design of control schemes. Only fairly recently has the idea of using engineered losses to actively control dynamics been explored, and has led to a spectacular amount of new applications. The mathematics underpinning these ideas is borrowed from fundamental quantum physics from a field known as non-Hermitian and PT-symmetric quantum theory. While the implementation of quantum dynamics generated by non-Hermitian Hamiltonians in waveguides has been a major success story over the last decade, many important aspects remain unexplored. In particular the application of explicitly time-dependent schemes that are of great importance in the absence of losses, has been little investigated, due to its nontrivial underlying mathematics. To propel the applications to the next level, it is imperative that the mathematical foundations of time-dependent non-Hermitian quantum systems are understood and made accessible to practitioners in physical applications. The main goal of this grant is to categorise and then exploit the rich mathematical nature of systems described by non-Hermitian Hamiltonians with explicitly time-dependent parameters. This is a challenging task, since the quantum adiabatic theorem, that provides the foundations of most Hermitian systems, breaks down in the non-Hermitian case. Recently we were able to make substantial progress for specific model systems, using the language of dynamical systems and tools from geometry and group theory. We will build on this to realise the vision of providing the mathematical foundations for next generation non-Hermitian waveguide applications.