How to find optimal quantum states for optical micromanipulation and metrology in complex scattering problems: tutorial

The interaction of quantum light with matter is of great importance to a wide range of scientific disciplines, ranging from optomechanics to high-precision measurements. A central issue we discuss here, is how to make optimal use of both the spatial and the quantum degrees of freedom of light for characterizing and manipulating arbitrary observable parameters in a linear scattering system into which suitably engineered light fields are injected. Here, we discuss a comprehensive framework based on a quantum operator that can be assembled solely from the scattering matrix of a system and its dependence on the corresponding local parameter, making this operator experimentally measurable from the far field using only classical light. From this, the effect of quantum light in the near field, i.e., in the vicinity of the target object, can be inferred. Based on this framework, it is straightforward to formulate optimal protocols on how to jointly design both the spatial shape and the quantum characteristics of light for micromanipulation as well as for parameter estimation in arbitrarily complex media. Also, the forces of the quantum vacuum naturally emerge from this formalism. The aim of our tutorial is to bring different perspectives into alignment and thereby build a bridge between the different communities of wave control, quantum optics, micromanipulation, quantum metrology, and vacuum physics.