From cavity QED with quantum gases to optomechanics

We study the nonlinear coupled dynamics of ultra-cold quantum gases trapped in the light field of high Q optical resonators. In the very low temperature limit the quantum nature of both, light and ultra-cold matter play equally important roles. Using the dynamically generated entanglement and properly designed measurements procedures of the light field allows controlled preparation of many-body atomic states as e.g. atom number squeezed states or Schroedinger cat states. If one traps the particles inside the optical cavity, one can create a optical potential, which is a quantized and a dynamical variable itself. In addition it mediates controllable long range interactions. The self-consistent solution for light and particles the includes new classes of quantum many-body states as super-solid states and polaron like excitations. In the deep trap limit the collective coupling of the particles and the field can be tailored to reproduce a wide range of optomechanic Hamiltonians with linear, quadratic or even higher order couplings in an environment very close to zero temperature.