Space-time crystals and analogue Black Holes in Bose-Einstein condensates

We focus on investigating the many-body physics of Bose-Einstein condensates of atoms and light. The many-body phenomena that we discuss in this thesis contains two different cases. The first case is the space-time crystal in a Bose-Einstein condensate of atoms. We show the observation of a space-time crystal using ultra-cold atoms, where the periodic structure in both space and time are directly visible in the experimental images. The underlying physics in our superfluid can be described ab initio and allows for a clear identification of the mechanism that causes the spontaneous symmetry breaking. Our results pave the way for the usage of space-time crystals for the discovery of novel nonequilibrium phases of matter. Moreover, we investigate the formation of the space-time. We theoretically model this space-time crystal that contains the high-order axial mode driving by the radial breathing mode. Furthermore, we provide the comparisons of the theoretical prediction, the experiment and the numerical simulation for the mode function, the flux and then the mode frequency. We obtain accurate agreements between the experiment, the numerical simulation and our theoretical prediction. The other case is the analogue Schwarzschild black hole in condensates of light. By etching a hole in the mirrors or by placing a scatterer in the center of a cavity, we can create a sink for light. In a Bose-Einstein condensate of photons this sink results in the creation of a so-called radial vortex, which is a two-dimensional analogue of a Schwarzschild black hole. We theoretically investigate the Hawking radiation and the associated greybody factor of this Schwarzschild black hole. In particular, we determine the density-density and velocity-velocity correlation functions of the Hawking radiation, which can be measured by observing the spatial correlations in the fluctuations in the light emitted by the cavity.