In this talk I will describe proposals for quantum simulations in Ion traps and BECs. It will be shown that the radial degree of freedom of strings of trapped ions in the quantum regime may be prepared and controlled accurately through the variation of the external trapping potential while at the same time its properties are measurable with high spatial and temporal resolution. This provides a new test-bed giving access to static and dynamical properties of the physics of quantum-many-body systems and quantum phase transitions that are hard to simulate on classical computers. Furthermore, it allows for the creation of double well potentials with experimentally accessible tunnelling rates and with applications in testing the foundations of quantum physics and precision sensing. A scheme for the study of methods for detecting Unruh-like acceleration radiation effects in a Bose-Einstein condensate in a 1+1 dimensional setup will be descried. In particular, the dispersive effects of the Bogoliubov spectrum on the ideal case of exact thermalization will be explained.