Bose-Einstein Condensates in Curved Space-Time – From Concepts of General Relativity to Tidal Corrections for Quantum Gases in Local Frames
Gabel, Oliver
Measuring effects of General Relativity and beyond in the gravitational field of the Earth is a main goal of current research and cutting-edge technology. These effects, predicted by Einstein’s theory, already play an important role in everyday life, for example in enabling for precise positioning and time keeping in global satellite navigation systems, such as GPS and GALILEO. The breakthrough experimental realisation of Bose-Einstein condensation in 1995, some 70 years after Einstein’s prediction, has since established matter-wave interferometers in laboratories worldwide, in which laser pulses are used to coherently split, reflect, and recombine a Bose-Einstein condensate. While already enabling highly accurate quantum sensors for technological applications as accelerometers, gyroscopes, and gravity gradiometers, matter-wave interferometers with their unprecedented potential sensitivity are highly anticipated to serve as formidable quantum probes of fundamental physics. For this reason, concentrated international efforts are currently under way to develop this promising technology into robust and sensitive instruments. The German QUANTUS collaboration is at the forefront of this development, having demonstrated the first Bose-Einstein condensates and matter-wave interferometers in free fall, and having recently achieved the very first BEC in space on the sounding-rocket mission MAIUS-1 in early 2017. As its long-time goal, QUANTUS is aiming at a quantum test of Einstein’s famous Equivalence Principle, which is at the heart of General Relativity as a geometric theory of gravity. In this context, it is relevant to develop a precise description of free fall in Earth’s gravity beyond the usual Newtonian approximation, and thus to take into account the full reality of curved space-time in terms Einstein’s theory of General Relativity. In this thesis, we take a grand tour of the relevant concepts of Special and General Relativity and eventually apply these to the modelling of free falling quantum gases. We base our description on the experimentally relevant local inertial and non-inertial frames which we can think of as moving along with experiments in free fall, for example in a drop tower, or in satellites orbiting the Earth, such as the International Space Station ISS. Our main tool are Fermi normal coordinates attached to these frames, which provide a local curvature expansion around flat space-time that exhibits local tidal effects, and can thus be seen as an expansion around the Equivalence Principle. Being fairly under-represented in the literature, we extensively discuss these Fermi coordinates, as well as the so-called Riemann normal coordinates on which they are built. In particular, we provide a new combinatorial interpretation for the complicated polynomials in the Riemann tensor and its derivatives, which arise in the expansion for the tetrads and the metric in these coordinates. We finally apply these methods to the mean-field description of free falling Bose-Einstein condensates in the gravitational field of the Earth. Modelling the space-time curvature around our planet in terms of the Schwarzschild metric, we explicitly calculate the metric in Fermi coordinates for local inertial frames in free fall along purely radial geodesics, which approximates the experimental situation in a drop tower, as well as along circular equatorial geodesics which can be used to model the situation on satellites, such as the ISS. We then use these metrics in the non-linear Klein-Gordon equation which can be seen to generalise the usual Gross-Pitaevskii equation to curved space-time. Performing the non-relativistic limit, we obtain the different local tidal-type Newtonian and relativistic corrections and discuss their orders of magnitude.
•Jan 1, 2019•2 citations