We analyze the evolution of a degenerate quantum gas (bosons and fermions) falling in Earth's gravity during long times $(10\phantom{\rule{0.3em}{0ex}}\mathrm{s})$ and over large distances $(100\phantom{\rule{0.3em}{0ex}}\mathrm{m})$. This models an experiment that is currently performed by the QUANTUS Collaboration at ZARM drop tower in Bremen, Germany. Starting from the classical mechanics of the drop capsule and a single particle trapped within, we develop a quantum field theoretical description for this experimental situation in an inertial frame, the corotating frame of the Earth, as well as the comoving frame of the drop capsule. Suitable transformations eliminate noninertial forces, provided all external potentials (trap, gravity) can be approximated with a second-order Taylor expansion around the instantaneous trap center. This is an excellent assumption, and the harmonic potential theorem applies. As a first application, we study the quantum dynamics of a cigar-shaped Bose-Einstein condensate in the Gross-Pitaevskii mean-field approximation. Due to the instantaneous transformation to the rest frame of the superfluid wave packet, the long-distance drop can be studied easily on a numerical grid.
We analyze the evolution of a degenerate quantum gas (bosons and fermions) falling in Earth's gravity during long times (10s) and over large distances (100m). This models an experiment that is currently performed by the QUANTUS Collaboration at ZARM drop tower in Bremen, Germany. Starting from the classical mechanics of the drop capsule and a single particle trapped within, we develop a quantum field theoretical description for this experimental situation in an inertial frame, the corotating frame of the Earth, as well as the comoving frame of the drop capsule. Suitable transformations eliminate noninertial forces, provided all external potentials (trap, gravity) can be approximated with a second-order Taylor expansion around the instantaneous trap center. This is an excellent assumption, and the harmonic potential theorem applies. As a first application, we study the quantum dynamics of a cigar-shaped Bose-Einstein condensate in the Gross-Pitaevskii mean-field approximation. Due to the instantaneous transformation to the rest frame of the superfluid wave packet, the long-distance drop can be studied easily on a numerical grid.
In this thesis, we study the dynamics of ultracold quantum gases with respect to micro-gravity experiments, discuss interferometric applications and aspects on collective scattering. We describe the dynamics of an ultracold quantum gas in a long-distance free-fall experiment, which is of relevance for the present QUANTUS experiment. We develop the quantum field theoretical description of a trapped, interacting degenerate quantum gas in a drop experiment in an inertial frame, the corotating frame of the Earth and the comoving frame of the drop capsule. This formalism provides us with an efficient description of the experiment, especially for numerical studies. Moreover, we review the ideas of gravitational sensing with the help of an interferometric setup. We show that in case of a spatially quasi-homogeneous condensate the same results as in single-atom interferometry can be obtained even in the presence of a nonlinear time evolution. A decline of the contrast in an interferogram is found, though. In addition, we propose a rotational sensor based on a superposition of a non-rotating state and a quantized vortex in a Bose-Einstein condensate. As an application for an interferometric experiment in a micro-gravity environment, we establish a number filter for matter waves. Our number filter allows for a number stabilization, i.e. after passing the filter, the number uncertainty is less than before. This can be achieved by using a nonlinear matter wave interferometer for a two-component BEC. An asymmetric splitting is required in order to observe the effect. Finally, we examine the collective scattering of a superfluid droplet impinging on a two-component Bose-Einstein condensate. We demonstrate that this mesoscopic scenario matches the microscopic setup for Feshbach scattering of two particles. We obtain resonant scattering phase shifts from a linear response theory. We find an energy-dependent transmission coefficient that is controllable between 0 and 100%.
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.
After a brief introduction to the static Gross–Pitaevskii (GP) differential equation for the ground state of a Bose–Einstein condensed gas and an example of its use in describing a vortex array in a trapped condensate, a recent generalization is reviewed. By starting from the Bogoliubov–de Gennes equations for superfluid fermions, an integral equation results. This is shown to contain the static GP equation as a limiting case, by means of a low-order gradient expansion of the order parameter. The time-dependent GP approximation is then presented and illustrated by a numerical example concerning drop emission under gravity from a gaseous condensate in an optical lattice. Further selected topics of present interest relating more generally to the dynamics of trapped superfluid gases concern the effects of temperature on collective excitations in a Bose gas and the fingerprints that these excitations in a gas of paired fermions contain of the crossover from a Bardeen–Cooper–Schrieffer correlated assembly to a Bose–Einstein condensate of composite bosons. Finally, some directions are pointed in which interaction of first-principles theory with experiment should prove fruitful.
The large-scale expansion dynamics of quantum gases is a central tool for ultracold gas experiments and poses a significant challenge for theory. In this work we provide an exact reformulation of the Gross-Pitaevskii equation for the ultracold Bose gas in a coordinate frame that adaptively scales with the system size during evolution, enabling simulations of long evolution times during expansion or similar large-scale manipulation. Our approach makes no hydrodynamic approximations, is not restricted to a scaling ansatz, harmonic potentials, or energy eigenstates, and can be generalized readily to noncontact interactions via the appropriate stress tensor of the quantum fluid. As applications, we simulate the expansion of the ideal gas, a cigar-shaped condensate in the Thomas-Fermi regime, and a linear superposition of counterpropagating Gaussian wave packets. We recover known scaling for the ideal gas and Thomas-Fermi regimes, and identify a linear regime of aspect-ratio preserving free expansion; analysis of the scaling dynamics equations shows that an exact, aspect-ratio invariant, free expansion does not exist for nonlinear evolution. Our treatment enables exploration of nonlinear effects in matter-wave dynamics over large scale-changing evolution.
We investigate the dynamics of magnetically trapped boson-fermion mixtures at very low temperature. Using the mean field approach, we derive the Gross-Pitaevskii equation for the Bose-Einstein condensate wavefunction. We apply the Thomas-Fermi approximation and the hydrodynamic approach to the degenerate Fermi gas and obtain an equation of motion for its number density perturbation. It is remarkable that coupled time-evolution equations for the mixture can be regarded as an extension of the Zakharov equations, a model for the Langmuir waves in plasma.
Ultra-dilute self bound droplets in Bose-Einstein condensates have been of great interest since their first prediction in 2015.The self-bound nature of these bosonic gases combined with their extremely low densities make them a prime target of investigation for both experimentalists and theoreticians.In this thesis work dipole-dipole interacting gases are simulated using the Gross-Pitaevskii equation applying a fourth order, split-operator imaginary time method in two dimensions.First, the systems are studied at collapse for dipoles aligned with a magnetic field that tilts the dipoles in the two-dimensional plane.For some angles the dipole interaction becomes partly attractive which can lead to the collapse of the mean field solution.However, quantum fluctuations can stabilize the gas.Introducing the so-called Lee-Huang-Yang (LHY) correction, the system can be stabilized and self-bound droplets may form.Here, we investigate this stabilization for a quasi two-dimensional set-up. Popular science Ultra-cold droplets intrigue physicistsA hot topic within ultra-cold atom physics has been the recent discovery of droplets formed out of Bose-Einstein condensates, a specific form of a super cold and very dilute gas.The condensate is however very different from any gas encountered in everyday life.A normal gas can be thought of as atoms or molecules moving around in all directions.If you cool this gas, all the particles move slower until they either form a liquid or freeze and become a solid.This is not the case for the Bose-Einstein condensate.Instead they spread through space due to quantum mechanics and together they form one entity.This entity, the so called condensate, behaves like no classical gas, as it is mainly governed by quantum mechanics and not by the random chance of a lot of small particles moving about.To add to this strangeness, droplets now seem to form out of these condensates.The existence of these droplets was quite a surprise.Among experimentalists working with creating Bose-Einstein condensates the expected result was a collapse of the system if forces pushing the condensate together won against the forces pushing it apart.However, experimentalists revealed that some other internal force stabilized drops inside the condensate.Not only that, this force seemed to be able to create drops that could exist outside a "trap", which usually is a number of lasers and magnets that keep the condensate together.Some droplets could not be formed without a trap, but instead formed crystal structures, a very intriguing result.This has led to an excited mood within the field of cold atom physics.But what to expect from this anti-collapse force, mainly being described as quantum fluctuations, and the droplets it creates needs further study to find out.In this thesis work we simulate the anti-collapse force and possible droplet systems solving the underlying quantum-mechanical many-particle equations numerically.One system of interest is the very thin disc of condensate, or a quasi two-dimensional condensate.This system is excellent for rotating the condensate in.Why? Imagine having a piece of pizza dough that in this case represents a condensate.If you spin this dough as a skilled pizza maker does, the pizza will become flat due to rotational forces.The same goes for the condensate.As you rotate it it will become flat anyways and therefore having a model where our very cold gas is already flat makes it easier to simulate rotating systems.But before you start rotating your drops, you have to find the drops.And to find the drops you have to study the forces pushing the condensate together and the forces pushing it apart.Therefore we presently aim at studying possible droplet formation in these pizza crust shaped condensates, to make way for the study of rotational excitations.Initial results show some promise that droplets can be found in these quasi two-dimensional systems.