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Bose-Einstein Condensates in Curved Space-Time – From Concepts of General Relativity to Tidal Corrections for Quantum Gases in Local Frames

Gabel, Oliver

Published Jan 1, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.26083/tuprints-00008809

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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.
Inertial sensors based on matter-wave interferometry are currently approaching the precision and accuracy of state-of-the-art classical sensors. While these devices are often realised with ultracold but not Bose-condensed atoms as matter waves, employing Bose-Einstein Condensates (BEC) promises to overcome certain limitations, especially those related to the ensemble's expansion. The point-source like character of BECs also enables utilising spatial interference patterns to measure e.g. rotation rates in single-shot experiments. Matter wave-based inertial sensors are considered for experiments ranging from Gravitational Wave detection to tests of the Universality of Free Fall (UFF) to gain insight into the joint between Quantum Mechanics and General Relativity. In the scope of this thesis, matter-wave interferometry with BECs was demonstrated for the first time in a microgravity environment with the QUANTUS-1 apparatus. The same instrument was then employed as a quantum tiltmeter utilising a novel beam-splitting mechanism known as Bragg Double Diffraction. To this end, the QUANTUS-1 apparatus designed as a BEC instrument to be operated in the drop tower at ZARM at University of Bremen was equipped with optics and laser systems required for performing matter-wave interferometry based on Bragg Diffraction. The apparatus employs an atom chip to create BECs of around 10000 Rubidium 87 atoms within 15 s. With a Mach-Zehnder like interferometer scheme, spatial interference fringes were observed after a free evolution time in the interferometer of up to 677 ms. To achieve these long time scales, a method known as Delta-Kick Collimation (DKC) was adapted to slow the expansion of the BEC to a kinetic energy equivalent below 1 nK, and the atoms were transferred to a non-magnetic Zeeman state via an adiabatic rapid passage (ARP). A similar interferometer scheme with a newly developed beam-splitter mechanism known as Bragg Double Diffraction was used to measure the tilt of the instrument on ground with a precision of up to 4.4 AA rad. This thesis presents an overview of the apparatus including ground-based characterisations of all required experimental steps. Results from over 400 free fall experiments are evaluated for expansion studies of the BEC and matter-wave interferometry in microgravity. The time-evolution of first and second order Bragg Double Diffraction beam splitters is studied, and an interferometer sensitive to the tilt of the instrument is implemented. Based on this work, a gravimeter with a new launch mechanism comprising Bragg beam splitters and Bloch oscillations to enable atomic fountains in atom-chip based devices was developed. The microgravity experiments were adapted for the MAIUS-1 sounding-rocket instrument to create the first man-made BEC in outer space and study the feasibility of operating matter-wave interferometers on space-borne platforms. The results of this thesis lay the groundwork for future space-borne missions using matter-wave interferometry for precision measurements of inertial forces.
Ultra-cold quantum gases in space promise to boost the sensitivity of matter-wave interferometers. Applications of the latter extend from fundamental physics over the use in navigation to interdisciplinary applications such as geodesy, e.g. satellite gravimetry [1, 2]. Exploiting quantum gases for high-precision interferometry places high demands on their control and manipulation. We take benefit of various microgravity platforms such as the Bremen drop tower [3], the Einstein elevator in Hannover [4], sounding rockets [5,6] and the international space station [7] to advance the necessary methods. The DLR-mission MAIUS-1 demonstrated Bose-Einstein condensation and performed first interferometry experiments [4]. NASA’s Cold Atom Laboratory continues this research in orbit on the ISS [7]. In addition, atom interferometry is pursued in highly dynamic environments such as parabolic flights [8]. Starting from a rubidium Bose-Einstein condensate, recently lowest expansion energies have been achieved by us in the Bremen drop tower as required for extending atom interferometry over several seconds [9]. Extending these methods to quantum mixtures not only opens up new physics in the absence of buoyancy, but also faces challenges regarding their use for interferometry. Interferometers based on two chemical elements have been proposed for quantum tests of the equivalence principle on the ISS as well as on satellites. Currently we prepare a sounding rocket mission to investigate the simultaneous generation and manipulation of potassium and rubidium condensates [10]. Together with CAL [7], these experiments will prepare the DLR-NASA multi-user facility BECCAL for research on quantum gas mixtures and interferometry [11] as well as enhance the readiness level of methods required for STE-QUEST [12], a proposal for a satellite mission currently studied in an modified version within ESA’s VOYAGE 2050 program [13].
Owing to the low-gravity conditions in space, space-borne laboratories enable experiments with extended free-fall times. Because Bose–Einstein condensates have an extremely low expansion energy, space-borne atom interferometers based on Bose–Einstein condensation have the potential to have much greater sensitivity to inertial forces than do similar ground-based interferometers. On 23 January 2017, as part of the sounding-rocket mission MAIUS- 1, we created Bose–Einstein condensates in space and conducted 110 experiments central to matter-wave interferometry, including laser cooling and trapping of atoms in the presence of the large accelerations experienced during launch. Here we report on experiments conducted during the six minutes of in-space flight in which we studied the phase transition from a thermal ensemble to a Bose–Einstein condensate and the collective dynamics of the resulting condensate. Our results provide insights into conducting cold-atom experiments in space, such as precision interferometry, and pave the way to miniaturizing cold-atom and photon-based quantum information concepts for satellite-based implementation. In addition, space-borne Bose–Einstein condensation opens up the possibility of quantum gas experiments in low-gravity conditions1,2.
Extended free fall enables experiments with quantum gases unbiased by gravity. Apart from a 10m-fountain, we have access to various platforms comprising the Bremen drop- and gravi-tower, the Einstein Elevator, sounding rockets as well as NASA‘s cold atom laboratory (CAL) in orbit, all offering experiments with different durations and repetition rates. Atom chips turned out to successfully create Rubidium Bose-Einstein condensates on these platforms. Recently we have lowered the expansion energies of Bose-Einstein condensates to an unprecedented level corresponding to 38pK which makes them an excellent resource for interferometry. We exploit the latter to study delta-kick collimation as well to extend interferometry times in free fall. Atom chips also allow to investigate quantum mixtures in absence of bouyancy. For this purpose, we have been creating potassium 41 BECs by sympathetic cooling with rubidium. These experiments were continued to space by sounding rocket missions and the CAL. In future, the recently inaugurated Einstein elevator as well as gravitower will be important platforms to increase the experimental cycles to study quantum gas mixtures and high-precision interferometry as well as to test fundamental physics such as Einstein’s principle of equivalence and the nature of dark energy on ground in free fall.
Low gravity provides an outstanding basis for precision measurements in atom optics pursuing multi-disciplines in fundamental physics. On the other hand it leads to an utilization of ultracold quantum matter techniques in unique practical applications. We report on the first establishments of atom optical experiments in the gravitation-free conditions at the Drop Tower Bremen in Germany, a facility of the Center of Applied Space Technology and Microgravity (ZARM). The chosen drop tower has an easy access to low gravity on earth with a daily usage, a more than sufficient time of free fall for experiments in quantum regimes (about 4s in drop mode and about 9s in catapult mode) and a well quality of weightlessness of 10-6g. Our report demonstrates the results of the precursor pilot project ATKAT ("atom catapult") and the current pilot project QUANTUS ("quantum systems under weightlesness") . The pure technological experiment ATKAT could successfully realize a compact and robust setup for trapping and cooling neutral rubidium atoms ( 87Rb) in microgravity conditions. After ATKAT's test accomplishment we have started the parallel developed QUANTUS main pilot projekt to investigate quantum degenerated gases in free fall. The first Bose-Einstein condensate (BEC) in a weightless environment world wide and longest evolution times of the 87Rb BEC up to 1s could be achieved within the science experiment QUANTUS. In respect thereof our collaboration opens further opportunities in atom optics. At least we will give an outlook of the new QUANTUS II project and its supporting project PRIMUS ("precision interferometry with matter waves under weightlessness") aiming on a precise inertial sensor by using an atom interferometer under weightlessness.
Since the possibility of trapping and cooling neutral atoms, ultracold quantum gases have shifted boundaries in a growing field of modern physics based on the first observation of Bose-Einstein condensates in 1995 and appreciated by the Nobel Prizes in 1997 and 2001. The current developments in the domain of atom optics lead to an utilization of ultracold quantum matter techniques in unique practical applications as high-precision atomic clocks, atom interferometer technologies and inertial sensing instruments for gravity field mapping, underground structure detection, autonomous navigation, as well as precision measurements in fundamental physics. The expectations of even higher precision measurements can be performed by arbitrarily extending the time of unperturbed evolution of those quantum systems. In respect thereof weightlessness provides an outstanding basis for such applications and measurements. Motivated by these prospects, many national and international groups have initialized research programs aiming for compact, transportable and ruggedly designed atom-optical experiments, which might be launched in parabolic flights and space applications. Thanks to an easy access to low gravity on earth, realization of quantum degenerated gases in excellent microgravity conditions at the Drop Tower Bremen opened a new kind of perspectives on atom-optical experiments, e.g., to currently achieve longest expansion times of Bose-Einstein condensates within the QUANTUS pilot project (up to one second). Thus, ultracold quantum matter in an environment of weightlessness represents an emerging area of science in quantum engineering with an impressive potential for a future technology and multidisciplinary applications.