Comparing Matter Waves in Free Fall
Authors: J. Hartwig, D. Schlippert, E. M. Rasel
Affiliation: Institut für Quantenoptik and Centre for Quantum Engineering and Space-Time Research (QUEST), Leibniz Universität Hannover, Germany
Introduction to Einstein’s Equivalence Principle
Einstein’s general relativity is based on three fundamental building blocks: local Lorentz invariance, the universality of the gravitational redshift and the universality of free fall. The enormous importance of general relativity in modern science and technology merits a continuous effort in improving experimental verification of these underlying principles.
The universality of free fall is one of the oldest mechanical theories originally proposed by Galileo. Testing can be done by so called free fall experiments, where two bodies with different composition are freely falling towards a third gravitating body.
Figure 1: Goddard Spaceflight Center Laser Ranging Facility. Source: NASA
Amongst the most sensitive measurements of this principle is the Lunar Laser Ranging experiment, which compares the free fall of earth and moon in the solar gravitational potential . This measurement is only surpassed by torsion balance experiments based on the design of Eötvös . In addition, exciting new insights are expected from the MICROSCOPE experiment  that is planned to launch in 2016.
Figure 2: Torsion balance experiment as used in the group of E. Adelberger, University of Washington. Source: Eöt-Wash-Group
The emergence of quantum physics and our improved understanding of the basic building blocks of matter increases the interest scientists have in the understanding of gravity. How do gravity and quantum mechanics interact? What`s the connection between different fundamental particles and their mass? Is there a deeper underlying principle combining our fundamental theories? To comprehensively approach these questions a wide array of parameters must be analyzed. The way how certain test materials may act under the influence of gravity can either be parametrized using a specific violation scenario, like the Dilaton scenario by T. Damour , or by using a test theory such as the extended Standard Model of particles (SME) . Since the SME approach is not based on a specific mechanism of violating UFF it also does not predict a level to which a violation may occur. Instead, it delivers a model-independent approach to compare methodically different measurements and confine possible violation theories.
Table 1 states possible sensitivities for violations based on the SME framework for a variety of test masses and underlines the importance of complementary test mass choices are. Hence in comparison to classical tests, the use of atom interferometry opens up a new field of previously inaccessible test masses with perfect isotopic purity in a well-defined spin state. Quantum tests appear to differ from previous test also in a qualitative way. They allow to perform test with new states of matter, such as wave packets by Bose Einstein condensates being in a superposition state. The work presented here is just another early step in a quest to understand the deeper connections between the quantum and classical relativistic world.
Table 1: Sample violation strengths for different test masses linked to “Neutron excess” and the “total Baryon number” based on the Standard Model Extension formalism. The test mass pairs are chosen according to the best torsion balance experiment  and existing matter wave tests . An anomalous acceleration would be proportional to the stated numerical coefficients. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014).
Measuring accelerations with atom interferometry
Measuring accelerations with a free fall experiment is always achieved by tracking the movement of an inertial mass in free fall in comparison to the lab frame of reference. This is even true when the inertial mass in question needs to be described by a matter wave operating on quantum mechanics. Falling corner cube interferometers operating on this principle are among the most accurate measurements of gravity with classical bodies. They use a continuous laser beam to track the change of velocity of a corner cube reflector due to gravity in a Michelson interferometer with the corner cube changing one of the arm lengths. Acceleration sensors based on free falling matter waves use a similar principle.
Figure 3: A side view of the experimental setup with the two-dimensional (left side) and three-dimensional (right side) magneto-optical traps employed in [Phys. Rev. Lett. 112, 203002 (2014)].
The first demonstration of a true quantum test of gravity with matter waves was performed 1975 with neutrons in the Famous COW experiment . We will focus on atom interferometers using alkaline atoms, since they are most commonly used for inertial sensing and are also employed in the discussed experiment. Experiments of this kind were first used for acceleration measurements in 1992  and have improved in their performance ever since. The first test of the equivalence principle comparing two different isotopes was then performed in 2004 . Research on quantum tests is for example proposed at LENS in Italy , in Stanford in an already existing large fountain  and in the scope of the French ICE mission in a zero-g plane . All these initiatives show the high interest of testing gravity phenomena with quantum matters as opposed to classical tests.
In the case of atom interferometers, coherent beam splitting is performed by absorption and stimulated emission of photons. Which atomic transitions are used is dependent on the specific application but, in the case of alkaline atoms two photon transitions coupling either two hyperfine and respective momentum states (Raman transitions) or just momentum states (Bragg transitions) are employed. The point of reference for the measurement is then given by a mirror reflecting the laser beams used to coherently manipulate the atoms, since the electromagnetic field is vanishing at the mirror surface. This results in a reliable phase reference of the light fields and constitutes the laboratory frame. The role of the retroreflecting mirror is similar to the one of the mirror at rest in in the case of the falling corner cube experiment.
The atomic cloud acts as the test mass, which in an ideal case, is falling freely without any influence by the laboratory, except during interaction with the light fields employed as beam splitters or mirrors. During the interaction, the light fields drive Rabi-oscillations in the atoms between the two interferometer states |g> and |e> with a time 2τ needed for a full oscillation. This allows for the realization of beam splitters with a τ/2 pulse length resulting in an equal superposition of |g> and |e>. Mirrors can be realized the same way by applying the beam splitter light fields for a time of τ which leads to an inversion of the atomic state. These pulses are generally called π/2 (for the superposition) and π (for the inversion pulses) in accordance with the Rabi-oscillation phase. The simplest geometry used to measure acceleration is a Mach- Zehnder-like geometry. This is produced by applying a π/2-π-π/2 sequence with free evolution times T placed between pulses. The resulting geometry can be seen in Figure 4.
Figure 4: Space-time diagram of a Mach-Zehnder-like atom interferometer. An atomic ensemble is brought into a coherent superposition of two momentum states by a stimulated Raman transition (π/2 pulse). The two paths I+II propagate separated, are reflected by a pi-pulse after a time T and superimposed and brought to interference with a final π/2 pulse after time 2T. The phase difference is encoded in the population difference of the two output states.
During the interaction with the light fields, the lattice formed by the two light fields imprints its local phase onto the atoms. This results in an overall phase scaling with the relative movement between the atomic cloud and the lattice. Calculating the overall phase imprinted on the atoms results in first order term, Φ=a*T2*keff, where keff is the effective wave vector of the lattice and a is the relative acceleration between lattice and atoms. This immediately shows the main feature of free fall atom interferometry: the T2 scaling of the resulting phase. This is of particular interest for future experiments aiming for much higher free evolution times than currently possible. The phase Φ also shows another key feature. As the acceleration between atoms and lattice approaches zero, the phase also goes to zero, independently of the interferometry time T. This yields a simple way to determine the absolute acceleration of the atomic sample by accelerating the lattice until the lattice motion is in the same inertial reference frame as the freely falling atoms.
Lattice acceleration is achieved by chirping the frequency difference between the two laser beams used for the two photon transition. This transforms the measurement of a relatively large phase, spanning many thousand radians, to a null measurement. The signal produced is the population difference between the interferometer states |g> and |e> as a function of lattice acceleration and thus frequency sweep rate, α. The sweep rate corresponding to a vanishing phase directly leads to the acceleration experienced by the atoms according to lattice acceleration formula a=α/keff. Taking into account Earth’s gravitational field and a lattice wavelength of 780/2 nm (the factor of ½ is introduced due to the use of a two-photon transition) this leads to a sweep rate of around 25 MHz/s. The advantage of this method is that the acceleration measurement is now directly coupled to measurement of the wavelength of the light fields and frequencies in the microwave regime, which are easily accessible.
Figure 5: Determination of the differential acceleration of rubidium and potassium. The main systematic bias contributions do not change their sign when changing the direction of momentum transfer. Hence, the mean acceleration of upward and downward momentum transfer direction greatly suppresses the aforementioned biases. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)
|Figure 6: the wave nature of 87Rb and 39K |
atoms and their interference are exploited
to measure the gravitational acceleration.
Figure 7: Allan deviations of the single species interferometer signals and the derived Eötvös ratio. Source: D. Schlippert et al., Phys. Rev. Lett. 112, 203002 (2014)
The data presented in this work  was acquired in a data run that was ~4 hours long. By acquiring 10 data points per direction of momentum transfer, and species and then switching to the opposite direction, we were able to determine the Eötvös ratio of rubidium and potassium to a statistical uncertainty of 5.4 x 10-7 after 4096s; the technical noise affecting the potassium interferometer is the dominant noise source.
Taking into account all systematic effects, our measurement yields η(Rb,K)=(0.3 ± 5.4) x 10-7.
In our measurement, the performance was limited both by technical noise and the limited free evolution time T. In order to improve these parameters, we are currently extending the free fall time in our experiment. Furthermore, in an attempt to increase the contrast of our interferometers and thus the signal-to-noise ratio, we are working on implementing state preparation schemes for both species.
We expect to constrain our uncertainty budget (which currently is on the 10 ppb level for the Eötvös ratio) on the ppb level and below through the use of a common optical dipole trap applied to both species. By using Bose-Einstein-condensed atoms, we gain the ability to precisely calculate the ensembles, as well as carefully control the input state. This technique will also be able to reduce uncertainty factors linked to the transverse motion of the cloud, in addition to spatial magnetic field and gravitational field gradients.
Improving the precision of a true quantum test into the sub-ppb regime is the focus of current research. For example we are currently planning a 10m very long baseline atom interferometer (VLBAI) in Hannover . In the framework of projects funded by the German Space Agency (DLR), we moreover develop experiments that are suitable for microgravity operation in the ZARM drop tower in Bremen and on sounding rocket missions .
Parallel to the development done in the LUH and at a national level, we are also involved in projects on an international level looking into extending the frontier of atom interferometry and especially the test of the equivalence principle. A major project investigating the feasibility of a space borne mission is the STE-Quest Satellite Mission proposed by a European consortium including nearly all major research institutions working in the field of inertial sensing with atom interferometry, as well as a variety of specialist of other fields . This mission is aimed towards doing a simultaneous test of the equivalence principle with two rubidium isotopes and a clock comparison with several ground based optical clocks, pushing the sensitivity to the Eötvös ratio into the 10-15 regime.
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