A Compact Gravimeter Based On An Atom Chip

From Left to Right: Martina Gebbe, Sven Abend, Matthias Gersemann, Holger Ahlers, Hauke Müntinga; (top right) Claus Lämmerzahl, (bottom right] Ernst M. Rasel.

Authors: Sven Abend¹, Martina Gebbe², Matthias Gersemann¹, Holger Ahlers¹, Hauke Müntinga², Enno Giese^3,4, Naceur Gaaloul¹, Christian Schubert¹, Claus Lämmerzahl², Wolfgang Ertmer¹, Wolfgang P. Schleich^3,5, Ernst M. Rasel¹

Affiliation:
¹Institut für Quantenoptik, Leibniz Universität Hannover, Germany,
²ZARM, Universität Bremen, Germany,
³Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, Germany,
⁴Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, Canada,
⁵Texas A&M University Institute for Advanced Study (TIAS), Institute for Quantum Science and Engineering (IQSE) and Department of Physics and Astronomy, Texas A&M University, College Station, Texas, USA,

Introduction to cold-atom gravimetry
The precise knowledge of gravity is important in the fields of geodesy, geophysics and metrology. Detecting small variations in the local gravitational force, for example, can provide information about the presence of mineral resources. Tectonic movements or volcanic activities can also be deduced by this method. Satellites allow the observation of the gravitational field of the whole planet yielding important parameters for climate studies such as sea-surface heights [1] or ice masses [2]. Moreover, gravity is a force that cannot be shielded and, thus, influences the measurement of other standard units.

Absolute classical gravimeters are based on tracing the free fall of macroscopic corner-cubes with laser interferometers, while the lab frame serves as inertial reference. Quantum inertial sensors operate similarly using freely falling atoms as test masses. They are based on the wave-particle duality stating that particles also have a wave nature and can therefore show diffraction and interference.

Prior to performing atom interferometry atoms have to be trapped and cooled close to absolute zero by a combination of magnetic fields and laser light. Room temperature atoms with velocities of hundreds of meters per second would almost immediately collide with the walls of the experimental chamber and leave no time to perform experiments. After release from the trap laser pulses direct the falling atoms along two different paths and let them recombine. The acceleration of free fall can then be read out from the interference pattern of the atomic wave functions.

Nowadays, commercially available quantum gravimeters feature an accuracy better than one part in 10⁸ of gravity [3,4] and are competitive to classical sensors. Their sensitivity could be improved by cooling the atoms further down until they reach an ultracold state of matter, called a Bose-Einstein condensate (BEC) where only a single quantum state is occupied. The size of a BEC is about 100 times smaller than typical millimeter-sized laser cooled clouds commonly used in atomic gravimeters. Such an ultracold cloud allows a more precise control of the atomic position and reduces the error due to spatially nonuniform laser pulse intensities.

In our paper [5] we have demonstrated an atom-chip fountain gravimeter, using a BEC source with an intrinsic sensitivity of one part in 107 of gravity, where all operations can be performed in a volume of less than a one centimeter cube.

Mach-Zehnder atom interferometer in free fall

In an atom interferometer beam splitting is based on the diffraction by an optical lattice generated by two counter-propagating laser beams. In our case, Bragg diffraction is employed, which can be described by the absorption and stimulated emission of a photon. During this process the atom receives an additional momentum kick. The duration and amplitude of the Bragg laser pulse determine the amount of atoms diffracted to another momentum state.

The Mach-Zehnder interferometer starts with a beam splitter pulse that gives a velocity kick with 50% probability resulting in an equal superposition of two atomic momentum states (see Figure 1(b)). After a free evolution time of T, the momentum states are inversed by a mirror pulse. At a time 2T a final pulse recombines the wave packets and an interferometer signal consisting of two output ports separated by a velocity difference is created. The atom number in each of these ports is detected via imaging with a CCD camera. An acceleration such as gravity leads to a path difference between the two interferometer arms and influences the population of the output ports. In principle, each atom participates independently in the interferometer. Large ensembles, however, are important to obtain detectable and statistically significant signals.

In such a free fall interferometer an important feature is the T square scaling of the phase shift, which directly links to the sensitivity. Hence, a gravitational sensor considerably benefits from an extension of the interferometer time. Unperturbed evolution times in the order of seconds can only be reached in very large devices [6], or on microgravity platforms [7]. In a compact, earth-bound device, however, the atom cloud can only fall a short distance, which limits T to the order of milliseconds.

Figure 1: (click on the image to view with higher resolution) Atom-chip-based gravimeter (a) and spacetime trajectories of an atom cloud in a Mach-Zehnder interferometer without (b) and with relaunch (c). Adapted from Reference [5]..

Measuring gravitation with a compact atom-chip setup

Figure 2: Centimeter-sized atom chip used for BEC generation.

BEC-based inertial sensors have usually been large, laboratory-sized experiments [9]. Our apparatus features a compact setup, provided amongst others by the use of a centimeter-wide atom chip [9,10], which allows a fast, robust and efficient generation of BECs (Figure 2). Inside a small vacuum chamber we repeatedly produce up to 15,000 condensed Rubidium-87 atoms at a temperature of 50nK [8, 11]. Each experimental cycle consisting of BEC generation and performing atom interferometry takes about 15 seconds.

The atom chip is not only used for the whole state preparation of the BEC but also acts as a reference mirror inside the chip gravimeter (see Figure 1(a)). It is oriented horizontally and retroreflects the light pulses coming from an upwards-pointing laser. The interferometer region extends about 1 cm below the atom chip. Since the atoms are freely falling along the optical lattice, its velocity has to be constantly increased in order to match the gravitational acceleration of the atoms. If lattice and atomic acceleration are identical, only one state entering the interferometer is detected at the output. This means, our interference signal, the normalized population assumes a minimum independent of the pulse interval T. Local gravity is than deduced from the optical lattice acceleration which can be measured very precisely.

Figure 3(a) shows interference signals in dependence of the lattice acceleration for pulse intervals of T=1,3 and 5ms in a dropped interferometer. The acceleration of free fall was determined to be g = (9.8134 ± 0.0006) m/s². After repeating the measurements with the maximal interrogation time T=5ms over 8h we achieved a relative precision of one part in 10⁵ of gravity (see Figure 3(b)).

Figure 3: (click on the image to view with higher resolution) Measuring gravitational acceleration in a single fall interferometer. (a) Normalized atom number in one output port of the MZI for T = 1, 3, 5 ms depending on the acceleration of the Bragg lattice. (b) Allan deviation of the acceleration improves with the square root (black line) of the integration time t reaching a precision of 1 part in 10⁵ of gravity after 8 hours. Source: Reference [5].

In order to improve the sensitivity of our interferometer we developed a simple and efficient fountain sequence illustrated in Figure 1(c). At the bottom of the detection zone the atoms are caught and tossed upwards with an optical lattice generated by the same laser light beams that are used for interferometry. This way, we are able to enhance the total free evolution time by a factor of three, without increasing the free-fall baseline of the experiment.

The interferometer starts directly after the relaunch and can be extended to T=25ms. Its intrinsic sensitivity equals one part in 10⁷ which represents a 20-fold increase in comparison to the simple fall setup. Due to the environmental conditions and the absence of any vibrational isolation we were not able to observe fringes any more. However, we did a statistical analysis of the data and determined a high interferometric contrast of 80%.

Conclusion and outlook

In conclusion, we demonstrated the first miniaturized atom-chip fountain gravimeter without and with relaunch. The new fountain scheme leads to extended interferometer times without changing the compact volume of a one centimeter cube. The sensor is currently limited by the large vibrations in our system, which we aim to suppress in the future.

The current sensitivity of 1 part in 10⁷ of gravity is about two orders of magnitude lower compared to state-of-the-art sensors. However, with an advanced apparatus [12], which produces 10⁵ atoms in a BEC per second, and small improvements in the measurement scheme, we believe an intrinsic sensitivity of one part in 10⁹ is feasible. At the same time, further miniaturization can be done by using for example a pyramidal-shaped retroreflector [13] that reduces the size of the laser system. All these improvements open up the route to a backpack-sized device for high-precision absolute gravimetry utilized in remote locations.

References:
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Comparing Matter Waves in Free Fall

[From Left to Right] J. Hartwig, D. Schlippert, E. M. Rasel

Authors: J. Hartwig, D. Schlippert, Ernst 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 [1]. This measurement is only surpassed by torsion balance experiments based on the design of Eötvös [2]. In addition, exciting new insights are expected from the MICROSCOPE experiment [3] 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 [4], or by using a test theory such as the extended Standard Model of particles (SME) [5]. 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 [6] and existing matter wave tests [7]. 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 [8]. 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 [9] and have improved in their performance ever since. The first test of the equivalence principle comparing two different isotopes was then performed in 2004 [10]. Research on quantum tests is for example proposed at LENS in Italy [11], in Stanford in an already existing large fountain [12] and in the scope of the French ICE mission in a zero-g plane [13]. 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*k_eff, where k_eff 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/k_eff. 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.

Our data

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.

In order to test the universality of free fall, we simultaneously chirp the Raman frequencies to compensate for the accelerations a(Rb,±)(g) and a(K,±)(g) experienced by rubidium and potassium that were previously identified (Figure 6). Here, the observed phase shift exhibits contributions due to additional perturbations, such as magnetic field gradients. We make use of a measurement protocol based on reversing the transferred momentum (upward and downward directions ±). This technique makes use of the fact that many crucial perturbations do not depend on the direction of momentum transfer. Thus, by computing the half-difference of the phase differences determined in a single momentum direction, phase shifts induced by, e.g., the AC-Stark effect or Zeeman effect, can be strongly suppressed [14].

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 [15] 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.

Outlook

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 [16]. 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 [17].

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 [18]. 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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