Beam splitting with stimulated Raman transitions

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Rotation measurements and the CASI ex-periment

Before introducing the CASI experiment, possible gyroscope applications are in-troduced. This is followed by an overview of gyroscope technologies.

Rotation measurements

An important application of rotation measurements is navigation. If the six mo-ments of inertia of a vehicle, which are rotation and acceleration in all three direc-tions, are tracked during the movement of the vehicle, it is possible to deduce its position relative to a given starting position at any time without external informa-tion (e.g. by the Global Positioning System, GPS). The demanded performance of the rotation measurement depends on the type of movement in, for example, aircrafts, ships, or submarines. The best resolutions are in the range of 10− 6 rad/s to 10−8 rad/s. Furthermore, this type of application sets high demands on the robustness of the measurement device and requires a large dynamic range.
Rotation measurements of typically smaller dynamics but higher precision are desirable in the field of geodetics and geophysics. Here, measurements of Earth rotation variations in orientation (polar motion) and magnitude (Length-Of-Day, LOD fluctuations) give an insight into tidal effects of the Earth as a whole and into internal dynamics of the body of our planet. Relative changes of 10 ppb of the Earth rotation rate of ΩE = 7.27 • 10−5 rad/s demand for rotation resolutions in the regime of 10−12 rad/s and below [41].
An even higher sensitivity is required for the measurement of the Lense-Thirring effect [42]. This so-called frame dragging effect was deduced by Lense and Thirring from general relativity. It predicts a rotation rate induced by rotating masses such as the Earth. On a satellite in an Earth orbit, this effect is expected to induce a rotation rate in the range of 10−15 rad/s.

Gyroscope technologies

The first demonstration of an Earth rotation measurement was realized in 1861 by Foucault. It consisted in showing the rotation of the plane of a pendulum’s oscil-lation. It was also Foucault who invented and constructed the first mechanical gyroscope, based on the work of Bohnenberger [43]. Such a device makes use of the conservation of angular momentum. The rotation axis of a rotating body suspended in a gimbal mount serves as a stable reference axis. This technique was used in navigation in order to track the orientation of e.g. ships and submarines over long times. By tracking the deduced trajectory it is then possible to determine the ship’s position without external reference. The technology has been pushed to its limits in the realization of superconducting magnetically suspended spherical gyroscope bodies in the Gravity Probe B project. Here, it was possible to mea-sure the a drift rate induced by the Lense-Thirring effect with an uncertainty of 19 % [44].
The Sagnac effect observed in 1913 enables the realization of gyroscopes based on interferometry. This effect describes the influence of rotations on the output phase of an interferometer featuring an area A that is enclosed by the interfer-ometer paths. While this was first shown for light coming from a mercury arc lamp by Georges Sagnac one century ago [5], this effect occurs for a wave of any nature. The phase shift on the output of the interferometer scales directly with the projection of a rotation Ω onto the enclosed interferometer area as well as with the particle energy E ΦΩ = 4πE Ω•A. (1.1)
Here, h is the Planck constant and c is the speed of light. Based on this effect it was possible to develop passive and active laser gyroscopes [45]. Passive laser gyroscopes are nowadays typically realized in fiber spools. This allows for a large enclosed area and thereby for a large interferometer scaling factor in a relatively compact device. Fiber gyroscopes with resolutions in the lower 10−6 rad/(s Hz) regime and high long term stabilities are today employed in so-called IMUs (“In-tertial Measurement Units”). These devices consist of three of these gyroscopes and a three-axis accelerometer. They are used in navigation of, for example, air-crafts and rockets. An even higher resolution is reached using active ring laser gyroscopes. Typical resolutions of some 10−7 rad/(s Hz) are realized for transportable instruments. Quasi-transportable instruments for the use in submarines even reach sensitivities of a few 10−8 rad/(s Hz). For their application in geodesy and geophysics, these kind of devices are scaled up to a maximum enclosed single-loop area of 834 m2 [46]. A resolution of 5 • 10−10 rad/s is reached within one second of measurement time, but the resolution is limited by scaling factor insta-bilities. The highest resolution of a large ring laser gyroscope is up to date the measurement of the annual wobble and the Chandler wobble of Earth with the aid of the Grossring G [47]. The best sensitivity was reported in [48] for this device with 16 m2 enclosed area of 2 • 10−11 rad/(s Hz).
The development of atom interferometers enabled the realization of gyro-scopes based on the interference of neutral atoms. The striking argument for the use of this technology is given by the Sagnac phase (1.1). The comparison of the photon energy for a laser wavelength λ = 633 nm and an atomic rest mass of a 87Rb atom results in a factor of Φatom ≈ 1011. (1.2) Φlight
This advantage is put into perspective as it is much more demanding to realize large enclosed areas in atom interferometers compared to light interferometers. Also, the effective atom flux is typically much lower than the used effective photon flux. Nonetheless, atom interferometry enables rotation measurements with gyroscopes that enclose much smaller areas while keeping a scaling factor comparable to light interferometry. The first and up to now most sensitive realization of an atomic gyroscope based on two atom interferometers with counter-propagating atomic beams and an area of 22 mm2 enclosed by each interferometer was built at Stanford 10−10 rad/(s Hz) and 2 • 10−11 rad/s after 2000 s of integration [49]. However, the use of thermal beams required a large apparatus. The separation of the first and last beam splitter application amounted to 2 m. A much more compact device was constructed at SYRTE in Paris where cesium atoms were trapped and cooled in magneto-optical traps prior to be launched on parabolic trajectories. The slow atomic forward velocity on steep parabolas enabled the application of all beam splitter light field pulses within the width of one laser beam with a diameter of 48 mm. For an enclosed area of 3.8 mm2 the sensitivity of this cold atom gyroscope was 2.4 10−7 rad/(s√ 1 • Hz) and, after an integration of approximately 1000 s, • 0−8 rad/(s√ [29]. Hz) A second gyroscope at Stanford University is based on laser-cooled atoms in a vertical fountain configuration with a pulse sequence that requires the gravitational acceleration to form the atomic trajectories. For this device, a sensitivity of 8.8 • 10−8 rad/(s√ was derived with an enclosed area Hz) of about 19 mm 2 [50]. A similar vertical fountain gyroscope is currently under* construction at SYRTE. The project aims for areas as large as 11 cm2 and for resulting sensitivities in the range of 10−9 rad/(s Hz).

 The CASI experiment

In the framework of the CASI (“Cold Atom Sagnac Interferometer”) project, a cold atom gyroscope consisting of two atom interferometers has been constructed. The project is motivated by the HYPER proposal [51], which aimed for the ap-plication of atomic gyroscopes for the measurement of the Lense-Thirring effect in a spaceborne device. Moreover, the compact design featuring an interferometer baseline of below 15 cm with an enclosed interferometer area of 19 mm2 makes the device suitable for high resolution local rotation measurements for applications in geophysics.
The two interferometers are based on the interference of laser-cooled 87Rb atoms on their free fall on flat parabolic trajectories using stimulated Raman transi-tions. As depicted in figure 1.2, these transitions are induced by phase stable laser fields that are applied in a counter-propagating configuration in three spa-tially separated interaction zones. The experiment has been presented in previous works [52, 53, 54, 55, 56]. Key to realizing the large area is the control of the relative pointing of the beam splitter light field in the three interaction zones. A technique for the relative pointing alignment is presented in [56, 57]. This led to the realization of the two atom interferometers with an enclosed area of 19 mm2 each. The short term rotation sensitivity of 5.3 • 10−7 rad/(s Hz) was found to be limited by environmental perturbations, and by the noise of the interferometer signal detection.
In this work, the implementation of a rotation measurement based on alternat-ing area orientation is presented. Such a technique was reported e.g. in [49]. In combination with an ellipse fit method, this allows to study the long term stability of the gyroscope signal. The relative wave front alignment coupled with mis-matches in the overlap of the two interferometers is identified as main drift source.
We have employed a monitoring technique for tracking the cloud overlap in one dimension enabling an effective improvement of the rotation sensitivity of more than one order of magnitude after integration, which is presented in chapter 3.

Short range forces and the FORCA-G experi-ment

In this section, the field of application of the FORCA-G experiment is introduced before giving some examples for short range force measurement experiments. A brief introduction to the FORCA-G experiment is given thereafter.

Atom-surface interactions

The scope of the FORCA-G experiment is the measurement of forces between an atom and a macroscopic surface in ranges of atom-surface separations below the millimeter, which are difficult to access with measurement techniques based on macroscopic bodies. In this regime, deviations from the Newtonian gravitational potential are predicted by theories that aim for a unification of the standard model and gravitation [58]. These deviations are generally parametrized by a Yukawa type potential of the form U = UN 1 + αer/λ (1.3) with UN the Newtonian gravitational potential, α the strength and λ the typical range of the deviation. Measurements in the range of tens of µm down to several hundreds of nm may give new insights on the predicted models by either measuring a deviation from the Newtonian potential or setting new limits on such deviations and thereby excluding regions on the α-λ-plane of the Yukawa parametrization as depicted in figure 1.3.
The challenge in the detection of such deviations is the domination of electro-dynamic forces in this regime which, up to now, have not been measured with an uncertainty of better than a few percent. These forces arise from the dipole-dipole interaction of an atom close to a conducting surface with its image charge. At very small distances, this interaction is instantaneous and results in the so-called Van-der-Waals-London (VdWL) force. In slightly larger distances, the so-called Casimir-Polder (CP) effect appears [60], which is the retardation effect of this dipole-dipole interaction. Assuming a perfectly conducting surface, the force is given by [61] FCP = 3~cα0 , (1.4) 2πd5 where α0 is the static polarizability of the atom and d is the atom-surface distance. The transition between these two regimes can be understood via the Heisenberg uncertainty relation ∆E∆t ≤ ~/2: The size of the Van-der-Waals-London poten-tial ∆EV dW L sets a limit onto the travel time of the electromagnetic interaction photon between the atom and the surface. Inserting an interaction photon energy ~ωDD and ∆d = c∆t (d being the atom-surface separation distance) into the uncer-tainty relation formula yields a maximum distance in the range of the interaction wavelength. For the 87Rb atom in the vicinity of a perfectly conducting surface, the transition from the VdWL force scaling with d−4 to the CP regime scaling with d−5 is found around 100 nm [62]. Furthermore, at distances of one order of magnitude larger, thermal effects result in a third regime, in which the force scales as in the VdWL case with d−4.

Short range force measurements

The direct measurement of VdWL and CP forces between neutral atoms and macroscopic surfaces has been carried out so far in different ways. For exam-ple, it is possible to determine the CP potential by reflecting matter-waves on a macroscopic surface. This so-called quantum reflection is demonstrated in [63] for metastable neon atoms reflected on a silicon and a BK7 glass surface in grazing incidence and in [64] for ultra-cold 23Na atoms in normal incidence on a silicon surface showing qualitative agreements with the predicted force as a function of the atom-surface separation. The possibility of a quantitative measurement the Casimir-Polder force is reported in [65] for quantum reflection of 3He atoms on the surface of a gold single-crystal resulting in resolutions for Casimir-Polder forces of a few percent.
Quantitative measurements of the Van-der-Waals interaction were realized with a technique involving the reflection of atoms on a short range potential that is shaped by the use of an evanescent wave. In [66], the reflection of 85Rb atoms that are dropped from a magneto-optical trap onto a mirror surface is investigated. The evanescent wave that allows to add a repulsive potential to the attractive CP potential, is created by total reflection of a laser beam inside the mirror. This ex-periment showed an uncertainty of 30 %, a clear agreement with the exact quantum electrodynamic (QED) or the electrostatic calculation could not be determined. A similar set-up recently led to the measurement of Casimir-Polder forces in dis-tances of 160 to 230 nm [67]. An ultra-cold 87Rb ensemble was accelerated in a combined magnetic-dipole trap and reflected on an evanescent wave on a glass prism. The results showed best agreement with the full QED calculation in the transition regime between the CP and VdWL regimes.
A slightly different approach was presented in [68]. A Bose-Einstein condensate (BEC) of 87Rb is trapped in a magnetic trap near a dielectric surface. By tracking the center of mass motion of the BEC oscillating in the magnetic trap, it is possible to measure CP forces in atom-surface separation distances of 6 to 12 µm with an uncertainty of down to 10 %. The impact of thermal black body radiation in this regime was already mentioned in [68] and later demonstrated in [69].
The first demonstration of the measurement of atom-surface interactions via the phase shift in an atom interferometer is presented in [70] for distances of down to 10 nm. One arm of a Mach-Zehnder type interferometer based on diffraction of Na atoms on material gratings is guided through a narrow dielectric cavity. The measurement of short range forces in distances of several µm is approached, among others, by trapped atom interferometry in a vertical unidimensional opti-cal dipole lattice. The acceleration of this lattice given by the local gravitational acceleration induces so-called Bloch oscillations predicted by quantum mechanics (see chapter 2). The frequency of this process, the so-called Bloch frequency, scales with the acceleration. Measuring this frequency therefore allows for a local acceler-ation measurement and, if carried out in the vicinity of a macroscopic surface, for a measurement of perturbations of the acceleration induced by short-range forces. The acceleration measurement far from a surface has been realized in different ways. In [37] this is done via the determination of the atomic momentum after release from the potential, which consists in a direct measurement of the Bloch os-cillation frequency by releasing the atoms after different holding times. A different approach is presented in [71], where the lattice depth is modulated and the spread of the atomic cloud is measured in-situ. This enabled a measurement of the local gravitational acceleration with a resolution of parts in 10−7 within a measurement time of about 1 h. In all cases, the 1D-lattice is created via the retro-reflection of a laser beam. The realization of the acceleration measurement close to the surface of the mirror used for retro-reflection results in a good control for the mapping of short range forces in multiples of the spatial lattice periodicity.

The FORCA-G experiment

Within the FORCA-G project a measurement of short range forces will be realized based on Raman atom interferometry. Laser-cooled 87Rb atoms are trapped in the anti-nodes of a vertical dipole lattice with a periodicity of λl/2 = 266 nm. This gives access to measurements far from the mirror surface, in the regime of a few µm and even below the µm. The target precision of CP measurements in the range of a few µm is below the percent. This would allow for exclusions of non-Newtonian forces in the α-λ-plane as depicted in figure 1.3.
The measurement principle is an atom interferometer measurement on an atomic spatial superposition of two different lattice sites as depicted in figure 1.4. This superposition is created by coherently inducing tunneling between lattice sites us-ing stimulated Raman transitions. After a certain evolution time, the two lattice sites can be coupled again resulting in an atomic interference. The phase of this trapped atom interferometer then depends on the potential difference between the two lattice sites. In the realization far from the mirror surface, the potential gradi-ent is given by the gravitational acceleration. In the vicinity of the mirror surface, the short range forces are measured as a perturbation of the latter.
Subject of this work is the demonstration of the process of Raman induced tunneling in our vertical 1D-lattice in a proof-of-principle experiment. The in-terferometer measurements that are realized far from the mirror surface allow to study the short and long term stability of the acceleration measurement leading to an estimation of the interferometer resolution for short range force measurements.
The Raman laser induced tunneling process far from the mirror surface was first demonstrated in [72] and described in more detail in [73]. Spectroscopic properties of the tunneling process were then studied in [74] and the realized atom interfer-ometer measurements were presented in [75]. The experimental studies presented in this work were mostly obtained during the year of my research stay within the framework of the jointly supervised PhD program. A detailed description of the interferometer measurements realized after my stay are presented in another dissertation [76].

Table of contents :

1 Introduction 
1.1 Rotation measurements and the CASI experiment
1.1.1 Rotation measurements
1.1.2 Gyroscope technologies
1.1.3 The CASI experiment
1.2 Short range forces and the FORCA-G experiment
1.2.1 Atom-surface interactions
1.2.2 Short range force measurements
1.2.3 The FORCA-G experiment
1.3 Outline
2 Theoretical tools for atom interferometry 
2.1 Beam splitting with stimulated Raman transitions
2.1.1 Two-level system
2.1.2 Coherent coupling with stimulated Raman transitions
2.2 Coupling momentum states
2.2.1 Mach-Zehnder like interferometer for inertial sensing
2.3 Coupling Wannier-Stark states
2.3.1 Atoms in an optical 1D-lattice
2.3.2 Inter-site coupling
2.3.3 Wannier-Stark spectroscopy and interferometry
2.4 Sensitivity function formalism
2.4.1 Interferometer noise estimation
2.5 Application to the 87Rb atom
2.5.1 Frequency shifts
2.6 Interferometer signal and measurement sensitivity
2.6.1 Sensitivity, stability and noise
2.7 Interferometer phase shifts
2.7.1 Non-inertial phase shifts in a Mach-Zehnder interferometer
2.7.2 Phase shifts in Wannier-Stark interferometry
3 Atom interferometer gyroscope 
3.1 Rotation measurements with the CASI gyroscope
3.2 Impact of the Raman wave fronts on the interferometer signal
3.2.1 Contrast reduction
3.2.2 Rotation phase offset
3.3 Experimental realization
3.3.1 Apparatus
3.3.2 Laser system
3.3.3 Vibration isolation platform
3.3.4 Optics for the coherent manipulation of the atoms
3.3.5 Computer control and data acquisition
3.4 Measurement sequence
3.4.1 Atom trapping, cooling and launching
3.4.2 Preparation of the interferometer state
3.4.3 Beam splitter pulse application
3.4.4 State selective fluorescence detection
3.4.5 Measurement set-up
3.4.6 Rotation phase read out
3.5 Long term stability
3.5.1 Drift sources
3.5.2 Relative beam splitter alignment and cloud overlap
3.5.3 Atom gyroscope sensitivity
3.6 Conclusions
4 Atomic short range force sensor 
4.1 Accelerometry in a trapped atom interferometer
4.2 Experimental realization
4.2.1 Apparatus
4.2.2 Laser system
4.2.3 Mixed trap and Raman beam set-up
4.2.4 Lattice laser frequency stabilization
4.2.5 Computer control and data acquisition
4.3 Measurement sequence
4.3.1 Trapping and cooling
4.3.2 State preparation
4.3.3 Raman laser and microwave pulse application
4.3.4 State selective fluorescence detection
4.3.5 Compensation of differential AC-Stark shifts
4.4 Wannier-Stark spectroscopy and interferometry
4.4.1 Wannier-Stark spectra and coherent coupling
4.4.2 Influence of the transverse optical dipole trap
4.4.3 High resolution Bloch frequency measurements
4.5 Bloch frequency measurement stability
4.5.1 Mixed trap differential light shift fluctuations
4.5.2 Raman laser differential light shift fluctuations
4.5.3 Symmetrized WSR interferometer
4.5.4 Vibrations of the apparatus
4.6 Set-up modifications and stability improvement
4.6.1 Modifications
4.6.2 Improved Bloch frequency measurement stability
4.7 Coherent atom elevator
4.7.1 Coherent atomic transport using Bloch oscillations
4.7.2 Experimental realization
4.7.3 First results
4.8 Conclusions
5 Outlook 
5.1 Atom interferometer gyroscope
5.2 Atomic short range force sensor
Appenix A: Allan standard deviation
Appenix B: Lattice laser frequency stabilization
Bibliography 
List of Figures
List of Tables
Acknowledgements
List of Publications
Curriculum Vitae

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