Sensitivity function approach: phase noise transfer function and pulses finite duration corrections

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New experimental setup for the determina-tion of h/mRb

The principle of our determination of h/mRb is based on the measurement of the recoil velocity vr of an atom, i.e. the velocity an atom acquires when it absorbs 1We should note that in 2011, the errorbars of the Harvard measurement and the h/mRb deter-mination overlapped. The situation changed with CODATA fundamental constants reevaluation in 2014 and corrections in the computation of ae.

NEW EXPERIMENTAL SETUP FOR THE DETERMINATION OF H/MRB5

a photon. As a photon carries a momentum ~k, where k is its wavevector, and through momentum conservation, we obtain:
For a plane wave in vacuum, we have k = 2πν/c, where ν is the frequency of the laser that drives the momentum transfer process. Thanks to the technology of frequency combs[Diddams, 2000], ν can be known with a better precision than the recoil velocity. Then, the combined measurement of vr and of ν allows for the determination of h/m.
The recoil velocity is measured through the measurement of the acquired ve-locity by the atom upon the photon absorption in a differential velocity sensor based on the compensation of Doppler effect.
With the absorption of a single photon, the atom ends in an excited state and decays incoherently through spontaneous emission. In order to avoid this effect, we use two-photon transitions in a Λ scheme: the atom absorbs a photon from a laser field and decays through stimulated emission in a second laser field. The process is now coherent and the atomic motion controlled. We place ourselves in counterpropagating configuration for the two laser fields, such that the two-photon transitions are sensitive to Doppler effect.
First, Raman transitions (see section 2.1) couple the two hyperfine states of the 87Rb atom and are used to build the differential velocity sensor. With these transitions, the recoil velocity vr ∼ 5.9 mm • s−1 translates into a Doppler shift of ∼ 15 kHz.
With 30 ms interferometric interrogations, where the effective duration on the velocity measurement is TRamsey = 10 ms, the Ramsey duration, we typically ob-tain precision on the recoil velocity of 1 1 ∼ 0.1 Hz, which corresponds to a relative precision of ∼ 7•10−6 on the recoil velocity. This corresponds to a velocity sensitivity of ∼ 40 nm•s−1. The duration of interrogation imposes the use of laser cooled atoms.
Secondly, Bloch Oscillations (BO) (see section 2.3) transfer an even number of recoils to the atoms while leaving their internal state unchanged. We typically transfer a 1000 recoils, which corresponds to an enhancement in the measurement performance up to ∼ 7 • 10−9. Moreover, the atoms acquire a velocity of 6 m • s−1 with the BO process. With the 30 ms interrogation duration above, the magnetic field should be controlled on a few tens of centimeters.
This is one of the major improvement of the new experimental setup, described in section 3.3.3. The excellent magnetic field control that extends over 45 cm allows for the reduction of this contribution to systematic errors, which was one of the largest in the latest measurement of our team[Bouchendira, 2011].
This also allowed us to increase the interrogation duration with TRamsey = 20 ms, thus doubling the sensitivity of our determination. Combined with a bet-ter control of vibration noise induced by a simpler vibration isolation setup, we exhibited unprecedented statistical uncertainty on h/m (see chapter 5).
The other major error source of the latest measurement is related to the gaus-sian beam correction to the wavevector k. Indeed, the photon wavevector is per-fectly known only in the case of a plane wave in vacuum. In order to reduce the effect, an as such the error, we increase the waist of the laser beams as they tend to a plane wave.
However, the intensity of the beam scales as P/w2 with P the laser power. In order to maintain the intensity constant and as such increase the laser power, our team has worked with a new laser technology based on high power lasers at 1.5 µm that are doubled in PPLN crystals[Andia, 2015b]. This technology, now fully deployed on the new setup, allows for larger beams but also increased the stability of the setup.
Finally, as the α/ae comparison promises to yield useful information in the search for new physics, the new experimental setup is also intended to run next generation interferometers with even increased sensitivity[Cladé, 2009]. Such in-terferometers are very sensitive to intensity inhomogeneities induced by the atomic motion in the laser beams.
In order to implement them, the setup is equipped with a Bose Einstein Con-densate (BEC) production setup in an optical dipole trap. Indeed, BEC sources present a smaller velocity spread and as such reduced sensitivity to intensity in-homogeneities. The setup has been described in [Jannin, 2015a] and [Courvoisier, 2016], and the effect of interactions in a BEC on an interferometer modeled in [Jannin, 2015b]. In this work we have improved the BEC production process and observed experimentally the effect of atom-atom interactions (see section 4.1).

Outlook of this manuscript

Aiming at providing precise determinations of the fine structure constant α, our work aims to a new determination with competitive uncertainty in order to pro-vide an alternative value to the comparison α/ae in order to confirm the recent discrepancy. While working on the installation of the interferometry laser system of our new experimental setup, we have worked on the long term BEC use.
In order to present this work, the remaining of the manuscript is divided in two parts and five chapters. The first part contains two chapters and is focused on the presentation of the experiment:
– In chapter 2, we will explain the fundamental concepts underlying our experiment: stimulated Raman transitions, atom interferometry and Bloch Oscillations.
– Chapter 3 presents the implementation of these concepts on the experiment.
The second part presents the results that we have obtained during this thesis work:
– In chapter 4, we discuss the difficulties that arise when using either BEC as an atom source or atoms at the output of an optical molasse. This discussion allows us to defend our choice of atomic source for the h/m measurement.
– Chapter 5 presents the exact protocol for the measurement of the h/m ratio and the data analysis associated. We also discuss the statistical performance of the setup, which has not been reached in another experiment to the best of our knowledge.
– In chapter 6, we detail our study of systematic error sources. This study, although yet incomplete, shows that the current measurement campaigns can allow to reach a competitive uncertainty.
A few tools are needed in order to measure the recoil velocity of an atom. The aim of this chapter is to present in detail these tools, and their associated concepts. In a few words, we simply need to transfer photon momenta to atoms and measure the velocity they have acquired.
We start by studying stimulated Raman transitions which are two-photon tran-sitions in a Λ scheme: a particle is subjected to two laser beams, absorbs a photon from one, and emits through stimulated emission another photon in the second laser beam, thus reaching its final state.
Immediately following, counterpropagating Raman transitions, a particular case, will be detailed. This will allow us to introduce the concepts of atom interferom-etry. In the frame of this thesis, this part is essential as it will introduce a core part of our work.
Finally, we will present the phenomenon of Bloch Oscillations which is respon-sible for the transfer of a large number of recoils to the atoms.

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Stimulated Raman transitions

General considerations

Two-photon transitions have proven to be extremely useful in spectroscopy. They may be used to either suppress or enhance Doppler effect. Using Raman tran-sitions, we are able to control the atomic wavefunction between two electronic ground states, which allows us to reach coherent manipulation of the atoms quan-tum states.
In order to present the features of these transitions, we assume that we can describe atoms through a three level system with two ground states |αi, |βi and an excited state |ei. Its internal Hamiltonian can be written:
Hinternal = ~ωHF S |βi hβ| + ~ωe |ei he| , (2.1.1) with the energy of |αi taken as reference and ωe > ωHF S as the couplings be-tween the ground states and the excited state are induced by optical transitions. The notation ωHF S is related to the hyperfine structure of the atom (see next paragraph).
An energy level diagram of a Raman transition is displayed on figure (2.1, left). Two laser fields ω1, ~k1 , ω2, ~k2 couple the excited state |ei to the ground states. A realistic atomic model would assume that the excited state spontaneously decays. In order to prevent such a process, the one-photon laser coupling is detuned from resonance by an amount Δ (Δ ∼ 60 GHz in our experiment), such that the excited state is not populated.
The geometric configuration of the beams is given by the two beam wavevectors ~ ~ . At this point, we make no assumptions on the relative orientations of k1 and k2 the beams. As we shall see in section 2.1.3, two particular configurations are of interest: the two beams are aligned and travel whether in the same direction (co-propagating) or in opposite directions (counterpropagating).

Table of contents :

1 General Introduction 
1.1 Redefinition of the kilogram – New SI
1.2 Test of the Standard Model
1.3 New experimental setup for the determination of h/mRb
1.4 Outlook of this manuscript
I The experiment 9
2 Fundamentals 
2.1 Stimulated Raman transitions
2.1.1 General considerations
2.1.1.1 Justification of the three level atom
2.1.1.2 Resonance condition
2.1.2 Quantum mechanical description of the coupling
2.1.2.1 Transition probabilities
2.1.2.2 Selection rules
2.1.3 Co- and counter-propagating transitions
2.2 Atom Interferometry
2.2.1 Ramsey sequences
2.2.2 Atom interferometry: Mach-Zehnder configuration
2.2.3 Interferometry in a gravitational field
2.2.3.1 Classical limit
2.2.3.2 Laser phases
2.2.3.3 Perturbative Lagrangians
2.2.4 Ramsey-Bordé in differential velocity sensor configuration
2.2.5 Sensitivity function approach: phase noise transfer function and pulses finite duration corrections
2.2.5.1 Definition
2.2.5.2 Preliminary treatment
2.2.5.3 Pulses finite duration
2.2.6 Conclusion
2.3 Bloch Oscillations
2.3.1 Optical lattice potential
2.3.2 States in a periodic potential
2.3.3 Principle of Bloch Oscillations
2.3.4 Bloch Oscillations in the tight binding regime
2.3.5 Application to the h/m measurement
3 Experimental implementation
3.1 Atomic sample production
3.1.1 Vacuum cell
3.1.2 Atom trapping laser system
3.1.2.1 Cooling laser system
3.1.2.2 Manipulation
3.1.3 Magneto-optical traps
3.1.3.1 Optical molasses operation
3.1.3.2 Residual magnetic field compensation
3.1.4 Absorption imaging
3.1.4.1 Principle
3.1.4.2 Operation
3.1.5 Evaporative cooling
3.2 Interferometry lasers
3.2.1 Amplification and doubling technology
3.2.2 Raman Lasers
3.2.3 Bloch Lasers
3.2.3.1 Relative frequency control
3.2.3.2 Absolute Frequency control
3.2.4 Vertical path description
3.2.5 Interferometry preparation
3.2.6 Frequency stabilization
3.3 Operation of the experimental setup
3.3.1 Sequence programming
3.3.2 Light sheets detection
3.3.3 Operation and characterization
3.3.3.1 Magnetic field measurement
3.3.3.2 Gravity gradient measurement
II Results
4 Atom sources comparison
4.1 Optimization of the Bose Einstein Condensates production process
4.1.1 Sequence for the production of a BEC
4.1.1.1 Dipole trap loading
4.1.1.2 Genetic algorithm
4.1.2 Atom-atom interactions phase shift
4.1.2.1 Spin distillation
4.1.2.2 Microwave-based Ramsey interferometry
4.1.2.3 Interaction in the context of a Ramsey sequence
4.1.2.4 Conclusion
4.2 Resonance mismatch during light pulses
4.2.1 Phase shifts
4.2.1.1 Effect of diagonal terms in the coupling
4.2.1.2 Conclusion
4.2.2 Application to atom interferometry
4.2.2.1 Counterpropagating Raman transitions
4.2.2.2 Effect in a Ramsey-Bordé interferometer
4.2.2.3 Velocity dependence
4.2.2.4 Calibration of the intensity variation
4.2.3 Conclusion
5 Determination of the h/mRb ratio 
5.1 Experimental protocol
5.1.1 Gravity elimination
5.1.2 Raman inversion technique
5.1.3 Fringe extraction procedure
5.1.4 Discussion
5.1.5 Raman wavevector variation
5.2 Statistical performance
6 Analysis of systematic effects 
6.1 Introduction
6.1.1 General considerations
6.1.2 Description of the trajectories
6.1.2.1 Beginning of the interferometer
6.1.2.2 During the interferometer
6.2 Effect induced by the gradient of gravity
6.3 Effects on the atomic energy levels
6.3.1 Phase shift during Raman pulses
6.3.1.1 One-pulse phase shift
6.3.1.2 One-photon light shifts
6.3.1.3 Two-photon light shifts
6.3.2 Magnetic field inhomogeneities
6.4 Effect induced by the rotation of the Earth
6.5 Effects on the lasers wavevectors
6.5.1 Beams misalignment
6.5.2 Beam profile corrections
6.5.2.1 Ideal gaussian beam
6.5.2.2 Effective wavevector in a distorted wavefront
6.5.2.3 Effect on h/m
6.5.3 Laser frequencies
6.6 Phase shift in Raman radiofrequency chain
6.7 Current error budget and survey of systematic effects
Conclusion and outlooks

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