# Basic concepts for cold atom interferometry

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## Sensitivity function of a 4 light pulse interferometer

To describe how our system responds to external perturbation we rely on the sensitivity function. In this section I will describe how different types of perturbation (Raman laser phase noise, acceleration noise) modify the phase reading at the output of the interferometer.

### Laser phase sensitivity

We begin by studying the response of our system to an infinitesimal variation of the phase difference between the two Raman lasers, [44]. Such fluctuation generates a change P in the transition probability P = 1 2(1 + cos()) measured with the sensor.
If we assume the interferometer is operating in the middle of the fringe = 2 we can write the sensitivity function as: g = lim !0 (, t) .

#### Vacuum chamber – Atomic Fountain

The vacuum system is composed by four separated regions:
• 3D-MOT Trapping, cooling and launching of the atomic cloud.
• Interferometric zone Probing of the atomic matter-wave with Raman Pulses.
• Detection State population measurement by fluorescence.

2D MOT

The scope of the 2D-MOT is to generate a flux of pre-cooled atoms that is pushed toward the 3D-MOT system. The complete scheme of the 2D-MOT is described in [27]. In short, two laser beams are used for trapping the atoms, thanks to a system of beam splitter and retro-reflection mirrors, constraining the thermal vapor in two transverse directions. A third beam is then used as a pusher, moving the trapped cloud toward the center of the 3D-MOT where it is re-captured and cooled down. Two pairs of rectangular coils are positioned inside the structure and produce a magnetic field gradient of 20 Gcm−1[27]. The implementation of a 2D-MOT is useful to achieve a quick loading of the 3D-MOT.

3D MOT – Moving Molasses

The 3D-MOT is responsible for cooling and launching the atomic cloud. It is composed by 6 independent laser beams, trapping the atoms along three directions in space. The beams realize 3 pairs of counter propagating beams in a +/ − configuration, forming two tetrahedron pointing each other, trapping the atoms at the center of the structure. Two pairs of coils provide the necessary magnetic fields; one in an anti-Helmholtz configuration for creating a B-field gradient during the trap, one in an Helmholtz configuration for compensating the residual bias field from other magnetic sources.

Vibration Isolation Platform

All of the experiment rests on top of a Minus K platform. This platform has a resonant frequency around 0.5 Hz with a maximum load about 400 kg.
In Figure 3.12 we can see the transfer function of the Minus K in the 3 axis direction. Figure 3.12: Transfer Function Ground – Minus K. The vertical dashed red line indicates the nominal resonant frequency for the isolation platform. As we can see all three direction present a resonant peak close to this line. To adjust the position of such peak, on the horizontal direction can be done by adjusting the weights on top of it, while for the vertical direction Z, this adjustment are done by adjusting the stiffness of the internal springs.
Such graph has been calculated by comparing the signal from a seismometer situated on the platform and one situated on the ground. As expected a peak is present around the resonant frequency of the platform, around 0.5 Hz,shown with a red dashed line. A second peak is very prominent around 2Hz; this feature is being associated with a rotation mode of the experiment. This hypothesis can be confirmed by looking at the signal of the two seismometer when both of the sensors are attached to the experiment. By taking the half difference and the half sum of the signals from the two seismometers we can distinguish which vibration measurements come from rotations instead of common accelerations.

Interleaved Sequence

The sampling frequency of the experiment, even after the implementation of a joint scheme, still remain fairly low due to the long flight time of the atoms inside the interferometric region. While reducing the time of flight of the atoms is a possibility, this choice leads to a reduction of the scale factor since the sensitivity scales as T3. An improved scheme has been implemented [32] during the first year of my PhD, which relies on interleaving multiple joint sequence together. Exploiting this long flight time, we prepare and launch a cold-atom cloud every 2T/3, where 2T is the interrogation time having then 3 independent and joint atom interferometer. A visualization of triple joint interleaved can be seen in Figure 4.2.
Interleaving higher number of interferometers was not possible due to technical lim- Figure 4.2: Representation in time-space of 3 interleaved continuous atom interferometer, where each color represents a different joint sequence. itation given by the design of our sensor. Another limitation is the possibility to use only an odd number of interleaved sequence. This comes from the necessity to ramp the frequency of the Raman lasers to compensate for the Doppler effect. In a simple joint scheme, two consecutive interferometer share the first and last /2-pulse. This requires the Raman lasers to be at resonance for both cloud at the same time. With our design, this condition is satisfied by changing the sign of the ramp at the moment of the light pulse. This means we acquire measurements with alternating +keff and −keff momentum transfered. If we were to interleave two joint scheme, we would be in a condition where interferometers with the same sign of keff share a -pulse, see Figure 4.3. In this condition we would not be able to satisfy the resonance condition due to Doppler effect for both cloud at the same time. By extension a sequence 2n-interleaved would be equivalent to interleaving n-times a two joint scheme, meaning that with even number of interleaved sequence, the resonance condition cannot be satisfied for all interferometers. A solution would be to implement a double diffraction scheme where there is no Doppler effect.

1 Introduction
1.1 Cold atom inertial sensor
1.2 Sagnac based gyroscopes
1.3 Purpose of the thesis work
1.4 Plan of the Thesis
2 Basic concepts for cold atom interferometry
2.1 Raman transition and light pulses
2.1.1 Principles
2.1.2 Stimulated Raman Transitions
2.2 Atom optics
2.3 Mach-Zehnder Atom interferometry – 3 pulse scheme
2.3.1 Phase shift for a constant acceleration
2.3.2 Phase shift for constant rotations – Sagnac effect
2.4 4-pulse Atom Gyroscope
2.4.1 Constant acceleration – Zero sensitivity
2.4.2 Rotation Sensitivity – Sagnac area
2.5 Sensitivity function of a 4 light pulse interferometer
2.5.1 Laser phase sensitivity
2.5.2 Acceleration phase noise
2.5.3 Rotation phase noise
2.6 Conclusion
3 Experimental Set-Up
3.1 Lasers
3.1.1 Frequency chain
3.1.2 Cooling Laser system
3.1.3 Raman Laser system
3.2 Vacuum chamber – Atomic Fountain
3.2.1 2D MOT
3.2.2 3D MOT – Moving Molasses
3.2.3 Detection Region
3.2.4 Interferometric Region
3.2.5 Rabi oscillation
3.3 Vibration Isolation Platform
3.4 Rotation Stage
3.4.1 New Tilt Lock coil
3.5 Conclusion
4 Interleaved atom interferometry
4.1 Continuous operation
4.1.1 Joint measurement
4.1.2 Interleaved Sequence
4.2 Methods
4.2.1 Acquisition and processing based on seismometers
4.2.2 Real-time Compensation of vibration noise
4.2.3 Mid Fringe Lock
4.3 Sensitivity of the Gyroscope
4.3.1 Sensitivity with interleaved scheme
4.3.2 Interpretation of vibration noise averaging in a joint scheme
4.4 Measurements of weak dynamic rotation rates
4.4.1 How to apply weak dynamic rotation rate
4.4.2 Classical sensor
4.4.3 AI sensor
4.5 Conclusion
5 Scale Factor and bias of the Gyroscope
5.1 Gyroscope scale factor
5.1.1 Latitude estimation
5.1.2 Estimation of the initial bearing to north, N
5.1.3 Variation of interrogation time T
5.1.4 Proximity sensors
5.1.5 Estimation changing orientation by small angles d
5.1.6 Variation of the bearing to North using a rotation stage
5.2 Mirrors Alignment Bias
5.2.1 Interferometer contrast
5.2.2 Bias estimation
5.2.3 Mirrors alignment and Trajectory optimization
5.3 Conclusion
6 Non equal momentum transfer
6.1 Parasitic Interferometer
6.2 DC acceleration sensitivity and ramp optimization
6.2.1 Frequency ramp
6.2.2 Ramp optimization
6.3 Non Equal keff momentum transfer
6.3.1 Change of exchanged momentum modulus
6.3.2 Zero sensitivity to DC acceleration
6.3.3 Probability noise and ramp optimization
6.3.4 Sensitivity to rotation – Scale factor
6.4 Conclusion
7 Conclusion
A Estimation of visibility and amplitude noise
B Publications
Bibliography

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