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## Stimulated Raman Transitions

In our gyroscope we manipulate the atomic wave-packet by mean of stimulated Raman transitions [34, 35]. Using two laser fields, E1 and E2, set close to the D2 line of ~ ~ 6S1/2, F = 3E and |ei = Cesium atom, we connect the Cesium hyperfine states |gi = 6S1/2, F = 4 E .

Stimulated transition is a process that occurs when the atom, starting from one of the lower states e.g. |gi, absorbs one photon of energy ~ω1 from one laser field and emits a photon of energy ~ω2 with the second laser field, being transfered coherently to the final state |ei.

In order for this transition to take place, the two hyperfine states are coupled to an intermediate virtual level |i0i red-detuned by Δ from |ii = 6P3/2, F 0 = 3E. The detuning Δ, is introduced to minimize one photon transitions to the exited state, thus limiting spontaneous emission eﬀects that will decrease the coherence of the wave-packet. Spontaneous emissions processes decrease as 1/Δ2 while the coupling eﬃciency to the intermediate level |ii decreases as 1/Δ, therefore an optimum condition can be found in order to maximize both coherence and coupling.

In the case of counter-propagating laser beams, the two electromagnetic fields ~ ( ) E1 t and ~ ( ) have opposite direction, therefore the exchanged momenta, when the atom E2 t absorbs or emits a photons, have the same sign. The total momentum transfered to the atom then, will be ~ = (~ − ~ ) ≈ 2 ~ . ~keff ~ k1 k2 ~k1.

The stimulated Raman transition with counter propagating beams then, associate an external momentum state, | i or | + ~ i, to an internal state | i or | i, physically p~ p ~keff g e separating the two population into separate arms of the interferometer.

To describe the interaction between the atom and the light pulse, we start from studying the time-dependent Schrödinger equation associated with the system: d ˆ i~ dt Ψ(t) = H · Ψ(t) (2.1). The wave-function Ψ(t) for our three level system can be expressed as a linear combina-tion of the atom levels’ eigenstates: |Ψ(t)i = Ci(t) |ii + Ce(t) |ei + Cg(t) |gi (2.2).

ˆ ˆ ˆ (t) describes the complete Hamiltonian of the system with The operator H = H0 + V ˆ ˆ H0 as the non interactive Hamiltonian and with V (t) as the time dependent interaction between the atoms and the total electromagnetic field.

### Mach-Zehnder Atom interferometry – 3 pulse scheme

With a mirror and beam splitter pulse, we now have all the tools to create an atom interferometer. An optical Mach-Zehnder interferometer requires two beam splitters, and two mirrors. With atom interferometry we can achieve the same type of configuration, by using the correct series of pulses. We start with a π/2-pulse to physically separate the two arms, a single π-pulse to reflect then conclude the sequence with a π/2-pulse to recombine the wave-packets. We can estimate the phase in one of the output ports of the interferometer, by calculating the diﬀerence of accumulated lasers phase between the two arms: ΔΦ3P = (φ1 − φ2) − (φ2 − φ3) = φ1 − 2φ2 + φ3 (2.12). Here we introduce our sign convention for the imprinted phase; if the starting state is the ground state |gi, we apply a positive phase shift, +φ. On the contrary when we shine the light pulses on the exited state |ei, phase imprinted is negative, −φ.

#### Phase shift for a constant acceleration

In the previous section we explained how we are able to create a superposition of atomic states while also separating the two wave-packets in space. As we saw the separation is proportional to the eﬀective momentum exchanged but it also depends on the time between each light pulses. As the separation between the two wave-packets increases, the interferometer’s sensitivity to ‘inertial’ forces grows. For the case of a 3-pulse Mach-Zehnder, subjected to a constant acceleration field ~a we are able to generalize the ex-pression for the phase shift as: Φa = keff · (~x1 (0) − ~x2(T )) − (~x2(T ) − ~x3(2T )) ~ ~ (0) − 2~x2(T ) + ~x3(2T )) (2.13) = keff · (~x1 = ~ · 2 keff ~aT.

The displacement of the atom on each path, is obtained by doubly integrating the acceleration ~a, thus we obtain the phase to scale as T 2.

**Phase shift for constant rotations – Sagnac eﬀect**

As our Mach-Zehnder atom interferometer encloses a physical area ~, another famous interference phenomenon can be observed [19]. Sagnac experimentally proved that when light is split and recombined after enclosing a certain area, the interference pattern at the output changes depending on the rotation rate of the apparatus [19, 38]. Sagnac discovered that such phase shift is proportional to the area ~ enclosed by the two path, and to the rotation rate, Ω~. The general expression for Sagnac phase shift is: ΔΨΩ = 1 I Ω~ × ~x E · d~x ~c2 (2.14) 2E ~ ~ = ~c2 A · Ω. where E is the total energy of the particle used to perform interferometry.

It’s important to notice that Eq. (2.14), holds true for both purely optical interferometer and matter-waves ones [39, 40, 41]. In Eq. (2.14), lies the reason why performing Sagnac interferometry with cold atoms is more advantageous with respect to optical interferometers. In fact for equal enclosed area A, the energy ratio between an atom (Eat ≈ mc 2 ) and a photon (Eph = ~ω) gives: Eat ≈ 1011 (2.15) Eph.

**4-pulse Atom Gyroscope**

The four pulse atom interferometer uses a sequence of light-pulses (π/2 − π − π − π/2). The scheme could be seen as two Mach-Zehnder interferometer, where the output of the first is the input of the second, but with no last(first) pulse.

**Table of contents :**

**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 **