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Characterization and limits of a cold atom Sagnac interferometer

We present the full evaluation of a cold atom gyroscope based on atom interferometry. We have performed extensive studies to determine the systematic errors, scale factor and sensitivity. We demonstrate that the acceleration noise can be efficiently removed from the rotation signal allowing to reach the fundamental limit of the quantum projection noise for short term measurements. The technical limits to the long term sensitivity and accuracy have been identified, clearing the way for the next generations of ultra-sensitive atom gyroscopes.


I. INTRODUCTION
Inertial sensors are of interest in both science and industry. High precision sensors find scientific applications in the areas of general relativity [1], geophysics [2] and navigation [3].
In these fields, matter-wave interferometry is promising since it is expected to be an extremely sensitive probe for inertial forces [4]. In 1991, atom interferometry techniques were used in proof-of-principle demonstrations to measure rotations [5] and accelerations [6]. The first demonstrations of highly sensitive atomic gyroscopes using thermal beams [7,8] were obtained in 1997, followed by sensors with sensitivities at the state-of-the-art level [9,10].
For practical applications, cold atom interferometry is of fundamental interest thanks to its intrinsic stability and accuracy, as the measurement of inertial forces is realized with respect to the inertial frame of the free-falling atoms. The use of cold atoms allows better control of atomic velocity and interaction time, leading to a better accuracy in a much more compact instrument [11,12,13,14].
In this paper we present the full characterization of a gyroscope based on atom interferometry, sensitive to the Sagnac effect. Different parameters have been taken into account for the study: the short term noise, the stability of the systematic effects, the scale factor and its linearity. The apparatus uses Cesium atoms and Raman transition to manipulate the matter wave-packets. In our setup, we use a single Raman beam interacting with slow atoms, which makes the setup very versatile. Thus the experiment enables us to measure the full basis of inertia (three components of acceleration and rotation) with the same apparatus [11], making it suitable for applications such as inertial navigation. In this paper, we emphasize the possibility to measure the systematic effects and the scale factor accurately thanks to our polyvalent apparatus. In section II we describe the experimental setup and the measurement process. A detailed analysis of the different sources of systematic errors and tests of the scaling factor are presented in section III. Finally, the analysis of the stability of the rotation measurement and its main limitations are described in section IV.

II. APPARATUS
A global view of the experimental setup is presented in Fig. 1. The whole experiment is surrounded by two layers of mu-metal shield in order to reduce the impact of external This creates an interferometer of 80 ms total maximum interaction time. The atomic phase shift is then obtained from the population in each output port, which is measured by a fluorescence technique thanks to the state labelling of the interferometer output ports [15].
In this paper we will focus on the configuration based on vertical Raman beams. The use of two atomic sources allows us to discriminate between the acceleration along the vertical direction and the rotation around the y horizontal axis. The experiment is mounted on a horizontal rotating platform, which enables us to vary the projection of the Earth's rotation rate along the sensitive axis of the gyroscope.

A. Atomic preparation
Cesium atoms are loaded from a thermal vapor during 140 ms into two independent MOTs. After the MOT-coils are turned off, the atoms are kept in an optical molasses for 15 ms, to allow the stray magnetic field to decay. A frequency shift between the upper and the lower cooling beams is then applied, to launch the atoms thanks to the moving molasses technique. The independent control of the lower and upper cooling beam frequencies is achieved by passing through two different acousto-optic modulators (AOM). By applying a frequency shift of 3.2 MHz on the AOM controlling the lower cooling beams, the atoms are launched with a velocity of 2.4 m s −1 at an angle of 8 • with respect to the vertical direction. In addition, the atoms are cooled down to a temperature of about 1.2 µK in the molasses by chirping the mean frequency down to −15Γ, with respect to the closed transition Following this launching stage, the atoms are distributed among all Zeeman sub-levels of the |6S 1/2 , F = 4 state. In order to reduce the sensitivity to parasitic magnetic fields, atoms are selected in the sub-level m F = 0. For this purpose, a static magnetic field of 30 mG is applied in the z direction to lift the degeneracy of the Zeeman sub-levels. Atoms in |6S 1/2 , F = 4, m F = 0 are transferred to |6S 1/2 , F = 3, m F = 0 when passing through a micro-wave cavity. Any atoms remaining in |6S 1/2 , F = 4 are then removed by means of a pusher beam. After this preparation stage, we obtain typically 10 7 atoms in the |6S 1/2 , F = 3, m F = 0 ground state with a residual in the other states of less than 1%, for both sources.
B. Implementation of the interferometer

Three pulse interferometer
When the atoms arrive close to the apex of the parabolic trajectories, occurring at t ap = 244 ms after launch, the wave packets are split, deflected and recombined by stimulated Raman transitions [6] in order to realize the interferometer. Since the Raman beams are vertically oriented the interferometric area is created in the (x,z) plane (Fig. 2).
The output phase shifts ∆Φ A and ∆Φ B of the two interferometers ∆Φ A,B are composed of three terms which depend respectively on the acceleration a, the rotation Ω and the Raman laser phase differences of the three pulses [16]: In the vertical Raman configuration studied here, these contributions are written as a function of the vertical acceleration a z , the horizontal rotation Ω y and the Raman laser phase differences φ i , i = 1, 2, 3: where k eff is the effective wave-vector of the Raman beam. The rotation phase shifts ∆Φ A,B Ω measured by the two interferometers are related to the horizontal velocities V A,B x and have opposite signs for the two sources. The use of two counter-propagating sources allows to discriminate between the acceleration and rotation phase shifts [9].

Raman lasers
In order to drive Raman transitions between |6S 1/2 , F = 3, m F = 0 and |6S 1/2 , F = 4, m F = 0 , two counter-propagating laser beams, with a frequency difference of 9.192 GHz, are required. These two optical frequencies are generated by two extended cavity laser diodes [17] emitting at λ = 852 nm. The first laser is locked by frequency comparison to the MOT repumper beam with a detuning of 350 MHz with respect to the |6S 1/2 , F = 3, m F = 0 ↔ |6P 3/2 transition. A second laser is phase-locked to the previous one by comparing the beat note between the two beams with a microwave reference [18].
In order to get sufficient power to drive the transitions, the two laser beams are injected into a common semiconductor tapered amplifier (EYP-TPA-0850-01000-3006-CMT03) [19].
The power ratio between the two lasers is adjusted close to 0.5 in order to cancel the effect of the AC Stark shift (see part III C 2). This ratio is then finely tuned by means of Raman spectroscopy on the cold atom samples. After amplification, the two lasers have the same polarization and are guided to the atoms through the same polarizing fibre. At the output of the fiber, the beam is collimated with an achromatic doublet lens of 240 mm focal length, giving a diameter at 1/e 2 of 35 mm. The intensity at the center of the beam is 20 mW cm −2 .
The two counter-propagating beams are obtained thanks to a retro-reflected configuration in which the two frequency beam passes through the vacuum chamber and is reflected by a mirror, crossing a quarter-wave plate twice (Fig. 1). The quarter-wave plate is set in such a way that retro-reflected polarizations are orthogonal with respect to the incident ones. In this manner, counter-propagating Raman transitions are allowed while co-propagating ones are forbidden. This retro-reflected configuration limits the parasitic effects induced by the wave-front distortions, which are critical in order to achieve good accuracy and long term stability (see part III C 4).
Moreover, since the atoms are in free fall, the frequency difference between the two atomic levels is Doppler shifted in the vertical direction by ω D = k eff · g(t − t ap ), which depends on the gravity g. In order to satisfy the resonance condition during the whole atomic flight, the frequency difference between the two Raman lasers is chirped thanks to a Direct Digital Synthesizer (DDS). Additionally the interferometers are realized with a delay of 5 ms with respect to the apex of the trajectories in order to avoid a null Doppler shift during the π pulse.  states. The transition probability P is deduced from the fluorescence signals obtained on the two photodiodes (s 1 and s 2 ) and is written: P = s 1 s 1 +s 2 .

Detection noise analysis
The noise affecting the fluorescence signals can be separated into three main contributions [20]. The first is related to power or frequency fluctuations of the probe beams that induce a noise which scales linearly with the total number of atoms N A and N B in each source. The second contribution consists of a technical noise related to the detection system (photodiode dark current and amplifier noise) that gives a contribution independent of the total number of atoms in the probe. Finally, the quantum projection noise (QPN) gives a fundamental limit of the measurement [21] and scales as P A,B (1 − P A,B )N A,B 1/2 . These independent sources of noise can be added quadratically, giving a variance of the transition probability: In order to determine the parameters α and γ when working at one side of a fringe of the interferometer, we use a single Raman laser pulse giving an average transition probability close to 0.5. In such a case, the noise detection properties are similar to those of the interferometer without being sensitive to the interferometer phase noise. In practice, the Raman laser power is reduced so that the Raman pulse duration τ = π/Ω Rabi limits the average transition probability to P A,B = 0.5 by velocity selection. By this means, the measurement is made almost insensitive to any power fluctuations of the Raman beam.

Noise Fitted parameters
Laser  The detection noise σ P is evaluated as a function of the number of detected atoms, which is changed by varying the loading time of the MOTs (see Fig. 4). The data are scaled as a function of the fluorescence signal S = s 1 + s 2 , which is proportional to the actual atom number N A,B = ηS. The two detection systems clearly show identical noises for the two atomic sources. A fit of the experimental data has been realized by eq. 3 in which the scaling of the number of atoms is a free parameter. This leads to a fit with three parameters (a, b, c), related to α, γ and η, which are reported in Table I. For the usual experimental parameters, the number of atoms detected corresponds to S = 7000 and the signal to noise ratio is then limited by the quantum detection noise. In this case, the parameter b allow the determination of the actual number of atom detected for each source, giving N = 3.6 × 10 5 atoms, which is in agreement with additional measurements realized by absorption.

D. Measurement process
Interference fringe patterns are scanned by taking advantage of the Raman laser phase control. Indeed, by adding a laser phase offset δϕ n between the second and the third pulses of the n th measurement, the atomic phase measured evolves as ∆Φ laser = φ 1 − 2φ 2 + (φ 3 + δϕ n ). In practice, the phase increment δϕ n is applied to the micro-wave signal used as a reference to phase-lock the two Raman lasers. The fringe patterns, shown in Fig On the other hand, because of the thermal expansion of the atomic cloud, all the atoms do not experience the same laser power in the gaussian Raman beam, and therefore do not experience the same Rabi frequency.
The transition probability measured at the output of the interferometer is written as a function of the following terms: an offset M A,B , a contrast C A,B , a phase shift due to inertial effects ∆Φ A,B I (acceleration and rotation) and a laser phase shift ∆Φ laser : The phase shift accumulated by the two interferometers can then be deduced by fitting each fringe pattern with the equation 4. The acceleration and rotation phase shifts are then discriminated by calculating respectively the half sum and half difference of the two fitted phase shifts: In the general case, the interferometric phase shift can always be extracted from four points: With the retro-reflected configuration, atoms are submitted to four laser waves which To sum up, the experimental sequence consists in acquiring transition probability measurements on the two sources, alternately for each direction of the wave vector ±k eff . The fit is processed afterwards and we infer a measurement of the inertial phase shifts for each set of eight acquisition points.

III. SYSTEMATIC ERRORS AND SCALE FACTOR
A. Characterization of the atomic trajectories A crucial point, when measuring a differential phase shift between two interferometers [9,22,23], is the overlap of the two atomic trajectories. Indeed, if the trajectories are perfectly overlapped, many systematic effects cancel out, as explained in paragraph III C.
Consequently, the two cold atom sources were realized with particular care. In this section, we study both the overlap and the stability of the two trajectories.

Overlap of the atomic trajectories
When using Raman laser beams oriented in the vertical direction, the atomic trajectories have to be overlapped in the orthogonal plane (xy) since the Raman laser system remains We directly map the two trajectories in the interferometer zone thanks to the Raman laser beams, with an movable aperture of diameter 5 mm. By maximizing the transfer efficiency of a Raman pulse, we deduce the actual position of each atomic cloud in the xy plane at a given time. Fig. 6 shows the two trajectories which are overlapped to better than 0.5 mm over the whole interferometer zone. The measurement resolution is limited by the spatial extension of the atomic clouds (5 mm FWHM at the apex). In order to perform inertial force measurements, we need to know accurately the scale factor and the systematic errors, which link the actual rotation and acceleration quantities to the measured phase shifts ∆Φ rot and ∆Φ acc . In order to determine these two parameters for rotation, we change, in a controlled way, the rotation rate measured by the gyroscope.
In addition, taking advantage of the single Raman laser beam pair, the interaction time T can be changed continuously from T = 0 to 40 ms. The systematic error and scale factor measurements were performed for various interaction times, giving a test of the quadratic scaling of the rotation phase shift with T .

Dependence on the rotation rate
The first test of the rotation scale factor consists in checking the proportionality between the rotation phase shift and the rotation rate. For this purpose, the orientation θ between the sensitive axis of the gyroscope and the East-West direction, is changed to measure different horizontal projections of the Earth rotation rate, Ω y = Ω h sin θ. At the Observatoire de Paris, located at the latitude λ = 48 0 50 ′ 08 ′′ , the horizontal rotation rate Ω h is 4.8 × 10 −5 rad s −1 .
The whole device is placed on a rotation mount, which determines the orientation with a relative accuracy of 50 µrad.
In Fig. 8 the rotation phase shift is plotted with respect to the orientation of the gyroscope. The experimental data are fitted by a sinusoidal function ∆Φ = ∆Φ er Ω + A (0) sin(θ − θ 0 ). The magnitude A (0) is used to calibrate the rotation scale factor: we find A (0) / (Ω T sin λ) = 15124 ± 12 rad rad −1 s −1 . Moreover this measurement allows to deduce the systematic error of the rotation signal: ∆Φ er Ω = 28.3 ± 0.7 mrad. The same data, when plotted as a function of the rotation rate, are well aligned with a slope equal to the scale factor. Non-linearities appear as deviations to the straight line and can be evaluated by fitting with a quadratic term. Their relative contributions are below 10 −5 in the range of measurement.

T scaling of the rotation phase shift
In order to test the T 2 scale factor dependance, the experimental setup is oriented at θ ≃ ± 90 • , for which the sensitivity of the gyroscope to the Earth rotation is maximum.
We also take advantage of the fact that, for these two orientations, an error in the orientation angle θ has a second order effect on the rotation phase shift. Therefore we write: where Φ er rot (T ) is the systematic error on the rotation signal for a given interaction time T . The half difference of the signal measured at θ = +90 • and θ = − 90 • (Fig. 9(a)) gives a test of the scaling of the rotation phase shift as T 2 . It shows an excellent agreement with the expected behavior. In addition, the systematic error which affects the rotation signal is deduced from the half sum of the signals for the two orientations. The evolution of the error phase shift is displayed as a function of the interaction time in Fig. 9

Quadratic Zeeman effect
The first expected source of systematic error in the interferometer phase shift comes from the Zeeman shift induced by the magnetic field. In order to limit its impact, the interferometer is realized between the two m F = 0 Zeeman sub-levels, whose energy difference evolves quadratically with the magnetic field as: ∆ mag = K (2) B 2 , with K (2) = 427.45 Hz G −2 for Cesium atoms. In addition, thanks to the symmetric features of the interferometer, the phase shift is not sensitive to a constant frequency shift. However, a magnetic field gradient breaks the symmetry and gives rise to a phase shift. Assuming a linear gradient δ bx along the atomic trajectories the phase shift induced is:

AC Stark shift
We study the effect of a frequency shift induced by the AC Stark shift ∆ AC on the interferometer phase shift [25]. This frequency shift between the two hyperfine levels can be cancelled by setting a proper intensity ratio between the two Raman beams. However, if the ratio is flawed, a phase shift ∆Φ AC is induced on the interferometer. Assuming the AC Stark shift to be constant during each laser pulse gives: where ∆ are the respective effective Rabi frequencies [26]. In our experimental setup, the two Raman lasers are provided by a single optical fiber and are retro-reflected, ensuring the stability of the intensity ratio between the two lasers throughout the Raman beam.
Equation (8)  The sensitivity of the rotation signal to the AC Stark shift was characterized by changing the intensity ratio between the two Raman lasers so as to induce a frequency shift of ∆ AC = 10 kHz. A sensitivity of 5 mrad kHz −1 was measured. This AC Stark shift is however independent of the effective wave vector k eff so it can be rejected by the k-reversal method, considering that the fluctuation of ∆Φ AC is slow compared with the repetition rate of the measurement. Using this technique, we measure a residual error due to the AC Stark shift at the level of 0.1 mrad kHz −1 corresponding to a sensitivity to this effect reduced by a factor of 50. When running the interferometer, we directly optimize the AC Stark shift on the atomic signal with an accuracy better than 500 Hz, leading to a residual systematic error below 0.05 mrad which is negligible.

Two-photon light shift
As explained above the retro-reflected Raman beams couple the ground state where Ω (i) eff is the effective Rabi frequency and δ (i) D the Doppler shift for the i th pulse. This phase shift depends on k eff and cannot be cancelled out with the k-reversal method.
It is shown that this shift is similar for the two interferometers and induces an error on the acceleration signal of only 12 mrad while remaining below 0.3 mrad on the rotation signal.

Wave-front distortion
The atomic phase shift depends on the effective laser phases φ i imprinted on the atomic wave at the moments of the three pulses. Because of a non-uniform laser wave-front, the phase shift measured by each interferometer depends on the position of the atomic cloud in the Raman wave-front (x i , y i ) at the i th pulse and can be written as: The spatial variations of the laser phase along the wave-front induce a phase shift on the interferometric measurements. In our setup, the use of a retro-reflected configuration allows to reduce the number of optical elements and hence to decrease the aberrations between the two opposite Raman beams. Therefore, the wave-front defects are mainly induced by the Raman window, the quarter wave plate and the retro-reflection mirror, which affect only the reflected beam.
When the trajectories of the two atomic clouds are perfectly overlapped, the wave-front defects are identical for both interferometers, which is equivalent to a constant acceleration.
If the paths are not perfectly overlapped, the phase shifts due to these wave-front defects are not identical and appear as an error on the rotation signal when subtracting the phase shifts of the two interferometers [28]. This is illustrated in Fig. 11, which represents the positions of the atomic clouds at the times of the three pulses π/2 − π − π/2. The error on the rotation signal results from the half difference of the two wave-front distortion phase A first estimation of this effect was performed by measuring the wave-front distortion induced by a single Raman window with a Zygo wave-front analyzer [29]. From these measurements, we deduce a wave-front phase shift of 20 mrad on the rotation signal for an interaction time of 2T = 80 ms, which is consistent with the error measured on the rotation signal. A second method was implemented to estimate this effect by moving the In Fig. 12, the rotation phase is displayed as a function of the delay compared to the usual laser sequence, and translated into trajectory shift. For small deviations, the phase shift due to wave-front defects can be linearized. Therefore we infer the sensitivity of the rotation measurement to a relative displacement along the x direction between the two sources by performing a linear fit of the data. The sensitivity obtained is:

Conclusion on systematic errors
We studied the relative contributions of different sources of parasitic phase shifts. The dominant contribution to the systematic effects has been identified as coming from the wavefront distortions. Indeed, an independent evaluation of its contribution (∼ 20 mrad) is in agreement with the actual parasitic phase shift (28.3 mrad) for 2T = 80 ms ( Fig. 9(b)).
This effect becomes significant when the interrogation time exceeds 30 ms. We attribute this to the growth of wave-front distortions on the edges of the windows. Using a larger window or the centers of three separated windows for the three pulses would limit its impact.

IV. SENSITIVITY
The bias stability of the acceleration and rotation signals was studied by orienting the area of the interferometer in the East-West direction so that the rotation rate measured is zero. Consequently, it is possible to measure a phase shift by setting the interferometers on

A. Acceleration measurements
The acceleration signal is deduced from the sum of the two interferometer phase shifts.
This sensitivity is mainly limited by the residual vibrations of the platform, as for the atomic gravimeter [30]. The standard deviation ( Fig. 15(a)) of the acceleration signal shows an improvement of the sensitivity proportional to τ −1/2 as expected. By integrating our measurement over 5000 s, we reach a sensitivity of 10 −8 m s −2 , which is close to our atomic gravimeter characteristics presented in [30]. The difference between the sensitivity of the two apparatuses is explained by our slightly lower repetition rate and interrogation time. In order to reach this sensitivity, the noise contribution from vibrations is filtered out thanks to a passive isolation platform (nanoK 350BM-1). The residual noise is further reduced by a correlated measurement performed with a low noise seismometer (Guralp T40) [30].
Additionally, variations of the gravity g due to tidal effects are computed from a model provided by tide parameters extracted from [31] and subtracted from the signal in order to infer the long term stability of the sensor.

B. Noise on the Rotation signal
The rotation phase shift is extracted from the difference between the signals of the two interferometers. The Allan standard deviation is plotted in Fig. 15 The Allan standard deviation of the rotation signal at one second is limited by the quantum projection noise evaluated in section II C 2. In order to confirm this point, we perform measurements for different numbers of atoms by changing the loading time of the two MOTs. Assuming that the detection noise is independent for the two interferometers A and B, its impact on the standard deviation σ Φ of the rotation phase shift yields: The contrasts of the two interference fringes are identical for the two interferometers, and denoted by C. The coefficients α and γ are related to the detection features and were determined in section II C 2. In order to characterize the contribution of the detection noise, it is convenient to plot the rotation phase noise versus the number of atoms. For a given loading time, the number of trapped atoms is different for the two atomic sources A and B, so we define a reduced atom number N = N A N B N A +N B . Thus equation (12) becomes: In Fig. 14 the rotation noise (blue squares) is displayed versus reduced atom number.
The data correspond to the Allan standard deviation at 1 second of the rotation phase shift

V. CONCLUSION
We carried out the characterization of a cold atom gyroscope in terms of sensitivity, systematic errors and scale factor. A study of the scale factor demonstrated excellent linearity and stability, limited by the resolution due to the drift of the systematic effects. This first study of the limits of a gyroscope using cold atoms has already demonstrated a sensitivity at the level of the best commercial optical gyroscopes (fiber and ring laser gyroscopes). In other work, a 400 times better short term performance was demonstrated using an atomic beam interferometer [9]. However, considering both at short and long term sensitivity, our gyroscope is 3 times less sensitive than the best atomic one [10]. Moreover, this work has clearly identified the limits to the sensitivity, pointing the way to further improvements.
The short term sensitivity was dominated by the quantum projection noise thanks to the use of a double-interferometer, which perfectly cancels the phase shift due to parasitic vibrations. The main contribution to the drift is related to the fluctuations of the atomic trajectories. When coupled to the Raman wave-front distortions, these fluctuations also limit the long term stability of the rotation measurements. More generally, similar effects from wave-front distortions should appear in the other kinds of dual cold atom interferometers, based on molasses techniques, such as the gravity gradiometers [22,23], or on in tests of the universality of free fall by comparing the accelerations of two clouds of different species [33,34].
Finally, these limits are not fundamental and can be improved by at least one order of magnitude through various improvements. The parasitic shifts due to wave-front distortions can be reduced by improving the quality of the optics and the stability of the launch velocities. Furthermore, their impact in terms of rotation rate can be reduced by modifying the geometry. Indeed, atoms can be launched in straighter trajectories with a higher longitudinal velocity as in Ref. [14] or by using the four pulse configuration previously demonstrated in Ref. [11], with a longer interaction time. In both cases, the area of the interferometer is significantly increased while keeping the phase shift due to wave-front distortions almost constant. Long term performance should then be improved to reach an expected level below 10 −10 rad s −1 , as achieved with giant ring laser gyroscopes [35], opening the way to new fields of application for atomic gyroscopes. Each point corresponds to a measurement achieved for a given angle ε.
vectors. The short term sensitivity to rotation is 5.5 × 10 −7 rad s −1 Hz −1/2 . The results are similar to those obtained on the vertical axis, taking in account the shorter interaction time T and the reduced contrast (20% instead of 30%).

Test of the separation
The horizontal configuration is well adapted to the measurement of the rejection of the acceleration phase shift on the rotation measurement. Indeed, as the interferometer is realized in the horizontal plane, it is possible to induce a large controlled change of the acceleration by tilting the device by an angle of ε with respect to the horizontal direction. The interferometer then measures a residual component of the gravitation g given by: ∆Φ a = k eff g sin ε T 2 (A1) By tilting the interferometer plane over a range of 0.5 mrad, we change the acceleration phase shift from ∆Φ a = −45 rad to 25 rad. Fig. 16 displays the rotation phase shift as a function of the acceleration induced on the interferometer. The measurements exhibit a very small slope of 1.5 × 10 −4 . Thus the effect of the acceleration on the rotation signal is cancelled at a level better than 76 dB.