# New concepts of inertial measurements with multi-species atom interferometry

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## Abstract

In the field of cold atom inertial sensors, we present and analyze innovative configurations for improving their measurement range and sensitivity, especially attracting for onboard applications. These configurations rely on multi-species atom interferometry, involving the simultaneous manipulation of different atomic species in a unique instrument to deduce inertial measurements. Using a dual-species atom accelerometer manipulating simultaneously both isotopes of rubidium, we report a preliminary experimental realization of original concepts involving the implementation of two atom interferometers, first, with different interrogation times and, second, in phase quadrature.

## 1 Introduction

Cold atom interferometry has allowed, in the last decades, the development of extremely sensitive and accurate inertial sensors for measuring the gravity acceleration [1], Earth’s gravity gradient [2], or rotations [3]. They appear very promising for a wide range of applications such as inertial navigation [4], geodesy [5], natural resource exploration [6], or fundamental physics [7, 8, 9, 10, 11, 12, 13]. To address most of these applications, an important research effort is made to make these cold atom instruments more adapted to withstand operational constraints [14, 15, 19]. In addition to improve these instruments in terms of compactness and robustness, it is of major concern to extend their dynamic measurement range. When shot-to-shot acceleration variations are small compared to the dynamic measurement range, defined here as one half of the interferometer fringe spacing, cold atom gravimeters allow to retrieve the gravity acceleration *g* with a resolution of \(\approx 10^{-9}\) g at a repetition frequency of few Hz. This dynamic measurement range lies in the range 0.2–3 \(\upmu\)g for state-of-the-art cold atom gravimeters [16, 17, 18], much less than the acceleration level of roughly 50 \(\upmu\)g that could be encountered in a typical laboratory environment, 50 *mg* that could be encountered in a plane [15], and 0.3 g that could be encountered in a ship [19]. To allow the operation of these atomic sensors in a laboratory environment, they are typically used on a passive [20, 21] or active vibration isolation platform [22, 23, 24] or set directly on the ground by combining the atomic signal with that of a conventional accelerometer [15, 19, 25, 26]. To our knowledge, only a few onboard demonstrations of inertial measurements have been reported whether in an elevator [21], on a plane [15], or on a boat [19]. For the elevator demonstration, the interrogation time of the atom interferometer is reduced significantly to increase the fringe spacing, and therefore, the dynamic measurement range, but at the cost of a drastic loss of the sensitivity measurement. For the plane and boat demonstrations, the atom interferometer is coupled to a conventional accelerometer which helps to determine, for each measurement cycle, the fringe index on which the atom interferometer is operating. In this last configuration, the use of a conventional accelerometer often limits the performance of the instrument either because of the lower performance intrinsic noise of the conventional accelerometer, its non-linearity, the lack of knowledge of its transfer function, or because of a mismatch between the localization of the conventional accelerometer measurement point and that of the atom interferometer.

As an alternative or in complement to vibration isolation platforms and conventional accelerometers, we propose and study in this article new concepts of an atomic accelerometer dedicated to onboard applications. These concepts could be interesting in the future for increasing the dynamic measurement range of the atomic instrument and allowing the atomic interferometer to operate always in its linear regime [25], leading to a maximum sensitivity whatever the measured acceleration. The instrument configurations that will be presented do not rely on external devices making our instrument self-independent and only relying on atom interferometry. This work is in straight line with the one reported earlier by Sorrentino et al. [27], where two separated atomic clouds of \(^{87}\)Rb are simultaneously operated in a gravity gradiometer configuration, which opened the way to this kind of self-independent atomic sensors. Here, we study innovative concepts based on atom interferometry with different atomic species which allows simultaneous operation of several atomic clouds at the same spatial position. Up to now, the few experiments dealing with multi-species atom interferometry were dedicated to test the universality of free fall [28, 29, 30, 31, 32, 33] and their interests to improve single species atomic inertial measurements were never discussed to our knowledge. The principle of the experiment is the following. The atomic species are trapped and cooled simultaneously in the same vacuum chamber, sharing the same laser beams and constituting a whole unique embedded atomic cloud. The atomic species are then free falling simultaneously, feeling, therefore, the same inertial forces, and during which they are interrogated through atom interferometry by the same laser beam. At the end of the free fall, each atomic species is detected and provide complementary acceleration measurements that allow to improve performances reached by standard single species atom interferometers. Using different atomic species in the same vacuum chamber is of great interest, since each atom-light interaction for manipulating one atomic species should not disturb significantly the other atomic species. Indeed, each laser line dedicated to manipulate one atomic species is seen far of resonance regarding the other atomic species.

## 2 Presentation of the multi-species concepts

### 2.1 Introduction

*a*along the interferometer’s laser direction of propagation [1]:

*T*.

*P*to be in a particular atomic state which is a sinusoidal function of the interferometric phase:

*A*are, respectively, the offset and the amplitude of the fringes. \(\varDelta \varPhi _{\text {op}}\) is a very well-known and controlled-phase shift added by the operator. For instance, in an atom gravimeter, a frequency chirp \(\alpha\) is added to the interferometer’s laser frequency to compensate the Doppler shift induced by gravity

*g*, such as \(\varDelta \varPhi _{\text {op}}=2\pi \alpha T^{2}=-k_{\text {eff}}\cdot g\cdot T^{2}\).

The sinusoidal response of the atom interferometer is responsible of two important drawbacks when the phase shift \(\varDelta \varPhi\) is extracted.

*P*could correspond to a large number of acceleration values (see Fig. 1):

Second, the sensitivity of the interferometer depends on the value of the total phase shift and, consequently, on the measured acceleration. When \(\varDelta \varPhi +\varDelta \varPhi _{\text {op}}=n\pi\) (*n* being an integer), i.e., when the interferometer works at the top or at the bottom of the fringes, the sensor becomes highly insensitive, because the derivative of the measured probability *P* with respect to \(\varDelta \varPhi\) is zero (see Fig. 1). In a low vibration environment, cold atom gravimeters are usually forced to operate at mid-fringe by adjusting \(\varDelta \varPhi _{\text {op}}\) at each cycle to ensure a maximum sensitivity of the instrument [1]. For vibration levels exceeding the measurement range of the atom interferometer, a solution consists in compensating the vibrations in real time thanks to a conventional accelerometer allowing the atom interferometer to stay at mid-fringe [26].

### 2.2 First concept

The first configuration relies on the implementation of two simultaneous atom interferometers with two different interrogation times \(T_1\) and \(T_2\) (see Fig. 2a). The atom interferometer with the highest interrogation time \(T_2\) presents a greater sensitivity but conversely a smaller measurement range. On the other hand, the atom interferometer with a lower interrogation time \(T_{1}\) benefits from an increased measurement range, by a factor \(\left( T_{2}/T_{1}\right) ^{2}\), but, at the expense of a degraded sensitivity, following the same scaling factor. Associating both interferometer signals allows to benefit simultaneously from a high sensitivity and a high measurement range. Note that, for a given measurement range fixed by the atom interferometer with \(T_1\), this configuration allows also to benefit from a higher accuracy thanks to the interferometer with the highest interrogation time \(T_2\). Indeed, the impact of a significant number of systematic effects scales inversely with the interrogation time.

### 2.3 Second concept

## 3 Experimental demonstrations

### 3.1 Experimental setup

We report here preliminary experimental demonstration of these two configurations. The experimental setup is similar to the one described in Ref. [34]. With this setup, approximatively \(6\times 10^{8}\), respectively, \(8\times 10^{8}\), atoms of \(^{87}\)Rb, resp. \(^{85}\)Rb, are simultaneously loaded from a background vapor into a 3D Magneto-Optical Trap (MOT) in 250 ms. The atoms are then further cooled down in an optical molasses phase allowing the atoms to reach a temperature of \(\sim\) 2 \(\mu\)K. The atoms are then selected in the magnetic insensitive state \(|F=1,m_{F}=0\rangle\) for \(^{87}\)Rb and \(|F=2,m_{F}=0\rangle\) for \(^{85}\)Rb thanks to a microwave pulse and then dropped over a distance of \(\sim 6\) cm. For both atomic species, a Mach–Zehnder-type interferometer is implemented, consisting in three equally spaced Raman laser pulses \(\pi /2-\pi -\pi /2\). Note that the interrogation times can be varied for each species to a maximum value of \(T=47\) ms. The Raman laser beam is retro-reflected on a mirror which acts as the reference for the inertial measurement. Since both atom interferometers share the same Raman laser beam, the two measurement axis are, therefore, rigorously the same. Finally, the atomic probabilities are successively measured for each isotope by fluorescence detection. The whole sequence is performed at a repetition rate of 4 Hz. The instrument is installed on a vibration isolation platform to clearly resolve the atomic fringes and minimize the interferometer phase noise which will allow us to analyze more simply the potential of the presented concepts. Despite an apparent complexity, the dual-species experiment stays rather simple to implement and not significantly more complex than a single species instrument. All the required laser lines necessary for both isotopes manipulation are synthesized thanks to phase modulation of a single laser source [34]. During the cooling stage, the carrier frequency is tuned on the \(^{87}\)Rb cooling transition. The \(^{85}\)Rb cooling line and both repumper lines are generated by three modulation frequencies injected into the phase modulator as following: f\(_{\text {cooling}}^{~85}\) = f\(_{\text {carrier}}\) + 1.126 GHz, f\(_{\text {repumper}}^{~87}\) = f\(_{\text {carrier}}\) + 6.568 GHz, f\(_{\text {repumper}}^{~85}\) = f\(_{\text {carrier}}\) + 1.126 + 2.915 GHz. Phase modulation is also used for generating the laser lines during the interferometric sequence. Both Raman pairs are generated by directly injecting the Raman difference frequencies associated with both isotopes (i.e., 6.834 GHz for \(^{87}\)Rb and 3.035 GHz for \(^{85}\)Rb), making the carrier frequency common to both Raman pairs. The Raman pair corresponding to \(^{87}\)Rb is red detuned by 0.59 GHz with respect to the excited hyperfine state \(F'=2\), and therefore, the one corresponding to \(^{85}\)Rb is red detuned by 1.86 GHz with respect to \(F'=3\). Compared to the one species instrument, the dual-species instrument requires only one additional microwave source fitting in a standard 3U 19 inch rack.

### 3.2 First concept

The first dual-species operating mode that we study in this article allows to increase the measurement range of a single species atom accelerometer. For the experimental realization of this operating mode, we set the interrogation time for \(^{87}\)Rb to \(T_{87}=20\) ms and the one for \(^{85}\)Rb to \(T_{85}=47\) ms. The central \(\pi\)-pulse is kept common for both isotopes (see Fig. 3). This feature allows to keep the global symmetry of the entire interferometric sequence to cancel out roughly impact of light shifts on one atomic species due to the laser lines used to address the other isotope.

The dual-atom interferometer output should be impacted by light shifts, first, in the same way as the single species interferometers, due to the interaction of the atoms with their associated Raman laser (Type I), and, second, by the interaction of the Raman laser of one atomic species with the other atomic species (Type II). In this article, we focus on proof-of-principle experimental demonstrations of the presented concepts without studying light shift or other potential effects that could impacts the dual interferometer accuracy or stability. Nevertheless, the one photon light shift of type I and II should be efficiently rejected by the well-known reversing technique of the Raman laser wave vector [1]. For the concept of interferometers with two different interrogation times, light shift of type II should be further eliminated due to the symmetry of the embedded Mach–Zehnder schemes, sharing the same central pi Raman pulse. For this concept, the interferometer with lower T is not impacted, since the \(\pi\)/2 laser pulses of the other atomic species are temporally outside the interferometer and the central pi pulse should impact both paths of the interferometer in the same way. The interferometer with higher T should be impacted only weakly, since the effect of the first \(\pi\)/2 pulse on both interferometers paths should be canceled by the effect of the last pi/2 pulse. Concerning the two-photon light shift, the two-photon detuning of is too large to expect significant contributions at least regarding terrestrial applications.

*a*can be considered constant and equal to the gravity acceleration

*g*. Note that applying this frequency chirp \(\alpha\) is equivalent to generating a constant acceleration of the atoms with respect to the Raman laser. The results are shown in Fig. 4a. The \(^{85}\)Rb interferometer alone (red circles) does not allow to extract \(\alpha\) in the full range because of ambiguity on the determination of the fringe indexes (see Eq. 3). On the other hand, the \(^{87}\)Rb interferometer alone (black squares), with a smaller

*T*, allows to extract \(\alpha\) without any ambiguity on the entire variation range. Taking into account both signals (blue–grey circles) allows to overcome the ambiguity by keeping among the possible values of \(\alpha\) given by the \(^{85}\)Rb interferometer the value which is closest to that given by the \(^{87}\)Rb interferometer.

*P*), coming mainly in our experiment from the frequency noise of the laser beam, which could lead to a bad determination of the \(^{85}\)Rb interferometer fringe indexes, as is shown in Fig. 5. For the \(^{85}\)Rb interferometer noise, the main contribution comes from the microwave phase noise allowing the generation of the second Raman laser line. In Fig. 5, \(n_{85}\) (Fig. 5a) and \(s_{85}\) (Fig. 5b) correspond to the fringe indexes of the \(^{85}\)Rb signal used in Eq. 3 as

*n*and

*s*. The green circles give for each measurement point the values of the fringe indexes of the \(^{85}\)Rb signal to determine unambiguously \(\alpha\). These indexes are determined using the \(^{87}\)Rb signal. To evaluate the efficiency of this determination and the impact of the \(^{87}\)Rb interferometer noise on it, we have plotted in Fig. 5 in orange line the fringe indexes values that would be determined with a noiseless \(^{87}\)Rb signal, using a sine function fitting the \(^{87}\)Rb output.

*T*is decreased, the impact of the atomic signal noise is increased with an acceleration noise scaling as \(T^{-2}\). We can see, for instance, in Fig. 6, the poor efficiency of the fringe index determination with \(T_{85} = 47\) ms and \(T_{87}=10\) ms due to the noise of the \(^{87}\)Rb signal. In this case, there is no sensitivity gain with the dual-atom interferometer compared to a single species interferometer while, in the ideal situation of a noiseless \(^{87}\)Rb signal for the determination of the \(^{85}\)Rb fringe indexes, the expected gain sensitivity thanks to a dual-species interferometer reaches now a factor 17. Even removing measurement points near the extrema does not lead to an improvement of the sensitivity gain, because the noise of the \(^{87}\)Rb signal leads now to determination errors exceeding one fringe of the \(^{85}\)Rb signal.

*A*is the fringe amplitude defined in Eq. 2. In our experiment, the SNR is typically of \(\sim 50\). The range of \(\alpha\) variation is taken similar to the one chosen for the experimental demonstration with \(T_{85} = 47\) ms and \(T_{87} = 20\) ms. For the experimental results, in the dual-atom interferometer, the \(^{87}\)Rb signal is only used to determine the fringe indexes of the \(^{85}\)Rb interferometer. The simulation results are reported in Fig. 7. We find no gain for low SNR, meaning that the atom interferometer with low interrogation time provides equivalent performance compared to the dual-atom interferometer. However, as the SNR is further increased, the dual AI provides a better performance reaching an optimal gain of roughly a factor 3.8 for a SNR \(\sim 350\). For higher SNR, the gain decreases. This behavior is attributed to the presence of fringe extrema in the \(^{85}\)Rb atom interferometer signal with higher interrogation time. Extraction of \(\alpha\) near those fringe extrema lowers the global sensitivity of the dual-atom interferometer, whereas the single-atom interferometer, with a lower interrogation time, has no fringe extrema and keeps then a higher global sensitivity. Note that, for the experimental SNR of \(\sim 50\), the simulations foresee a sensitivity gain of \(\sim 2.5\), in agreement with that obtained experimentally.

### 3.3 Second concept

*P*such as \(\frac{2-\sqrt{2}}{4}<P<\frac{2+\sqrt{2}}{4}\). Selecting from both atom interferometer outputs and, at each cycle, the measurement corresponding to this linear range gives birth to a fully linear sensor. We can also notice that the correlation between both signals leads to extend the measurement range by a factor 2 compared to a single-species sensor. This is clearly visible in Fig. 9 showing the extraction of \(\alpha\) for a one atomic species interferometer (red circles) and for the phase-shifted dual-species interferometer (blue–grey circles).

Estimation of the sensitivity gains obtained by coupling the linear ranges of two interferometers in phase quadrature compared to a single-species interferometer, as a function of the signal-to-noise ratio (SNR)

SNR | Sensitivity gain |
---|---|

10\(^2\) | 2.7 |

10\(^3\) | 4.8 |

10\(^4\) | 8.5 |

The gain in sensitivity is given in Table 1. We notice that the higher the signal-to-noise ratio, the higher the gain in sensitivity is. Indeed, when the SNR increases, i.e., when the noise decreases for a given fringe amplitude, the uncertainty on the acceleration estimation decreases much more for points at mid-fringe than for points at the top or at the bottom of the fringes. Typically, in our experiment, the SNR is \(\sim 50\), and we expect then a sensitivity gain of \(\sim 2.7\). Putting this analysis into perspective, we can assume higher performance atom interferometer exhibiting SNR approaching \(10^4\) [35] for which a significant gain in sensitivity, higher than a factor 8, could be reached using this phase-shifted dual-species interferometer mode.

## 4 Conclusion

In conclusion, we have presented and made the first analysis of new concepts of cold atom inertial sensor using multi-species atom interferometry. These original configurations allow an improvement of the measurement range or sensitivity compared to the standard configurations of a single-species atomic accelerometer. For the first configuration, involving interferometers with different interrogation times, the low interrogation time atom interferometer noise limits clearly the achievement of ambitious performances. The second configuration remains very promising to get a fully linear atom interferometer and should be analyzed experimentally more deeply. Note that even if the use of a classical accelerometer is still necessary for the operation of a cold atom accelerometer, this last configuration seems still very interesting to implement. Indeed, this would improve the sensitivity of the global instrument and mitigate the required performance noise of the classical accelerometer. Manipulating simultaneously different atomic species in the same instrument offers new perspectives to be further studied. Using three atomic species such as \(^{87}\)Rb, \(^{85}\)Rb, and \(^{133}\)Cs [36] would allow, for instance, to implement simultaneously the two concepts presented in this article, to develop an inertial sensor without deadtimes [37] or a simultaneous multiaxis inertial sensor. A strong point of using different atomic species in the same instrument compared to a single species experiment is the ability to increase the number of complementary measurements without disturbance that could arise form spontaneous emission, the lasers manipulating one species being far of resonance relative to the other species.

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