Concept study and preliminary design of a cold atom interferometer for space gravity gradiometry

We start by calculating the phase shift of a single atom interferometer, and more specifically its dependence on the position and velocity of the atomic source, which will be relevant in the differential measurement of the gradiometer. The interferometer is linked to the frame of a satellite, orbiting at a fixed altitude with a constant orbiting frequency $\Omega_{\mathrm{orb}}$. The measurement axis is taken in the orbital plane, initially along z. We take into account the rotation rates of the satellite $\Omega_{x,y,z}$, which will allow to discuss the cases of nadir pointing, for which $\Omega_y = \Omega_{\mathrm{orb}}$ and $\Omega_{x,z} \simeq 0$, and inertial pointing, for which $\Omega_{x,y,z} \simeq 0$. The calculation of the atom trajectories is performed in the satellite frame, hence one needs to rotate the gravity gradient tensor. This is performed by applying a product of three orthogonal rotations, starting with the rotation along y , whose amplitude eventually largely dominates in Nadir pointing. As long as the two other rotations are small enough, this correctly deals with the influence of the satellite rotation at the leading orders.

We also consider to implement compensation systems for the two following physical quantities:

• the rotation of the satellite, in order to obtain mirrors with a fixed orientation in the frame of the atoms. This configuration can be obtained by tilting the two first and last retroreflecting mirrors by angles $\pm\theta=\pm \Omega_y T$, where $\Omega_y$ is the rotation rate along y , the cross-track axis, as displayed in figure 2. As we will see below, this configuration removes the sensitivity of the interferometer to centrifugal and Coriolis accelerations;
• the phase shift induced by the gravity gradient, as recently proposed in [50]. This relies on an adequate change by $\delta k$ of the Raman wavevector at the second pulse. This method has recently been demonstrated in [51, 52].

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The trajectories of the atomic wavepackets along the arms of the interferometer are determined analytically by solving the Euler–Lagrange equations using a power series expansion as a function of time t, as in [53]. Subsequently, the phase shift of a single interferometer can be calculated out of the positions of the center of the atomic wavepackets at the different Raman laser pulses using the following formula [54]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \phi = 2k_1\cdot r_A-4k_2\frac{r_B+r_C}{2}+2k_3\frac{r_D+r_{D2}}{2}. \label{formula1} \nonumber \end{align} \tag{ 1 }$$

We consider here a $\pi/2-\pi-\pi/2$ double diffraction interferometer, such as displayed later on figure 3, with effective wave numbers k_1,2,3, corresponding to two photon transitions. ($A, B, D$) (resp. ($A, C, D2$)) are the positions of the centers of the partial wavepackets at the time of the three pulses along the upper (resp. lower) arm of the interferometer. This formula is valid for Hamiltonians at most quadratic in position and momentum [54, 55], which is the case considered here; terms beyond this approximation can be treated as proposed in [56, 57].

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Table 1 presents the dominant terms contributing to the phase of a single interferometer and to the separation between the wavepackets at the output of the interferometer, listed with respect to their scaling on initial coordinates and velocities of the atoms in the satellite frame $(x,y,z,v_x,v_y,v_z)$. Here, T denotes the free evolution time between pulses, $k = 4\pi/(780~\mathrm{nm})$ the effective wave number for a two-photon Raman transition, T_zz the gravity gradient. Typical values for the relevant parameters are $\Omega_{\mathrm{orb}} = 1.2~\mathrm{mrad~s^{-1}}$ (for an altitude of about 250 km), $T = 5~\mathrm{s}$, $T_{zz} = -2.7\cdot10^{-6}~\mathrm{s}^{-2}$ and $k=16.1\cdot10^{6}~\mathrm{m}^{-1}$. Only the dominant terms, which scale as T² or T³ depending on the terms, are listed here. For the sake of simplicity, we take here $\Omega_x=\Omega_z=0$. We consider here different cases: the case where the rotation rates $\Omega_{\mathrm{orb}}$, $\Omega_y$ and $\Omega_m$, the equivalent rotation rate of the mirrors given by $\Omega_m=\theta/T$, are different, and the cases where the rotation rates and/or the gravity gradient are compensated ($\Omega_m=\Omega_y$ and/or $\delta k=kT_{zz}T^2/2$).

Table 1. Leading terms in the phase of a single interferometer.

	General case	Compensated rotation	Compensated gradient and rotation, Nadir pointing
	Any $\Omega_{orb,y,m}$	$\Omega_m=\Omega_y$	$\Omega_m=\Omega_y=\Omega_{\mathrm{orb}}$
	$\delta k=0$	$\delta k=0$	$\delta k=kT_{zz}T^2/2$
x	$kx(T_{xx}(5\Omega_y-4\Omega_m-\Omega_{\mathrm{orb}})+4(\Omega_y-\Omega_m)\Omega_y^2$ $+T_{zz}(\Omega_{\mathrm{orb}}-\Omega_y))T^3$	$kx(T_{xx}-T_{zz})(\Omega_{y}-\Omega_{\mathrm{orb}})T^3$	0
y	0	0	0
z	$2kz(T_{zz}-\Omega_m^2+\Omega_y^2)T^2$	$2kzT_{zz}T^2$	0
$v_x$	$4kv_x(-\Omega_m+\Omega_y)T^2$	0	0
$v_z$	$2kv_z(T_{zz}-\Omega_m^2+4\Omega_m\Omega_y-3\Omega_y^2)T^3$	$2kv_zT_{zz}T^3$	0
$v_y$	0	0	0
Separation
$\Delta x$	$4 \hbar k T^2(\Omega_m-\Omega_y)/m_{\mathrm{Rb}}$	0	0
$\Delta y$	0	0	0
$\Delta z$	$2\hbar kT^3(T_{zz}-(\Omega_m-3\Omega_y)(\Omega_m-\Omega_y))/m_{\mathrm{Rb}}$	$2 \hbar kT^3 T_{zz}/m_{\mathrm{Rb}}$	0

In the absence of gravity gradient compensation (second column of table 1), the contribution of the gravity gradient to the single interferometer phase shift is given by $\Phi=2kzT_{zz}T^2$. This leads to a differential phase of $\Delta\Phi=2kDT_{zz}T^2$ between two interferometers separated along z by the distance D. For D  =  0.5 m, and the parameters above, this phase shift is as large as 1087 rad.

We assume here that the individual phase measurements are performed with a sensitivity limited by detection noise at the quantum projection limit, for which $\sigma_\Phi=1/\sqrt{N}$, N being the number of detected atoms. For 10⁶ detected atoms at the output of each interferometer, we obtain an Allan standard deviation of the gravity gradient fluctuations of $\sigma_{T_{zz}}=\sqrt{2}/2\sqrt{N}kDT^2=3.5$ mE/shot. This limit corresponds to a perfect contrast, so that in the presence of non negligible dephasing, the sensitivity will be degraded by a factor 1/C, where C is the interferometer contrast (here assumed identical for the two interferometers).

We first discuss the case where $\Omega_m=0$, i.e. without rotation compensation. The gravity tensor measurement is biased by a contribution in $\Omega_y^2$ due to centrifugal accelerations. The interferometer phase features a Sagnac phase term $4 \, kv_x \, \Omega_y \, T^2$, and the separation between the two partial wavepackets at the output of the interferometer along x is $4 \, \hbar k \, T^2 \, \Omega_y/m_{\mathrm{Rb}}$. This leads to a reduction of the interferometer contrast due to dephasing when averaging the Sagnac phase across the velocity distribution. Equivalently, the contrast is reduced when the separation is comparable to the coherence length of the atomic wavepackets. For the resulting loss of contrast to be negligible, one needs $\sigma_v\ll 1/ \left ( 4k\Omega_yT^2 \right)$. This corresponds to temperatures $T_{\rm emp} \ll T_0$, where T₀ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_0=\frac{m_{\mathrm{Rb}}}{16k_{\rm B}k^2\Omega_y^2T^4} \, \nonumber \end{align} \tag{ 2 }$$

where $k_{\rm B}$ is the Boltzmann's constant. For Nadir pointing, in which $\Omega_y=\Omega_{\mathrm{orb}}=1.2~\mathrm{mrad~s^{-1}}$, a temperature lower than T₀  =  3 fK is required, which is well below what can be achieved with current technology. This limit would also apply for measurements along the x-axis, which is also impacted by the large rotation rate along the y -axis, but not for measurements along the y -axis. Thus, as such, gravity gradient measurements in Nadir configuration can only be performed along one axis.

Also, in a similar manner, and in the absence of gravity gradient compensation, gravity gradients put an additional requirement on the atomic temperature, arising from the dependence of the phase versus $v_z$. This requirement is given by $T_{\rm emp} \ll T_1$, where T₁ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle T_1=\frac{m_{\mathrm{Rb}}}{4k_bk^2T_{zz}^2T^6} \, . \nonumber \end{align} \tag{ 3 }$$

For $T_{zz} = -2.7\cdot10^{-6}~\mathrm{s}^{-2}$, $T_1\sim100$ pK. Such a limit is compatible with the ultralow atomic temperatures reached thanks to Delta Kick collimation techniques (of the order of a few tens of pK), as discussed in section 3.

Measuring the three axes in Nadir configuration requires a compensation scheme for the large rotation along y , which can be realized by tilting the Raman mirrors, as discussed above. This corresponds to the second case (third column) in table 1, where the Sagnac phase, as well as the centrifugal acceleration, are canceled if the angles set on the mirrors are perfectly tuned ($\Omega_m=\Omega_y$). On the other hand, if the pointing of the satellite is inertial (and no rotation compensation is applied), rotation rates up to $\Omega_0=6~\mu\mathrm{rad~s^{-1}}$ can be accepted, for the same temperature limit of 100 pK. This makes gravity measurements along three orthogonal directions possible in inertial pointing mode (with flat mirrors). Equivalently, this translates into the same limit for the maximum rotation rate mismatch between $\Omega_m$ and $\Omega_y$ in the case of imperfect rotation compensation in the Nadir pointing mode: $\delta\Omega=\Omega_m-\Omega_y \ll \Omega_0$.

Finally, the last column in table 1 shows that for a properly tuned rotation rate for Nadir operation ($\Omega_{y}=\Omega_{\mathrm{orb}}$), for properly tilted mirrors ($\Omega_m=\Omega_y$), and for a properly set change in the Raman wavevector at the second Raman pulse ($\delta k=kT_{zz}T^2/2$), all higher order terms in the interferometer phase, as well as the separation of the two wavepackets at the output of the interferometer, are cancelled. This implies that the main sources of loss of contrast are suppressed.

With the gravity gradient compensation scheme applied, the gravity gradient contribution to the single interferometers phases is (on averaged) removed. This corresponds to offsetting to zero the differential gradiometric phase. Nevertheless, fluctuations of the measured differential phase $\sigma_{\Delta\Phi}$ (either due to actual gravity gradient fluctuations, or equivalently to detection noise) will correspond to a noise on the determination of the gravity gradient of $\sigma_{T_{zz}}=\sigma_{\Delta\Phi}/2kDT^2$. The sensitivity of the gradiometer measurement, for quantum projection noise limited detection noise with 10⁶ atoms, is thus unchanged: $\sigma_{T_{zz}}=3.5$ mE/shot.

5.1.1. Inertial case.

5.1.2. Impact of imperfect rotation compensation in the Nadir case.

Table 2 lists the dominant terms in the development of the single interferometer output phase, for Nadir pointing and in the case where the compensation of the rotation is not perfect ($\Omega_m \neq \Omega_y$).

Table 2. Residue of the phase of a single interferometer, for compensated gravity gradient and non perfect rotation compensation, corresponding sensitivity for differential measurements, and phase dispersion. Dominant terms are considered here.

	Terms	Differential phase (in rad)	Phase dispersion (in rad)
	$\Omega_m=\Omega_y+\delta\Omega_m$	$\delta\Omega_m=10^{-6}$ rad s−1
	$\delta k=kT_{zz}T^2/2$	$\Omega_y-\Omega_{\mathrm{orb}}=10^{-6}$ rad s−1
		$\delta x=10^{-6}$ m	$\sigma_x=0.1$ mm
		$\Delta z=D=0.5$ m	$\sigma_z=0.1$ mm
		$\delta v_x=10^{-6}$ m s−1	$\sigma_v=98~\mu$m s−1
		$\delta v_z=10^{-6}$ m s−1	$\sigma_v=98~\mu$m s−1
x	$-kx((T_{xx}-T_{zz})(\Omega_{\mathrm{orb}}-\Omega_{y})$ $+4\delta\Omega_m (T_{xx}+\Omega_y^2))T^3$	$-8.3\times10^{-9}$	$-8.3\times10^{-7}$
z	$-4kz\delta\Omega_m\Omega_yT^2$	−0.944	$-1.9\times10^{-4}$
$v_x$	$-4kv_x\delta\Omega_mT^2$	$-1.6\times10^{-3}$	−0.157
$v_z$	$4kv_z\delta\Omega_m\Omega_yT^3$	$9.4\times10^{-6}$	$9.2\times10^{-4}$
Separation		Separation
		(in m)
$\Delta x$	$4 \hbar k \delta\Omega_m T^2/m_{\mathrm{Rb}}$	$1.2\times10^{-6}$
$\Delta z$	$4\hbar k\Omega_y\delta\Omega_m T^3/m_{\mathrm{Rb}}$	$6.8\times10^{-9}$

We calculate in table 2 the impact of an error in the tilt of the mirrors of $\delta \theta=5~\mu$rad, which corresponds to a mismatch $\delta\Omega_m=10^{-6}$ rad s⁻¹. We find a phase error on the differential acceleration of  −944 mrad, due to residual centrifugal accelerations. To keep the error below 1 mrad, an uncertainty in the tilt (of the two mirrors) of 5 nrad is thus required. Conversely, this corresponds to an uncertainty in the knowledge of the rotation rate along y of 1 nrad s⁻¹. These are very stringent requirements.

Table 3 gives the requirements on the source parameters (velocities and positions) to keep the phase error below 1 mrad, in the case of non perfect compensation of the rotation, and for non perfect Nadir pointing. We assume here a knowledge of the rotation rate along y at the level given above. The corresponding requirements are found to be largely manageable. One should nevertheless keep in mind that these requirements scale with the amplitudes of the mismatches $\delta\Omega_m$ and $\Omega_y-\Omega_{\mathrm{orb}}$.

Table 3. Requirements on the relative initial positions and velocities for the phase error to remain below 1 mrad, in the case of non perfect rotation compensation $\delta\Omega_m=10^{-6}$ rad s⁻¹ and non perfect Nadir pointing $\Omega_y-\Omega_{\mathrm{orb}}=10^{-6}$ rad s⁻¹.

x	$\delta x < 0.12$ m
z	$\delta z < 0.5$ mm
$v_x$	$\delta v_x < 0.6~\mu$m s−1
$v_z$	$\delta v_z < 106~\mu$m s−1

The interferometer configuration required to measure T_xx is identical to the one studied before, as for any measurement axis in the orbital plane, the rotation rate along y  needs to be compensated. The conclusions and requirements derived in the previous sections thus apply, provided the role of x and z are exchanged. As for the cross-track (y ) axis, fixed and parallel mirrors can be used. This simplifies the laser setup design and relaxes the constraints on the control of the beam alignment, such that the target sensitivity and accuracy can easily be met when measuring gravity gradients along the y -axis. This was the configuration considered in [58].

While the requirements on the control and knowledge of $\Omega_{x,z}$ can be met with current technologies, using for instance fiber optic gyroscopes of the Astrix class, the ones of the rotation rate $\Omega_y$, and of the mismatch of the mirror with respect to the ideal tilt, are very stringent, and cannot presently be met, even with the best space qualified gyroscopes. Instead we propose to use the mathematical properties of the gravity tensor, and its null trace, to estimate $\Omega_y$, or at least its fluctuations.

The phase signals of the three gradiometers are given by:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \delta\Phi_x &= 2kD(T_{xx}-T_{xx}^{\mathrm{ef\,\!f}}-\Omega_{m,x}^2+\Omega_y^2+\Omega_z^2)T^2\nonumber \\ \delta\Phi_y &= 2kD(T_{yy}-T_{yy}^{\mathrm{ef\,\!f}}+\Omega_x^2+\Omega_z^2)T^2\nonumber \\ \delta\Phi_z &= 2kD(T_{zz}-T_{zz}^{\mathrm{ef\,\!f}}-\Omega_{m,z}^2+\Omega_y^2+\Omega_x^2)T^2 \nonumber \end{align*}$$

with $T_{ii}^{\mathrm{ef\,\!f}}=2\delta k_{ii}/kT^2$ and $\delta k_{ii}$ the change in k applied in the direction i.

Assuming the $T_{ii}^{\mathrm{ef\,\!f}}$ are tuned so as to null the output phases, we have

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle T_{xx}^{\mathrm{ef\,\!f}} &= T_{xx}-\Omega_{m,x}^2+\Omega_y^2+\Omega_z^2\nonumber \\ T_{yy}^{\mathrm{ef\,\!f}} &= T_{yy}+\Omega_x^2+\Omega_z^2\nonumber \\ T_{zz}^{\mathrm{ef\,\!f}} &= T_{zz}-\Omega_{m,z}^2+\Omega_y^2+\Omega_x^2. \nonumber \end{align*}$$

Summing the three equation and exploiting the null trace relation $T_{xx}+T_{yy}+T_{zz}=0$, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum T_{ii}^{\mathrm{ef\,\!f}} = -\Omega_{m,x}^2-\Omega_{m,z}^2+2(\Omega_x^2+\Omega_y^2+\Omega_z^2). \nonumber \end{align} \tag{ 4 }$$

For $\Omega_x$ and $\Omega_z$ well below 10⁻⁶ rad s⁻¹ (or sufficiently well determined with gyroscopes), and assuming that the mirror tilts are left unchanged, fluctuations of $\Omega_y$ can be determined with an uncertainty limited by the combined sensitivities of the three gradiometers.

Writing $\Omega_y=\Omega_{y0}+\delta\Omega_y$, where $\Omega_{y0}$ is a reference value close to $\Omega_{\mathrm{orb}}$, we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sum T_{ii}^{\mathrm{ef\,\!f}} \simeq -\Omega_{m,x}^2-\Omega_{m,z}^2+2\Omega_{y0}^2+4\Omega_{y0}\delta\Omega_y. \nonumber \end{align} \tag{ 5 }$$

The uncertainty in the evaluation of $\delta\Omega_y$ is finally given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta(\delta\Omega_y)=\frac{\sqrt{3} \, \sigma_T}{4 \, \Omega_{\mathrm{orb}}} \nonumber \end{align} \tag{ 6 }$$

where $\sigma_T$ is the sensitivity of each gradiometer.

For $\sigma_T=3.5$ mE and $\Omega_{\mathrm{orb}}=1.17$ mrad s⁻¹, we find $\Delta(\delta\Omega_y)=0.9$ nrad s⁻¹, which meets the requirement derived in section 5.1.2.

This determination can in turn be used to correct the measurement along x and z from centrifugal accelerations. Neglecting as before terms related to $\Omega_x$ and $\Omega_z$, this yields the following equations:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle T_{xx} &= T_{xx}^{\mathrm{ef\,\!f}}+\Omega_{m,x}^2-\frac{1}{2}\sum T_{ii}^{\mathrm{ef\,\!f}}+\Omega_{m,x}^2+\Omega_{m,z}^2\nonumber \\ T_{yy} &= T_{yy}^{\mathrm{ef\,\!f}}\nonumber \\ T_{zz} &= T_{zz}^{\mathrm{ef\,\!f}}+\Omega_{m,z}^2-\frac{1}{2}\sum T_{ii}^{\mathrm{ef\,\!f}}+\Omega_{m,x}^2+\Omega_{m,z}^2 \nonumber \end{align*}$$

and finally

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle T_{xx} &= \frac{1}{2}(T_{xx}^{\mathrm{ef\,\!f}}-T_{yy}^{\mathrm{ef\,\!f}}-T_{zz}^{\mathrm{ef\,\!f}}+\Omega_{m,x}^2-\Omega_{m,z}^2)\nonumber \\ T_{yy} &= T_{yy}^{\mathrm{ef\,\!f}}\nonumber \\ T_{zz} &= \frac{1}{2}(T_{zz}^{\mathrm{ef\,\!f}}-T_{xx}^{\mathrm{ef\,\!f}}-T_{yy}^{\mathrm{ef\,\!f}}+\Omega_{m,z}^2-\Omega_{m,}^2). \nonumber \end{align*}$$

Exploiting the null trace to correct for the centrifugal acceleration actually also decreases the uncertainty in the gradiometric measurement along x and z by a factor $2/\sqrt{3}=1.15$.

Note that we have considered here only the gravity tensor component along the measurement axis, the term related to gravity gradient compensation by a wavevector change at the second pulse, and centrifugal terms due to rotation rates and tilted mirrors. Coriolis terms as well as higher order terms in the gradiometer phases would need to be taken into account when their amplitude is large enough, typically larger than 1 mrad (see tables 2 and 3 for the scaling and amplitude of some of these terms, and the constraints on the source parameters we derive to keep them below that limit). In any case, provided the uncertainty on these terms is lower than the instrument noise floor (about 1 mrad), the method remains valid and allows for determining the rotation rate along y  with the required uncertainty of 1 nrad s⁻¹.

The interferometer geometry is based on the double diffraction technique demonstrated in [29]. The Raman beams, of wavevectors $\vec {k}_{1}$ and $\vec {k}_{2}$, are brought together onto the atoms before being retroreflected on mirror(s), leading to the existence of two pairs of counterpropagating Raman beams, with opposite effective wavevectors $\pm\hbar\vec {k}_{\mathrm{ef\,\!f}}$, where $\vec {k}_{\mathrm{ef\,\!f}}=\vec {k}_{1}-\vec {k}_{2}$. Both pairs are resonant when the motion of the atoms is perpendicular to the laser beams, as no Doppler shift lifts the degeneracy between the resonance condition between them.

Similarly to diffraction by stationary optical waves, the coupling with the two Raman lasers pairs lead to populating several orders of diffraction. In our model, we consider the first 5 coupled atomic states $\vert\,j\rangle$ ($j=-2\cdots2$). These states correspond to diffraction orders 0, $\pm 1$ and $\pm 2$: they are linked to the interaction of the atoms with 2 or 4 photons following the two directions $\pm\hbar\vec {k}_{\mathrm{ef\,\!f}}$. They also differ by their internal states (0 and $\pm 2$ correspond to $\vert F=1\rangle$, $\pm\,1$ to $\vert F=2\rangle$) as the Raman pairs couple different electronic states. Then, the evolution of the atomic quantum state during the pulse is calculated by solving the Schrödinger equation, generalizing the method of [59], based on adiabatic elimination of the excited state.

The double diffraction interferometer is realized using a three Raman pulses sequence $\frac{\pi}{2} - \pi - \frac{\pi}{2}$, as shown in figure 3, separated from each other by a free evolution time T. The three pulses are realized with the same laser power but with different duration corresponding respectively to $\tau - 2\tau - \tau$. The duration $\tau$ is defined with respect to the effective Rabi pulsation $\Omega_{\mathrm{ef\,\!f}}$, using the relation: $\tau =\pi / \left ( \sqrt{2} \, \Omega_{\mathrm{ef\,\!f}} \right)$, so that the first Raman pulse enables to split the wave-function of the atoms in coherent superposition between the two coupled states $\vert 0\rangle\longrightarrow\vert\pm 1\rangle$. During the first free evolution time T, the two arms of the interferometer separate. The second Raman pulse acts on each interferometer arms in order to deflect them (in our case $\vert\pm 1\rangle$ into $\vert\mp1\rangle$). The first and second Raman pulses can also populate other coupled states (such as $\vert\pm 2\rangle$), leading to parasitic paths which could interfere. We suppress these unwanted paths which are in the $\vert F=2\rangle$ state by pushing them away of the interferometer area using resonant laser beams, after each Raman pulse, as shown in figure 3. The non-deflected wave-packets after the middle pulse do not disturb the measurement process if we detect only the 'interference signal' (see figure 3), which is possible in our configuration due to the large distance between the different wave-packet trajectories.

After the second $\pi$-Raman pulse, the two wave-packets $\vert\mp1\rangle$ get closer and overlap after a second free evolution time T. Finally, the last $\frac{\pi}{2}$ Raman pulse recombines these atomic wave packets, realizing thus a Mach–Zehnder type interferometer.

The interferometer phase, which corresponds to the difference between the phase shifts accumulated by the two interferometer arms, is finally extracted from the measurement of the transition probability $P=N_1/(N_1+N_2)$, where N₁ and N₂ are respectively the number of atoms detected in the hyperfine states $\vert F=2\rangle$ (corresponding to the atoms in state $\vert 0\rangle$) and $\vert F=1\rangle$ (corresponding to the sum of the atoms in states $\vert \pm 1\rangle$).

With this geometry, the interferometer phase is given by:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta\Phi &= \left[\phi_{\downarrow}(\vec {r}_{A}) - \phi_{\downarrow}(\vec {r}_{C}) + \phi_{\uparrow}(\vec {r}_{C}) - \phi_{\uparrow}(\vec {r}_{D})\right] \nonumber \\ & - \left[\phi_{\uparrow}(\vec {r}_{A}) - \phi_{\uparrow}(\vec {r}_{B}) + \phi_{\downarrow}(\vec {r}_{B}) - \phi_{\downarrow}(\vec {r}_{D})\right] \label{Analytic_PaseShift} \nonumber \end{align} \tag{ 7 }$$

where $\vec {r}_{A},\vec {r}_{B},\vec {r}_{C}$ and $\vec {r}_{D}$ are the center of mass positions of the atomic wave-packets at the different locations $A, B, C$ and D represented in the figure 3, and $\phi_{\uparrow}$ (resp. $\phi_{\downarrow}$) the phase difference between the $k_{\uparrow}$ (resp. $k_{\downarrow}$) pair of Raman lasers. Equation (7) generalizes equation (1) and allows to account for differences between the wavevectors of the counterpropagating Raman lasers pairs.

Using this model for the interferometer, we have developed Monte-Carlo simulations of the space-borne gravity gradiometer depicted in figure 1. We average the contribution to the output signals of a large ensemble of atoms, randomly drawing their initial positions and velocities in Gaussian distributions, and calculating the evolution of their wave-function as well as their classical trajectory along the two interferometer paths. The initial mean longitudinal velocity of the atoms is $4v_{\rm rec}$, and the rms initial atomic position is taken to be 100 $\mu$m. The initial mean vertical velocity, ideally null, can be taken different from zero when we estimate the effect of an initial velocity drift along the direction of the lasers onto the interferometer. By simulating two interferometers at different initial positions, and computing the difference between the output phase shift, we simulate a cold atom interferometer gravity gradiometer.

To simulate the propagation of the atoms through the interferometer, we consider that the momentum kicks occur at the middle of the Raman pulses and we neglect the variations of their position during the pulses. The duration of the first Raman pulse is $\tau =\frac{\pi}{\sqrt{2}\Omega_{\mathrm{ef\,\!f}}}$, where $\Omega_{\mathrm{ef\,\!f}}$ is the Rabi angular frequency at the center of the Raman lasers, whose intensity profiles are Gaussian with identical waists w₀. Therefore, the effective Rabi angular frequency $\Omega_{\mathrm{ef\,\!f}}(\vec {r})$ seen by the atoms, as well as the phase shifts $\phi_{\uparrow\downarrow}(\vec {r})$, depend on their position in the Raman beams $\vec {r}$ at the time of the pulses.

The following section discusses the results from the simulation and derives the requirements.

6.3.1. Parallel retroreflecting mirrors.

We start by calculating the contrast and fraction of detected atoms as a function of the Rabi frequency and the temperature, for parallel Raman mirrors (which corresponds to the case where the measurement axis is cross-track, i.e. along y ). The results are displayed in figure 4 for temperatures ranging from 0.1 pK to 10 nK, and Rabi pulsation from 5 rad s⁻¹ to 10 Mrad s⁻¹. Here, the Raman laser beams are taken as Gaussian beams with a 5 mm waist size.

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In figure 4(a) we present the interferometer contrast with respect to $\Omega_{\mathrm{ef\,\!f}}$ for different temperatures. A plateau of contrast  >0.8 is observed for a range of Rabi pulsation between 10² and 10⁵ rad s⁻¹ and a temperature range between 10 and 100 pK. The corresponding fractions of detected atoms are displayed in figure 4(b), where a similar plateau is found, though smaller, corresponding to a range of $\Omega_{\mathrm{ef\,\!f}}$ between 5.10³ and 2.10⁵ rad s⁻¹ and a temperature range between 10 and 100 pK. These results confirm the expectations that at large effective Rabi pulsations (>10⁵ rad s⁻¹), the coupling to higher momentum states ($\vert\pm 2\rangle$ ...) leads to a loss of contrast. Also, when the temperature increases, the fraction of detected atoms decreases quickly due to the velocity selectivity of the Raman transitions. This motivates to work with the lowest possible temperatures.

Based on these calculations, we select for the rest of the simulations the following parameters: $\Omega_{\mathrm{ef\,\!f}}=40\times10^{3}$ rad s⁻¹ and a temperature of 100 pK, for which the contrast C is $\approx$80% and the fraction of detected atoms $\approx$80%. Temperatures below 100 pK range have already been obtained using the delta-kick collimation technique [60]. As for the Rabi pulsation, it corresponds to Raman pulses duration $\tau$ of the order of 55 $\mu$s similar to what is used in standard ground-based Raman interferometers [29].

The simulation also allows evaluating the effect of the finite size of the Raman beams onto the phase of the interferometers, due to the effects of curvature and Gouy phase. Figure 5 shows the calculated phase shifts as a function of the Raman laser waists in the 1–10 mm range, assuming identical waists for the three beams, located at y   =  0, in between the two interferometers at $y=\pm25$ cm. The temperature is T  =  100 pK and the effective Rabi pulsation 40 krad s⁻¹. The retroreflecting mirrors are placed at y   =  +0.4 m.

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Figure 5 shows the calculated phase shifts for each individual interferometers, displayed as open squares and circles, and their difference, displayed as full triangles. We find phase shifts that decrease quickly with increasing waist sizes, which are not suppressed in the differential measurement. The resulting systematic effect on the gradiometer phase is lower than 1 mrad for waists larger than 4 mm, and was found to be dominated by the impact of the residual curvature of the wavefront rather than the Gouy phase. This motivates the choice made above of a waist of 5 mm.

The relative positions of the Raman laser waists were then varied in order to evaluate the effect of their positions on the differential phase shift of the gravity gradiometer, and define an error margin for the adjustment of the waist position of the Raman laser beams. The positions of the waists of the three incoming Raman beams were randomly drawn in the range $\pm$10 m (resp. $\pm$50 m) with respect to the y   =  0 position at the middle of the two interferometers. The corresponding gradiometer phase shifts were found to vary respectively within $\pm$0.5 mrad (resp. within 6 mrad) around the average value of 0.5 mrad. In order to keep the differential phase shift  <1 mrad, the relative y -positions of the Raman laser waists should thus be in the range of $\pm$10 m. The Rayleigh length of a Gaussian beam of 5 mm waist being 100 m, this corresponds to a maximum radius of curvature of 1 km at 10 m from the waist, which is well within the measurement capabilities of state of the art wavefront sensors.

The model was also used to evaluate the impact of other effects, such as light shifts, residual mean Doppler shifts. In particular, we calculated for single interferometers a residual sensitivity to the mean initial velocity along y . The sensitivity amounts to 0.05 mrad per $\mu$m/s of mean velocity drift for our parameters (Rabi frequency $\Omega_{\mathrm{ef\,\!f}} = 40$ krad s⁻¹), and increases when decreasing the Rabi frequency. A control of the relative initial vertical velocity between the two atomic clouds at the input of each interferometer better than 20 $\mu$m s⁻¹ is thus required to keep the phase error below 1 mrad.

6.3.2. Tilted retroreflecting mirrors.

In the tilted mirror configuration, finite size effects are expected to have a stronger impact, as the positions of the atomic clouds are (symmetrically) offset with respect to the centre of the Raman beams by about 1.5 mm at the first and third Raman pulses. Figure 6 presents the calculated contrast (left) and fraction of detected atoms (center) as a function of the beam waist. As expected, smaller contrasts and fraction of detected atoms are found with respect to the parallel configuration (63% of contrast instead of 80% for a waist of 5 mm). The effect of the curvature onto the gradiometer phase is displayed on figure 6-right), where a waist larger than 8 mm is required to keep the phase error below 1 mrad. We finally chose a waist of 1 cm and evaluated the impact of the Rabi frequency and temperature onto the contrast and fraction of detected atoms. We found similar behaviours as before, with plateaus in the trends with respect to the Rabi frequency, with significantly higher contrast (92 %) and fraction of detected atoms (95%) for a temperature of 100 pK.

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Figure 7 shows the general architecture of the vacuum system for T_zz and T_xx. The main dimensions of the vacuum chamber are adapted to the displacement of the atomic clouds. The architecture of the BEC chamber, where the ultra-cold atomic source is produced, is inspired by the solution designed for STE-QUEST ATI [61] but without the dipole trap. In the 2D-MOT chamber, a beam of pre-cooled atoms is created by a two dimensional MOT. This pre-cooled beam of atoms is formed out of the background gas pressure created by a reservoir. In the BEC chamber this atom beam is captured by a three dimensional MOT and the atoms are then transferred in a purely magnetic trap. The magnetic fields for the traps are created by a combination of a three-layer atom chip and comparably small magnetic coils. In the magnetic trap, the atomic cloud is compressed and then cooled via RF-evaporation. The chip is parallel to the plane of the atomic clouds displacements. The transport beams launch the atoms into the cold atom interferometer (CAI) chamber. The atom cloud keep a horizontal momentum of $4 \hbar k$ during the whole sequence until they reach the detection chamber.

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Four mirrors are fixed inside the vacuum system: one for the two vertical Bloch lattices and three tilted reference mirrors for the interferometer to compensate for the rotation of the satellite which can create bias terms in the output phases and a loss of contrast. The relative angle between two consecutive reference mirrors is  ∼7 mrad corresponding to the mean rotation rate of the satellite. The two mirrors for the $\pi/2$ pulses are fixed on piezoelectric tip-tilt mounts to allow the fine control of the relative angle between the three reference mirrors [25]. A dynamic range of $\pm$$30~\mu$rad and an accuracy of 10 nrad is needed, which is slightly beyond the state-of-the-art of this technology and requires custom development. The impact of these actuators on the power budget is not negligible and their power consumption needs to be optimized. For the T_yy CAI, a single reference mirror is used for the splitting and the three interferometer pulses instead of three independent reference mirrors and no tip-tilt mount is needed.

During the interferometer the atom clouds pass through the CAI to finally reach their respective detection chambers. Figure 7 zooms in the detection zone. The idea is to wait for the atom cloud to exit the interferometer chamber to avoid parasitic light due to fluorescence. A shutter is placed at the entrance to completely block the scattered photons to reach the CAI chamber. A double diffraction $\pi$ pulse is applied to bring the diffracted states back at the center. Spatial fringes on the atomic population are observed with a CDD camera.

Spatially resolved detection prevents the contrast loss determined by the inhomogeneous dephasing due to initial velocity and position distribution and allows the extraction of information on velocity dependent phase shifts [62, 63]. We consider at first a point-like atomic source, to evaluate the effects of the satellite angular rotation $\Omega_{y,y,z}$ when $\Omega_m=0$ on the final fringe pattern thanks to the ballistic expansion of the atomic ensemble during the long interrogation time. The remaining phase terms are those related to the initial velocities of the atoms ($v_x, v_y, v_z)$, and including Sagnac terms and the effect of the vertical component of the gravity-gradient T_zz:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \phi(v_x, v_y, v_z)= 4 k \left ( v_x \Omega_y + v_y \Omega_x \right) T^2 + 2 k v_z \left ( T_{zz} - 3\Omega_y^2 \right) T^3. \label{eq:phase_velocity} \nonumber \end{align} \tag{ 8 }$$

The x–y  cross-section of the final density distribution is shown in figure 8 for different values of $\Omega_y$ and when $\Omega_x=\Omega_z=1 \times 10^{-6}$, and $T_{zz}=-2.7 \times 10^{-6}$ s⁻². The signals are calculated with a bias phase to have the top of a fringe at $(x,y,z)=(0,0,0)$; in the case of a small residual radial velocity, and an interferometer signal spanning over only a fraction of a fringe period, a suitable phase shift can be applied to the interference pattern in order to center the signal at half fringe to increase the phase sensitivity. The increasing angular rotation along the y  axis determines an increasing spatial frequency for the fringes along the x axis. The effect of T_zz is to spread the phase by $\approx$600 mrad along the z direction. When z is chosen as observation direction, the interference pattern encodes the angular velocities along the x and y  direction as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \phi(x,y)= 2 k T \left ( \Omega_y x + \Omega_x y \right). \nonumber \end{align} \tag{ 9 }$$

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Where the two angular velocities can be obtained with a 2D fit on the atomic fringes. The fringe spacing is inversely proportional to the projection of the angular velocity on the x–y  plane, i.e. $\Omega_{x-y}=\left ( \Omega_x^2 + \Omega_y^2 \right){}^{1/2}$, and it is equal to $\pi/ \left ( k T \Omega_{x-y} \right)$. In order to resolve the interferometer fringes, the CCD camera has to have enough resolution; the requirement becomes more demanding as the satellite angular rotation increases. For example, when $\Omega_{x-y}=\Omega_{{\rm orb}}$ the fringe spacing is equal to 33 $\mu$m, and imaging 4 mm over 1024 pixels will lead to 3 pixels for each fringe.

The initial spatial distribution of the atomic cloud determines the blurring effect on the interferometer fringes; the resulting signal is obtained by calculating the convolution between the probability distribution obtained for the point-like source case with the initial spatial distribution of the atomic ensemble. In figure 9 is shown how the fringes signal worsens when the initial cloud is considered as an isotropic normal distribution along the three directions, with a standard deviation equal to 150 $\mu$m. The fringe contrast is further decreased because of the density distribution integration along the observation direction required for the imaging. For $\Omega_y \ll 10$ mrad s⁻¹ the phase spread along the vertical direction would mainly be due to $\Gamma_{zz}$ if it would not be compensated (see equation (8)) and would be $\approx$600 mrad over the final size of the atomic cloud.

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The position noise (jittering) for the initial atomic cloud center translates in first approximation into a displaced fringe pattern at each shot at the detection, resulting in an additional blurring if averaging consecutive fringe patterns. However, we expect that fringe patterns relative to each interferometric sequence will be analyzed individually at each shot and that such displacements will be extracted out of their fits, as the specific position of the center of the fringe pattern envelope is determined by the cloud center position. If averaging over several realizations would be used, such displacements could thus easily be removed, in order to prevent any related blurring.

Other effects, not taken in to account here, contribute to further reduce the phase sensitivity, like the quantum projection noise (QPN) due to the finite number of atoms detected by each pixel of the CCD, and the technical noise determined by the detection technique [64]; these effects must be evaluated to define the requirements for the instrument adopted for the detection.

Finally, note that the above results also apply to the case where the Raman mirrors are tilted to compensate for rotation, provided one replaces $\Omega_y$ with $\Omega_y-\Omega_m$.

The architecture of the laser system including the frequency/power distribution is depicted on figure 10. It is based on telecom technology combined with second harmonic generation (SHG) [65]. The main specifications of the laser are detailed in table 4.

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Table 4. Main specifications of the laser source for T_zz. Each specification is determined for one type of function. For instance the 'Raman' column is for both the atom motion and the $\pi/2$ interferometer pulses since the specifications are identical. The required power is for a single Raman beam. Since they are used simultaneously, this value need to be multiplied by the number of beams in the total power budget. The two values for the laser power correspond respectively to one optical frequency. Disruptive lines can be produced if we use a phase modulator to create the second optical frequency. $\omega_{R}$ is the recoil frequency. The difference about the laser parameters for T_xx and T_yy are the frequency detuning range (($-17\,150$;$-14\,650$) MHz) and the laser power (22/22 mW) of the interferometer $\pi$ pulse.

Specifications	Units	2D MOT	3D MOT	Raman	Bloch lattice	Push	Detection	Raman $\Delta k_{z}$
Wavelength	nm	780.24	780.24	780.24	780.24	780.24	780.24	780.24
Power	mW	200/20	100/10	45	200	5	5	30/30
Frequency	MHz	−20	[−20;  −120]	−3400	$+100\,000$	∼0	∼0	[$+19\,100$;$+24\,100$]
Linewidth	MHz	0.5	0.5	<0.01	<6	1	0.1	<0.01
Frequency accuracy	kHz	<100	<100	<100	Not critical	<100	<100	<25
Second frequency	NA	$F=1 \rightarrow F'=2$	$F=1\rightarrow F'=2$	6834  +  $\omega_{R}$	[0.075;  −3.087]	No	6568	6834-3 $\omega_{R}$
Polarization	dB	20	20	30	20	20	30	30
Power stability	NA	<$1\%$	<$1\%$	<$0.1\%$	<$1\%$	<$0.1\%$	<$1\%$	<$0.1\%$
Disruptive lines	NA	Tolerated	Tolerated	No	No	No	No	No

A reference laser supplies an absolute optical frequency corresponding to the atom transition of Rubidium (⁸⁵Rb crossover transition $|F=3\rangle\rightarrow|F'=3(4)\rangle$). Then all the laser frequencies are servo locked compared to this reference thanks to a beatnote and a lock-in electronics box. The laser system is composed of four other blocks corresponding to the functions of table 4: one block for the cooling and the detection, two subsystems for the Raman transitions (one for the $\pi$ pulse of the interferometer, one for the other pulses), and the last part corresponding to the implementation of the Bloch lattice. The wave vector modification of the $\pi$ pulse interferometer $\delta k$ implies an additional laser system. Indeed, the laser system to generate $\pi$ and $\pi/2$ pulses of the atom interferometer cannot be the same because the frequency difference is too large and they are used simultaneously. The reference frequency for the $\pi$ Raman pulse is produced using a phase modulator (PM 2) with a tunable microwave frequency reference (MWFR). The reference frequency for the Bloch lattice laser is the fifth harmonics at  +50 GHz of a frequency comb generated using a phase modulator (PM 1) with a DRO operating at 10 GHz.

The laser source for the cooling part is a standard DFB diode. A phase modulator (PM 3) with a DRO at 6.8 GHz creates the repumping frequency used for laser cooling in the 2D MOT. This beam is then amplified by an erbium-doped fiber amplifier (EDFA). Frequency doubling from 1560 nm to 780 nm is accomplished via SHG in a periodically-poled lithium niobate waveguide (PPLN-WG), where the confinement of the optical mode leads to high intensity and thus high efficiency. A similar architecture is implemented for the 3D MOT, the detection and the push beam except for the laser diode (external cavity diode laser). A fibered splitter is used to separate the beams for the three functions.

The configuration of the Raman laser source is based on the generation of the two frequencies by distinct external cavity diodes (ECDL). The two lasers are phase locked thanks to a beat note on a photoconductor at the output of the laser system to avoid phase noise along the optical paths. We duplicate the Raman laser system for the interferometer $\pi$ pulse which has a specific detuning and power compared to the $\pi/2$ pulses due to the shift of the wave vector $\delta_{k}$. The required frequency difference between the $\pi$ pulse and the $\pi/2$ pulses to compensate for T_zz is  +25 GHz. Since the gravity gradient is not constant during the full orbit and the full mission, this frequency shift has to be tunable over a range of $\pm$2.5 GHz. Moreover, the idea is to actively compensate for the phase shift due to the variation of the gravity gradient, and extracting the measurement from the needed frequency shift. This requires a control of the frequency shift with a relative accuracy of 10⁻⁶ to be compliant with the specification of 5 mE per shot for the gravity gradient measurement.

The laser source for the Bloch lattice subsystem is a DFB diode. To create the two frequencies, the beam is split into two paths and a different RF frequency is supplied on each fibered acousto-optic modulators (AOMs). The two beams are then recombined on the free space combination bench.

ZERODUR$^{\unicode{174}}$ free-space platforms hold AOMs and mechanical shutters to switch on and off the output light. An assembly of polarizing cubes and waveplates splits and recombines the beams. Two laser sources with optical frequencies separated by the clock transition of Rubidium 6.8 GHz are combined for the Raman pulses. The polarization are crossed and the two output of the polarizing cubes are exploited in order to avoid power losses. A beat-note is done on a photoconductor and the second laser source is phase locked on the first one.

The mirrors discussed in this section retroreflect the beamsplitting light fields. They serve as a reference for the effective wave fronts of the light fields used to coherently manipulate the atoms. Ideally, these effective wave fronts are flat and smooth over all atom-light interaction zones to avoid spurious phase contributions. This condition implies a parallel alignment of the surfaces in the cross-track direction, and well aligned tilts in the other two directions which compensate for rotations [62, 66, 67] and lead to effective parallel wave fronts seen by the atoms. The retro reflection setup suppresses inhomogeneity of the incoming light field, but imperfections in the ideally flat mirror surface directly affect the effective wave front. Wave front distortions are a major contribution to the uncertainty in precision atom interferometers [10–12, 66]; this noise can be reduced using colder atoms [11, 68], and further minimized with condensed [26] or even collimated atoms [40, 46, 60]. These considerations motivate the detailed discussion for the gradiometer. We assess the requirements on the mirror in three steps: (i) tilts, (ii) defocus and Gouy phase, (iii) higher order distortions, and propose a technical implementation. The following assessments treat a Gaussian distribution with an initial standard deviation of the position of $400~\mu\mathrm{m}$, and of the velocity of $100~\mu\mathrm{m~s^{-1}}$ for an atomic ensemble of 10⁶ atoms.

For assessing the impact of relative tilts of the effective wave fronts, we follow the description of [69]. To first order, and for small angles $\alpha$, a tilt implies a change in the distance of the atoms to the retro reflection mirror of $d\cdot\sin{\alpha}$. Here, d denotes the distance between the pivot point and the projection of the position of the atoms onto the mirror surface in direction of the beam splitting light field. Although the choice of d has a strong impact onto the phase of a single atom interferometer, it is suppressed in the differential signal. This suppression depends on the overlap of the trajectories parallel to the mirror surfaces [70] which itself depends on the precision of the launching mechanism, that transfers the atoms into the interferometer region, and ultimately on the shot-noise limited position and velocity uncertainty of the atomic ensembles. For our gradiometer, we have to control phase noise, and can neglect an unknown, but stable phase shift. Figure 11 shows an example of a phase shift induced by the relative tilt between two effective wave fronts and its suppression in the differential signal when assuming a shot noise limited differential starting position and velocity (cross-track direction). From the slope of the curves we derive the requirement to keep relative tilts below $25~\mu$rad ($0.5~\mu$rad) if the relative position jitter between two subsequent cycles is below $2~\mu$m ($100~\mu$m), which is at or above the shot-noise limit (<2 $\mu$m). This corresponds to a peak to valley of the mirror surface of 250 nm (5 nm) over a region with a diameter of 1 cm. The same requirements are valid for the other two axes (along track and nadir) when interpreting the relative mirror tilts as the deviation from the ideal alignment for compensating the rotations.

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In a simplified model [11], an acceleration signal is contaminated by $\delta\,a=\sigma_v^2/R$ for an expansion rate $\sigma_v$ of the atomic ensemble, assuming a Gaussian distribution, and a radius R of a static effective wave front curvature. In our more detailed model the wave front curvature changes as the beam propagates, and the effective wave front is generated by subtracting the retro reflected beam from the incoming one [71]. We propagate the beam parameters via ABCD matrices [72], determining the wave front curvature and Gouy phase depending on the longitudinal position inside the light field. The wave front of the incoming beam depends on the numerical aperture of the fiber, a free propagation, the focal length of the lens for collimating the beam, and another free propagation to the atoms. The wave front of the retro reflected beam depends on the numerical aperture of the fiber, a free propagation, the focal length of the lens for collimating the beam, a free propagation to the mirror (past the atoms), the focal length of the lens describing the mirror, and a free propagation back to the atoms. We take the different positions of the atoms on the upper and lower trajectories of the interferometer into account, and calculate the phase for both interferometers forming the gradiometer. Our model also enables the evaluation of cases where the atoms are off center with respect to the beam. In addition, it introduces a weighting according to the Gaussian intensity profile, since transition probabilities at the edge of the beam degrade because of reduced Rabi frequencies, and atoms at these positions contribute less to the average phase.

We initially randomize the values for longitudinal positions of the optics in the beam (standard deviation of 5 $\mu$m–20 $\mu$m), the focal length of the collimation lens (standard deviation of few 10 $\mu$m), the velocities and positions of the atomic ensembles (shot noise limited standard deviation of 0.1 $\mu$m s⁻¹, and 0.4 $\mu$m, respectively), and the curvature of the lens modelling the mirror (standard deviation of 300 m) to slightly deviate from the ideal value. For the mirror, we assume a mean curvature radius of R  =  5600 m. This corresponds to a peak to valley of $\lambda$/20 for $\lambda=780$ nm and diameter of the mirror of 6 cm and leads to a reflected wave front with a curvature radius of R  =  2800 m. We individually scan the focal lengths of the collimation lens and the mirror for each interaction zone and keep other parameters fixed to evaluate the impact on the phase of the interferometer for approximate beam waists of 2.5 mm, 5 mm, and 10 mm. An example curve is shown in figure 12. In addition, we take the standard deviation of the phase for 1000 shots with randomized values for the optics positions in longitudinal direction (standard deviation of few $\mu$m), the positions of the beam splitting axes in transverse direction (standard deviation 100 $\mu$m), and the velocities and positions of the atomic ensembles (shot noise limited).

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For the cross-track direction, with an approximate waist of 2.5 mm (5 mm, 10 mm), we estimate a phase noise per shot in the gradiometer signal of about 1 mrad (0.2 mrad, 0.05 mrad) and a phase bias of about 4 mrad (4 mrad, 1 mrad) or less for a deviation of the focal length of the collimation lens from the optimum of $\leqslant$$5~\mu$m (40 $\mu$m, 40 $\mu$m) or less and a deviation of the focal length of the mirror of $\leqslant$600 m. For waists above 2.5 mm, the Gouy phase dominates the phase excursion in a single atom interferometer. Although it still dominates the residual phase shift in the differential signal for the gradiometer in our simulation, it is sufficiently suppressed.

For simulating the other two axes, we add displacements to the positions of the atoms with respect to the beam center as required by the tilted-mirror configuration. Again using an approximate waist of 2.5 mm (5 mm, 10 mm), we estimate a phase noise per shot in the gradiometer signal of about 3 mrad (1 mrad, 3 mrad) and a phase bias of about 10 mrad (5 mrad, 1 mrad) or less for a deviation of the focal length of the collimation lens from the optimum of $\leqslant$$5~\mu$m (40 $\mu$m, 40 $\mu$m) or less and a deviation of the focal length of the mirror of $\leqslant$600 m. Here, the quality of the input collimation dominates the residual shift for waists of 2.5 mm and 5 mm, and for 10 mm the dependence on the Gouy phase is qualitatively similar to the previous case. To push the noise of the configuration with a waist of 10 mm below 1 mrad, the standard deviation of the positions of the beam splitting axes in transverse direction has to be limited to 25 $\mu$m.

Consequently, this validates our choice of a mirror with a curvature radius of about R  =  5600 m, and sets the requirement for minimum beam waist of 5 mm.

To simulate higher order distortions, we follow the approach of [12, 70, 73] by parameterizing the atomic trajectories and calculating the local phase shifts imprinted by the atom-light interactions which are affected by imperfect mirror surfaces, represented by reference objects. Here, we do not propagate the wave fronts, and simply assume the effective wave fronts to be a copy of the surface inhomogeneities. The three reference objects have a peak-to-valley figure of about $\lambda$/20 ($\lambda$/75) and a root mean square of  ∼$\lambda$/100 ($\lambda$/150) over the whole surface (central region, diameter 1 cm). Due to reflection, these values are doubled in the simulation. We take the standard deviation of 10 averages with 400 shots each for randomized positions (velocities) of an atom with a standard deviation of 400 $\mu$m (100 $\mu$m s⁻¹). This procedure is repeated for four different center positions of the reference objects and two different sequences of the reference objects to rule out readings from particularly good or bad spots. The simulation returns an estimate for the differential phase noise of 10 mrad per shot, consistent between the individual intializations. This is a factor of 10 above the requirements. Since the results indicate uncorrelated noise between the two single interferometers contributing to the gradiometer signal, we expect these results to hold for the tilted mirror configuration.

The simulations imply the requirements of having a local peak-to-valley figure of $\leqslant\lambda$/1000, and a local root mean square of $\leqslant\lambda$/1000 (∼1 nm). For designing appropriate optics, the simulation can be adapted by replacing the reference objects with two blanks and an object whose surface is defined by a single Zernike polynomial [74], and determining the required pre-factors.

Summarizing these assessments, the requirements are a peak-to-valley figure of $\lambda$/20 for a mirror with a diameter of 6 cm, a peak-to-valley figure of $\lambda$/1000 (∼1 nm) in the central region with a diameter of 1 cm, a root mean square of $\lambda$/1000 (∼1 nm) in the central region with a diameter of 1 cm, and a maximum relative tilt of the effective wave fronts in the beam splitting zones below 0.5 $\mu$m. When assuming a fused silica substrate for the mirrors, temperature gradients have to be limited below 63 K/m to avoid distortions which violate the requirements above.

Mirrors in gravitational wave detectors as VIRGO [75, 76] and LIGO [77, 78] are based on fused silica substrates coated by ion beam sputtering with SiO₂ and Ta₂O₅ or TiO₂. Here, the surface quality of the substrate dominates the inhomogeneity after coating. Test substrates with a diameter of 48 mm and a thickness of 12 mm were polished to a peak-to-valley figure below 1 nm and a root mean square of 0.2 nm [79–81]. In LIGO, a test sample of a coated substrate reached a roughness below 0.5 nm root mean square, and the requirement on controlling the curvature of the mirror at the level of $\pm$2 m implies a control of about 1 nm in the peak-to-valley figure. For our case with light at 780 nm, the coating has to be adapted, corresponding to a different thickness of the layers of SiO₂ and Ta₂O₅ or TiO₅ without qualitatively changing the coating procedure [82, 83]. Manufacturers are available [82, 84–86].

Since these values are compatible with our requirements, we propose to use the same mirror technology for the gradiometer. For the cross-track axis we propose a single mirror covering all three atom-light interaction zones for passive stability with a size of 6$\, \times \,$ 6$\, \times \,$ 31 cm³, leading to a mass of 2.5 kg, and for the other two axes three separate mirrors with a diameter of 6 cm and a thickness of 1.5 cm. The mirrors should be mounted inside the vacuum system to avoid additional distortions by viewports. In the axes with tilted mirror configuration, the outer ones require motorized mirror mounts for initial alignment and adjustment to the actual rotation rate.

7.5.1. Requirements.

For the interferometer region a static bias field of e.g. B  =  100 nT aligned with the Raman lasers is applied to provide a quantization axis for the atoms. External magnetic fields need to be attenuated below this level by passive shielding and temporal and spatial fluctuations of the magnetic fields need to be suppressed since they cause phase shifts due to the second order Zeeman effect and bias in the measurement of the gravity gradient.

With a gradient in the magnetic field $\nabla B$ and the quadratic Zeeman shift in energy $\delta E=hKB^2$ with K  =  575 Hz G⁻¹, the atoms accelerate by $\delta a = \frac{-2hK}{m} B\nabla B$. Given the same constant spatial gradient in both interferometers, the differential acceleration between the two interferometers then is $\Delta(\delta a)=\frac{-2hK}{m}(\nabla B){}^2 \Delta z$. This results in a bias on the gravity tensor

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta T_{zz}=\frac{\Delta (\delta a)}{\Delta z} = \frac{-2hK}{m}(\nabla B)^2. \nonumber \end{align} \tag{ 10 }$$

In order to reduce the bias on the gradiometer phase below 1 mrad, the magnetic field gradient needs to be reduced below 60 nT m⁻¹.

Similarly, a difference of the time averaged field $\langle B \, \rangle$ between the first and second half of the pulse sequence leads to a bias in the phase measurement of $\Delta \phi = 4 \pi K B \Delta \langle B \, \rangle T$. Thus with B  =  100 nT a field fluctuation by $\Delta \langle B \, \rangle=14$ nT causes a phase shift of 1 mrad. This is then subject to common mode suppression in a differential measurement depending on the spatial and temporal correlations of the fields at the two interferometers.

All of the above requirements need to be met, as the satellite moves in Earth's magnetic field of $B_{{\rm earth}} \approx 40~\mu$T, causing a change of the field component projected on the interferometer axis of up to 60 nT s⁻¹. Thus, an efficient mumetal magnetic shield is required to provide a suitable magnetic field environment [87].

7.5.2. Magnetic shield design.

To provide shielding of the interferometer against external magnetic fields, we consider a passive multilayer mumetal shield. The attenuation of static or slowly varying external fields is described by the shielding factor

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S = \frac{B_{\rm outside}}{B_{\rm inside}}, \nonumber \end{align} \tag{ 11 }$$

defined by the ratio of the total residual magnetic field inside the shield to the total initial magnetic field outside. The magnetic field requirements derived above thus translate into a required shielding factor of S  >  1000.

To verify that our design complies with this requirement, we performed simulations based on finite-element modeling (FEM), similar to [88]. The modeled mumetal shield was placed inside a static magnetic field $B_{{\rm earth}}=40$ $\mu$T and the residual field at the interferometer region was calculated to determine the shielding factor and the resulting magnetic field gradient along the interferometer area. Starting from a cylindrical design the best trade-off between shielding effectiveness and dimension and mass was found with elliptical shields. The three-layer magnetic shield design, shown in figure 13, consists of one overall outer layer and two inner layers for the 2D-MOT/BEC chamber and the CAI chamber respectively. The gaps between the layers were set to 20 mm and each layer has a thickness of 1 mm.

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The FEM simulation results in shielding factors of $S_{x,y,z} (49\,700,9900,4800)$ for the interferometer area, which complies with the requirement. For the magnetic field gradient we observe a maximum value of $\Delta B_{x,y,z} = (0.74,2.7,4.6)$ nT m⁻¹, also fulfilling the requirement. The latter values are determined from the maximum observed differences of the total residual fields over the two interferometer regions and thus represent an upper limit to the gradient. The actual performance of the shielding, in particular residual magnetic field gradients from within the shielding, will of course depend on the fabrication and details of the design such as location of necessary feedthroughs which are not yet included here. Thus, we take these results only to confirm that a three-layer shield of this size and mass should be capable to meet the requirement in principle. A detailed design will then have to add specific measures to further reduce residual magnetic field gradients such as magnetic probes and compensation coils e.g. as done in [89].

Here we consider a satellite in nadir-pointing configuration with a payload of three differently orientated instruments to measure the gravity gradient in each spatial direction: T_xx, T_yy and T_zz. Each of the three instruments is divided into three functional units: the physics package, the laser system and an electronics unit as depicted in the functional diagram of figure 14. In table 5, we give a breakdown of the mass and power budget for the single T_zz instrument into these three units. The mass and power for the T_xx and T_yy sensors slightly differs due to the additionally required tip-tilt mirror. This budget is derived from the design presented in section 7 and based on state of the art technology of cold atom experiments in microgravity [26, 46].

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Table 5. Mass and power budget for a T_zz instrument including the tip-tilt mirror design. A component margin of 20% on mass and power is included here.

System	Mass (kg)	Power (W)	Size (mm)
Physics package	160.4	128.4	(1052 $\times$ 444 $\times$ 805)
Laser system	51.1	104.2	(300 $\times$ 300 $\times$ 400)
Electronic system	52.4	607.9	(300 $\times$ 300 $\times$ 1000)
Total	264.0	840.9	(1052 $\times$ 750 $\times$ 805)

From the basic vacuum chamber design in figure 7, we estimate the size of the physics package for a single axis instrument to be (1052$\, \times \,$ 444$\, \times \,$ 805) mm ³ including the elliptical magnetic shield design, discussed in section 7.5.2. An arrangement of all three instruments including lasers and electronics then results in an approximate estimate of the payload size of (1054$\, \times \,$ 1054$\, \times \,$ 1600) mm ³. This allows for an elongated satellite shape similar to that of GOCE with a front surface slightly larger than 1 m², which is important to minimize the residual atmospheric drag of the satellite. The power and mass of a full three-axis instrument including a $20\%$ margin is estimated at 785 kg and 2940 W. However we expect that ongoing technology development should allow for a significant reduction of these values.

In order to represent the chain of the measurement process realistically, we carried out this study through a closed-loop simulation in the time domain. This approach is flexible and provides the possibility to precisely quantify the errors in terms of gravity field solutions.

A general overview of the closed-loop simulation workflow is given in figure 15. On the one hand, the simulator takes a gravitational model and the noise-free time series of the satellite orbit, angular velocity and attitude, and, on the other hand, the spectral or statistic characteristics of the sensors' noise, to synthesize realistically degraded observables, namely the estimated orbit, angular velocity, attitude and gradients. These synthesized observables are finally used to derive a gravity field model. The comparison of the estimated gravity field model with the original one enables then to precisely characterize the error in the frequency and spatial domain.

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The Earth's global gravity field is usually expressed in terms of a spherical harmonics (SH) series [90] as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq_gravity_field_model} V (r,\theta,\lambda) = \frac{GM}{R} \sum_{n=0}^{N} (\frac{R}{r})^{n+1} \sum_{m=0}^{n} \left[ \bar{C}_{nm} \cos~m\lambda + \bar{S}_{nm} \sin~m\lambda \right] \bar{P}_{nm}(\cos \theta), \nonumber \end{align} \tag{ 12 }$$

where GM is the gravitational constant of the Earth, R is the radius of Earth, $(r,\theta,\lambda)$ are spherical coordinates of a point on the Earth surface (r radius, $\theta$ co-latitude, $\lambda$ longitude), $n,m$ denote SH degree and order, N denotes the maximum degree of the model expansion (in theory, the maximum degree is $\infty$), $\bar{P}_{\rm nm} (\cos \theta)$ are fully normalized associated Legendre functions, and $\bar{C}_{\rm nm}, \bar{S}_{\rm nm}$ are the normalized SH coefficients, which are the unknowns of the gravity field solution.

The gravity gradients $V_{ij}$ are point-wise measurements of the second-order partial derivatives of the gravity potential. They are usually delivered in the instrumental reference frame, e.g. the gradiometer reference frame (GRF), while the Earth's gravity field model is expressed in the earth-fixed reference frame (ERF). We thus have to transform the gravity gradients and the gravity field model to the same reference frame by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq_Func_model} V_{ij} = R \frac{\partial^2 V}{\partial x_i \partial x_j} R^T , \nonumber \end{align} \tag{ 13 }$$

where R represents the rotation matrix between different reference frames. This equation represents the observational equation for the gravity gradients in the frame of gravity field recovery.

Due to the large amount of observations and the large number of unknowns, it forms a large-scale and over-determined linear equation system for the determination of the gravity field model. The classic least-squares (LS) adjustment is applied to solve this linear equation system, which poses a great numerical challenge because of the high computational requirements in terms of both time and memory. The computation of this part is mainly accomplished on the clusters of Leibniz universität IT services (LUIS)⁹.

Three kinds of observations are required for gravity field recovery, including satellite's orbit, attitude and gravity gradients. Orbit data is mainly used to geolocate other observations, while attitude data is necessary for the setup of the rotation matrix between different reference frames. The gravity gradients are the primary observations for the retrieval of the SH coefficients.

8.3.1. Synthesis of noise-free data.

Since the gravity field signal attenuates quickly with altitude, it is important to have the satellite's orbit as low as possible. For the sake of rigorous comparisons to GOCE, we have chosen 71 d of the GOCE orbit, from 2 March to 10 May, 2013. During this period, the orbit had an average altitude of 239 km. Note that this altitude is lower than the standard one of GOCE, typically of 259 km. The time series of the orbit coordinates have been interpolated at 1 Hz from the GOCE L2 dynamic orbit data sampled at 0.1 Hz using spline functions. Based on the orbit data, the noise-free gravity gradients are computed using the gravity field model 'Eigen-6c4' [91] up to SH degree and order of 360.

The generation of the angular velocity $\boldsymbol{\Omega}$ is indispensable for the determination of attitude and rotations. This can be done by two steps: first, the instantaneous angular velocity $\omega_{{\rm LORF}}$ of the local orbital reference frame (LORF) with respect to the inertial reference frame (IRF) has to be computed from the orbital position and velocity, and then assigned to the component $\omega_y$ so that the main rotation of the instrument frame, i.e. GRF, at the orbital frequency is indeed about the y -axis; second, a residual zero-mean angular velocity $\delta \boldsymbol{\Omega} = (\delta \omega_x, \delta \omega_y, \delta \omega_z){}^T$ is added so that we finally have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{angular_velocity} \boldsymbol{\Omega} = \left( \begin{array}{@{}l@{}} \delta \omega_x \nonumber \\ \omega_{\rm LORF} + \delta \omega_y \nonumber \\ \delta \omega_z \end{array}\right) . \nonumber \end{align} \tag{ 14 }$$

8.3.2. Synthesis of noisy data.

We assumed a zero-mean, normally distributed error with a standard deviation of 2.6 cm added to the orbit initial coordinates. This precision was typically achieved with GOCE [92].

We assumed that three star-trackers and a gyroscope are on board of the satellite for the determination of the angular velocity. Both kinds of sensors give estimates of the three components of the angular velocity, which are then optimally combined in the frequency domain using a Wiener filter like in the processing of the angular velocity of GOCE, see [93]. This method requires to have a model of the PSD of the noise of both sensors. Thereafter, the noise models of both sensors were assumed, with their spectral characteristics shown in figure 16. For more details on this aspect, we refer to [58]. The errors of the gravity gradients are the sum of the gradiometer noise and the error due to the correction of the centrifugal terms. An additive noise with a white behaviour is assumed for the gradiometer. The amplitude of the PSD is assumed as 5 mE (Hz^1/2)⁻¹. The centrifugal terms are the square of the angular velocities whose noise has been discussed above.

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Global gravity field models were finally recovered from the synthesized noisy observations. They were resolved up to SH degree and order of 240, including 58 077 unknowns. As pointed out previously, we could have the diagonal gradient components, i.e. $V_{xx}, V_{yy}, V_{zz}$, obtained for the nadir pointing mode with compensation on rotations. We thus derived three component-wise gravity field solutions as well as a combined solution from all three components. To evaluate the performance of these models, both the true errors (the differences between the recovered SH coefficients and the input background model, i.e. Eigen-6c4) and the formal errors (the accompanied standard deviations of the parameters, obtained in the LS adjustment) are analyzed.

The formal errors of the component-wise and combined solutions are displayed in figure 17. It is shown that these gradient components are sensitive to different parts of the gravity field. For instance, $V_{xx}$ is more sensitive to lower order zonal and near-zonal coefficients, i.e. around order zero, but less sensitive to higher order coefficients. In this regard, $V_{yy}$ exactly complements $V_{xx}$ with the inverse sensitivity. It contributes mainly to non-zonal coefficients, i.e. the sectorial coefficients. The reason for the inverse and complementary contribution is related to the orientation of the corresponding gradiometer pairs. Compared to $V_{xx}$ and $V_{yy}$, $V_{zz}$ is sensitive to all orders of the coefficients, according to Laplace's equation, $V_{zz} = - (V_{xx} + V_{yy})$. The contribution of $V_{zz}$ is identical to the combination of $V_{xx}$ and $V_{yy}$. This could be demonstrated indirectly by comparing the $V_{zz}$ component solution with the combined solution, where both show quite similar patterns. Here, we would also like to mention that the zonal and near-zonal coefficients for all solutions are determined in a degraded performance. This is attributed to the GOCE orbit, which leaves polar gaps where no observations are available [94].

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The contribution analysis indicates that the one-axis nadir pointing mode shows natural deficits for determining the Earth's gravity field. Only a part of the gravity field signal can be precisely retrieved from the $V_{yy}$ component. However, when the compensation of the rotation was applied in this mode, the other two components $V_{xx}$ and $V_{zz}$ can be obtained with a comparable high accuracy. This three-axis mode can thus integrally capture the gravity field signal.

The degree medians of the true errors for all recovered models are shown in figure 18. The degree medians that are more robust with respect to the degraded coefficients are used to represent the error amplitudes for each degree. And for a better understanding, the errors are expressed in terms of geoid height. The solutions from $V_{yy}$ in the one-axis and the three-axis mode show the same performance, as the same signal-to-noise ratio (SNR) of the observations has been assumed. Similarly, the error curve of the $V_{xx}$ component is at a comparable level as $V_{yy}$. However, the solution from $V_{zz}$ is about twice better than those of $V_{xx}$ and $V_{yy}$. This is mainly due to the double power of signal, see more discussion in [58]. To sum up the contribution of all components, the combined solution shows the best performance. It is much better than the component-wise solutions of $V_{xx}$ and $V_{yy}$ but only marginally better than the $V_{zz}$ component solution. The combination of all components by optimal weighting indicates that the combined solution is dominated by the $V_{zz}$ solution.

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We analyze now to what extent the CAI gradiometer concept can outperform GOCE when assuming a nominal gradiometer noise of 5 mE (Hz^1/2)⁻¹. To this end, we extrapolate the error of these models from 71 d to 8 months assuming the error is purely stochastic and reduces as $\sqrt{t}$ where t is the time of integration. To gain an idea of the GOCE solution error, we compute a gravity field model based on the three diagonal gravity gradients of the whole GOCE mission period (November 2009–October 2013, about 47 months). Since the comparison concerns only the contribution of the gradiometer, we have not taken into account the GOCE high-low satellite-to-satellite tracking data that is mainly responsible for the recovery of the low-degree gravity field coefficients.

The 8 month solution for the three-axis nadir pointing mode is better than the GOCE solution for the whole mission period, as shown in figure 19. We can thus conclude that an 8 month mission at an altitude of 239 km and using a three-axis CAI gradiometer in the nadir pointing mode with a nominal white noise of 5 mE (Hz^1/2)⁻¹ would outperform the full GOCE mission and eventually yields a more precise gravity field model. Nonetheless, in this comparison, it should be kept in mind that the GOCE satellite was most of its lifetime at a altitude higher than 239 km, about 3 months at 239 km and 5 months lower than 239 km. This fact shows that it is technically possible to fly a satellite at an altitude as low as 239 km for a duration of 8 months.

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For a fair comparison, we have also plotted the GOCE solution for the 8 months where the satellite was at an altitude equal or lower than 239 km. Again, the 8 month CAI gradiometer in the three-axis nadir mode at a constant altitude yields a better solution despite the fact that the 8 month GOCE solution is partly based on gravity gradients measured at a lower altitude.