1. INTRODUCTION

Since the first demonstration of cold atom interferometric inertial sensors [1,2] in the 1990s, light-pulse atom interferometers (AIs) have become very stable and extremely accurate sensors for the measurements of fundamental constants, such as the gravitational constant [3] or the fine structure constant [4,5] and inertial forces like gravity acceleration [6,7], Earth’s gravity gradient [8,9], or rotations [10–12], finding a variety of applications in geodesy, geophysics, metrology, inertial navigation, and more. In addition, high sensitivity AIs have become promising candidates for laboratory tests of general relativity [13], especially for gravitational wave detection [14,15], short-range forces [16], and tests of the weak equivalence principle [17]. For all applications, high-resolution measurements must take into account high-order terms in the phase shift to increase the sensor’s accuracy and sensitivity.

Phase shift calculations in light-pulse AIs in the presence of multiple gravito-inertial fields have already been deeply investigated [18–21]. However, the phase shift formulations are not always convenient when one wants to use them for practical applications, such as the development of atomic sensors that could be selectively sensitive to a particular inertial field, such as rotation and acceleration, as well as any cross terms. In this paper we derive a novel formulation for phase shift calculations in $N$-light-pulse AIs considering the case of two-path interferences in the presence of multiple gravito-inertial fields. The formulation presented in this paper could be of practical interest both for the understanding of the contribution of inertial terms to the phase shift, as well as for applications in the development of selective inertial sensors. This simple and exact formulation is obtained starting from an exact analytical phase shift formula valid for any pulse sequence and taking into account gravity, the gravity gradient, and rotations [22].

The paper is organized as follows. In Section 2, we introduce the general framework of our calculations and give a simple formulation of the phase shift up to $T^k$ dependences. We explicitly derive the phase shift up to order $k=4$ in the presence of multiple gravito-inertial fields. In Section 3, we give a physical analysis of the leading coefficients of the phase shift obtained with our formulation. In Section 4, we consider the effects of wave vector change and photon recoil on the atomic phase shift in the presence of inertial forces. In Section 5, we give the expression of the total phase shift for any $N$-light pulse AI according to our formulation. We then apply our formula to the case of a Mach–Zehnder atomic interferometer and evaluate the phase shift terms of our compact cold rubidium atom gravimeter. We show that our results are consistent with previous work of Dubetsky and Kasevich, Antoine and Bordé, and Wolf and Tourrenc [19,22,23]. In Section 6, we demonstrate the benefit of our formulation when searching for a particular selective inertial atomic sensor. We give theoretical examples of $N$-light-pulse AIs dedicated to the photon recoil measurement.

2. CALCULATIONS

A. Notations

We consider $N$-light-pulse AIs in which atoms undergo two-photon transitions (Raman transition or Bragg pulse) and where the two laser beams are counterpropagating or copropagating with pulse sequences only separated in time. We consider AIs in the limit of short pulses. A two-photon transition is treated as a single photon transition with effective frequency $\omega ={\omega }_1-{\omega }_2$ and effective wave vector $k=|{\mathrm{k⃗}}_1-{\mathrm{k⃗}}_2|$ corresponding to the difference in frequency and wave vector, respectively. In our formalism, $2M$-photon Bragg transitions (with $M$ being the Bragg diffraction order, $M\ge 1$) could be considered by simply giving to the effective wave vector a magnitude $Mk$. In the following calculations, we will consider $M=1$, unless otherwise specified. Thus, a Bragg transition will be equivalent to a Raman transition if one does not consider the quantum internal state. We will predict whether the internal quantum state changes (Raman pulse) or not (Bragg pulse) if necessary.

1. Time and Laser Field Notations

We first consider AIs for which light pulses are all equally separated by time $T$ [extension to the case of an arbitrary pulse sequence is possible (see Section 3.B)]. The AIs are cut into as many slices as there are interactions.

Hence, the time of the $i$th light pulse is defined as

The effective frequencies and wave vectors will be considered different for each pulse in order to compensate for Doppler shift. Hence, the effective laser frequency is chirped linearly to maintain the resonance condition for each light pulse, as in the case of gravity, and the resonance condition will be satisfied.

The effective wave vector $k_i$ and frequency ${\omega }_i$ of the two-photon transition of the $i$th light pulse are defined as

We introduce an effective laser phase as

2. Interferometric Variables

Any of the AIs that we consider will consist of time sequences of the following two kinds of pulses [24]:

One path of an $N$-pulse AI is described by a vector $\mathrm{ϵ⃗}=({ϵ}_1,{ϵ}_2,\cdots ,{ϵ}_i,\cdots ,{ϵ}_N)$, where ${ϵ}_i$ accounts for the angular splitting induced by the atom–light interaction at time $t_i$, with ${ϵ}_i=+1$ and ${ϵ}_i=-1$ for an upward and downward momentum transfer $\mathrm{\Delta }{p}_i={ϵ}_i\hslash k_i$, respectively, and where $k_i$ is along the $z$ direction. When the atom remains in the same momentum state, ${ϵ}_i=0$. Examples of a pulse beam splitter (i.e., $\pi /2$ pulse) and a mirror pulse (i.e., $\pi$ pulse) are given in Fig. 1.

 

Fig. 1. Recoil diagram of a two-photon transition: the two-photon transition (Raman or Bragg) couples two momentum states. During the process, the momentum transfer $\mathrm{\Delta }{p}_i={ϵ}_i\hslash k_i$ at time $t_i$. (a) Beam splitter pulse: the atom is put in a coherent superposition of two momentum states. (b) Mirror pulse: the atom’s momentum state is changed with momentum $\mathrm{\Delta }p$. Solid and dashed lines correspond to two different momentum states corresponding to the same internal quantum state (Bragg pulse) or different internal quantum states (Raman transition). The two paths are described with vectors $(\mathrm{ϵ⃗})$ and $({\mathrm{ϵ⃗}}^{\prime })$.

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We will consider two-path AIs. Each path is described by vectors $G_k$ (upper path) and $G_k^{\prime }$ (lower path), which take into account the two-photon pulse sequence:

We introduce

 

Fig. 2. Space–time recoil diagram in the absence of gravity of a Mach–Zehnder interferometer consisting of pulse sequence $\pi /2-\pi -\pi /2$.

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B. Atom Interferometer Phase Shift Calculation

The phase shift in a light-pulse AI is often calculated in three steps. In the first step, one identifies the different arms of the interferometer and calculates the phase shift due to the atom–light interaction and the free propagation of the atoms along each independent arm. In the second step, one finds the difference between the phase shift obtained in each arm. Finally, in the third step, a phase shift due to the spatial separation of the atoms is added. To calculate the total phase shift, one can use the Feynman path integral formalism [25], considering atoms as plane waves. One limitation of this approach is that it is difficult to perform when one wants to account for simultaneous gravito-inertial effects, such as gravity, rotation, the gravity gradient, and their interplay, as addressed in this paper.

In this context, the starting point of our approach is based on the atom optics ABCD formalism developed by Bordé [21]. This method allows one to consider atoms as wave packets and to calculate the exact interferometric phase shift arising from multiple inertial effects.

1. N-Light-Pulse AI Phase Shift Derivation

For any time-dependent Hamiltonian at most quadratic in position $z$ and momentum $p$, which is the case for essentially all atom interferometric applications, the exact phase shift formula for an $N$-light-pulse AI with wave vector $k$ along the $z=\frac{\mathrm{k⃗}}{k}·\mathrm{R⃗}$ direction is [22]

The right-hand side of Eq. (7) has two separate contributions. The first summation term accounts for the inertial phase shift, which we will denote as $\mathrm{\Delta }{\mathrm{\Phi }}_{\text{inertial}}$ with $z_i$ (and $z_i^{\prime }$) being the position of the wave packet at time $t_i$ of the upper path (and lower path), respectively, and where one assumes no offset in the initial positions of the wave packet, $\mathrm{\Delta }{z}_1=z_1-z_1^{\prime }=0$. This inertial phase shift depends on the classical midpoint position of the atoms at time $t_i$. The second summation term is a time-dependent laser phase shift that we will denote as $\mathrm{\Delta }{\mathrm{\Phi }}_t$. According to Eq. (4), the time-dependent laser phase shift for any AI consisting of an $N$-light-pulse can be expressed as follows:

Thus, we will mainly focus on the inertial phase shift calculation. For simplicity, we will first assume no wave vector change ($\mathrm{\Delta }k=0$), as well as no photon recoil effect. These high-order terms will be taken into account in Section 4. Thus, the absence of the recoil effect leads to $z_i=z_i^{\prime }$ in Eq. (7), which can be written as

In our method, we calculate the inertial phase shift by making a Taylor expansion in the power of $T$ of the $z$ position of the wave packet with respect to the first light pulse ($t_1=0$) of the interferometer, assuming $z_1=0$:

After some algebra, the inertial phase shift is given by

One can rewrite Eq. (11) considering the action of any gravito-inertial fields on the atoms located at position $\mathrm{R⃗}$, leading to the final expression

C. Phase Shift due to Gravito-Inertial Fields

In this section, we use our formulation to derive the phase shift when the atom is submitted to simultaneous time-independent gravito-inertial fields like gravity, the gravity gradient, and rotation. We consider the same two cases (A and B) treated in the previous work of [22].

In case A, the AI is fixed to the Earth frame ${\mathcal{R}}^{\prime }$, rotating with rotation rate $\mathrm{\Omega ⃗}$ with respect to the inertial reference frame (i-frame), $\mathcal{R}$ (see Fig. 3). In case B, the AI is fixed to a rotating platform of rotation rate ${\mathrm{\Omega ⃗}}^{\prime }=\mathrm{\Omega ⃗}$ for convenience.

 

Fig. 3. Nonrotating inertial geocentric reference frame $\mathcal{R}$ (i-frame) defined by the fixed stars having its origin at the center of the Earth and Earth reference frame ${\mathcal{R}}^{\prime }$ with local coordinate system $(x,y,z)$, where the $z$-axis is chosen to point away from the Earth’s center, with rotation rate components $\mathrm{\Omega ⃗}=(0,\mathrm{\Omega }\cos \lambda ,\mathrm{\Omega }\sin \lambda )$.

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To give the phase shift for each case (A and B), we calculate the ${\mathcal{F⃗}}_k={(\frac{d^k\mathrm{R⃗}}{d{t}^k})}_{t=t_1}$ functions of Eq. (12). For $k=\mathrm{0,1}$ these functions simply correspond to initial atomic position ${\mathrm{R⃗}}_1$ and velocity ${\mathrm{v⃗}}_1$, respectively. To calculate higher order derivatives ($k\ge 2$), we use the classical equations of atomic motion given by

In order to calculate ${\mathcal{F⃗}}_k={(\frac{d^k\mathrm{R⃗}}{d{t}^k})}_{t=t_1}$ ($k\ge 2$), one has to perform the time derivation of Eq. (13). Moreover, one can notice that the time derivation of Eq. (14) strictly depends on case A or B. In order to provide a unified treatment, we introduce parameter $\eta$ to distinguish between both cases. Assuming $\eta =1$ for case A and $\eta =0$ for case B, one obtains

One interesting feature of Eq. (15) is that it takes into account both cases A and B simultaneously. Choosing $\eta =0$ or $\eta =1$ and calculating successive higher order time derivatives using Eq. (13) and its derivatives and substituting in Eq. (12) allows us to calculate the global inertial phase shift for any $N$-light-pulse AIs.

1. Application

Considering the rotating Earth frame, one can define ${(\frac{d\mathrm{R⃗}(t=t_1)}{dt})}_{{\mathcal{R}}^{\prime }}={\mathrm{v⃗}}_1^{\prime }$ as the initial velocity of the atoms and acceleration

Calculating the inertial phase shift up to terms proportional to $T^2$ ($k=2$) leads to

Equation (17) exhibits the total inertial phase shift to the second order in the time separation between pulses. In this case, one can see that ${\mathcal{F⃗}}_2$ is only a function of constant acceleration $\mathrm{a⃗}$, initial velocity ${\mathrm{v⃗}}_1^{\prime }$, and constant rotation $\mathrm{\Omega ⃗}$.

In Table 1, we extend the phase shift calculation up to terms proportional to $T^4$ for whatever the case is (A or B). In this case, cross terms between the gravity gradient $\overrightarrow{\overrightarrow{\mathrm{\Gamma }}}$, acceleration $\mathrm{a⃗}$, and rotation $\mathrm{\Omega ⃗}$ appear in the phase shift expression.

Table 1. Phase Shift Terms up to the Order $k=4$

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One can see from Table 1 that the coupling between the gravity gradient and the rotation is case dependent. When the AI is fixed to a mobile platform, there is a coupling between the gravity gradient and the rotation, whereas this coupling vanishes in case A when the sensor is fixed to the Earth frame.

Moreover, Table 1 highlights the benefit of our formulation, where the phase shift terms up to $T^k$ dependences appear as a simple product between the inertial fields and a specific $\mathrm{\Delta }{G}_k$ coefficient.

The physical meaning of $\mathrm{\Delta }{G}_k$ coefficients will be given in Section 3 for well-known AIs. Moreover, one would like to emphasize that this formulation may also address the case of an optical corner cube gravimeter, such as the FG-5 [26], by simply nulling the laser chirp $\mathrm{\Delta }\omega$ in Table 1 and considering a two-pulse interferometer with $\mathrm{\Delta }\mathrm{ϵ⃗}=(\mathrm{0,1})$.

3. ANALYSIS AND LINK WITH WELL-KNOWN AIS

In this section, we give a physical interpretation of the leading terms $\mathrm{\Delta }{G}_k$ of the phase shift, which are directly related to the two-photon light pulse sequence of the interferometer through Eq. (6).

A. Impact of $\mathrm{\Delta }{G}_k$ Coefficients and AI Symmetry

Assuming no inertial effects and focusing on the first $\mathrm{\Delta }{G}_0$ $(k=0)$ term of the phase shift, one can show that this term gives information about the interferometer’s closure in momentum space. This information is related to the space–time geometry of the atom interferometer.

Searching for the first $N$-light-pulse interferometer closed in momentum space (i.e., $\mathrm{\Delta }{G}_0=0$) with $\mathrm{\Delta }{ϵ}_1=1$, one finds out that $N\ge 2$. For $N=2$, the simplest atom interferometer closed in momentum space is depicted in Fig. 4(a). It is the atomic clock configuration, also called the Ramsey–Raman interferometer, consisting of two pulses $(\pi /2,t_1=0)-(\pi /2,t_2=T)$ separated by a free precession time $T=t_2-t_1$. One can see that this interferometer is time antisymmetric, meaning that $\mathrm{\Delta }{ϵ}_1=-\mathrm{\Delta }{ϵ}_2$. This closure in momentum space can be expressed as

 

Fig. 4. Space–time recoil diagrams in the absence of gravity. (a) Ramsey interferometer consisting of two $\pi /2$ pulses separated by a free precession time $T=t_2-t_1$. The interferometer is closed in momentum space $\mathrm{\Delta }{G}_0=0$. (b) The Mach–Zehnder interferometer consisting of pulse sequence $(\pi /2,t_1=0)-(\pi ,t_2=T)-(\pi /2,t_3=2T)$. This single-loop interferometer is closed in both momentum and position space. Its sensitivity to acceleration is proportional to the space–time area enclosed by the loop. (c) Double-diffraction atom interferometer: two pairs of Raman beams with effective wave vectors $\pm k_i$ are shined simultaneously on the atoms at three moments in times separated by time $T$. DDP1 and DDP3 are double-diffraction pulses that act as splitters, whereas DDP2 acts as a mirror pulse transferring momentum $\mathrm{\Delta }p=\mp 2\hslash k_i$ to $\pm \hslash k_i$ wave packets. The interferometer is totally symmetric, and the scale factor is improved by a factor of 2. In cases (b) and (c), the phase shift does not depend on the initial velocity of the atoms.

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The $\mathrm{\Delta }{G}_1$ $(k=1)$ coefficient of the phase shift is related to the closure in space–time of the atom interferometer. In the Ramsey–Raman interferometer ($\mathrm{\Delta }{G}_1=1$), the total phase shift is dependent on the initial atomic velocity and on the time-dependent phase shift term. In order to build interferometers closed in both momentum and position space, one needs to look for $N\ge 3$ pulse interferometers. Considering a three-pulse interferometer, one has to solve for

For $\mathrm{\Delta }{ϵ}_1=1$, one finds the well-known single-loop, Mach–Zehnder configuration [27,28] consisting in the pulse sequence $(\pi /2,t_1=0)-(\pi ,t_2=T)-(\pi /2,t_3=2T)$. The interferometer is depicted in [Fig. 4(b)]. Considering two-photon Raman transitions, the first $\pi /2$ pulse puts the atom in a coherent superposition of ground and excited states and acts as a beam splitter by transferring $\hslash k$ momentum to the wave packet, making the transition to the excited state. Then, the two wave packets propagate freely during time $T$, drifting apart with relative momentum $\hslash k$. A mirror pulse is applied at time $T$, interchanging the ground and excited states and reversing their relative momenta. Finally, the two wave packets drift back toward each other during time $T$ and a final beam splitter pulse is applied at time $2T$, recombining the two wave packets and interfering them. The interferometer’s closure in position space can be related to the time symmetry through

Solving the set of equations in Eq. (19) and assuming $(\max |\mathrm{\Delta }{ϵ}_1|=2M,M\ge 1)$, any three-pulse AI closed in momentum and position space obeys

For $M=1$, one finds the double-diffraction scheme using, for example, two-photon Raman transitions [27] as depicted in Fig. 4(c). In this configuration, two pairs of Raman beams simultaneously couple the initial ground state with zero momentum $|g,0\hslash k⟩$ to the excited state with two symmetric momentum states, $|e,\pm \hslash k⟩$. The interferometer is time and space symmetric as ${ϵ}_i=-{ϵ}_i^{\prime }$. This spatial symmetry can be expressed for any $N$-pulse AI as

The separation between the atomic wave packets leaving the first beam splitter is increased by a factor of two, leading to a difference of momenta between the two arms of $\mathrm{\Delta }p=2\hslash k$, thereby increasing the space–time area of the apparatus by a factor of two. One can see that for these two configurations closed in position space, the phase shift does not depend on the initial atomic velocity to first order (i.e., when one neglects inertial forces).

When looking at four-pulse AIs, assuming $\mathrm{\Delta }{G}_1=\mathrm{\Delta }{G}_0=0$ one finds the Ramsey–Bordé interferometer [29–32] and the double-diffraction Ramsey–Bordé interferometer ($M=1$) depicted in [Figs. 5(a) and 5(b)]. Finally, cancellation of $\mathrm{\Delta }{G}_{0,1,2}$ coefficients can be obtained with a five-pulse AI geometry, where the middle pulse is not shined. This interferometer [depicted Fig. 5(c)], is the well-known double-loop interferometer consisting of the pulse sequence $(\pi /2,t_1=0)-(\pi ,t_2=T)-(\text{No}\text{pulse},t_3=2T)-(\pi ,t_4=3T)-(\pi /2,t_5=4T)$. This double-loop interferometer is well suited when one wants, for example, to build a sensor insensitive to homogeneous acceleration for rotation or direct gravity gradient measurements [8]. In Table 2, we give the absolute values of the phase coefficient $\frac{\mathrm{\Delta }{G}_k}{k!}$ for the interferometers described above.

Table 2. Examples of Phase Shift Coefficient Values $\frac{\mathrm{\Delta }{G}_k}{k!}$ for Usual Atomic Interferometers

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Fig. 5. Space–time recoil diagrams in the absence of gravity. (a) Symmetric four-pulse Ramsey–Bordé interferometer. (b) Ramsey–Bordé interferometer in a double-diffraction scheme ($M=1$). (c) Five-pulse double-loop interferometer closed in both momentum and position space. In this configuration, $\mathrm{\Delta }{G}_2=0$, making the interferometer insensitive to homogeneous acceleration.

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Fig. 6. Space–time diagram of a double-loop interferometer with nonidentical loops. Assuming $t_1=0$ and total interrogation time $t_4=1$, in order to eliminate all $t^3$ terms (i.e., $\mathrm{\Delta }{G}_3=0$) to the phase shift, one finds $t_{\mathrm{2,3}}=\frac{\sqrt{5}\mp 1}{4}$. Solid and dashed lines correspond to momentum states separated by $\hslash k$.

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Considering the simple expression of $\mathrm{\Delta }G$ given in Eq. (23), one could think of finding any interferometer configurations (i.e., any sets of vector $\overrightarrow{\mathrm{\Delta }{ϵ}_i}$) for which the interferometer could be selectively sensitive to gravity, acceleration, the gravity gradient, or rotation by fixing the value of a particular $\mathrm{\Delta }G$ coefficient to zero and solving

B. Arbitrary Temporal Pulse Sequence

Our approach remains consistent when considering unequally spaced in time laser pulses. In this general case, Eq. (6) has to be rewritten, including the instant $t_i$ of the $i$th laser pulse leading to

Consequently, $\mathrm{\Delta }{G}_k$ coefficients are now time dependent but still related to the two-photon pulse sequence of the AI through a so-called Vandermonde matrix $V^{(t_1,t_2,\dots ,t_i,\dots ,t_N)}$ with time-dependent coefficients.

Considering a four-pulse double-loop interferometer $\mathrm{\Delta }\mathrm{ϵ⃗}=(1,-2,-2,-1)$, one can look for solutions in time $t_i$ of the laser-pulse sequence assuming $\mathrm{\Delta }{G}_{k=0,1}=0$. Applying Eq. (23) with respect to $V^{(t_1,t_2,t_3,t_4)}$ leads to

The detailed calculations are given in Appendix A considering two cases and assuming $t_1=0$ and $t_4=1$ in dimensionless unit.

In the first case, we assumed $\mathrm{\Delta }{G}_2=0$. If one refers to Table 1, in this case, the phase-dependent terms which scale quadratically with time $t$ are eliminated. This is interesting when one wants to measure gravity gradients or rotations. It then comes out that the pulse sequence is $\{t_2=\frac{1}{4}{t}_4;t_3=\frac{3}{4}{t}_4\}$, leading to $\mathrm{\Delta }{G}_3=\frac{3}{16}{t}_4^3$. This pulse sequence corresponds exactly to the case of the identical double-loop interferometer found in Section 3.A when one was not varying the interpulse time [Fig. 5(c)]. In the second case, we assumed $\mathrm{\Delta }{G}_3=0$, thereby eliminating all phase terms scaling as $t^3$. This case is interesting when one wants to increase the accuracy to acceleration measurement by eliminating $T^3$ corrections to the phase. We represent in Fig. 6 the pulse sequence $\{t_2=\frac{\sqrt{5}-1}{4};t_3=\frac{\sqrt{5}+1}{4}\}$ corresponding to a nonidentical double-loop interferometer leading to $\mathrm{\Delta }{G}_2=(1-\frac{\sqrt{5}}{2})t_4^2$ as was found in the previous work of [19], where phase shift calculations were done using density matrix formalism.

4. EFFECT OF WAVE VECTOR CHANGE AND PHOTON RECOIL ON THE PHASE SHIFT

A. Wave Vector Change

Considering the wave vector change, one has to add a correction to Eq. (11) leading to

B. Recoil Phase Shift

The two-photon transitions contribute to transfer two photon momenta to the atoms. Thus, the inertial phase shift is modified and one needs to account for $z_i\ne z_i^{\prime }$. Assuming no rotation and no inhomogeneous acceleration (i.e., the gravity gradient), one can simply express the position of the atoms under free fall at time $t_i$ considering the recoil velocity term $\hslash k/m$ for an atom of atomic mass $m$ as

 

Fig. 7. Space–time diagram showing the Mach–Zehnder interferometer paths in the presence of gravity acceleration. Solid lines: upper and lower interferometer paths ($z_i^{\prime }$ and $z_i$). Dashed line: classical midpoint positions of the atoms taking into account the recoil effect. Solid line (in blue): parabola in the absence of photon recoil. The trajectories are plotted assuming $k_1=k_2=k_3$.

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Considering the two-photon momenta transfer, the inertial phase shift cannot be approximated by Eq. (9) and one has to account the recoil effect contribution of Eq. (28). Hence, the inertial phase shift is

The first term of Eq. (30) is the well-known recoil shift associated with the kinetic energy of an atom absorbing a photon of momentum $\hslash k$, and the two last terms depend on the wave vector change $\mathrm{\Delta }k$ to the first and second order, respectively. These terms may account for corrections to the gravity measurement when one considers the variation of the laser frequencies to compensate for the Doppler shift of freely falling atoms [23].

One can generalize the recoil phase shift to higher orders in the time separation between pulses by simply writing

Table 3. Recoil Phase Shift Terms up to the Order $T^4$ ($k=4$) Assuming $\mathrm{\Delta }k=0$

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In Table 4, we give the $\mathrm{\Delta }{A}_k/k!$ terms for usual interferometers.

Table 4. Examples of Recoil Phase Shift Coefficient Values $\frac{\mathrm{\Delta }{A}_k}{K!}$ for Usual Atomic Interferometers

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One can note that contrary to the Ramsey–Bordé–Chu AI [33] consisting of two pairs of $\pi /2$ pulses, where the second pair of $\pi /2$ pulses propagate in the opposite direction to the first pair, the Ramsey–Bordé interferometer [Fig. 5(a)] used to determine the fine structure constant through a measurement of the atom recoil velocity in [4,5] is not sensitive to the recoil shift at the first order in time $T$ between pulses (i.e., $\mathrm{\Delta }{A}_1=0$), although recoil-velocity-dependent terms appear as corrections to the phase in power of $k\ge 3$ in the time separation between pulses. This main difference comes from the additional kinetic energy term acquired by the atoms in one arm of the Ramsey–Bordé–Chu interferometer, inducing a photon recoil dependence of the phase. However, in order to measure the atomic recoil in the Ramsey–Bordé interferometer, one has to add photon recoil to the atoms in between the two sets of beam splitters. This can be done by implementing a sequence of Bloch oscillations as in [4,5], thus leading to a significant sensitivity increase in the recoil measurement.

Moreover, recoil phase shift terms presented in Table 3 are responsible for not perfectly closed AIs due to coupling between the atomic recoil with rotations and gravity gradient forces. Focusing on the Mach–Zehnder AI used as a gravimeter with vertically propagating optical pulses (along the $z$-axis), this means that the classical trajectories of the upper path and the lower path will not exactly intersect at the final beam splitter. The separation distance between the wave packets at the last pulse is given by $\mathrm{\Delta }z=\partial \mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{rec}}/\partial k_1\approx \frac{\mathrm{\Delta }{A}_3}{3!}\frac{\hslash k_z{\mathrm{\Gamma }}_{zz}}{2m}{T}^3$ when one only looks at the contribution of the gravity gradient force.

Considering a ${}^{87}\mathrm{Rb}$ atom interferometer ($\lambda =780\mathrm{nm}$, $T=1\mathrm{s}$, $k_z=\frac{4\pi }{\lambda }$), one finds a separation between the wave packets of $\mathrm{\Delta }z\simeq 18\mathrm{nm}$. This separation distance is not a problem when using $\mu \mathrm{K}$-temperature thermal cloud sources obtained from usual optical molasses. Nevertheless, this separation phase shift could become an issue for long-baseline AIs (typically $T>5\mathrm{s}$), where BEC-based AIs are required [34].

C. Total Phase Shift

The total phase shift of an $N$-light-pulse AI is simply given by

One finds that

5. APPLICATION TO THE MACH–ZEHNDER AI

As an application, we use our formulation to calculate the total phase shift $\mathrm{\Delta }{\mathrm{\Phi }}_{MZ}$ of the well-known Mach–Zehnder AI. According to our formulation, the temporal symmetries of the interferometer implies $\mathrm{\Delta }{A}_k=\mathrm{\Delta }{G}_k$; $\mathrm{\Delta }{B}_k=\mathrm{\Delta }{G}_{k+1}$; $\mathrm{\Delta }{C}_k=0$. Considering Eq. (36), this leads to a simple expression of the phase shift in terms of the ${\mathcal{F⃗}}_k$ function:

As an application, we evaluate phase terms with parameters corresponding to our transportable cold ${}^{87}\mathrm{Rb}$ atom gravimeter based on a $(\pi /2-\pi -\pi /2)$ two-photon Raman pulse sequence, as described in [35]. The time between pulses is $T=48\mathrm{ms}$ with initial velocity components ${\mathrm{v⃗}}_1^{\prime }=(v_{x,y}=v_{\perp },v_z)$, where $v_{\perp }\simeq 1,2\mathrm{cm}/\mathrm{s}$ correspond to the transverse velocity of the atoms and $v_z\simeq 15\mathrm{cm}/\mathrm{s}$ corresponds to the vertical velocity at the first light pulse. All phase terms are calculated considering the sensor fixed in the local Earth frame defined in Section 2. Results are given in Table 5.

Table 5. Contribution to the Phase for Atomic Gravimeter of Ref. [35] in the Rotating Earth Frame^a

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Comparison of our Mach–Zehnder AI phase shift determination can be made with previous work of Dubetsky and Kasevich, Antoine and Bordé, and Wolf and Tourrenc [19,22,23] with a simple change of variable, considering case A or B of Section 2. As an example, in the work of Antoine and Bordé [22], the wave vector change is not considered ($\mathrm{\Delta }k=0$) and gravity is defined as

6. ORIGINAL $N$-LIGHT-PULSE AI SENSOR

Our simple formulation allows one to seek for original $N$-light-pulse AIs, where the phase shift dependences to specific inertial effects could be canceled by simply searching solutions to Eq. (23) and zeroing specific $\mathrm{\Delta }{G}_k$ coefficients. We give hereafter two theoretical examples of such AIs that would be dedicated to the measurement of the photon recoil.

A. Photon Recoil Measurement AIs

Direct and sensitive recoil frequency measurements using a four-pulse Ramsey–Bordé–Chu AI can be used to determine the fine structure constant [33,36]. Considering Tables 1, 3, and 4, the atomic phase shift up to the power 2 in time $T$ in the rotating Earth frame (i.e., Case A, $\eta =1$) is

One can see from Eq. (40) that the phase shift is sensitive to inertial forces.

In order to make measurements only sensitive to the recoil shift, conjugate interferometer geometries are used to remove the sensitivity to local gravitational acceleration and, moreover, simultaneous conjugate interferometers help reject common-mode vibrational noise [36] that may affect the interferometer sensitivity. However, for example, the gravity gradient effect, which scales cubically with time $T$ (see Table 1), cannot be totally rejected as the two conjugate interferometers are separated in space.

We give hereafter an illustration of how to use our simple formulation to find a theoretical $N$-pulse AI scheme sensitive to the recoil frequency and independent of gravity acceleration, the gravity gradient, and Earth’s rotation.

1. Example 1: Measurement Independent of Gravity

One can look for an AI that would measure the photon recoil independent of the local gravity acceleration. In this case, one would have to solve Eq. (23) assuming $\mathrm{\Delta }{G}_{k=\mathrm{0,1},2}=0$ and $\mathrm{\Delta }{A}_1\ne 0$. If one considers equal time $T$ between the two-photon laser pulses, and assuming $\max |\mathrm{\Delta }ϵ|=2$, one finds the $N=6$-light-pulse sequence of Fig. 8, where we recall that the two-photon wave vector is not addressed in terms of direction. Nevertheless, one can see that for the third and fourth pulses, one needs to realize a double-diffraction scheme. The beam splitters of the interferometer are denoted as $S$.

 

Fig. 8. Space–time recoil diagram of the six-pulse atom interferometer sensitive to recoil phase shift and insensitive to gravity acceleration. Black line: (upper path) $\mathrm{ϵ⃗}=(1,-1,-1,1,1,-1)$. Red line: (lower path) ${\mathrm{ϵ⃗}}^{\prime }=(0,0,1,-1,0,0)$. $S$, beam splitter pulse; DDP, double-diffraction pulse.

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In Table 6, we give the phase shift coefficient values of the interferometer.

Table 6. Phase Shift Coefficients of the Six-Light-Pulse Atom Interferometer of Fig. 8

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One can verify that the total phase shift to the first order in $T$ is

2. Example 2: Measurement Independent of Gravity, the Gravity Gradient, and Earth’s Rotation

We looked for an AI configuration insensitive to the gravity gradient and Earth’s rotation. For this case, we solved $\mathrm{\Delta }{G}_{k=\mathrm{0,1},\mathrm{2,3}}=0$ with $\mathrm{\Delta }{A}_1\ne 0$. Considering, for simplicity, optical pulses equally spaced with time $T$ and assuming $\max |\mathrm{\Delta }ϵ|=2$, we found several eight-light pulse AI configurations. We give two of these configurations in Figs. 9 and 10. In Tables 7 and 8, we give the phase shift coefficient values of the two degenerate interferometer configurations. One can see that the main difference appears in the $\mathrm{\Delta }{A}_3/3!$ coefficient, where rotation and the gravity gradient are coupled to the atomic recoil, leading to a difference of almost an order of magnitude in between the two coefficients.

Table 7. Phase Shift Coefficients of the Eight-Light-Pulse Atom Interferometer of Fig. 9

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Table 8. Phase Shift Coefficients of the Eight-Light-Pulse Atom Interferometer of Fig. 10

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Fig. 9. Space–time recoil diagram of the eight-pulse atom interferometer sensitive to recoil phase shift and insensitive to Earth’s rotation, gravity acceleration, and its gradient. Black line: (upper path) $\mathrm{ϵ⃗}=(1,-1,-1,1,1,-1,-1,1)$. Red line: (lower path) ${\mathrm{ϵ⃗}}^{\prime \prime }=(0,1,-1,0,0,-1,1,0)$. $S$, beam-splitter pulse; M, mirror pulse; DDP, double-diffraction pulse.

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Fig. 10. Space–time recoil diagram of an eight-pulse atom interferometer sensitive to recoil phase shift and insensitive to Earth’s rotation, gravity acceleration, and its gradient. Black line: (upper path) $\mathrm{ϵ⃗}=(1,-1,0,0,0,0,-1,1)$. Red line: (lower path) ${\mathrm{ϵ⃗}}^{\prime }=(0,1,0,-1,-1,0,1,0)$. $S$, beam-splitter pulse; M, mirror pulse.

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These theoretical AI schemes consider perfect beam splitter pulses. However, practically, the use of nonperfect beam splitters induce many two-wave interferences, leading to spurious phase shifts and possible contrast loss [37]. This multiple two-wave interference effect is not treated with our theoretical framework. Nevertheless, from a practical point of view, experimental strategies to mitigate this effect could be developed, such as the use of a pushing beam at resonance with the atoms to suppress parasitic paths in the interferometer [31].

7. CONCLUSION

In this work, we have presented a novel formulation of the phase shift in multipulse atom interferometers considering two-path interferences in the presence of multiple gravito-inertial fields. We showed, considering constant acceleration and rotation and starting from an exact phase shift formula obtained in [22], that coefficients in the leading terms of the phase shift up to $T^k$ dependences could be simply expressed as the product between a so-called Vandermonde matrix and a vector characterizing the two-photon pulse sequence of the AI.

We showed that this formulation could be of practical interest when one wants to use atom interferometers as a selective sensor of some specific inertial field. As an application, we presented theoretical examples of original AIs that could be dedicated to photon recoil measurements independent of rotation, gravity, and the gravity gradient. We therefore think that this formulation could benefit the community.

A noteworthy attribute of our work is that time-dependent accelerations or rotations could be easily included in our formulation, thereby allowing one to look for AI configurations where time-dependent rotations would be minimized or suppressed as one knows their detrimental effects on atomic gradiometer performances [34]. Finally, multiple-beam atom interferometers have demonstrated to be of interest for precision measurements of physical quantities [38,39]. Therefore, a possible improvement of this work would be to consider the effect on the interferometer’s phase shift of the greater number of interfering paths when dealing with multipulse scheme atom interferometers.