We report on an original and simple formulation of the phase shift in -light-pulse atom interferometers. We consider atomic interferometers based on two-photon transitions (Raman transitions or Bragg pulses). Starting from the exact analytical phase shift formula obtained from the atom optics ABCD formalism, we use a power series expansion in time of the position of the atomic wave packet with respect to the initial condition. The result of this expansion leads to a formulation of the interferometer phase shift, where the leading coefficient in the phase terms up to dependences () in the time separation between pulses can be simply expressed in terms of a product between a Vandermonde matrix and a vector characterizing the two-photon pulse sequence of the interferometer. This simple coefficient dependence of the phase shift reflects very well the atom interferometer’s sensitivity to a specific inertial field in the presence of multiple gravito-inertial effects. Consequently, we show that this formulation is well suited when looking for selective atomic sensors of accelerations, rotations, or photon recoil only, which can be obtained by simply zeroing some specific coefficients. We give a theoretical application of our formulation to the photon recoil measurement.
© 2016 Optical Society of America
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  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 , especially for gravitational wave detection [14,15], short-range forces , and tests of the weak equivalence principle . 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 -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 .
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 dependences. We explicitly derive the phase shift up to order 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 -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 -light-pulse AIs dedicated to the photon recoil measurement.
We consider -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 and effective wave vector corresponding to the difference in frequency and wave vector, respectively. In our formalism, -photon Bragg transitions (with being the Bragg diffraction order, ) could be considered by simply giving to the effective wave vector a magnitude . In the following calculations, we will consider , 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 [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 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 and frequency of the two-photon transition of the 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 :
- • pulses that will play the role of matter-wave beam splitters.
- • pulses that will act as matter-wave reflectors (or mirrors).
One path of an -pulse AI is described by a vector , where accounts for the angular splitting induced by the atom–light interaction at time , with and for an upward and downward momentum transfer , respectively, and where is along the direction. When the atom remains in the same momentum state, . Examples of a pulse beam splitter (i.e., pulse) and a mirror pulse (i.e., pulse) are given in Fig. 1.
We will consider two-path AIs. Each path is described by vectors (upper path) and (lower path), which take into account the two-photon pulse sequence:
We introduce2 the case of the well-known two-path Mach–Zehnder atomic interferometer. Focusing on one exit port of the interferometer, one obtains for the upper path and . Consequently, one finds . One could have chosen the other exit port of the interferometer, which would not affect .
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 , 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 . 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 and momentum , which is the case for essentially all atom interferometric applications, the exact phase shift formula for an -light-pulse AI with wave vector along the direction is 
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 with (and ) being the position of the wave packet at time of the upper path (and lower path), respectively, and where one assumes no offset in the initial positions of the wave packet, . This inertial phase shift depends on the classical midpoint position of the atoms at time . The second summation term is a time-dependent laser phase shift that we will denote as . According to Eq. (4), the time-dependent laser phase shift for any AI consisting of an -light-pulse can be expressed as follows:6).
Thus, we will mainly focus on the inertial phase shift calculation. For simplicity, we will first assume no wave vector change (), 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 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 of the position of the wave packet with respect to the first light pulse () of the interferometer, assuming :
After some algebra, the inertial phase shift is given by6).
One can rewrite Eq. (11) considering the action of any gravito-inertial fields on the atoms located at position , leading to the final expression12) in the next section.
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 .
In case A, the AI is fixed to the Earth frame , rotating with rotation rate with respect to the inertial reference frame (i-frame), (see Fig. 3). In case B, the AI is fixed to a rotating platform of rotation rate for convenience.
To give the phase shift for each case (A and B), we calculate the functions of Eq. (12). For these functions simply correspond to initial atomic position and velocity , respectively. To calculate higher order derivatives (), we use the classical equations of atomic motion given by
In order to calculate (), 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 to distinguish between both cases. Assuming for case A and for case B, one obtains
One interesting feature of Eq. (15) is that it takes into account both cases A and B simultaneously. Choosing or 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 -light-pulse AIs.
Considering the rotating Earth frame, one can define as the initial velocity of the atoms and acceleration
Calculating the inertial phase shift up to terms proportional to () 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 is only a function of constant acceleration , initial velocity , and constant rotation .
In Table 1, we extend the phase shift calculation up to terms proportional to for whatever the case is (A or B). In this case, cross terms between the gravity gradient , acceleration , and rotation appear in the phase shift expression.
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 dependences appear as a simple product between the inertial fields and a specific coefficient.
The physical meaning of 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 , by simply nulling the laser chirp in Table 1 and considering a two-pulse interferometer with .
3. ANALYSIS AND LINK WITH WELL-KNOWN AIS
In this section, we give a physical interpretation of the leading terms of the phase shift, which are directly related to the two-photon light pulse sequence of the interferometer through Eq. (6).
A. Impact of Coefficients and AI Symmetry
Assuming no inertial effects and focusing on the first 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 -light-pulse interferometer closed in momentum space (i.e., ) with , one finds out that . For , 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 separated by a free precession time . One can see that this interferometer is time antisymmetric, meaning that . This closure in momentum space can be expressed as
The coefficient of the phase shift is related to the closure in space–time of the atom interferometer. In the Ramsey–Raman interferometer (), 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 pulse interferometers. Considering a three-pulse interferometer, one has to solve for
For , one finds the well-known single-loop, Mach–Zehnder configuration [27,28] consisting in the pulse sequence . The interferometer is depicted in [Fig. 4(b)]. Considering two-photon Raman transitions, the first pulse puts the atom in a coherent superposition of ground and excited states and acts as a beam splitter by transferring momentum to the wave packet, making the transition to the excited state. Then, the two wave packets propagate freely during time , drifting apart with relative momentum . A mirror pulse is applied at time , interchanging the ground and excited states and reversing their relative momenta. Finally, the two wave packets drift back toward each other during time and a final beam splitter pulse is applied at time , 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 , any three-pulse AI closed in momentum and position space obeys
For , one finds the double-diffraction scheme using, for example, two-photon Raman transitions  as depicted in Fig. 4(c). In this configuration, two pairs of Raman beams simultaneously couple the initial ground state with zero momentum to the excited state with two symmetric momentum states, . The interferometer is time and space symmetric as . This spatial symmetry can be expressed for any -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 , 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 one finds the Ramsey–Bordé interferometer [29–32] and the double-diffraction Ramsey–Bordé interferometer () depicted in [Figs. 5(a) and 5(b)]. Finally, cancellation of 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 . 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 . In Table 2, we give the absolute values of the phase coefficient for the interferometers described above.
Considering the simple expression of given in Eq. (23), one could think of finding any interferometer configurations (i.e., any sets of vector ) for which the interferometer could be selectively sensitive to gravity, acceleration, the gravity gradient, or rotation by fixing the value of a particular coefficient to zero and solving23) in Section 6.
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 of the th laser pulse leading to
Consequently, coefficients are now time dependent but still related to the two-photon pulse sequence of the AI through a so-called Vandermonde matrix with time-dependent coefficients.
Considering a four-pulse double-loop interferometer , one can look for solutions in time of the laser-pulse sequence assuming . Applying Eq. (23) with respect to leads to
The detailed calculations are given in Appendix A considering two cases and assuming and in dimensionless unit.
In the first case, we assumed . If one refers to Table 1, in this case, the phase-dependent terms which scale quadratically with time are eliminated. This is interesting when one wants to measure gravity gradients or rotations. It then comes out that the pulse sequence is , leading to . 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 , thereby eliminating all phase terms scaling as . This case is interesting when one wants to increase the accuracy to acceleration measurement by eliminating corrections to the phase. We represent in Fig. 6 the pulse sequence corresponding to a nonidentical double-loop interferometer leading to as was found in the previous work of , 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 . 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 considering the recoil velocity term for an atom of atomic mass as7, we represent the space–time recoil interferometer paths and the classical midpoint position of the atoms in a Mach–Zehnder geometry with pulse sequence .
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 is26) in the absence of photon recoil, and the second term is the recoil phase shift contribution that can be expressed in the time separation between pulses as
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 , and the two last terms depend on the wave vector change 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 .
One can generalize the recoil phase shift to higher orders in the time separation between pulses by simply writing3, assuming for simplicity. Nevertheless, phase shift terms can be obtained by calculating and coefficients.
In Table 4, we give the terms for usual interferometers.
One can note that contrary to the Ramsey–Bordé–Chu AI  consisting of two pairs of pulses, where the second pair of 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 between pulses (i.e., ), although recoil-velocity-dependent terms appear as corrections to the phase in power of 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 -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 when one only looks at the contribution of the gravity gradient force.
Considering a atom interferometer (, , ), one finds a separation between the wave packets of . This separation distance is not a problem when using -temperature thermal cloud sources obtained from usual optical molasses. Nevertheless, this separation phase shift could become an issue for long-baseline AIs (typically ), where BEC-based AIs are required .
C. Total Phase Shift
The total phase shift of an -light-pulse AI is simply given by
One finds that1 and 3. One can now calculate the total phase shift of any -light-pulse AI for any case (, ) of Section 2.
5. APPLICATION TO THE MACH–ZEHNDER AI
As an application, we use our formulation to calculate the total phase shift of the well-known Mach–Zehnder AI. According to our formulation, the temporal symmetries of the interferometer implies ; ; . Considering Eq. (36), this leads to a simple expression of the phase shift in terms of the function:
As an application, we evaluate phase terms with parameters corresponding to our transportable cold atom gravimeter based on a two-photon Raman pulse sequence, as described in . The time between pulses is with initial velocity components , where correspond to the transverse velocity of the atoms and 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.
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 , the wave vector change is not considered () and gravity is defined as19], one has to consider the rotating Earth frame with initial velocity of the atoms with acceleration 16). Finally, in the work of Wolf and Tourrenc , who studied the effect of wave vector change (), one has to consider our formulation in the case of in an inertial frame in the absence of rotation. In all cases, we found all phase shifts to be consistent with our novel formulation.
6. ORIGINAL -LIGHT-PULSE AI SENSOR
Our simple formulation allows one to seek for original -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 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 in the rotating Earth frame (i.e., Case A, ) is36], to enhance the sensitivity to the recoil shift by a factor .
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  that may affect the interferometer sensitivity. However, for example, the gravity gradient effect, which scales cubically with time (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 -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 and . If one considers equal time between the two-photon laser pulses, and assuming , one finds the -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 .
In Table 6, we give the phase shift coefficient values of the interferometer.
One can verify that the total phase shift to the first order in 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 with . Considering, for simplicity, optical pulses equally spaced with time and assuming , 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 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.
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 . 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 .
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 , that coefficients in the leading terms of the phase shift up to 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 . 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.
APPENDIX A: CALCULATION OF SECTION 3
A. Derivation of Section 3.B
We assume a pulse AI with . Starting from Eq. (25), one finds
Assuming and and vector , one finds
The authors thank Marc Himbert (LCM-Cnam) and Michel Lefebvre (ONERA) for this fruitful collaboration between the two institutes.
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