Representation-free description of atom interferometers in time-dependent linear potentials

In this article we present a new representation-free formalism, which can significantly simplify the analysis of interferometers comprised of atoms moving in time-dependent linear potentials. We present a methodology for the construction of two pairs of time-dependent functions that, once determined, lead to two conditions for the closing of the interferometer, and determine the phase and the contrast of the resultant interference. Using this new formalism, we explore the dependency of the interferometer phase on the interferometer time T for different atom interferometers. By now, it is well established that light pulse atom interferometers of the type first demonstrated by Kasevich and Chu (1991 Phys. Rev. Lett. 67, 181–4; 1992 Appl. Phys. B 54, 321–32), henceforth referred to as Mach–Zehnder (MZ) atom interferometers, have a phase scaling as T2. A few years ago, McDonald et al (2014 Europhys. Lett. 105, 63001) have experimentally demonstrated a novel type of atom interferometer, referred to as the continuous-acceleration bloch (CAB) interferometer, where the phase reveals a mixed scaling which is governed by a combination of T2 and T3. Moreover, we have recently proposed a different type of atom interferometer (Zimmermann et al 2017 Appl. Phys. B 123, 102), referred to as the T3-interferometer, which has a pure T3 scaling, as demonstrated theoretically. Finally, we conclude that the CAB interferometer can be shown to be a hybrid of the standard MZ interferometer and the T3-interferometer.


Introduction
Atom interferometers [1][2][3] and in particular light pulse atom interferometers [4,5] were first demonstrated in 1991. Since then different technologies and interferometer schemes have emerged, driven by the demand of high-sensitivity devices for precision measurements. In this article we compare three atom interferometers sensitive to linear accelerations by means of a novel formalism, which provides us with an intuitive picture for the scaling of their particular interferometric phase shift with respect to the interferometer time T 7 .
Despite the already demonstrated exquisite sensitivity of an atom interferometer, there are various ways to push it even further. The interferometer sensitivity usually depends on the mass of the particles in the interferometer [64], so using heavier atoms (e.g. rubidium or cesium instead of sodium) is one straightforward way for improvement. The sensitivity of the interferometer with respect to a constant acceleration or rotation also depends on the area enclosed by the interferometer [65][66][67]. This is due to the fact that light pulses not only create a coherent superposition of electronic states, but also transfer momentum. As a consequence, one finds the corresponding phase contribution to scale as the enclosed space-time area. Therefore, there have been several demonstrations of large-area interferometers by using composite and other more exotic pulses inducing a large momentum transfer [68][69][70][71][72][73][74][75].
For an interferometer in the conventional Mach-Zehnder (MZ) configuration, for instance, the enclosed area scales as T 2 , where T is the total time of the interferometer. However, instead of increasing the effective momentum transferred by the pulses, there is an alternate way to improve the sensitivity of an interferometer by changing the interferometer scheme in order to achieve a different scaling of the phase with respect to T. McDonald et al [76] have demonstrated an interferometer measuring the gravitational acceleration with a combination of a T 2 and T 3 scaling, with the potential to achieve even higher-order scalings. Moreover, we have proposed an interferometer [77] with a pure T 3 scaling as acceleration sensor being sensitive to magnetic field gradients. Indeed, these three setups serve as motivation for this article: rederiving their scaling behavior by means of a representation-free description for interferometers in time-and state-dependent linear potentials.
Interestingly, also rotations and gravity gradients [78,79] can lead to phase contributions, which are proportional to T 3 . However, a discussion of these effects goes beyond the scope of this article.

An intuitive representation-free description of atom interferometers
In this article we develop a novel formalism based on the ideas and results presented in [80][81][82], valid for atoms moving in any time-and state-dependent linear potential. In order to determine the phase and contrast of an atom interferometer, the complete time evolution is usually split into pieces describing the time-evolution resulting from its basic building blocks: beam splitters, mirrors, and free propagation. This is generally the case within the path-integral approach [65], the ABCD formalism [83][84][85], and even in representation-free approaches based on operator methods [80,86].
Within a representation-free formalism the resulting product of the corresponding unitary operators may lead to very complicated expressions, owing to the rise of nested commutators within the calculation as the Hamiltonian governing the evolution of the atoms does usually not commute with itself at different times. In contrast, our formalism involves time-dependent functions instead of operators. The action of the different building blocks directly enter into these functions, which immediately determine the complete time-evolution. Additionally, in our formalism momentum and phase changes imprinted by laser pulses and other external potentials are both described in a coherent way.
Essentially, we present a methodology to construct only two pairs of time-dependent functions w which we associate with the upper and lower branch of the interferometer, respectively. Two closing conditions are determined solely by the difference δF(t)≡F u (t)−F l (t) and lead to general expressions for the phase and contrast of the interferometer in terms of δF(t), and δω(t)≡ω u (t)−ω l (t).

Outline
First, we use in section 2 a representation-free approach [80-82, 86, 87] and introduce beam splitters, mirrors, and free propagation in terms of unitary operators. We show how to combine these building blocks in order to obtain a general expression for the phase and contrast of the interference pattern. Next, in section 3 we present the details of our formalism describing atom interferometers exposed to time-and state-dependent linear potentials. Using the resulting schematic rules, we paint a new description of atom interferometers in section 4.
Here we briefly describe the three types of atom interferometers mentioned before: (a) the MZ interferometer, (b) the T 3 -interferometer, and (c) the continuous-acceleration bloch (CAB) interferometer, focusing on the different experimental realizations that lead to their particular T scaling. For each interferometer type, we use our formalism to calculate the T dependence of the phase of the interferometer, thereby showing the versatility and utility of our formalism. As a result of our analysis, we are also able to show that the CAB interferometer can be viewed as a hybrid between the MZ interferometer and the T 3 -interferometer, which we discuss in section 5 before concluding in section 6.
In order to keep our article focused on the essential ideas, we present detailed calculations in four appendices. In appendix A we derive effective Hamiltonians corresponding to the upper and lower branch of the atom interferometer. Next, we determine in appendix B the dynamics induced by these Hamiltonians and analyse in appendix C the contrast and phase shift resulting from the interference of the two interferometer branches. Finally, we discuss in appendix D the phase shift of the CAB interferometer in detail.

Atom interferometers
The basic working principle of an atom interferometer is to coherently split the atomic wave packet into several components that probe the environment-especially the action of certain potentials such as the gravitational one-and to finally recombine the different components. After recombination the interference signal contains information about the interactions which have occurred within the interferometer.
In order to describe the atoms in our interferometer, we introduce an effective two-level system consisting of the two orthogonal states ñ |1 and ñ |2 . Moreover, including the center-of-mass motion of the atoms in the state ñ |k (k=1, 2) described by y ñ | ( ) t k , we arrive at characterizing the entire atomic matter-wave at time t.
The three key steps in atom interferometry are (i) the initial preparation, (ii) the implementation of a certain interferometer sequence, and (iii) the final read-out. Typically, in step (i) only the state ñ |1 is initially populated and the read-out in step (iii) also measures the population of this state. The interferometer sequences discussed in this article consist of two beam splitter pulses being separated by a period of free propagation, which may be interrupted by a certain number of mirror pulses.

Building blocks of atom interferometers 2.1.1. Free propagation
During free propagation the atom experiences no transitions between the two states ñ |1 and ñ |2 . Therefore, the corresponding Hamiltonian being the identity operator in the subspace spanned by these states. Here m denotes the atomic mass, andr andp are the position and momentum operators of the atomic center-of-mass motion, respectively. Generally, the atom might probe two different potentials which we both assume to be linear in position and are determined by the time-dependent functions ( ) t F k and ω k (t) corresponding to the state ñ |k (k=1, 2). The term proportional to F k (t) may describe the interaction of the atom with a linear gravitational potential or the interaction of a magnetic dipole moment corresponding to the state ñ |k with a magnetic field having a constant gradient [54,77]. In the latter case the potentials (ˆ) V t r, k become state-dependent if the mean values of the magnetic dipole moments corresponding to the two internal states differ. Specific examples of F k (t) appear later in sections 3 and 4.
On the other hand, the term proportional to ω k (t) in equation (4) may include the energy of an internal atomic state and additional time-dependent contributions due to, for instance, the interaction of magneticsensitive states with a homogeneous magnetic field. However, in the interaction picture with respect to ω k (t) determined by the unitary operator , can be absorbed in the time-evolution of the wave function during the free propagation. Here t i denotes the initial time before applying the first beam splitter pulse. Hence, in this picture equation (2) for the Hamiltonian reads leading to the evolution operator between the times t a and t b . Here we have introduced the time evolution operatorˆ( ) E t t , kk b a corresponding to the Hamiltonian with k=1, 2.

Beam-splitter and mirror pulses
In this section we describe perfect beam-splitter and mirror pulses for an atom interferometer. As it is often done [80], we assume that these pulses are so short that the atomic center-of-mass motion during the pulses may be neglected. As a result, these pulses act instantaneously at a given time t j . Indeed, the interaction of an atom with a near resonant electromagnetic field may fulfill this requirement and thus offers an experimental realization for beam splitters (p 2-pulse) and mirrors (p-pulse). However, we note that the consequences of a finite pulse duration are discussed in [88,89]. In order to describe the atom-light interaction we include the effects of a possible momentum transfer ÿk and a laser phase f(t j ). Using the effective wave vector k and the effective phase depending on the laser frequency w L , we obtain [80,88,90,91] the unitary beam splitter operator . The operator  ( ) t j creates an equally weighted superposition of the two states ñ |1 and ñ |2 and is determined by the unitary operator = - Here the prefactor 1 2 in equation (10) ensures the unitarity of the operator  ( ) t j . A perfect mirror instead should create a perfect inversion of the populations of the states ñ |1 and ñ |2 . This case is described by the unitary operator with  ( ) L t j being defined by equation (11). It is important to emphasize that these ideal beam-splitter and mirror pulses imprint additional phases and lead to displacements in momentum of ±ÿk without distortion of the center-of-mass wave packet.

Interferometer sequence
We have now all the ingredients at hand to construct the atom interferometers discussed in detail in section 4. Indeed, each interferometer consists of a period of free evolution (t i , t 1 ), an opening beam splitter pulse applied at time t 1 , followed by a number of n−1 periods (t 1 , t 2 ), (t 2 , t 3 ), K, (t n−1 , t n ) of free evolution, being separated by n−2 mirror pulses applied at times t j ( j=2, K, n−1), and a closing beam splitter pulse applied at time t n , followed by a free evolution period (t n , t f ), as depicted in figure 1.
In order to describe such an interferometer sequence with instantaneous laser pulses, we use the building blocks presented by equations (7), (10), and (12), and obtain the total evolution operator , , of this interferometer with the initial time t i <t 1 and the final time t f >t n . Now we use the facts that the free evolution operator  , equation (7), is diagonal and the operator  , equation (12), is anti-diagonal in the ñ ñ {| | } 1 , 2 -basis. In this case, the operator describing the time evolution of the atom being prepared in ñ |1 and detected in ñ |1 , is determined by the sum of the evolution operators for the upper branch and for the lower branch of the interferometer with an even number of pulses. Here we have introduced the notation With equations (10) and (12) we recast equations (15) and (16) in terms of the operator  ( ) L t j as , , , , Similarly, for an odd number of pulses we obtain instead , ,  It is important to recall that the operatorsÛ u andÛ l are unitary by construction and describe the time evolution from t i to t f for the upper and lower branch of the interferometer, respectively.
Next, we relate these operatorsÛ u andÛ l to the observables such as the contrast and phase of the interferometer. Assuming an atom being prepared in the state ñ |1 corresponding to the initial state where y ñ | i describes the initial state of the center-of-mass motion, the probability P 11 (t f ) to observe atoms in the state ñ |1 after the closing beam splitter pulse (at t=t n ), as shown in figure 1, is thus given by Applying equation (14), we arrive at i l u i i of the operator productˆ † U U l u determined by the initial state y ñ | i of the center-of-mass motion. In the case that the atom is prepared in state ñ |1 , but measured in state ñ |2 , the probability reads Figure 1. A one-dimensional sketch of a space-time diagram of a general two-branch interferometer for a two-level atom having the states ñ |1 (dashed blue line) and ñ |2 (solid red line), and being exposed to a time-and state-dependent linear potential. The atom interacts with two beam splitter pulses at times t 1 and t n andn 2 mirror pulses at times t j ( = ¼ - 3. An intuitive picture of the interferometer

Interferometer contrast and phase
In the preceding section we have shown that the contrast C and the interferometer phase δf are both determined by the matrix element y y á ñ |ˆˆ| † U U i l u i , equation (26). The operatorÛ b , equations (19)- (22), describes the time evolution along the upper (b=u) and lower (b=l) branch of the interferometer. As shown in appendix A, this operator can be associated with the effective Hamiltonian determined by the time-dependent functions ω b (t) and F b (t), accordingly. In appendix B we obtain the timeevolution operatorˆ( ) U t t , b i in terms of these functions, and find in appendix C a general expression for the operator product Here Δf(t f ) is a phase factor and is the displacement operator with arguments P and R being determined in equation (29) by the time-dependent vectors Indeed, the vectors δp(t) and δr(t) correspond to the relative displacements in momentum and position between the two classical trajectories associated with the two branches of the interferometer, respectively [82].
In order to maximize the interferometer contrast independently of the initial state y ñ | i , we require that the operatorD in equation (34) is the identity operator, which leads us to the closing conditions of the interferometer, that is or the relative displacements in momentum and position between the two branches should vanish at the final time t f . In this case we speak of a closed interferometer.
Consequently, the interferometer phase We emphasize that the interferometer phase δf is independent of the initial state y ñ | i for a closed interferometer. Moreover, forF being time-independent, the phase contribution δf 1 , equation (39), is proportional to the enclosed area between the upper and lower branch of the interferometer in the space-time diagram, shown in figure 1. In the case that the two interferometer branches cross in space, it is the signed area which is relevant. Needless to say that the general result derived in appendix C agrees with the considerations presented in [81,82] which are valid for both Raman and Bragg diffraction. In the special case of a closed interferometer, equation (37) coincides with the result obtained within the semi-classical description [67].

Schematic rules
Now we are in the position to summarize our scheme in order to obtain the contrast and phase of the general interferometer presented in figure 1.
The key ingredient of our method is the construction of only two pairs of time-dependent functions corresponding to the upper and lower branch of the atom interferometer, accordingly, as depicted in figure 2. Here a dashed blue line indicates the state ñ |1 while a solid red line corresponds to the state ñ |2 . Since the initial state is assumed to be prepared in the state ñ |1 , equation (23), both branches start from the dashed line. The procedure to obtain the functions ω b (t) and F b (t) (b=u, l), as well as the contrast and phase of the interferometer consists of four steps.
Step 1: Each function ω b (t) is the sum of all contributions arising from position-independent potentials and consists of summands 8 and Φ(t j ) is defined by equation (9). The difference δω(t), equation (40), of the functions ω b (t) for the upper and lower branch determines the contribution δf 0 , equation (38), to the interferometer phase.
Step 2: Each function F b (t) is the sum of all contributions (see footnote 6) presented in figure 2 originating from (i) the free evolution between pulses applied at the times t=t j and t=t j+1 , that is P j j 2 1 (solid red line) for an atom in the state ñ |1 or ñ |2 , respectively, and (ii) the momentum kick ÿkδ(t−t j ) at the transition ñ  ñ | | 1 2 (change from dashed to solid line), or −ÿkδ(t−t j ) at the transition ñ  ñ | | 2 1 (change from solid to dashed line).
Step 4(a): If the interferometer is closed, the contrast C=1. Find the function¯( ) t F defined by equation (41), which determines together with δr(t) the contribution δf 1 , equation (39). The interferometer phase δf, equation (37), is the sum of δf 0 and δf 1 . 8 Here δ(t) denotes the Dirac delta function and P( , is the rectangular function of unit height on the interval (a, b) as defined in equation (43).
Step 4(b): If the interferometer is not closed, then the contrast C is given by equation (34) and depends on δr(t f ), δp(t f ), as well as the initial state y ñ | i . Its phase δf is derived in appendix C.1.

A comparison of three atom interferometers
In this section we apply our formalism introduced in section 3 to the MZ interferometer [4,5,54,65,80,84,88,92], the T 3 -interferometer [77], and the CAB interferometer [76]. We extend the analysis of these interferometers discussed briefly in [77]. As a result of our formalism, we present a simple explanation of the different scalings of the interferometer phase with respect to the total interferometer time T. In particular, we show that the CAB interferometer is a combination of the MZ and the T 3 -interferometer. Moreover, we discuss the different experimental techniques used to imprint forces, namely Raman transitions, magnetic field gradients, Bragg transitions, and Bloch oscillations.
In order to obtain a coherent description of these interferometers, we introduce the following family of functions vanishing outside the interval (a, b) with a<b and c≡(a+b)/2, which are based on the Dirac delta function δ(t).
• The rectangular function and F b (t) (b=u, l). Each function ω b (t) is a sum of all contributions arising from position-independent potentials and contains (see footnote 6) the scalar terms mΦ m (t)δ(t−t j ), as defined by equations (9) and (42). Moreover, each function F b (t) is a sum of (i) contributions P + ( | ) t t t F , k j j 1 describing the free evolution of the atom in the state ñ |k (for k=1, 2), where ñ |1 and ñ |2 correspond to a dashed blue and solid red line, and (ii) the momentum kicks +ÿkδ(t−t j ) and −ÿkδ(t−t j ) for the transition ñ  ñ | | 1 2 and ñ  ñ | | 2 1, respectively. Here j=1, K n, where n is the number of pulses.
• The triangular function t with the height (b−a)/2.
• The piece-wise quadratic function

MZ interferometer 4.1.1. Description
We briefly review the physics underlying the basic MZ atom interferometer, emphasizing the points salient to this article. In particular, we discuss the origin of the scaling of the interferometer phase with T 2 . A standard MZ type atom interferometer is depicted in figure 3(a). For this interferometer type we focus on the case of driving Raman transitions between two ground electronic states, labeled ñ |1 and ñ |2 . Atoms entering the interferometer have usually been prepared in the state ñ |1 . The atoms are subject to a beam-splitter pulse applied at time t 1 =0 which creates a coherent superposition of states ñ |1 , shown by a blue dashed curve in figure 3(a), and ñ |2 , depicted by a red solid curve. Subsequently, the atoms evolve in the dark for a time T/2. Then they experience a mirror pulse at t 2 =T/2, which causes the states to 'flip', that is ñ  ñ | | 1 2 and ñ  ñ | | 2 1 . The atoms evolve again in the dark for another time T/2. Finally, the two arms of the interferometer are recombined with a final beam splitter pulse at t 3 =T.
In the case of the MZ interferometer both states ñ |1 and ñ |2 are exposed to the same linear time-independent potential leading to |1 and ñ |2 . The atom interacts with two short laser π/2-pulses (at t=0 and t=T) and a π-pulse (at t=T/2), where T is the total interferometer time. During each pulse the atom absorbs (emits) the momentum ÿk in combination with the transition (55), and (c) position δr(t), equation (56), as a function of time t.
for the functions F 1,2 (t) in equations (4), (6), and (8). This is the case for atoms being exposed to (i) the gravitational field [4,5] with the constant gravitational acceleration g, where F 0 =mg, and (ii) the magnetic field z z z 0 with constant gradient ∇ z B z and the unit vector e z in the z-direction 9 . The latter case leads to F 0 =μ z ∇ z B z e z , when the atom has the same z-component of the mean value of the magnetic dipole moment μ z =μ 1z =μ 2z with m m º á ñ |ˆ| k k k (k=1, 2) for the states ñ |1 and ñ |2 , as presented in [54]. As shown below, the relative displacement in momentum δp(t) between the two branches of the interferometer is piece-wise constant, figure 3(b), whereas the relative displacement in position δr(t) is piecewise linear in t, figure 3(c), leading to an interferometer phase scaling as T 2 .

Analysis of the MZ interferometer
According to Step 1 of our procedure (section 3.2), as well as to figures 2 and 3(a), we arrive at for the upper branch together with for the lower branch of the MZ interferometer determining the difference of ω u (t) and ω l (t). We make use of equations (38) and (42), and arrive at describing the phase contribution arising from the position-independent potentials. Next, we follow Step 2 of our analysis and identify according to figures 2 and 3(a) the functions for the upper and for the lower branch of the MZ interferometer having the difference Next, we check the closing conditions for this interferometer. The difference δF(t), equation (54), gives rise to the time-dependent vectors for the relative displacement in momentum shown in figure 3(b), and in position  (55) and (56), we immediately prove that 9 Here we use the notation  = = ¶ ¶ ( ) for the derivative of the z-component of the magnetic field B=B(r) along the z-direction at the origin r=0. This derivative is assumed to be small compared to B 0 , such that  where L is the total length of the interferometer. Moreover, we note that the form of the magnetic field given by equation (47) is an approximate one. Indeed, according to the Maxwell equation  = · B 0, which is valid everywhere, a non-zero value of ∇ z B z induces non-zero values of ∇ x B x and ∇ y B y , such that ∇ x B x +∇ y B y =−∇ z B z , where B x and B y are the components of B along the x-and y-axis. However, in the limit of   | | L B B z z 0 the magnetic field B given by equation (47) is approximately directed along the z-axis.
for t>T, that is the MZ interferometer is closed. Thus, the interferometer contrast C=1 and the interferometer phase is defined by equation (37), where δf 0 is given by equation (51).
In order to determine δf 1 , we consider the average (52), and F l (t), equation (53). According to equation (39) and by using equation (56) we arrive at the second contribution to the interferometer phase In the case of the gravitational field F 0 =mg with the gravitational acceleration g, we obtain for T=2T 1 the familiar result kgT 1 2 [4,5] for the phase δf 1 , where T 1 is the time between the first and second laser pulse. Moreover, the combination of a gravitational field and a magnetic field of constant gradient ∇ z B z yields F 0 =mg+μ z ∇ z B z e z , and we obtain the result of [54].

T 3 -interferometer 4.2.1. Description
We now move to the T 3 -interferometer [77] having an interferometer phase scaling purely as T 3 .
The key idea of this interferometer is to apply constant linear potentials with different magnitude to each of the two states ñ |1 and ñ |2 . For this purpose we have suggested [77] to use two atomic states ñ |1 and ñ |2 having different mean values of the magnetic dipole moment m 1,2 and expose them to the magnetic field B, equation (47) The T 3 -interferometer, as depicted in figure 4(a), consists of two beam-splitter pulses at the times t 1 =0 and t 4 =T, while the two mirror pulses act at the times t 2 =T/4 and t 3 =3T/4, where T is again the total interferometer time. The two mirror pulses are necessary to close the interferometer. Moreover, since copropagating Raman pulses are used as beam-splitter and mirror pulses, we assume that there is no transfer of photon momentum to the atoms, that is we set k=0.
As shown in figure 4(b), the relative displacement in momentum δp(t) between the two arms of the interferometer is now piece-wise linear in t, whereas the relative displacement in position δr(t) presented in figure 4(c) is piece-wise quadratic in t, leading to a total interferometer phase scaling as T 3 .

Analysis of the T 3 -interferometer
According to Step 1 of our procedure (section 3.2) and to figures 2 and 4(a), we obtain the functions for the lower branch of the T 3 -interferometer, which determine the difference By using equation (38), we obtain the first contribution to the interferometer phase being determined by the position-independent potentials, where the last term −π is a result of the even number of pulses used in this interferometer. Next, we follow Step 2 of our procedure and obtain according to figures 2 and 4(a) the functions  Figure 4. (a) Space-time diagram of the T 3 -interferometer for an atom having two states ñ |1 and ñ |2 and interacting with two short laser π/2-pulses (at t=0 and t=T) and two π-pulses (at t=T/4 and t=3T/4), with T being the total interferometer time. We assume that during the light pulses there is no transfer of photon momentum to the atoms. Instead, the two states ñ |1 (dashed blue line) and ñ |2 (solid red line) are exposed to two different linear potentials. The z-component of the relative displacements in (b) momentum δp(t), equation (73), and (c) position δr(t), equation (74), for the T 3 -interferometer as a function of the time t. The area enclosed by the two interferometer branches in the space-time diagram is the same as the one of the dashed triangle.
for the upper branch, together with for the lower branch of the T 3 -interferometer, leading to the difference , , 72 where we have used equations (62) and (63). Next, we verify the closing conditions given by equations (35) and (36). Indeed, by applying equations (31) and (32) we obtain the relative displacement in momentum shown in figure 4(b), and in position presented in figure 4(c). As evident from figures 4(b) and (c), and equations (73) and (74), we obtain for t>T, that is the T 3 -interferometer depicted in figure 4(a) is closed. Hence, its contrast C=1 and its phase df df df = + ( ) 76 0 1 is defined by equation (37), where δf 0 is given by equation (69). Now, we determine the phase contribution δf 1 and thus consider the average (70), and F l (t), equation (71), where we have used equations (62) and (63). According to equation (39) together with δr(t), equation (74), we obtain the phase contribution , and ΔF=m(a 2 −a 1 ) e z , where a k e z (k=1, 2) denotes the acceleration of the atom in the state ñ |k and T=4T 1 , with T 1 being the time between the first two laser pulses.
Moreover, it is interesting to note that the functions δF(t), equation (72), δp(t), equation (73), and δr(t), equation (74), are identical to the ones presented in the context of a Stern-Gerlach interferometer [93][94][95], where the time-dependence of these functions is not induced by applying mirror pulses, but by changing the direction of the magnetic field gradient at the time T/4 and 3T/4. In this scheme [93][94][95] only the case F 0 =0 has been considered, and thus the cubic scaling of the interferometer phase did not occur in the analysis of the Stern-Gerlach interferometer.

CAB interferometer 4.3.1. Description
As pointed out in section 4.1, it is the fact that the relative displacement in position δr(t) is piece-wise linear in t that leads to a T 2 scaling in a standard MZ interferometer. Motivated by this argument, McDonald et al [76] suggested and implemented an interferometer in which the relative displacement δr(t) between the two branches is piece-wise proportional to α 0 t 2 +β 0 t. Under these conditions, the interferometer phase scales as α 1 T 3 +β 1 T 2 with α i and β i (i=0, 1) being constant. The interferometer scheme is similar to the MZ style interferometer, that is it employs the same pulse sequence with beam splitter pulses at t 1 =0 and t 3 =T, separated by a mirror pulse at time t 2 =T/2. However, during the free evolution between the pulses, the state ñ |1 is exposed to a time-dependent linear potential.
In this work [76], the beam splitter and mirror pulses are created by Bragg diffraction resulting in a momentum transfer ±nÿk 1 , where n is the diffractive order and ÿk 1 is the momentum transfer for a first-order imprinted by the Bloch lattice could be reinterpreted as the action of an effective linear potential during the time interval (0, T/4), where n b is the number of the Bloch oscillations imprinted during the time T b . Subsequently, as indicated by the arrows in figure 5(a), the direction of the acceleration Δa induced by the Bloch lattice is inverted and the sequence is repeated for the interval (T/4, T/2). The lattice is moving sufficiently fast so that the atoms in the momentum state labeled by ñ |2 experience only a time-averaged potential from the Bloch lattice which imparts no additional acceleration. In the following we will not consider the additional phase shift imprinted by this potential.
At time t=T/2 a Bragg π-pulse acts as mirror pulse and the same acceleration sequence is now applied to the other branch of the interferometer. A final p 2-pulse closes the interferometer.
As a result, atoms in the state ñ |2 only experience a constant (background) linear potential described by whereas the time-dependent linear potential experienced by atoms in the state ñ |1 is determined by where Here we neglect the loading time T r and the deceleration time T f . Figure 5. (a) Space-time diagram of the CAB interferometer for an atom having the states ñ |1 and ñ |2 , and interacting with two short laser π/2-pulses (at t=0 and t=T) and a π-pulse (at t=T/2). The state ñ |1 (blue dashed line) is accelerated by a Bloch lattice, where the arrows indicate the direction of acceleration. However, the state ñ |2 (solid red line) is exposed to a constant linear potential. The z-component of the relative displacements in (b) momentum δp(t), equation (88), and (c) position δr(t), equation (89), for the CAB interferometer as a function of the time t.

Analysis of the CAB interferometer
According to Step 1 of our procedure (section 3.2) and figures 2 and 5(a), we obtain the time-dependent functions ω u (t), equation (48), for the upper path and ω l (t), equation (49), for the lower path identical to the ones of the MZ interferometer discussed in section 4.1 as the same pulse sequence is used. Thus, we also arrive at the same phase contribution for the CAB interferometer arising from the position-independent potentials. Next, we follow Step 2 of our procedure and according to figures 2 and 5(a), we identify the functions for the upper branch together with for the lower branch of the CAB interferometer, where F 2 and F 1 (t) are given by equations (81) and (82), respectively. The difference of F u (t) and F l (t) yields 10 We observe that δF(t) is a combination of the corresponding expressions for the MZ interferometer (section 4.1), equation (54), and the T 3 -interferometer (section 4.2), equation (72).
In the next step, we check the closing conditions equations (35) and (36) for the CAB interferometer. According to equations (31) and (32), we obtain the time-dependent vectors for the relative displacement in momentum shown in figure 5(b) and in position presented in figure 5(c). As a result of equations (57) and (75), we immediately obtain for t>T, resulting from the fact that both the MZ and the T 3 -interferometer are closed. Hence the CAB interferometer is closed as well. Thus, its contrast C=1 and its phase is defined by equation (37), where δf 0 is given by equation (84).
In order to determine the phase δf 1 we consider the average of F u (t), equation (85), and F l (t), equation (86). In contrast to the average functionsF for the MZ interferometer, equation (59), and the T 3 -interferometer, equation (77), which are constant in the interval 0<t<T, due to the time-dependence of F 1 (t), also the average¯( ) t F is now time-dependent during the interferometer sequence. However, as shown in appendix D, due to the symmetries of δr(t), equation (89), and¯( ) t F , only the constant term F 0 in F 2 , equation (81), and F 1 (t), equation (82), will contribute to the phase shift δf 1 arising from the linear potentials. According to equation (39) we thus obtain the second contribution 10 Here we use , and P( | ) b a b , will not contribute to any integral independent of their exact value.
to equation (91), which is a combination of the corresponding phases for the MZ interferometer, equation (60), and the T 3 -interferometer, equation (78).
In the case of T=2T 1 , F 0 =ma, k=2n k 1 , where T 1 denotes the time between the first two pulses and Δ F is defined by equation (83) with τ 1 ≡T b /n b , this expression agrees with the one presented in [76].
It is remarkable, that even in this special case of a time-dependent average function¯( ) t F , equation (92), the phase δf 1 is again proportional to the enclosed area between the two branches in the space-time diagram of the CAB interferometer.

Discussion
Section 3 represents one of the main features of this article-a methodology to analyze an atom interferometer consisting of two branches, assuming the atoms are exposed to linear potentials. More specifically, we formulated the rules for constructing two pairs of time-dependent functions corresponding to the upper and lower branch of the interferometer, which depend on the pulse protocol and the (possible state-dependent) potentials the atoms move in. Once the potentials and the pulse sequence are specified, the functions δω(t)=ω u (t)−ω l (t), δF(t)=F u (t)−F l (t), and The function δF(t) defines the relative displacements in momentum δp(t) and position δr(t) of the two interferometer branches. Thus, in order to fulfill the closing conditions given by equations (31) and (32), an experimental control of the physical quantities determining δF(t) is necessary. Additionally, it is important to emphasize that the closing conditions are independent of the functions δω(t) and¯( ) t F , and hence the interferometer can be used to measure the physical quantities governing these functions. Under the condition that the interferometer is closed, its contrast equals unity and the functions δω(t),¯( ) t F , and δr(t) are used to determine the phase, equations (37), (38), and (39), of the interferometer, which is in this case independent of the initial state. The power of our methodology is shown in section 4.

Review of the three atom interferometers
In section 4, we presented the analysis of three interferometers: (a) the MZ interferometer, (b) the T 3 -interferometer, and (c) the CAB interferometer. For all three cases, we derived the expression for δF(t) given by equations (54), (72) and (87), respectively, and used them to show that the interferometers are all closed. Closing occurs when both the relative displacement in momentum δp(t) and position δr(t) between the two branches of the interferometer vanishes for t>T, as demonstrated by equations (57), (75), and (90), respectively.
Of greater significance are the phases of these interferometers. In all three cases, the interferometer phase is written as the sum of two contributions: the phase δf 0 determined by the laser phases and the positionindependent potentials, and the phase δf 1 due to the motion of the atoms in a linear potential.
We first focus on the expressions for the phase δf 0 . For each type of interferometer this phase is given by equation (51), (69), and (84), accordingly, and is reproduced here again for convenience: where a superscript indicates the corresponding interferometer. It is clear that the contribution for the CAB interferometer is identical to the one of the MZ interferometer, being a discrete second-order derivative of the function Φ(T), that is as T→ 0. However, for the T 3 -interferometer we obtain instead a discrete third-order derivative of Φ(T), that is as T→ 0. It is interesting to note that the scaling with respect to T, and even the absolute value of the coefficients in equations (97) and (98) agree with the values of the integrals (61) and (79) determining the phase δf 1 for the MZ and the T 3 -interferometer, respectively. We now turn to a discussion of the contribution δf 1 of each interferometer, equations (60), (78), and (93), respectively, which we reproduce here for the ease of comparison: By comparing equations (99), (100) and (101), we immediately see that the CAB interferometer can be thought of as a hybrid between the standard MZ interferometer and the T 3 -interferometer. The phase for the CAB interferometer is the sum of a contribution that scales as T 2 , which is identical to the MZ T 2 -phase, and a contribution that scales as T 3 , which is identical to the T 3 -term of the T 3 -interferometer. However, it is important to bear in mind that the force difference DF is caused by the application of a magnetic field gradient in the case of the T 3 -interferometer, whereas it is generated by a Bloch lattice for the CAB interferometer.

Interpretation of the interferometer phase
Finally, we turn to an interpretation of the phase of the interferometer induced by the linear potentials. Examining equation (39), we observe that the contribution δf 1 is the integral of the product d ( ) ( ) t t F r for t between t i and t f . In the case of¯( ) t F being time-independent for t i < t < t f , as in the case of the MZ interferometer and the T 3 -interferometer, this term is proportional to the enclosed area between the two branches in the spacetime diagram of the interferometer displayed in figures 3(a) and 4 (a). However, even in the special case of the CAB interferometer where¯( ) t F is time-dependent, the phase contribution δf 1 is again proportional to the enclosed area of the interferometer in the space-time diagram, as discussed in section 4.3.2 and shown in figure 5(a).
The connection between the phase and the area enclosed by the interferometer in a space-time diagram was also analyzed within the semi-classical approach [67]. Our results are derived using only knowledge of the functions δF(t), δω(t), and¯( ) t F and are completely consistent with the ones obtained in this reference. Moreover, it is important to highlight that it is the signed area, which is relevant for the calculation of the interferometer phase. For example in the Butterfly geometry [86] the signed area vanishes, and the resulting interferometer phase becomes insensitive toF, assumingF is time-independent. Hence, this device might be used to probe the influence of higher-order effects such as gravity gradients.
More generally, we can consider an interferometer having the pulse protocol p p --¼--p p 2 2 , where the first and the last two pulses are each separated by the time T 1 and all other pulses have a separation time of 2T 1 . In the case that the force difference δF is only determined by the momentum transfer ±ÿk induced by the laser pulses, the dependence of the interferometer phase onF is determined by the parity of the number of pulses. For an odd number of pulses the interferometer phase depends onF, whereas for an even number of pulses it is insensitive to a constant mean forceF.
Intuitively, one might expect that the interferometer phase always scales with the enclosed area for a closed interferometer and arbitrary time-dependent linear potentials. However, since the closing conditions of the interferometer are only determined by δF(t), interferometers which are closed for a time-independent functionF will also be closed in the case that¯( ) t F is time-dependent. Therefore, the interferometer phase will in general not scale as the enclosed area. Nevertheless, it is interesting to ask the question which information about the timedependent potentials determining¯( ) t F will be obtained by different interferometer schemes.

Limitations of our approach
In principle, we can consider within our approach additional effects such as finite pulse durations, gravity gradients, and rotations. However, in the case of finite pulse durations, both the pulse profile and the atomic motion during the atom-light interaction have to be taken into account, as discussed in [88,89]. As a result, equations (10) and (12) for the beam-splitter and mirror operators would have to be modified: (i) the mirror operator, equation (12), will not have anti-diagonal form and instead contains also non-vanishing diagonal elements. This corresponds to the case that after each pulse new branches appear and it is impossible to describe the interferometer only based on two branches, as presented in our article. (ii) It is not any more possible to interpret the action of the pulses as being induced by effective linear potentials. Indeed, in addition to a phase shift and a displacement of the atomic center-of-mass wave packet in momentum as it is the case for a perfect beam-splitter or mirror, the resulting complicated potentials also lead to a state-dependent distortion of the center-of-mass wave packet. The additional interferometer branches give rise to interferometer outputs depending on the initial state of the atomic center-of-mass motion as well as the parameters of the pulse. We emphasize that in most cases the resulting equations for an actual experimental setup can only be solved numerically.
Moreover, we point out that the representation-free approach presented in [86] valid for state-independent gravity gradients and rotations has already been generalized in [82] to state-dependent quadratic potentials. In the future it would be of interest to consider these effects within our simplified treatment.

Conclusion
In conclusion, we have considered atom interferometers exposed to time-and state-dependent linear potentials. We have studied the scaling of the output phase with T, the total interferometer time, for three types of atom interferometers-the MZ, the T 3 -, and the CAB interferometer. To aid in the analysis, we have first developed a new formalism based simply on functions that were derived from the knowledge of the potentials in which the atoms move and the pulse sequence used for the interferometer. Our formalism leads directly to expressions for closing conditions in both momentum and position, which allow to immediately determine if an interferometer is closed. Additionally, the phase and contrast can also be obtained. We used this method to determine expressions for the phase of the three types of interferometers mentioned. In particular, we derived the wellknown result that the phase of the MZ interferometer scales as T 2 . Our results also confirm the ones presented in [77]. Here the phase for an interferometer using magnetically sensitive states with different mean values of the magnetic dipole moment and being exposed to an applied magnetic field gradient scales as T 3 . Finally, we showed that the recently demonstrated CAB interferometer [76] is a hybrid between the standard MZ interferometer and the T 3 -interferometer.
where δ(t) denotes the Dirac delta function and Now we are in the position to consider the time-ordered product of the unitary operators  L andÊ kk , which determines the operatorsÛ u andÛ l .
A.1. Effective Hamiltonians for an even number of pulses In the case of an even number of pulses the operatorsÛ u andÛ l are given by equations (19) and (20). Using equations (7), (8), (11) and (A.1), we arrive at the Hamiltonians for the upper branch, and  with an infinitesimally small positive time t ε and j=1, K, n.

A.2. Effective Hamiltonians for an odd number of pulses
In the case of an odd number of pulses we arrive at a slightly modified result, namely with F(t) being a time-dependent vector and ω(t) a time-dependent scalar function. We represent the corresponding time evolution operator as a product [82,83,96] of the exponent of the phase factor f(t), the displacement operator depending on the displacements in position R(t) and momentum P(t), and the evolution operator for a free particle. By following the approach presented in [82], we derive the equations for the time-dependent functions R(t), P(t) and f(t) resulting from the Schrödinger equation  where a dot denotes the derivative with respect to t and we have used the identity The relation [97] = +  As a result, by using equations (B.5), (B.7), (B.10), and (B.13), we arrive at the relation^^ for the unknown functions f(t), R(t), and P(t).
By comparing the coefficients in front of the operatorsp andr, as well as the remaining non-operator valued contribution, we obtain Note that equations (B.15) and (B.16) correspond to the classical equations of motion for a particle of mass m having the momentum P(t) and position R(t) and being exposed to the time-dependent force F(t).
Using the initial conditions  Here F b (t) and ω b (t) are the time-dependent functions for the upper (b=u) and lower (b=l) branch of the interferometer, as discussed in section 3. In order to evaluateˆ † U U l u we again follow the approach presented in [82].   In the next step, we use the identity  Indeed, δr(t) is shown in figure 5(c) and consists of the functions L( | ) t T 0, , defined by equation (44), and W( | ) t T 0, , defined by equation (45), which are even with respect to the point t=T/2. Moreover, as depicted in figure  for the second contribution to equation (91).