Quantum-clock interferometry has been suggested as a quantum probe to test the universality of free fall and the universality of gravitational redshift. In typical experimental schemes, it seems advantageous to employ Doppler-free E1–M1 transitions which have so far been investigated in quantum gases at rest. Here, we consider the fully quantized atomic degrees of freedom and study the interplay of the quantum center-of-mass (COM)—that can become delocalized—together with the internal clock transitions. In particular, we derive a model for finite-time E1–M1 transitions with atomic intern–extern coupling and arbitrary position-dependent laser intensities. We further provide generalizations to the ideal expressions for perturbed recoilless clock pulses. Finally, we show, at the example of a Gaussian laser beam, that the proposed quantum-clock interferometers are stable against perturbations from varying optical fields for a sufficiently small quantum delocalization of the atomic COM.

## I. INTRODUCTION

Light-pulse atom interferometry (LPAI) has demonstrated its versatility in a myriad of applications: Starting from measuring gravitational acceleration^{1–3} and rotation^{4} to field applications^{5–8} and mobile gravimetry,^{6} the measurement of Newton's gravitational constant^{9} as well as the so-far most accurate determination of the fine structure constant.^{10,11} In the last decade, there have been proposals for mid-band gravitational wave detection,^{12–14} complementary to LIGO/VIRGO and LISA, and recently, construction has started on first prototypes, which might be sensitive to ultra-light dark matter signals^{15–17} and serve as testbeds for gravitational wave antennas^{18–20} based on atom interferometry.

These advancements have paved the way to perform tests on the fundamental physical principles underlying today's best physical theories with high precision atomic sensors.^{21–24} On the other hand, ever-increasing precision goals require an upscaling of the interferometers' spacetime areas. For that reason, several very-large baseline projects are currently being planned globally, hoping to reach the kilometer scale: AION-km in the UK,^{25} MAGIS-km in the USA,^{18} MIGA/ELGAR in Europe,^{26,27} and ZAIGA in China.^{28}

A side beneficiary of these endeavors will be new long baseline tests of the Einstein equivalence principle, encapsulated in its three pillars^{29} consisting of local Lorentz invariance, the universality of free fall (UFF), and local position invariance, which, in turn, contains the universality of gravitational redshift (UGR) and universality of clock rates (UCR). Together, these principles form the backbone of general relativity.^{30,31} All aspects of the equivalence principle have proven to be extremely resilient to experimental challenges over an extremely large regime ranging from the microscopic to the cosmic scale.^{32–44}

UFF, in particular, has been tested via LPAI by comparison of the free fall rates of different atomic isotopes and species^{24,42,45} as well as for different internal states^{46} of the same atomic species. However, LPAI using quantum clocks as initial states has been shown^{47} to be insensitive to UGR violations in a linear gravitational field without additional internal transitions during^{48–51} the interferometer. Recently, two different LPAI schemes were introduced for UGR and UFF tests by Roura^{49} and Ufrecht *et al.*^{52} In contrast to the original proposals by Zych *et al.*^{53} and Sinha *et al.*^{54} to detect general relativistic time dilation by the interference of quantum clocks in a gravitational field, both proposals are predicated on the essential step of initializing the atomic clock inside the interferometer in order to unequivocally isolate such a signal. Without this crucial step, one is stuck with the no-go result.^{47} Therefore, the scheme of Roura^{49} needs a superposition of internal states to gain UGR sensitivity (and being insensitive to UFF violations); the alternative approach of Ufrecht *et al.*^{52} does not require superpositions of internal states (as seen by the laboratory frame). In turn, it becomes sensitive to both UGR and UFF violations. Similarly, other proposals^{29,51} can test different aspects of local position invariance like UCR.

Ideally, one would like to initialize an atomic clock inside the interferometer without disturbing the center-of-mass (COM) motion of the atomic test masses, which serve as inertial reference. Hence, recoilless internal transitions are strongly beneficial or might even be necessary since they can ease the experimental constraints and the implementation significantly. In this study, we examine recoilless transitions implemented via two-photon E1–M1 couplings, i.e., two-photon transitions consisting of one electric dipole (E1) and one magnetic dipole (M1) transition. This type of two-photon process has previously been investigated for Doppler-free two-photon spectroscopy^{55,56} and for the application in optical atomic vapor clocks^{57,58} without COM motion. In contrast to these previous studies, we will consider the full quantum nature of all atomic degrees of freedom—internal and COM. Due to the quantized nature of the COM degrees of freedom, one would *a priori* expect that the LPAI phase shift suffers from the corresponding delocalizing light–matter interaction.^{59} In particular, we find after incorporating COM motion that additional branch-dependent phases as well as momentum kicks, and thus branches, appear when considering realistic spatial laser profiles. The effect of phase shifts due to finite-time pulses with Doppler-shifted detunings for different classical Rabi frequencies has been investigated by Gillot *et al.*^{60} for Mach–Zehnder-type atom interferometers. Here, in contrast, the effects arise due to position-dependent Rabi frequencies in addition to the finite pulse time and structured laser beams when considering the quantized atomic COM. However, we show that the protocols of Roura^{49} and Ufrecht *et al.*^{52} are resilient to leading order effects in the induced COM spread when compared to the interferometer size. Nonetheless, our results can serve as a guide when such or similar corrections need to be accounted for or modeled in future high precision experiments.

### A. Overview and structure

Our article is structured as follows: In Sec. II, we will recapitulate the two interferometer schemes presented by Roura^{49} and Ufrecht *et al.*^{52} and put them into the context of the dynamical mass energy of composite particles. In Sec. III, we will introduce an idealized model for E1–M1 transitions using plane waves for the electromagnetic field, as an intermediate step but often serving as the foundation in the literature,^{55,57,58,61} and that takes into account the quantized atomic COM and finite pulse times. The internal structure of the atom will be described by a three-level system that can be reduced to an effective two-level system using adiabatic elimination. To achieve the absorption of two counter-propagating photons, a specific polarization scheme is needed.^{57,58} We will verify explicitly that standard Rabi oscillations are recovered to lowest order, canceling any quantum COM delocalization effects. In Sec. IV, we will extend these results for E1–M1 transitions by taking into account position-dependent laser intensities. The generalized *π*- and $ \pi / 2$-pulse operators will be obtained in Sec. IV A for the experimentally relevant case of a Gaussian laser beam to lowest order. In Sec. V, we will come back to the two interferometer schemes^{49,52} and analyze the implications due to the finite pulse times and position-dependent Rabi frequencies, in particular their impact on the phase and visibility of the interferometers. We conclude with a summary, discussion, and contextualization of our results in Sec. VI.

## II. UGR AND UFF TESTS WITH QUANTUM CLOCK INTERFEROMETRY

Here, we will briefly review the two interferometer schemes^{49,52} employing quantum clocks to test UGR and UFF. In the following, we will denote the scheme proposed by Roura^{49} as scheme (A) and the one proposed by Ufrecht *et al.*^{52} as scheme (B). Before starting this discussion, we will introduce the relevant aspects of the dynamical mass energy (or mass defect) of atoms, which is the underlying connection to test the UGR and UFF in an interferometer with quantum clocks. We note that our introduction only serves as a sketch of the ingredients necessary for incorporating a description of dynamical mass energy perturbatively into atoms.

While Einstein's mass energy equivalence $ E = M c 2$ has been known for more than 100 years now, its impact on quantum interference due to the possibility of obtaining which-path information for composite particles with time-evolving internal structure has only recently been highlighted in the works of Zych *et al.*^{53} and Sinha *et al.*^{54} How dynamical mass energy manifests in a Mach–Zehnder atom interferometer was first sketched by Giulini^{48} in the context of the redshift debate.^{62–67} Based on these initial considerations, significant progress has been made. For a review of these initial discussions and proposed experiments beyond the ones discussed here,^{49,52} see, e.g., the works of Pikovski *et al.*^{68} or Di Pumpo *et al.*^{51} and references therein.

However, to the authors' knowledge, dynamical mass energy itself was already discussed in the works of Sebastian^{69,70} on semi-relativistic models for composite systems interacting with a radiation field. There, the author indicates that the appearance of these terms (including dynamical mass energy) is intimately linked to relativistic corrections to the COM coordinates first derived by Osborn *et al.*^{71,72} and Krajcik *et al.*^{73} over 50 years ago. The last few years have seen significant efforts and discussions devoted to providing first principles derivations from atomic physics of dynamical mass energy. Specifically, we refer to the works of Sonnleitner *et al.*^{74} and Schwartz *et al.*^{75,76} for systems with quantized COM motion, respectively, without and with gravity. A field theoretical derivation has recently been performed by Aßmann *et al.*^{77} Moreover, Perche *et al.*^{78,79} contain a discussion under which conditions and by which guiding principles, effective models for composite systems can be constructed in curved spacetime. Extensions examining the coupling of Dirac particles to gravitational backgrounds have recently also been discussed,^{78,80,81} yielding overall sensible but in the details slightly differing results in the weak-field limit. A general review discussing the issues and problems regarding such couplings of quantum matter to gravity is available in Giulini *et al.*^{82}

### A. A simple model for the dynamical mass energy of atoms

*M*moving in a weak gravitational field is prescribed

^{83}by the sum of the kinetic COM energy and its gravitational potential energy $ U ( R )$,

*M*of the particle corresponds to its rest mass here. Hence, only in going beyond this non-relativistic model, the dynamical nature of mass energy can become relevant.

^{68–70,75,76,84,85}and change the Hamiltonian. The most impactful change resulting from this is the insight that the total atomic mass is no longer just the sum of the rest masses of its constituent particles but contains a contribution from the internal Hamiltonian of the atom, as one would naively expect from mass energy equivalence. We incorporate this in our simple model by performing the replacement,

^{76}

^{47}

^{,}$ | | H \u0302 A | | / ( M c 2 ) \u2009 \u2243 \u2009 10 \u2212 11$. Thus, often we can assume the perturbative identification

^{47}$ M \u2212 1 ( 1 \u2212 H \u0302 A / ( M c 2 ) ) \u2243 M \u2212 1 ( 1 + H \u0302 A / ( M c 2 ) ) \u2212 1$ via the geometric series. Consequently, we can also replace $ M \u2212 1 \u21a6 M \u0302 \u2212 1$ in the terms in Eq. (1) describing the potential and kinetic energy. The overall Hamiltonian $ H \u0302 ( MD )$, including the mass defect, accordingly takes the following form:

*M*. In this form, the Hamiltonian directly embodies the equivalence of inertial and gravitational mass.

_{n}^{83}While we have omitted the coupling to external (electromagnetic) fields in all our considerations for simplicity, they are, in principle, instrumental to actually prepare and manipulate the atomic wave packet in experiments. These may be accounted for in an interaction Hamiltonian $ H \u0302 int$ added to Eq. (3) or Eq. (4), since the mass eigenstates are identical to the internal energy eigenstates except for an energy shift. The details of this interaction Hamiltonian can be quite complicated

^{74,76,77}when all corrections from the mass defect are included. However, to leading order, it consists of the standard electric or magnetic dipole transitions described by

*M*, such as the Röntgen term.

^{59}In conclusion, we arrive at the total model Hamiltonian (excluding higher order contributions for the electromagnetic field coupling),

While our introduction here can only serve as a sketch, motivated by mass energy equivalence, it turns out that the derivation of the intern–extern coupling can be made fairly rigorous,^{47,75–77,83,84,86} however, with serious gains in the theoretical complexity of the model depending on the setting as well as the starting point. Nevertheless, the basic premises and leading order results do not change significantly.

### B. Phase shift in a light-pulse atom interferometer

There are multiple methods available to calculate the phase shift in a LPAI. In simple cases, with quadratic Hamiltonians and for instantaneous beam splitter pulses, one can often rely on path-integral methods.^{87} However, path-integrals become quite unwieldy in case of non-quadratic systems as there are no or only few standard methods available for their solution.^{88} In these more involved cases, e.g., with multiple internal states and complicated external potentials involved, the Hamiltonian approach^{67,89,90} offers a more versatile toolbox. Moreover, phase-space methods^{91–93} are also available and sometimes helpful for interpretation.

^{29,67}

^{67}

^{,}$ O \u0302 21 = U \u0302 2 \u2020 U \u0302 1$ between the branches. The absolute value of this amplitude is the visibility $ V 21 = | \u27e8 \psi 0 | U \u0302 2 \u2020 U \u0302 1 | \psi 0 \u27e9 |$ of the interference signal, while the argument $ \Delta \varphi 21 = arg \u27e8 \psi 0 | U \u0302 2 \u2020 U \u0302 1 | \psi 0 \u27e9$ is the interferometer phase.

^{29,52,67}

In general, the situation in a realistic LPAI can be a bit more complex, and the overall signal $ I \varphi exit$ detected in an exit port results from the pair-wise interference of all paths through the interferometer contributing to the exit port. Practically, additional and often undesired paths can originate, e.g., from imperfect diffraction processes^{94,95} or perturbing potentials acting during the interferometer.

*m*directly leads to the overall exit port signal,

*relative path phases*weighted by the

*relative path visibilities*. In an (open) two-path interferometer, the sums terminate after two terms and are, thus, identical to Eq. (8).

### C. Interferometer phase, (classical) action, and proper time

^{47,51,67,87}and a subsequent non-relativistic expansion. The resulting expression is then quantized and introduced as governing action $S$ of an appropriate path integral for the particle. Afterward, one identifies the quantum mechanical phase

^{47}acquired along the trajectory via

^{87,88}being a valid approximation. This is due to the fact that only in the semi-classical limit, the dominant contributions to the path-integral come from the classical trajectories, resulting from solving the Euler–Lagrange equations for the (classical) Lagrangian.

^{88}Ultimately, this is what makes the identification between proper time and the action in Eq. (13) possible also for quantum particles but only in the semi-classical limit.

### D. UGR sensitive scheme (A)

^{49}shown in Fig. 1(a), initializes an atomic clock by a recoilless $ \pi / 2$-pulse so that the atoms that enter the interferometer in the ground state are in a 50:50 superposition of excited and ground state atoms after the clock initialization. Due to the atoms having a different mass $ M g , e$ in their respective internal ground and excited states, the Compton frequency $ \omega g , e$ becomes state-dependent. One can measure the frequency in the ground and excited state exit port between the two branches via the differential phase shift $ \Delta \varphi g , e$ and separate out the gravitational redshift by a double-differential measurement, i.e., calculating the phase difference $ \Delta \varphi \u2212$ between the excited and ground state exit port and performing two runs of the experiment with different initialization times

*T*

_{2}of the atomic clock,

*g*is the gravitational acceleration,

*k*is the wave number of the laser that drives the atoms onto the two branches,

_{p}*δT*is the separation time, and $ \Delta M = M e \u2212 M g$ is the mass difference due to the mass defect. Since the rest mass

*M*and the mean mass $ M \xaf$ are equivalent to our order of approximation, i.e., to order $ O ( c \u2212 2 )$,

^{47}we may identify the mean mass as

*M*.

### E. UGR and UFF sensitive scheme (B)

^{52}[cf. Fig. 1(b)] is sensitive to both the UGR and UFF. In contrast to scheme (A), it does not require a superposition of internal states. The sensitivity arises from the specific space–time geometry of the interferometer and a change of internal states so that the atoms are in the same state at equal times (in the laboratory frame). The total phase

*M*. This part of the total phase can be used for tests of UFF.

^{52}The proper time differences $ \Delta \tau n$ in each segment

*n*of the interferometer enter the phase proportional to the mass defect $ \Delta M$ such that it can be associated with the ticking rate of an atomic clock.

^{51}The $ \lambda \xb1$ indicates the internal state for each segment: $ \lambda \u2212 = \u2212 1$ for the ground state and $ \lambda + = + 1$ for the excited state. Since the sum $ \Delta \tau = \Delta \tau 1 + \Delta \tau 2 + \Delta \tau 3$ of the proper-time differences vanishes in this geometry, the proper-time difference in the middle segment can be written as $ \Delta \tau 2 = \u2212 ( \Delta \tau 1 + \Delta \tau 3 )$. Changing the internal state in the middle segment (associated with $ \Delta \tau 2$), the total phase becomes

### F. Common challenges

Both interferometer schemes presented earlier require the manipulation of the internal states during the interferometer sequence. While this manipulation can be achieved by (technically challenging) optical double-Raman diffraction^{96} in scheme (B), i.e., kicking the atoms and changing the internal states simultaneously, scheme (A) requires recoilless internal transitions. There are several reasons why one would like to avoid double-Raman diffraction: first of all, to drive this kind of transitions, one needs quite long laser pulses leading to finite pulse-time effects. Second, the single-photon detuning cannot be chosen arbitrarily large if one still wants to have significant Rabi frequencies. This constraint for the detunings leads to problems with spontaneous emission. Furthermore, double-Raman diffraction requires a high stability for the difference of the two laser frequencies during the pulse. Replacing the double-Raman diffraction by a momentum-transfer pulse, e.g., double-Bragg diffraction, a state-changing pulse could alleviate these issues. These recoilless transitions can be achieved by E1–M1 transitions where the atom absorbs two counter-propagating photons with equal frequency *ω* so that the total momentum kick caused by the two-photon transition vanishes. Such E1–M1 transitions were only investigated without (quantized) COM motion in the context of optical clocks.^{57,58,97} However, in atom interferometry, the COM motion plays a crucial role. Hence, its influence also needs to be included when modeling the pulses to account for possible corrections. This is the task of the rest of the manuscript.

## III. IDEALIZED MODEL FOR E1–M1 TRANSITIONS

In this section, we will derive an effective model for the finite-time E1–M1 transition processes during the LPAI schemes discussed in Sec. II for an atomic cloud in a gravitational potential, see Fig. 2. The cloud will be modeled as a fully first-quantized atom, including its quantized COM motion. In particular, this can be applied to general initial atomic wavepackets. We will assume, for now, that the electromagnetic fields of the laser beam are classical plane waves. We will explicitly show that, to the leading order, standard Rabi oscillations are recovered, since the COM dependence drops out for fields of constant intensity in space. The extension to realistic position-dependent laser intensities as well as finite pulse-time effects will be treated in Sec. IV.

### A. Model

*M*in a gravitational field along −

*Z*and via the dipole approximation, the Hamiltonian reads

**E**and

**B**, to be plane waves with frequency

*ω*for now. Note that we have neglected the mass defect, cf. Eq. (2), during the interaction with the laser since the pulse time

*t*is much smaller than the characteristic interferometer time

*T*. The internal-state dependent mass energy enters the phase via $ \Delta M c 2 \xb7 t / \u210f$ and $ \Delta M c 2 \xb7 T / \u210f$, respectively. The effects of the mass defect during the laser pulse compared to the effects during the rest of the interferometer sequence are, therefore, negligible. In particular, the $ O ( c \u2212 2 )$ correction of Eq. (3) is subdominant with respect to the dipolar interaction terms. Moreover, for the same reason, we have only retained the linear potential contribution from the gravitational potential energy and do not consider the higher order contributions due to gravity gradients and kinetic-energy to position couplings from Eq. (61). If necessary, they could be included perturbatively, similar to the optical potentials in Sec. IV.

^{98}via $ k L Z \u0302 ( t ) = k L Z \u0302 \u2212 k L g t 2 / 2$ will be compensated through chirping in the following.

### B. Adiabatic elimination

*δ*, the ancilla state gets populated by the electric dipole transition and depopulated by the magnetic dipole transition so fast that the ancilla state is only virtually populated, i.e., the probability of finding the atom in the state $ | a \u27e9$ is vanishingly small. To see this, we are forcing the atomic three-level system into a form where the ancilla state is separated from the other two by writing the Schrödinger equation as

*et al.*

^{99}for a treatment including weakly time-dependent detunings. We then obtain from Eq. (26) the differential equation

^{100,101}given by

*δ*, we can, thus, define adiabaticity parameters

^{101}Hence, we use a perturbative ansatz

^{102}otherwise, important terms of the form $ i d a g \xb7 E \u0302 i *$ and $ i \mu a e \xb7 B \u0302 i$ will be lost. Later, we will see that in a retro-reflective geometry and for the $ \sigma +$– $ \sigma \u2212$ polarization scheme, these terms lead to a doubling of the AC Stark shift.

### C. Doppler-free two-photon transitions

In order to obtain a Doppler-free interaction without momentum kicks, one has to eliminate the position-dependent terms, which can be done formally by setting the Rabi frequencies $ \Omega B 0 = \Omega E 1 = 0$ (or vice versa). This means, recalling Fig. 2 and the field configuration shown there, that the two-photon transition is driven by counter-propagating photons. Practically, this can be done by using a certain polarization scheme suppressing the unwanted single-photon transitions.^{55,57,58} To find the right polarization configuration, one has to apply the selection rules of single-photon dipole transitions. The selection rules for two-photon transitions can then be obtained by interpolating the sequential single-photon transitions.

#### 1. Selection rules and polarization scheme

In the following, we make use of the well-known dipole selection rules.^{103–109}

##### a. Electric dipole transitions

E1 transitions can only take place between two internal states with different parity, and the change of angular momentum has to be $ \Delta L = \xb1 1$.

##### b. Magnetic dipole transitions

M1 transitions can only take place between two internal states with the same parity. Therefore, the change of angular momentum has to be $ \Delta L = 0$.

However, in both cases (E1 and M1 transitions), the total angular momentum $ J = L + S$ has to change via $ \Delta J = 0 , \xb1 1$, while transitions from *J* = 0 to $ J \u2032 = 0$ are forbidden. Furthermore, conservation of angular momentum leads us to selection rules for the magnetic quantum number $M$, which changes depending on the polarization of the light: linearly polarized light does not change the magnetic quantum number, i.e., $ \Delta M = 0$, while positive (negative) circularly polarized light changes the magnetic quantum number via $ \Delta M = + 1$ ( $ \Delta M = \u2212 1$). Note that the distinction whether it is positive circular ( $ \sigma +$) or negative circular ( $ \sigma \u2212$) depends on the propagation direction and the quantization axis. Coming back to our setup displayed in Fig. 2, the selection rules for the change of angular momentum $ \Delta L$ are fulfilled since between $ | g \u27e9 = 1 S 0$ and $ | a \u27e9 = 3 P 1$ (the E1 transition) we have $ \Delta L = 1$ and between $ | a \u27e9 = 3 P 1$ and $ | e \u27e9 = 3 P 0$ (the M1 transition) we have $ \Delta L = 0$.

To suppress unwanted transitions, i.e., ensuring that the atom absorbs two counter-propagating photons, we use now a $ \sigma +$– $ \sigma \u2212$ scheme, where the two counter-propagating laser beams have positive (negative) circular polarization. The aforementioned selection rules together with this polarization scheme require $ d a g \xb7 E 0 \u2260 0$ while $ d a g \xb7 E 1 = 0$, and $ \mu a e \xb7 B 0 * = 0$ while $ \mu a e \xb7 B 1 * \u2260 0$, given the electric field satisfies $ E 0 \u221d \sigma +$ and $ E 1 \u221d \sigma \u2212$. Note that if the electric field has positive circular polarization, the corresponding magnetic field has negative circular polarization and vice versa.

*π*- or $ \pi / 2$-pulses. That is why finite pulse-time effects become important for E1–M1 transitions. Since the transition between $ | g \u27e9$ and $ | e \u27e9$ is forbidden for single-photon transitions, however, we can still neglect spontaneous emission.

*γ*as well as $ \omega AC ( + ) = ( | \Omega E 0 | 2 + | \Omega B 1 | 2 ) / ( 2 \Delta )$ the mean AC Stark shift and $ \omega AC ( \u2212 ) = ( | \Omega E 0 | 2 \u2212 | \Omega B 1 | 2 ) / ( 2 \Delta )$ the differential AC Stark shift. Note that the relative detuning

*γ*does not depend on the COM momentum but on the overall detuning

*δ*and the AC Stark shift. Thus, the overall detuning can be set in such a way that it compensates the AC Stark shift $ \omega AC ( \u2212 )$. After going into another interaction picture with respect to the mean detuning $ \gamma \xaf$, the new time evolution operator can be easily obtained by calculating the corresponding matrix exponential such that

*γ*. Since the transformations leading to this result are unitary transformations on the diagonal of the Hamiltonian, the transformed states are physically equivalent to the old ones.

Depending on the initial state, we observe the well-known Rabi oscillations between the ground and the excited state. For instance, if the atom is initially in the ground state, the probability to find the atom in the excited or in the ground state at time *t* is given, respectively, by

*γ*. The highest amplitude is achieved for a vanishing relative detuning, i.e., when the detuning

*δ*compensates the AC Stark shift. Increasing the relative detuning

*γ*leads to a decreasing amplitude and an increasing effective Rabi frequency $ \Omega eff$. For $ \gamma > | \Omega |$, it is no longer possible to achieve a 50:50 superposition of excited and ground state. Therefore, we have explicitly shown that taking into account finite pulse-time effects of beams with position-independent intensities—even with delocalizable atomic COM clouds—reproduces Rabi oscillations, connecting to known results in general contexts, see Weiss

*et al.*

^{61}where finite-time Raman transitions for COM momentum eigenstates were considered. In the following, we will investigate the generalization to arbitrarily structured laser beams.

## IV. FINITE PULSE-TIME EFFECTS OF ARBITRARY BEAMS

*ω*, the Doppler detuning $ \nu ( P \u0302 )$, and the (position-dependent) mean AC Stark shift $ \omega AC ( + ) ( R \u0302 )$ (expanded around the origin) via the definitions

_{k}*H*and obtain the new Hamiltonian

### A. Example: Fundamental Gaussian laser beam

*Z*

_{0}. In Fig. 4, we show in (a) an optical setup for this case as well as in (c/d) the corresponding interferometer sequences with the optimal placement of the E1–M1 clock transitions in them. In the following, we neglect beam distortion effects, assuming here a negligible influence of random phase and intensity noise. As shown by Bade

*et al.*,

^{110}there are, however, correlations of phase and intensity noise, which may prevent the averaging out even for many iterations of the interferometric experiments, and as such a proper inclusion of the distortion effects is left as necessary future work. Introducing the radial position operator

*Z*-axis, we can write the Gaussian

^{113}electromagnetic field in cylindrical coordinates. Thus

^{114}

*Z*-components by

*Z*and all radial components by

_{R}*w*

_{0}, i.e., the introduction of dimensionless operators $ Z \u0303 \u0302 = Z \u0303 \u0302 / Z R$ and $ \rho \u0302 = \u03f1 \u0302 / w 0$ as well as $ Z 0 = Z 0 / Z R , \u2009 \delta Z R = \delta Z R / Z R$, and $ \delta w 0 = \delta w 0 / w 0$, we can use the definitions of the single-photon Rabi frequencies, Eq. (38), and determine the operators that are present in the final Hamiltonian, Eq. (61), via their definitions, Eqs. (46), (52), and (55), expanded to the second order in these scaled parameters in terms of the Heisenberg trajectories of $ H \xaf \u0302$,

_{E}and Ω

_{B}in total analogy to Eq. (38). Furthermore, note that the extension to, e.g., Gillot

*et al.*

^{60}becomes quite clear in our example: quantization of the atomic COM and structured beam shapes not only change, to leading order, the detuning but also the Rabi frequency [in this case, the phase of the two-photon Rabi frequency, cf. Eq. (71b)]. Recalling Eqs. (38), (46), and (51), the effective kick due to the Gaussian beam shape is then given by

Symbol . | Description . | Order of magnitude . |
---|---|---|

w_{0} | Beam waist | ∼ $ 10 \u2212 2 \u2009 m$ (Ref. 112) |

Z _{R} | Rayleigh length | ∼ $ 10 3 \u2009 m$ |

$ \delta Z 0$ | Distance of interferometer arms | ∼ $ 10 \u2212 1 \u2009 m$ (Ref. 115) |

$ \delta w 0$ | Change of beam waist | ∼ $ 10 \u2212 4 \u2009 m$ |

$ \delta Z R$ | Change of Rayleigh length | ∼ $ 10 \u2009 m$ |

Symbol . | Description . | Order of magnitude . |
---|---|---|

w_{0} | Beam waist | ∼ $ 10 \u2212 2 \u2009 m$ (Ref. 112) |

Z _{R} | Rayleigh length | ∼ $ 10 3 \u2009 m$ |

$ \delta Z 0$ | Distance of interferometer arms | ∼ $ 10 \u2212 1 \u2009 m$ (Ref. 115) |

$ \delta w 0$ | Change of beam waist | ∼ $ 10 \u2212 4 \u2009 m$ |

$ \delta Z R$ | Change of Rayleigh length | ∼ $ 10 \u2009 m$ |

*ω*. The Heisenberg trajectories for the mean Hamiltonian Eq. (54) can be calculated via the Heisenberg equations of motion, Eq. (58), which leads to

_{k}*H*in Eq. (73). Since we do not go beyond the first order of $ \delta Z R$, the Hamiltonian of Eq. (73) is (quasi-) commuting at different times. Calculating the time-evolution operator,

*π*-pulse ( $ \pi / 2$-pulse), we obtain the generalized

*π*-pulse and $ \pi / 2$-pulse operators

*λ*is the wavelength of the laser beam, and $ k = \delta Z R e Z / Z R 2 \u2192 0$), they reduce to the well-known ideal

*π*and $ \pi / 2$-pulse operators, respectively,

#### 1. Discussion of the generalized *π* and *π*/2-pulse operators

*π*

*π*

##### a. Additional momentum kicks and branch-dependent phase

The displacement operators $ D \u0302$ and $ D \u0302 \u2020$ correspond to a transfer of momentum $ \xb1 \u210f k = \xb1 \u210f \delta Z R e Z / Z R 2$ and to an imprinting of a branch-dependent phase $ \Phi ( Z 0 ) = \u2213 \delta Z R Z 0 / Z R 2$, i.e., the transitions from ground to excited state and vice versa are accompanied by small additional momentum kicks and phase shifts. Note that although there are displacement operators present in the terms $ U \u0302 \pi , e e$ and $ U \u0302 \pi 2 , e e$, i.e., the atoms remaining in the excited state during the laser pulse, the momentum of the atom is identical before and after the pulse. There is only a momentum shift occurring during the interaction with the laser.

##### b. Action of $ U \xaf \u0302 ( \tau )$

*Z*-direction [recall Eqs. (52) and (72)] and a laser phase. Recall that we chose the overall detuning $ \delta = \u2212 \omega k \u2212 \omega AC ( \u2212 ) ( 0 )$ so that the recoil frequency

*ω*and the part of the mean AC Stark shift $ \omega AC ( + ) ( 0 )$ corresponding to the M1 transition in zeroth order is compensated in the mean Hamiltonian. Furthermore, the first order of the mean AC Stark shift vanishes for the $ TEM 00$ laser mode.

_{k}##### c. Additional branches

The *π* and $ \pi / 2$-pulses further contain operators of the forms $ sin ( \xi P \u0302 z / \u210f )$ and $ cos ( \xi P \u0302 z / \u210f )$, i.e., a splitting of the branches in opposite directions. Moreover, we see that the *π* and $ \pi / 2$-pulse operators do not transform all the atoms to the appropriate internal state (in contrast to the ideal case). Both the *π*- and $ \pi / 2$-pulses lead, therefore, to a splitting into four branches.

*T*is a characteristic time of the interferometer sequence, e.g., $ T = T 4 \u2212 T 2$ in Fig. 5 or $ T = T 3 \u2212 T 2$ in Fig. 6. Therefore, we will neglect branch splitting from now on and continue with the

*π*- and $ \pi / 2$-pulse operators, given by

Atom . | E1/ea_{0}
. | M1/μ_{B}
. | $ \lambda \u2009 ( nm )$ . | $ \Omega \u2009 ( Hz )$ . | $ t \pi / 2 = \pi / ( 2 \Omega ) \u2009 ( ms )$ . |
---|---|---|---|---|---|

Yb | 0.54^{116} | $ 2$^{116} | 1157^{117} | 150^{57,58} | 10.47 |

Sr | 0.15^{118} | $ 2$^{119} | 1397^{120} | 52.8^{57,58} | 29.75 |

## V. EFFECTS ON THE INTERFEROMETER PHASE FROM ADDITIONAL MOMENTUM KICKS

^{49}and (B)

^{52}(recall Sec. II). We assume ideal momentum kick operators

*π*and $ \pi / 2$-pulses, see Eq. (83) in Sec. IV. The evolution of the atom in the gravitational field from time

*T*to

_{k}*T*in between laser pulses in its ground/excited state can be described via

_{i}^{47,49,52}

*L*

^{2}-normalized Gaussian wave packet

### A. UGR tests using superpositions of internal states

^{49}the atoms entering the interferometer in the ground state are divided into two branches, and in the middle segment, a Doppler-free $ \pi / 2$-pulse is applied simultaneously on both branches to get a 50:50 superposition of excited and ground state atoms, i.e., the initialization of an atomic clock. However, considering finite pulse-time effects and the fact that the E1–M1 transitions are not Doppler-free anymore originating from the position dependency of the Rabi frequency, the modified trajectories of this interferometer are shown in Fig. 5. Nevertheless, we can still measure the intensity in the ground state and the excited state exit ports. Describing the evolution along the lower and upper trajectories, respectively, by

*T*]

_{i}*T*

_{2}, is given by

### B. UGR and UFF tests without superposition

The interferometer scheme (B)^{52} does not require superpositions of different internal states but a change of the internal state in the middle segment, i.e., a recoilless *π*-pulse. Again, we need two runs of the experiment: one with the initial ground state and one with the initial excited state. We found in Sec. IV A that the E1–M1 transitions are not perfectly recoilless for realistic beam shapes and finite pulse times. Likewise, the interferometer scheme in this section is modified by additional momentum kicks during the *π*-pulses, see Fig. 6.

^{52}can, thus, be reproduced immediately, i.e., the differential signals read

## VI. CONCLUSION

Very-large baseline atom interferometry is built upon exploiting the beneficial scaling of the interferometer signal in terms of the enclosed space–time area. However, due to the resulting long free-fall and interaction times, imperfections and perturbations act over much longer timescales and even small, accumulated effects can lead to a loss of visibility in the interference pattern in such devices. In case of (local) magnetic and gravitational field gradients,^{121,122} these effects have to be studied in detail for upcoming large baseline setups in addition to the already available results for, e.g., gravity gradients or rotations.^{4,90,92,123–125} In this spirit, our paper may be regarded as an extension of such studies to the case of E1–M1 transitions in quantum clock interferometry, performed under realistic conditions arising in large baseline atomic fountains.

Following the recent proposals of Roura^{49} and Ufrecht *et al.*^{52} for LPAI schemes that are sensitive to UGR and UFF tests, we have investigated recoilless clock transitions mediated by E1–M1 processes. For this purpose, we took into account the fully quantized atomic degrees of freedom, including the quantized—and possibly delocalizing—COM motion as well as the intern–extern coupling/mass defect. As a simple starting point, we considered electromagnetic plane waves in a $ \sigma +$– $ \sigma \u2212$ polarization scheme such that the atom absorbs two counter-propagating field excitations. However, since the M1 transitions are much weaker than E1 transitions in typical atoms, one needs quite large laser intensities and still long pulse times in an interferometer scheme containing E1–M1 transitions. Accordingly, finite pulse-time effects need to be included for a precise modeling. After an adiabatic elimination procedure, we found that the internal atomic dynamics yields standard Rabi oscillations, where the COM dependency drops out due to the fields having a constant intensity in space.

Furthermore, we also took into account that a realistic laser beam has a spatially dependent intensity, i.e., the Rabi frequencies become position dependent. Here, we considered the case of an atom falling through a general laser beam with arbitrary spatial coupling and finite pulse times and provided an expression for the dynamics of such an atom. In the exemplary case of a Gaussian laser beam, we derived *π* and $ \pi / 2$-pulse operators that generalize the standard expressions. In particular, our results show as the dominant effect additional momentum kicks $ \xb1 \delta Z R \u210f / Z R 2 e Z$ and an imprinting of a branch-dependent phase $ \Phi ( Z 0 ) = \u2213 \delta Z R Z 0 / Z R 2$ and a splitting of the branches to the next order. As one would expect, these expressions reduce then to the idealized operators when we neglect the laser profile and assume a strictly localized atomic COM wave function.

It is important to emphasize that the laser inhomogeneities result in corrections to the idealized plane-wave model already for very localized atomic wave packets. In the case of atomic clouds that are suffering from strong dispersion, e.g., for long interferometer times or strong atom–atom interactions, the assumption of large scale separation between atomic and laser extension no longer holds, and the efficiency of optical pulses is expected to be reduced even further; cf. for instance, the efficiency of a Bragg beam splitter with two E1 transitions.^{126}

Finally, we returned to the initial question of the paper and applied our results for the leading order finite pulse-time effects, i.e., the additional momentum kicks, to the proposed UGR and UFF tests of Roura, scheme (A),^{49} and Ufrecht *et al.*, scheme (B).^{52} In both cases, we derived the interferometer phases of the modified schemes and found that the additional momentum kicks are canceling each other out to leading order in the respective signal of interest. In the interferometer scheme (B),^{52} these effects cancel out already in the non-differential phase due to symmetry, while in the interferometer scheme (A),^{49} they are only canceled out in the double-differential phase.

## ACKNOWLEDGMENTS

We are grateful to W. P. Schleich for his stimulating input and continuing support. We are thankful to C. Ufrecht, E. Giese, T. Asano, F. Di Pumpo, and S. Böhringer as well as the QUANTUS and INTENTAS teams for fruitful and interesting discussions. A.F. is grateful to the Carl Zeiss Foundation (Carl-Zeiss-Stiftung) and IQST for funding in terms of the project MuMo-RmQM. The QUANTUS and INTENTAS projects are supported by the German Space Agency at the German Aerospace Center (Deutsche Raumfahrtagentur im Deutschen Zentrum für Luft- und Raumfahrt, DLR) with funds provided by the Federal Ministry for Economic Affairs and Climate Action (Bundesministerium für Wirtschaft und Klimaschutz, BMWK) due to an enactment of the German Bundestag under Grant Nos. 50WM2250D (QUANTUS+) and 50WM2178 (INTENTAS).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Gregor Janson:** Conceptualization (supporting); Formal analysis (lead); Investigation (lead); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review and editing (equal). **Alexander Friedrich:** Conceptualization (lead); Formal analysis (supporting); Investigation (supporting); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review and editing (equal). **Richard Lopp:** Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (equal); Supervision (equal); Validation (equal); Writing – original draft (supporting); Writing – review and editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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