Single-photon transitions are one of the key technologies for designing and operating very-long-baseline atom interferometers tailored for terrestrial gravitational-wave and dark-matter detection. Since such setups aim at the detection of relativistic and beyond-Standard-Model physics, the analysis of interferometric phases as well as of atomic diffraction must be performed to this precision and including these effects. In contrast, most treatments focused on idealized diffraction so far. Here, we study single-photon transitions, both magnetically induced and direct ones, in gravity and Standard-Model extensions modeling dark matter as well as Einstein-equivalence-principle violations. We take into account relativistic effects like the coupling of internal to center-of-mass degrees of freedom, induced by the mass defect, as well as the gravitational redshift of the diffracting light pulse. To this end, we also include chirping of the light pulse required by terrestrial setups, as well as its associated modified momentum transfer for single-photon transitions.

## I. INTRODUCTION

Atomic sensors, such as clocks^{1} or atom interferometers,^{2} gain increasing attention as an alternative for detecting gravitational waves,^{3–16} possible Standard-Model violations like dark matter^{17–26} (DM), or violations of the Einstein equivalence principle^{27–38} (EEP). While most of these proposals rely on internal atomic transitions, e.g., induced by (optical) single-photon transitions,^{39–42} they mainly focus on the dynamics *between* the times of interaction with the electromagnetic field. To complement these approaches, we study in this article the effects of gravity, chirping, and of a basic model for DM and EEP violations on single-photon transitions, focusing on the atom's dynamics *during* the interaction with electromagnetic fields.

Differential measurements between two spatially separated atomic clocks or light-pulse atom interferometers, proposed for tests of gravitational waves^{5,7–9} or DM,^{18,22,43,44} probe different points in spacetime. While a finite propagation speed of light already contributes to the phase of a single atom interferometer, for differential setups this effect becomes even more crucial due to the large spatial separations and is in fact key to those measurement schemes. As a deleterious side effect, differential laser-phase noise enters the signal^{7} when relying on two-photon transitions, such as Raman or Bragg diffraction commonly used^{45} for atom interferometers. This differential noise is suppressed using (optical) single-photon transitions, which can also drive optical atomic clocks in the Lamb–Dicke regime.^{39} In addition, some atom-interferometric tests of EEP rely^{27,29,30,33,34,37,38} on such transitions.

Even though some demonstrators for atom-interferometric gravitational-wave detection under construction in horizontal configurations^{10,13} rely on Bragg diffraction,^{5,11,45,46} a great number of the current vertical proposals^{12,14,16} are based on single-photon transitions to avoid laser-phase noise. In such terrestrial setups with very long baselines,^{31,47–49} the propagation of light as well as its gravitational redshift^{37} has to be considered. Moreover, because of gravity, also chirping^{50} is necessary to remain resonant during the diffraction process. In addition, possible DM fields may further modify the transition. Since other long-baseline setups plan on using single-photon transitions for EEP tests,^{51} possible EEP-violating fields also have to be included for a complete description of diffraction.

For the design and configuration of a differential atom-interferometric sensor, one has to decide on an atom species that has implications on atomic diffraction:^{52} Fermions offer weakly allowed clock transitions such that direct single-photon transitions without an auxiliary state are possible for fairly low laser powers. However, they suffer from spontaneous emissions and atom loss, resulting in an increase in shot noise and loss of coherence. Moreover, the increased cloud size and high expansion rate, caused by their fermionic nature, are detrimental for atom interferometers. Contrarily, the corresponding transitions of bosonic candidates have a clock lifetime limited by $ E 1 \u2212 M 1$ processes^{53,54} and offer the possibility to generate Bose–Einstein condensates with a low expansion rate. However, a direct excitation of such a transition with feasible laser powers is not possible, resulting in the need for magnetically induced transitions.^{40,41,52,55,56} An alternative for bosons is driving the intercombination line,^{42} although its lifetime is significantly shorter than for the clock line, limiting interrogation times and the spatial separation of atom-interferometric detectors.

Consequently, this article focuses on magnetically induced single-photon transitions. At the same time, by neglecting Stark shifts and by replacing the effective Rabi frequency by the corresponding actual one, our results can easily be transferred to direct single-photon transitions without magnetic fields. In our study, we include gravity,^{37} chirping,^{50} and a weakly coupled, ultralight, scalar dilaton field^{27,57–61} as a model for both DM and EEP violations. We consider relativistic effects like the coupling of internal energies to the center-of-mass (c.m.) motion of the atom, induced by the mass defect,^{62–66} which is necessary for a consistent modeling of many DM-detection schemes and EEP-test proposals. In Sec. II, we derive an effective two-level system for magnetically induced single-photon transitions including all perturbations discussed above. The resulting modified resonance condition, applying to both magnetically induced and direct single-photon transitions, is discussed in Sec. III. We study perturbatively the time evolution during a pulse in Sec. IV and discuss the effects on the phase of an atom after diffraction. We conclude the article in Sec. V. In Appendix A, we present technical details on adiabatic eliminations including time-dependent perturbations. Appendix B solves the time evolution in the Heisenberg picture, while Appendix C shows the phase resulting from the modified wave vector for non-ideal chirping.

## II. MAGNETICALLY INDUCED SINGLE-PHOTON TRANSITIONS

^{37}into a DM and an EEP-violating part. It depends in a one-dimensional model on c.m. position $ z \u0302$ and laboratory time

*t*and couples

^{27,57}to all particles of the Standard Model including electromagnetic fields and the constituents of atoms. As a consequence, electric and magnetic fields as well as the mass of the atom and its internal energies depend

^{36,37}on the dilaton. Since we focus on magnetically induced single-photon transitions for now, the laser does not couple the ground state $ | g \u27e9$ directly to the excited state $ | e \u27e9$ but to an ancilla state $ | a \u27e9$, which is strongly detuned by a frequency Δ from resonance, see Fig. 1. To transfer the population nonetheless, an additional static magnetic field, magnetically coupling the ancilla state to the excited one, is required.

^{40,41,55,56}The Hamiltonian for the three-level system takes the form

**as well as the magnetic dipole operator $ \mu \u0302 = \mu a e | a \u27e9 \u27e8 e | + h . c .$ interacting with a static, real-valued magnetic field $ B 0$. Here, the dipole transition element $ d a g$ is coupling the ground state to the ancilla state, while the magnetic dipole transition element $ \mu a e$ couples the excited state to the ancilla state. The c.m. momentum $ p \u0302$ and position $ z \u0302$ operators point in the**

*E**z*-direction of the laboratory frame, with $ [ z \u0302 , p \u0302 ] = i \u210f$. The time- and position-dependent dilaton field $ \u03f1 ( z \u0302 , t )$ is specified later. Since it couples to the mass-energy of the atom,

^{62–66}it also influences the motion of the atom in each internal state $ | j \u27e9$ through

*c*being the speed of light and the gravitational acceleration

*g*aligned with the

*z*-direction. We emphasize that the state-dependent mass $ m j ( \u03f1 )$ depends not only on the dilaton but also on the atomic state $ | j \u27e9$, i.e., it includes the mass defect

^{62–66}and introduces a coupling of internal states to c.m. operators. Furthermore, we assume

*α*is its chirp rate in units of an acceleration. Chirping is necessary to compensate for Doppler shifts and induce resonant transitions in gravity, as shown in Fig. 2, and therefore of particular relevance for vertical setups. The first term in Eq. (4) is the spatial mode function and gives rise to the momentum imparted on the atom via a displacement operator. Without taking into account any perturbations, it transfers a momentum $ \u210f k$ to the atom, where the dispersion relation $ k c = \omega L$ holds, in contrast to two-photon processes where the effective wave vector can be tuned independently from the transferred energy. This first term also includes a time-dependent modification from chirping, usually not present for two-photon transitions, as well as a gravitational redshift factor $ g z \u0302 2 / ( 2 c 2 )$ from the gravitational modification of the wave vector. Since chirping acts as an additional acceleration, it also appears as an accelerational redshift factor $ \alpha z \u0302 2 / ( 2 c 2 )$. Hence, the momentum transfer does not include a redshift modification proportional to $ z \u0302 2$ for perfect chirping $ \alpha = \u2212 g$. These perturbations can be derived from a gravitationally modified eikonal equation.

^{67}

### A. Rotating-wave approximation

^{68}by neglecting these quickly oscillating terms. The validity of this approximation is independent of the internal energies, instead a large temporal derivative of the laser phase $ | | \u210f \phi \u0307 L / [ d a g E 0 ( \u03f1 ) ] | | \u226b 1$ is required.

### B. Adiabatic elimination

^{68}a projector $ P \u0302$ with $ | \psi a \u27e9 = P \u0302 ( | \psi e \u27e9 , | \psi g \u27e9 ) T$. Since $ H \u0302 rot$ depends on time, $ P \u0302$ may also be time dependent, in contrast to the conventional treatment.

^{68–70}In Appendix A, we derive the Schrödinger equations for $ | \psi a \u27e9$ and $ ( | \psi e \u27e9 , | \psi g \u27e9 )$ and find the Bloch equation,

Note that for direct single-photon transitions without a magnetic field, one can directly use a two-level Hamiltonian in the form of Eq. (10) by assuming vanishing Stark shifts and replacing the effective Rabi frequency by the fundamental one. All other features, in particular the resonance condition discussed in Sec. III, remain of the same form.

## III. RESONANCE CONDITION FOR SINGLE-PHOTON TRANSITIONS

For discussing single-photon transitions, both magnetically induced and direct ones, one needs to analyze the resonance condition based on the (effective) Hamiltonian $ H \u0302 rot$, including chirping and perturbations. To this end, we split the dilaton field $ \u03f1 ( z \u0302 , t ) = \u03f1 DM ( z \u0302 , t ) +\u2009 \u03f1 EP ( z \u0302 )$ into DM and EEP-violating contributions. We consider an ultralight dilaton field expanded locally in the laboratory frame. Cosmological and galactic contributions to this field act as background resulting in DM, and local gravitational contributions sourced by Earth act as EEP violations.^{37} The DM part $ \u03f1 DM ( z \u0302 , t ) = \u03f1 \xaf 0 \u2009 cos ( \omega \u03f1 t \u2212 k \u03f1 z \u0302 + \varphi \u03f1 )$ includes the perturbative amplitude $ \u03f1 \xaf 0$, the frequency $ \omega \u03f1$, the wave vector $ k \u03f1$, and the phase $ \varphi \u03f1$ of the DM field. The EEP-violating part $ \u03f1 EP ( z \u0302 ) = \beta S g z \u0302 c \u2212 2$ contains the coefficient *β _{S}* that arises from the expansion of a source mass, e.g., Earth, in orders of the dilaton field. Similarly, we expand the internal-state-dependent mass of the atom $ m j ( \u03f1 ) = m j ( 0 ) [ 1 + \beta j \u03f1 ( z \u0302 , t ) ]$ around its Standard-Model value $ m j ( 0 )$, with linear expansion coefficient

*β*. We define the mass defect

_{j}^{66}as $ m e / g ( 0 ) = m \xaf \xb1 \u210f \omega e g / ( 2 c 2 )$, with mean mass $ m \xaf$ and energy difference between the internal states $ \omega e g = \omega e \u2212 \omega g$. Then, we insert the dilaton field and the mass defect into Eq. (2) and approximate all terms to first order in $ c \u2212 2 , \u2009 \u03f1 \xaf 0 , \u2009 \beta e / g$, and $ \omega e g / \omega \xaf$. Consequently, DM introduces time-dependent oscillations of the internal energies,

^{71}while the transferred momentum is modified according to Eq. (4) by chirping

^{50}and by the gravitational redshift

^{37}of the wave vector

*k*. The kinetic and potential energies include state-dependent modifications due to the mass defect and possible EEP violations.

*kgt*that can be compensated by the term $ \u2212 k \alpha t$ if the laser frequency is appropriately chirped.

We observe that modifications to Doppler shifts arising from the mass defect and the modified momentum transfer cannot be fully compensated. In summary, we find from the effective resonance condition that even in the case $ \alpha = \u2212 g$ and neglecting velocity selectivity, perturbative effects persist. As such, it causes a different time evolution during the pulse compared to the unperturbed case, and, thus, leads to phase shifts on diffracted atomic wave packets that scale with the pulse duration. We stress that these results are valid for both magnetically induced single-photon transitions and, with the replacements discussed above, also for direct ones.

## IV. PHASE SHIFTS BETWEEN DIFFRACTED WAVE PACKETS

Since the modified resonance condition affects the time evolution during the pulse, perturbations lead to additional phase shifts between diffracted and undiffracted atoms. In our study, we included as perturbations possible EEP violations, DM, chirping, the redshifted momentum transfer, and the mass defect. In the following, we analyze effects of these perturbations on the phase imprinted on atoms by the pulses.

^{72,73}we change into an interaction picture with respect to the unperturbed Rabi oscillation via the transformation $ U \u0302 \phi $ and then solve the time evolution of the remaining perturbations with a Dyson-series approach, see Appendix B for details. Since we consider weakly coupled, ultralight DM, we expect it to have a small Compton frequency and associated wave vector.

^{19}Therefore, we neglect its time dependence over the duration of a pulse and evaluate the field at the initial time of the pulse. This assumption results in the relation $ \u03f1 DM ( z \u0302 H , t ) \u2245 \u03f1 DM ( z \u0302 , 0 )$, with perturbative parameters $ \omega \u03f1 / \Omega , \u2009 k \u03f1 p \u0302 / ( m \xaf \Omega )$, and $ k \u03f1 g / \Omega 2 \u226a 1$. With this procedure, we arrive at the solution $ U \u0302$ for the time evolution in the Heisenberg picture, see Appendix B. Transformed back into the Schrödinger picture, we find the time evolution

In an atom interferometer that tests for dark matter or EEP violations, one seeks to bound the coupling parameters $ \beta \xaf , \u2009 \Delta \beta $, or *β _{S}*. While a measurement of the phases $ \varphi EP$ and $ \varphi DM$ may serve such a purpose, stricter bounds are given by the interferometer signal induced during propagation in between the diffracting pulses. Since the phases presented above are on the timescale of the pulse duration $ t \u223c 1 / \Omega $ that is much smaller than the interrogation time of the interferometer, we refrain from discussing the bounds implied by Eqs. (19b) and (19c) or their relative size.

Similar results for beam splitters with $ \Omega t = \pi / 2$ can be obtained from the time evolution in Appendix B. Since the effects of the perturbations presented here take place *during* the pulse, they will be suppressed for short durations.^{73} Such short pulse durations are limited by the effective Rabi frequency, but in contrast to Bragg diffraction^{45} no higher diffraction orders arise. Hence, in principle, a short pulse duration and a large effective Rabi frequency are possible, but they are limited by the coupling strength and the available laser power and magnetic field intensity.

For direct single-photon transitions without magnetic field,^{42} the Rabi frequency would be time- and position-dependent due to dilaton modifications, which are suppressed by $ \u03f1 / \Delta $ in the magnetically induced case. However, to first order in $ \u03f1 \xaf 0$ and $ c \u2212 2$, the Rabi frequency commutes with $ \nu \u0302 H$ and $ \nu \xaf \u0302 H$ even without magnetic field, so that our results immediately transfer to direct single-photon transitions.

## V. CONCLUSIONS

In this article, we described the light–matter interaction for single-photon transitions, including gravity and perturbations like DM, EEP violations, as well as a modified wave vector of light caused by gravity and chirping. As such, we derived an effective two-level system for single-photon transitions with a magnetic field. Considering the dynamics during the pulse shows that the resonance condition contains modifications from these perturbations, causing imperfect diffraction of atoms. Moreover, we showed that these perturbations result in additional phase contributions for atoms after diffraction. For example, in contrast to two-photon diffraction, chirping leads to a modified momentum transfer due to a fixed dispersion relation, which would even be present in the instantaneous-pulse approximation. In contrast, the gravitational redshift of light implies a modified momentum transfer for two-photon transitions.^{37} As we demonstrated, however, this effect is suppressed in single-photon transitions for perfect chirping.

We emphasize that our results can be immediately transferred to direct single-photon transitions without a magnetic field, starting directly from a two-level system and not an effective one. Then, effective quantities like the Rabi frequency are replaced by their counterparts, while no adiabatic elimination is necessary.

Vertical terrestrial proposals^{12,14–16,43,44,49} take place in gravity. Hence, chirping is crucial to stay on resonance, which becomes particularly relevant for very-long-baseline setups.^{31,47,48} As discussed here, chirping leads to a modified momentum transfer, which has not been considered in earlier works so far. Hence, by including all relevant perturbations consistently, we have performed the first step by focusing on the dynamics during the pulse. The next step, before turning to differential atom-interferometric schemes, would be to transfer our results to single atom interferometers with many pulses and to include also the dynamics between the pulses and the corresponding perturbations without electromagnetic interaction. Although the effects during the pulse are small in the context of atom interferometers for sufficiently short pulse durations, they might accumulate for multiple sequential pulses and large-momentum-transfer schemes^{42} planned for gravitational-wave and DM detectors.^{7,14} Our results will help to analyze also much more dominant effects between the pulses and will lead to a comprehensive description of differential atom-interferometric experiments.

## ACKNOWLEDGMENTS

We are grateful to W. P. Schleich for his stimulating input and continuing support. We also thank O. Buchmüller, A. Friedrich, J. Rudolph, and C. Ufrecht as well as the QUANTUS and INTENTAS teams for fruitful and interesting discussions. 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. 50WM1956 (QUANTUS V), 50WM2250D-2250E (QUANTUS+), as well as 50WM2177–2178 (INTENTAS). The projects “Metrology with interfering Unruh-DeWitt detectors” (MIUnD) and “Building composite particles from quantum field theory on dilaton gravity” (BOnD) are funded by the Carl Zeiss Foundation (Carl-Zeiss-Stiftung). The Qu-Gov project in cooperation with “Bundesdruckerei GmbH” is supported by the Federal Ministry of Finance (Bundesministerium der Finanzen, BMF). F.D.P. is grateful to the financial support program for early career researchers of the Graduate & Professional Training Center at Ulm University and for its funding of the project “Long-Baseline-Atominterferometer Gravity and Standard-Model Extensions tests” (LArGE). E.G. thanks the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) for a Mercator Fellowship within CRC 1227 (DQ-mat).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Alexander Bott:** Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). **Fabio Di Pumpo:** Conceptualization (equal); Investigation (supporting); Methodology (supporting); Project administration (equal); Supervision (equal); Validation (equal); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal). **Enno Giese:** Conceptualization (equal); Investigation (supporting); Methodology (supporting); Project administration (equal); Supervision (equal); Validation (equal); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

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

### APPENDIX A: ADIABATIC ELIMINATION

^{68}$ P \u0302$ such that $ | \psi a \u27e9 = P \u0302 | \psi e g \u27e9$. Note that $ H \u0302$ depends on time, so $ P \u0302$ may also be time dependent. Without this projector, we find the Schrödinger equations,

^{68}

### APPENDIX B: TIME EVOLUTION IN HEISENBERG PICTURE

^{73}by changing into a picture co-rotating with Ω via the transformation

*π*and $ \pi / 2$ pulses as mirrors and beam splitters, the typical interaction duration is proportional to $ \Omega \u2212 1$. We observe that the transformed Hamiltonian is perturbative on these timescales, i.e., $ | | H \u0302 \phi | | / ( \u210f \Omega ) \u226a 1$. Thus, we can use a first-order Dyson expansion to determine the time evolution in the co-rotating picture. Reverting the transformation $ U \u0302 \phi $ after performing the expansion yields the time evolution in the Heisenberg picture

t
. ^{j} | Coefficients $ \nu \u0302 ( j )$ of the detuning $ \nu \u0302 H$ . | Coefficients $ \nu \xaf \u0302 ( j )$ of the mean energy $ \nu \xaf \u0302 H$ . |
---|---|---|

t^{0} | $ ( \omega e g + \nu \u0302 k + \Delta \omega a c ) \u2212 \omega L + k \alpha z \u0302 c + \omega e g \omega \xaf ( m \xaf g z \u0302 \u210f \u2212 p \u0302 2 2 m \xaf \u210f \u2212 \omega k 4 )$ | $ \omega \xaf \beta \xaf \u03f1 DM ( z \u0302 , 0 ) + \omega \xaf \beta \xaf \beta S g z \u0302 c 2 \u2212 k p \u0302 4 m \xaf \omega e g \omega \xaf \u2212 \omega k 2 ( g + \alpha ) z \u0302 c 2$ |

$ + \omega \xaf \Delta \beta \u03f1 DM ( z \u0302 , 0 ) + \omega \xaf \Delta \beta \beta S g z \u0302 c 2 \u2212 { k p \u0302 2 m \xaf , ( g + \alpha ) z \u0302 c 2}$ | ||

t^{1} | $ \u2212 k ( g + \alpha ) ( 1 + p \u0302 2 m \xaf 2 c 2 \u2212 g z \u0302 c 2 ) + 2 \nu \u0302 k \alpha c + ( 2 \omega e g \omega \xaf + \Delta \beta \beta S ) g p \u0302 \u210f$ | $ \beta \xaf \beta S g p \u0302 \u210f + k g 4 \omega e g \omega \xaf + \omega k 2 ( \alpha c \u2212 g + \alpha c 2 p \u0302 m \xaf )$ |

t^{2} | $ \u2212 3 k g \alpha 2 c \u2212 \omega e g \omega \xaf m \xaf g 2 \u210f \u2212 \omega \xaf \Delta \beta \beta S g 2 2 c 2 + \nu \u0302 k g ( g + \alpha ) c 2$ | $ \u2212 \omega \xaf \beta \xaf \beta S g 2 2 c 2 + \omega k 2 ( \alpha 2 c 2 + g + \alpha c 2 g 2 )$ |

t^{3} | $ \u2212 k ( g + \alpha ) g 2 c 2$ | 0 |

t
. ^{j} | Coefficients $ \nu \u0302 ( j )$ of the detuning $ \nu \u0302 H$ . | Coefficients $ \nu \xaf \u0302 ( j )$ of the mean energy $ \nu \xaf \u0302 H$ . |
---|---|---|

t^{0} | $ ( \omega e g + \nu \u0302 k + \Delta \omega a c ) \u2212 \omega L + k \alpha z \u0302 c + \omega e g \omega \xaf ( m \xaf g z \u0302 \u210f \u2212 p \u0302 2 2 m \xaf \u210f \u2212 \omega k 4 )$ | $ \omega \xaf \beta \xaf \u03f1 DM ( z \u0302 , 0 ) + \omega \xaf \beta \xaf \beta S g z \u0302 c 2 \u2212 k p \u0302 4 m \xaf \omega e g \omega \xaf \u2212 \omega k 2 ( g + \alpha ) z \u0302 c 2$ |

$ + \omega \xaf \Delta \beta \u03f1 DM ( z \u0302 , 0 ) + \omega \xaf \Delta \beta \beta S g z \u0302 c 2 \u2212 { k p \u0302 2 m \xaf , ( g + \alpha ) z \u0302 c 2}$ | ||

t^{1} | $ \u2212 k ( g + \alpha ) ( 1 + p \u0302 2 m \xaf 2 c 2 \u2212 g z \u0302 c 2 ) + 2 \nu \u0302 k \alpha c + ( 2 \omega e g \omega \xaf + \Delta \beta \beta S ) g p \u0302 \u210f$ | $ \beta \xaf \beta S g p \u0302 \u210f + k g 4 \omega e g \omega \xaf + \omega k 2 ( \alpha c \u2212 g + \alpha c 2 p \u0302 m \xaf )$ |

t^{2} | $ \u2212 3 k g \alpha 2 c \u2212 \omega e g \omega \xaf m \xaf g 2 \u210f \u2212 \omega \xaf \Delta \beta \beta S g 2 2 c 2 + \nu \u0302 k g ( g + \alpha ) c 2$ | $ \u2212 \omega \xaf \beta \xaf \beta S g 2 2 c 2 + \omega k 2 ( \alpha 2 c 2 + g + \alpha c 2 g 2 )$ |

t^{3} | $ \u2212 k ( g + \alpha ) g 2 c 2$ | 0 |

j
. | Coefficients $ \eta j ( t )$ for diagonal elements $ \u27e8 n | U \u0302 | n \u27e9$ . | Coefficients $ \xi j ( t )$ for off-diagonal elements $ \u27e8 n | U \u0302 | m \u2260 n \u27e9$ . |
---|---|---|

0 | $ 2 \u2009 sin \u2009 \phi t$ | 0 |

1 | $ 2 \phi \tau \u2009 sin \u2009 \phi t$ | $ \u2212 2 \u2009 sin \u2009 \phi t + 2 \phi t \u2009 cos \u2009 \phi t$ |

2 | $ \u2212 4 \u2009 sin \u2009 \phi t + 4 \phi t \u2009 cos \u2009 \phi t + 4 \phi t 2 \u2009 sin \u2009 \phi t$ | $ 4 \phi t ( \u2212 sin \u2009 \phi t + \phi t \u2009 cos \u2009 \phi t )$ |

3 | $ \u2212 12 \phi t \u2009 sin \u2009 \phi t + 12 \phi t 2 \u2009 cos \u2009 \phi t + 8 \phi t 3 \u2009 sin \u2009 \phi t$ | $ 2 ( 4 \phi t 2 \u2212 6 ) ( \u2212 sin \u2009 \phi t + \phi t \u2009 cos \u2009 \phi t ) \u2212 4 \phi t 2 \u2009 sin \u2009 \phi t$ |

j
. | Coefficients $ \eta j ( t )$ for diagonal elements $ \u27e8 n | U \u0302 | n \u27e9$ . | Coefficients $ \xi j ( t )$ for off-diagonal elements $ \u27e8 n | U \u0302 | m \u2260 n \u27e9$ . |
---|---|---|

0 | $ 2 \u2009 sin \u2009 \phi t$ | 0 |

1 | $ 2 \phi \tau \u2009 sin \u2009 \phi t$ | $ \u2212 2 \u2009 sin \u2009 \phi t + 2 \phi t \u2009 cos \u2009 \phi t$ |

2 | $ \u2212 4 \u2009 sin \u2009 \phi t + 4 \phi t \u2009 cos \u2009 \phi t + 4 \phi t 2 \u2009 sin \u2009 \phi t$ | $ 4 \phi t ( \u2212 sin \u2009 \phi t + \phi t \u2009 cos \u2009 \phi t )$ |

3 | $ \u2212 12 \phi t \u2009 sin \u2009 \phi t + 12 \phi t 2 \u2009 cos \u2009 \phi t + 8 \phi t 3 \u2009 sin \u2009 \phi t$ | $ 2 ( 4 \phi t 2 \u2212 6 ) ( \u2212 sin \u2009 \phi t + \phi t \u2009 cos \u2009 \phi t ) \u2212 4 \phi t 2 \u2009 sin \u2009 \phi t$ |

### APPENDIX C: PHASE FROM MODIFIED WAVE VECTOR

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*Quantum Field Theory and Gravity: Conceptual and Mathematical Advances in the Search for a Unified Framework*

*General Relativity*

*Applications + Practical Conceptualization + Mathematics = Fruitful Innovation*