Atomic diffraction from single-photon transitions in gravity and Standard-Model extensions

Atomic sensors, such as clocks¹ 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 signal⁷ when relying on two-photon transitions, such as Raman or Bragg diffraction commonly used⁴⁵ 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 redshift³⁷ has to be considered. Moreover, because of gravity, also chirping⁵⁰ 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 $E 1 − M 1$ 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.

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.

For discussing single-photon transitions, both magnetically induced and direct ones, one needs to analyze the resonance condition based on the (effective) Hamiltonian $H ̂ rot$⁠, including chirping and perturbations. To this end, we split the dilaton field $ϱ ( z ̂ , t ) = ϱ DM ( z ̂ , t ) + ϱ EP ( z ̂ )$ 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 $ϱ DM ( z ̂ , t ) = ϱ ¯ 0 cos ( ω ϱ t − k ϱ z ̂ + ϕ ϱ )$ includes the perturbative amplitude $ϱ ¯ 0$⁠, the frequency $ω ϱ$⁠, the wave vector $k ϱ$⁠, and the phase $ϕ ϱ$ of the DM field. The EEP-violating part $ϱ EP ( z ̂ ) = β S g z ̂ c − 2$ 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 $m j ( ϱ ) = m j ( 0 ) [ 1 + β j ϱ ( z ̂ , t ) ]$ around its Standard-Model value $m j ( 0 )$⁠, with linear expansion coefficient β_j. We define the mass defect⁶⁶ as $m e / g ( 0 ) = m ¯ ± ℏ ω e g / ( 2 c 2 )$⁠, with mean mass $m ¯$ and energy difference between the internal states $ω e g = ω e − ω g$⁠. Then, we insert the dilaton field and the mass defect into Eq. (2) and approximate all terms to first order in $c − 2 , ϱ ¯ 0 , β e / g$⁠, and $ω e g / ω ¯$⁠. Consequently, DM introduces time-dependent oscillations of the internal energies,⁷¹ while the transferred momentum is modified according to Eq. (4) by chirping⁵⁰ and by the gravitational redshift³⁷ of the wave vector k. The kinetic and potential energies include state-dependent modifications due to the mass defect and possible EEP violations.

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 $α = − g$ 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.

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.

In an atom interferometer that tests for dark matter or EEP violations, one seeks to bound the coupling parameters $β ¯ , Δ β$⁠, or β_S. While a measurement of the phases $ϕ EP$ and $ϕ DM$ 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 $t ∼ 1 / Ω$ 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 $Ω t = π / 2$ 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 diffraction⁴⁵ 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 $ϱ / Δ$ in the magnetically induced case. However, to first order in $ϱ ¯ 0$ and $c − 2$⁠, the Rabi frequency commutes with $ν ̂ H$ and $ν ¯ ̂ H$ even without magnetic field, so that our results immediately transfer to direct single-photon transitions.

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 schemes⁴² 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.

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).

The authors have no conflicts to disclose.

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).

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