1 Introduction

The equivalence principle is the cornerstone of Einstein’s theory of general relativity and has been consistent with every experimental test [1–5]. Despite its success in elucidating gravitational interactions, a notable challenge arises in the majority of theoretical models aiming to unify gravitation with the other three fundamental interactions, as they often necessitate a departure from the equivalence principle [6–8]. The gravitational redshift (GR), predicted by general relativity, stands out as a significant phenomenon predicting that a clock positioned at a higher gravitational potential will tick at a faster rate than an identical clock at a lower potential. To experimentally verify this central prediction, various tests have been conducted utilizing advancements in the precision of atomic clocks. Recent experiments have taken advantage of space missions, comparing terrestrial atomic clocks with those in orbit [5, 9], and ground-based tests have also been performed, involving atomic clocks placed separately at the top and bottom of a tower [10]. These experiments have consistently yielded constraints on the GR test at a level of 10−5. Furthermore, noteworthy contributions to the GR test involve frequency gradient measurements across atomic ensembles at millimeter or centimeter scales within an optical lattice [11, 12]. All these experiments contribute valuable insights into the understanding and verification of the GR effect.

Atom interferometers play a pivotal role in discussions concerning the test of Einstein’s equivalence principle [2, 4, 13–18]. Merged as significant tools for both inertial sensing and fundamental physics research [19–22], atom interferometers are employed not only for precision applications in gravimetry [23–27] but also in tests of the universality of free fall (UFF) [13, 15–17] and local Lorentz invariance (LLI) [4, 14], other two basic assumptions of Einstein’s equivalence principle. While UFF and LLI have been scrutinized with remarkable precision, the GR test has encountered limitations in achieving comparable accuracy. GR, causing time dilation, manifests in the interference phase shift of an atom interferometer that compares phases accumulated along two spatially separated paths. This phase shift serves as an indicator of the proper time differences between the two branches. Expectations of GR effects in atom interferometers have prompted the development of specialized schemes [28–30]. Recent proposals utilize the recoil velocity of photons to introduce a centimeter-scale height difference in gravitational potential between the two branches. Despite these advancements, the precision of the GR test in atom interferometer remains at a relatively modest level, estimated at around 10−3 [30]. The primary challenge lies in the energetic proximity of internal states involved in the interference process, resulting in a weak phase difference imprinted by the GR effect. Overcoming this limitation is crucial for further advancements in precision testing with atom interferometry.

Given the imperative to enhance atom interferometer measurement schemes for the GR test, our work focuses on analyzing GR-related phase through a perturbation method shifts in a single-photon-based Mach-Zehnder atom interferometer and presents a scheme for the GR test. The geometry of a Mach-Zehnder atom interferometer can be conceptualized as two sequential coherent atomic clocks operating at the frequency of the atomic internal state. Hence, we term this configuration an atomic clock interferometer (ACI) in this paper. The clocks in an ACI are positioned at different heights due to the free-fall of atoms in a gravitational field, leading to a GR-induced phase shift. To further enhance the precision of the GR test for ACIs, we propose the use of a drop tower to obtain a larger scale of height difference and a free-fall laser device to suppress disturbance effects. The paper is structured as follows: Sect. 2 provides a brief introduction to the Hamiltonian of the atomic system, presenting interference phases for both two-photon-based and single-photon-based ACIs. In Sect. 3, the corresponding interpretations of interference phases are expounded, and a scheme for the GR test utilizing a single-photon-based ACI is discussed in detail, followed by a conclusion in Sect. 4.

2 GR-induced phase shifts in ACIs

An ACI can be conceptualized as two sequential coherent atomic clocks operating at the frequency of the atomic internal state and positioned at different heights, as illustrated in Fig. 1(a). The ACI measures the phase difference accumulated along two distinct spatial paths, wherein the GR introduces a contribution to this phase difference by influencing the ticking rates of internal atomic clocks based on the spatial path. To undertake a quantitative analysis of the interference phase shift, we delve into the Hamiltonian of the atomic system, considering low-order relativistic corrections. The energy of the atom is calculated as \(E= \sqrt{M^{2} c^{4} + P^{2} c^{2}}\), where

$$ M=m+ \frac{E_{i}}{c^{2}} $$
(1)

represents the total rest mass of an atom. According to the mass-energy equivalence, the total rest mass contains an equivalent mass of internal energy \(E_{i}\). The Hamiltonian under the laboratory reference frame can be expressed as [31]

$$ \begin{aligned} H&=M c^{2} + \frac{P^{2}}{2M} +Mgz\\ &\approx \biggl(m c^{2} + \frac{P^{2}}{2m} +mgz \biggr) + E_{i} \biggl(1- \frac{P^{2}}{2 m^{2} c^{2}} + \frac{gz}{c^{2}} \biggr) \end{aligned} $$
(2)

The first term in Eq. (2) indicates the external state of atoms accelerating uniformly in a gravitational field, while the second term represents the relativistic correction of the atomic internal state. Especially for the second term, \(\frac{gz}{c^{2}}\) is the modification of the GR, and \(- \frac{P^{2}}{2 m^{2} c^{2}}\) is the correction of the special relativity time dilation of internal dynamics. Treating the atomic internal state as a clock, the Hamiltonian of internal dynamics [32] can be written as \(H_{i} = E_{i} (1- \frac{P^{2}}{2 m^{2} c^{2}} + \frac{gz}{ c^{2}} ) \approx E_{i} \frac{d\tau}{dt}\), where \(\frac{d\tau}{dt}\) is the time-lapse coefficient of probe-time relative to coordinate-time.

Figure 1
figure 1

Treat the atomic internal state as a ticking clock. (a) The ACI functions as if two coherent atomic clocks are ticking at the frequency of the atomic internal energy level. These clocks are positioned at different heights, giving rise to interference phase shifts attributable to the GR. The ticking frequency of the clocks is related to the energy level of the atom. The energy level diagrams for (b) two-photon-based and (c) single-photon-based ACIs. In the process of atom interference in (b), an atom initially in the ground state \(\vert g \rangle\) interacts sequentially with two photons and eventually reaches another ground state \(\vert e \rangle\). The energy level difference between these two ground states is at the GHz level, equivalent to the frequency difference between the two lasers employed in the process. Conversely, in a single-photon-based ACI in (c), the atomic transition between two ground states \(\vert g \rangle \) and \(\vert e \rangle \) involves the absorption of just one photon. The energy level difference between the two ground states, precisely matching the laser frequency, is typically several hundreds of THz.

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2.1 Phase shift in two-photon-based ACIs

In a typical two-photon-based ACI, as outlined in Ref. [23], the interaction between atoms and lasers is based on a three-level system that encompasses two hyperfine ground states with energies \(E_{e}\) and \(E_{g}\), as illustrated in Fig. 1(b). The ideal time intervals between laser-atom interactions are denoted as exactly equal to T. However, accounting for the finite speed of light (FSL), these intervals are not precisely T (depicted in Fig. 2(a)). Consequently, photons interact with atoms in different paths at different moments, introducing an additional difference in the free evolution of the atomic wave packet in each path and resulting in an additional spatial separation at the exit of the interference. Moreover, the time delay associated with the FSL can influence the laser phase at the moment of interaction, introducing an additional phase shift transmitted from the lasers. Accordingly, the interference phase shift is calculated by dividing it into three parts: phase \(\Delta \varphi _{\mathrm{free}}\) is correlated with the free evolution of atomic wave packets along the classical paths, phase \(\Delta \varphi _{\mathrm{laser}}\) is induced through the atom-laser interaction, and phase \(\Delta \varphi _{\mathrm{separate}}\) arises from path separation at the exit of interference. This comprehensive breakdown allows for a detailed understanding and precise calculation of the interference phase shift in the ACI. The total interference phase shift can be written as \(\Delta \varphi _{\mathrm{total}} = \Delta \varphi _{\mathrm{free}} + \Delta \varphi _{\mathrm{separate}} + \Delta \varphi _{\mathrm{laser}}\).

Figure 2
figure 2

Space-time diagrams of ACIs. Atoms undergo interactions with three laser pulses to achieve interference. (a) represents the case of the FSL. The blue and grey dashes represent the atoms at different internal states, while the red dash represents the motions of the photons. (b) is the space-time diagram while introducing a non-zero acceleration of the laser device. The motion of the laser device is illustrated by the green solid line

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\(\Delta \varphi _{\mathrm{free}}\) is calculated by path integral method [33]. According to the Hamiltonian in Eq. (2), we obtain the Lagrangian as

$$ \begin{aligned} L &\approx -m c^{2} - E_{i} + \frac{m v^{2}}{2} \biggl( 1+ \frac{E_{i}}{m c^{2}} \biggr) - \biggl( m+ \frac{E_{i}}{c^{2}} \biggr) gz \\ &\equiv L_{0} + L_{1} \end{aligned} $$
(3)

where the internal-state-related Lagrangian \(L_{1} =- E_{i} \frac{d \tau}{ dt} \approx - E_{i} ( 1- \frac{v^{2}}{2 c^{2}} + \frac{gz}{c^{2}} )\) is considered as a perturbation of the external-state-related Lagrangian \(L_{0} = - m c^{2} + \frac{m v^{2}}{2} - mgz\). Accordingly, \(\Delta \varphi _{\mathrm{free}}\) can be calculated by

$$ \Delta \varphi _{\mathrm{free}} \approx - \frac{1}{\hslash} \int _{L_{0}} ( L_{0} + L_{1} ) \,dt= \Delta \varphi _{\mathrm{free}}^{\mathrm{external}} + \Delta \varphi _{\mathrm{free}}^{\mathrm{internal}} $$
(4)

Here, the free evolution phases associated with external and internal states are denoted as \(\Delta \varphi _{\mathrm{free}}^{\mathrm{external}}\) and \(\Delta \varphi _{\mathrm{free}}^{\mathrm{internal}}\), respectively. To simplify the calculation, we combine the free evolution phase shift related to the atomic external state \(\Delta \varphi _{\mathrm{free}}^{\mathrm{external}}\) with the phase shift introduced by the path separation \(\Delta \varphi _{\mathrm{separate}}\). For the closed loop of interference path formed by the combination of the free evolution of atomic external states and the path separation in the gravitational field [34–37], we have

$$ \begin{aligned} \Delta \varphi _{\mathrm{free}}^{\mathrm{external}} + \Delta \varphi _{\mathrm{separate}} &= \frac{m c^{2}}{\hslash} \oint \biggl( 1 - \frac{v^{2}}{2 c^{2}} + \frac{gz}{c^{2}} \biggr) \,dt \\ &\approx 0 \end{aligned} $$
(5)

Correspondingly, phase shift caused by the internal state during the free evolution is calculated as follows

$$ \begin{aligned} \Delta \varphi _{\mathrm{free}}^{\mathrm{internal}} &= \frac{1}{\hslash} \int _{L_{0}} E_{i} \biggl( 1 - \frac{v^{2}}{2 c^{2}} + \frac{gz}{c^{2}} \biggr) \,dt\\ &= k_{a} g T^{2} + \frac{v_{T}}{c} k_{a} g T^{2} \end{aligned} $$
(6)

Here, \(k_{a} = \omega _{a} /c\) represents the wave vector corresponding to the frequency difference \(\omega _{a}\) between the atomic hyperfine ground states, and \(v_{T} = v_{0} + v_{r} -gT\) is the average velocity of atoms in both paths at the moment of applying the π pulse with \(v_{r}\) the recoil velocity and \(v_{0}\) the atomic velocity at the beginning of the interference.

The first-order relativistic effect of the internal state in Eq. (6) is further subdivided according to the following derivations. As displayed in Fig. 1(a), the ACI is divided into two coherent clocks at different heights, and the induced phase is expressed as \(\Delta \varphi =- [ \int _{\mathrm{up}} \omega _{a} \frac{gz}{c^{2}} \,dt - \int _{\mathrm{low}} \omega _{a} \frac{gz}{c^{2}} \,dt ]\), which leads to a phase shift relating to the GR as

$$ \Delta \varphi _{\mathrm{redshift}} \approx - \frac{v_{T}}{c} k_{a} g T^{2} $$
(7)

For phase shift induced by the special relativistic time dilation, a perturbation calculation \(\Delta \varphi _{\mathrm{dilation}} =- \int _{L_{0}} E_{i} \frac{v^{2}}{2 c^{2}} \,dt\) is adopted to obtain the phase shift of this effect as

$$ \Delta \varphi _{\mathrm{dilation}} \approx - \frac{v_{T}}{c} k_{a} g T^{2} $$
(8)

The remaining phase shift of the first-order relativistic effect in \(\Delta \varphi _{\mathrm{free}}^{\mathrm{internal}}\) is \(\Delta \varphi _{\mathrm{E}} = \frac{3 v_{T}}{c} k_{a} g T^{2}\), which represents the phase shift of the FSL effect of atomic internal states.

\(\Delta \varphi _{\mathrm{laser}}\) comes from the phase transmission from lasers to atoms during laser-atom interaction. \(\Delta \varphi _{\mathrm{laser}}\) is calculated as

$$ \Delta \varphi _{\mathrm{laser}} = - \biggl[ \sum_{i} \pm \phi ( x_{i}, t_{i} ) \vert _{\mathrm{up}} - \sum_{i} \pm \phi ( x_{i}, t_{i} ) \vert _{\mathrm{low}} \biggr] $$
(9)

where \(\phi ( x_{i}, t_{i} )\) represents the phase of lasers at the position \(x_{i}\) and the moment \(t_{i}\), and the signs of + or − correspond to the atom gaining or losing momentum as a result of the interaction. Employing the average-path method [38], we denote the moments of interactions for the second and third laser pulses during the interference as \(T_{1} =T+\delta T_{1}\) and \(T_{2} =2T+\delta T_{2}\), respectively, where

$$\begin{aligned} \textstyle\begin{cases} \delta T_{1} \approx \frac{ ( v_{0} T+ \frac{\hslash kT}{2m} - \frac{g T^{2}}{2} )}{c} + \frac{ ( v_{0} + \frac{\hslash k}{2m} -gT ) ( v_{0} T+ \frac{\hslash kT}{2m} - \frac{g T^{2}}{2} )}{c^{2}}\\ \delta T_{2} \approx \frac{ ( 2 v_{0} T+ \frac{\hslash kT}{m} -2g T^{2} )}{c} + \frac{ ( v_{0} + \frac{\hslash k}{2m} -2gT ) ( 2 v_{0} T+ \frac{\hslash kT}{m} -2g T^{2} )}{c^{2}} \end{cases}\displaystyle \end{aligned}$$
(10)

As the two lasers propagate in opposite directions, they induce a phase expressed by

$$ \begin{aligned} \Delta \varphi _{\mathrm{laser}} &=- 2k_{2} c\cdot ( \delta T_{2} -2\delta T_{1} ) \\ &\approx 2k_{2} g T^{2} + \frac{3 v_{T}}{c} 2k_{2} g T^{2} \end{aligned} $$
(11)

where \(k_{2}\) is the wave vector of the laser reflected by a mirror. However, compensating for the Doppler shift in experiments requires applying a frequency chirp to the lasers. Considering frequency chirping rates of \(-\alpha _{1}\) and \(\alpha _{2}\) for two lasers, the additional interference phase shift caused by frequency chirp is

$$\begin{aligned} \begin{aligned} \Delta \varphi _{\mathrm{chirp}} &\approx - \alpha _{1} T^{2} + \frac{\alpha _{2}}{2} \bigl[ ( 2T+2\delta T_{2} )^{2} -2 ( T+2\delta T_{1} )^{2} \bigr]\\ &\approx - ( \alpha _{1} - \alpha _{2} ) T^{2} +2 \alpha _{2} \frac{(2 v_{0} +2 v_{r} -3gT)}{c} T^{2} \end{aligned} \end{aligned}$$
(12)

Therefore, the total FSL effect phase from the lasers in a two-photon-based ACI can be expressed as \([ \frac{3 v_{T}}{c} 2k_{2} g +2 \alpha _{2} \frac{(2 v_{0} +2 v_{r} -3 gT )}{c} ] T^{2}\), which includes the frequency chirp item. Based on the above calculations, the total phase shift is concluded as

$$\begin{aligned} \begin{aligned} \Delta \varphi _{\mathrm{total}} &\approx \bigl[ 2k_{2} g - ( \alpha _{1} - \alpha _{2} ) \bigr] T^{2} \\ &\quad + \biggl[ \frac{3 v_{T}}{c} 2k_{2} g+2 \alpha _{2} \frac{ ( 2 v_{0} +2 v_{r} - 3gT )}{c} \biggr] T^{2} \\ &\quad + k_{a} g T^{2} + \frac{v_{T}}{c} k_{a} g T^{2} \end{aligned} \end{aligned}$$
(13)

2.2 Phase shift in single-photon-based ACIs

A single-photon-based ACI usually employs the transition between two atomic internal states with a resonant frequency of hundreds of THz, resulting in a much larger \(k_{a}\) than that in a two-photon-based ACI (shown in Fig. 1(c)) [26, 39]. This distinction in energy level differences presents a key advantage of single-photon-based ACIs. The larger energy separation provides a more pronounced and easily detectable phase shift, enhancing the GR test accuracy of the interferometer. Referring to the above phase-shift calculation, the phase shift induced through the atom-laser interaction is written as

$$ \Delta \varphi _{\mathrm{laser}} = - \alpha _{1} T^{2} $$
(14)

Here, the phase item correlated with the FSL effect of lasers is eliminated due to the absence of \(k_{2}\). The internal-state-induced part of the free evolution phase shift is calculated as \(\Delta \varphi _{\mathrm{free}}^{\mathrm{internal}} = k_{a} g T^{2} + \frac{v_{T}}{ c} k_{a} g T^{2}\) similar to Eq. (6), and the phase shifts induced by the GR and the special relativistic time dilation are \(\Delta \varphi _{\mathrm{redshift}} = \Delta \varphi _{\mathrm{dilation}} \approx - \frac{v_{T}}{c} k_{a} g T^{2}\), identical with the above calculations for the two-photon-based ACI. Generally, a single-photon-based ACI is applied to our model to calculate the total phase shift as

$$ \Delta \varphi _{\mathrm{total}} \approx ( k_{a} g- \alpha _{1} ) T^{2} + \frac{v_{T}}{c} k_{a} g T^{2} $$
(15)

3 Scheme for the GR test

We propose a novel measurement scheme designed to detect the GR effect. In our setup, we employ a single-photon-based ACI, conducting measurements in a microgravity environment within the free-fall facility of a drop tower. Notably, the laser device responsible for generating the manipulation lasers is fixed within the free-fall chamber, falling simultaneously with the atoms in the experimental setup. This scheme offers significant advancements. Firstly, the GR-induced phase shift is amplified by utilizing a single-photon-based ACI because of the larger energy separation between the two ground states. Secondly, potential phase shifts induced by other sources like the FSL effect are effectively mitigated through the execution of the experiment in a microgravity environment within a drop tower. This approach enhances the precision of the measurement, providing a platform for studying the GR effect.

In a single-photon-based ACI, the GR effect is remarkably amplified. Despite the expresses for both single-photon-based and two-photon-based ACIs being summarized as \(\Delta \varphi _{\mathrm{redshift}} \approx - \frac{v_{T}}{c} k_{a} g T^{2}\) in Table 1, the energy difference between the two internal states experienced by an atom during the interference in a single-photon-based ACI is typically 5 orders of magnitude larger than that in a two-photon-based ACI [34]. Displayed in Fig. 3 with typical parameters are the main phase shifts for both types of ACIs. In two-photon-based ACIs, the GR effect is negligible compared to the comparatively small \(k_{a}\) of the hyperfine structures. Conversely, in single-photon-based ACIs in our calculation, the phase shift induced by the atom-laser interaction is calculated to be \(- \alpha _{1} T^{2}\) and the phase shift arising from the FSL effect of lasers equals zero because of the absence of \(k_{2}\) and \(\alpha _{2}\). This results in a GR effect comparable to the FSL effect of the internal state. However, it’s noteworthy that the non-relativistic phase shift term of \(( k_{a} g- \alpha _{1} ) T^{2}\), induced by atom-laser interaction and the evolution of internal states, still domains in a single-photon-based ACI and requires further suppressed.

Figure 3
figure 3

The interference phase shifts induced by different effects varying with the interrogation time T in ACIs. The GR-induced phase shift \(\frac{v_{T}}{c} k_{a} g T^{2}\) is negligible compared to other phase shifts in a two-photon-based ACI, which has typical parameters of \(2 k_{2} \sim 1.6\times 10^{7} /\mathrm{m}\), \(k_{a} \sim 142/\mathrm{m}\), \(g =9.8\ \mathrm{m}/ \mathrm{s}^{2}\). In a single-photon-based ACI, the GR-induced phase shift will increase by 5 orders of magnitude due to a magnified \(k_{a} \sim 10^{7} /\mathrm{m}\) in a typical experiment [28, 30]

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Table 1 A list of all the phase shift terms calculated above for both two-photon-based and single-photon-based ACIs, as well as for single-photon-based ACI in microgravity
Full size table

The primary competing terms to the GR-induced phase shift come from the non-relativistic phase shift item and the FSL effect of the internal state. In our proposed scheme, we analyze how these disturbance terms are suppressed. In the preceding discussion, the laser device responsible for generating laser is fixed, while the atoms experience free fall in the Earth’s gravitational. This implies that, in the Earth reference frame, both the initial velocity \(v_{\mathrm{laser}}\) and the acceleration a of the laser device during the interference are zero. Considering the initial velocity \(v_{\mathrm{laser}}\) and a uniform non-zero acceleration a of the laser device (as shown in Fig. 2(b)), the moments of interactions for the second and third laser pulses during the interference become \(T_{1} =T+\delta T_{1} + \delta T_{1} '\) and \(T_{2} =2T+\delta T_{2} +\delta T_{2} '\), respectively. Substituting \(T_{1}\) and \(T_{2}\) into the calculations above, we have the phase shifts as follows

$$ \textstyle\begin{cases} \Delta \varphi _{\mathrm{laser}} = - \alpha _{1} T^{2} + \frac{3 ( v' - g' T )}{c} k_{a} g' T^{2}\\ \Delta \varphi _{\mathrm{free}} = k_{a} g' T^{2} - \frac{2v_{T}}{c} k_{a} g T^{2} \\ \hphantom{\Delta \varphi _{\mathrm{free}}}\quad - ( \frac{v_{\mathrm{atom}}}{c} a - \frac{v_{\mathrm{laser}}}{c} g ) k_{a} T^{2} \end{cases} $$
(16)

where \(v' = v_{\mathrm{atom}} - v_{\mathrm{laser}}\) is the difference between the initial velocities of the atom and laser device at the beginning of the interference, and \(g' =g-a\) denotes the acceleration difference. This consideration introduces a nuanced perspective on the temporal aspects of the laser-atom interactions during the interference process. Here, the phase shifts arising from the GR and the special relativistic time dilation remain identical to those in the situation of zero acceleration of the laser device. However, according to Eq. (16), the FSL effect phase shift \(\frac{3 ( v' - g' T )}{c} k_{a} g' T^{2}\) is variant with the velocity and acceleration of the laser device. By contrast, the phase shifts arising from GR, as well as special relativistic time dilation, remain independent of the motion of the laser device. These shifts are exclusively related to the internal states experienced by atoms during the interference. Generally, the total phase shift in a single-photon-based ACI with a moving laser device is calculated as

$$ \begin{aligned} \Delta \varphi _{\mathrm{total}} &\approx \bigl( k_{a} g'- \alpha _{1} \bigr) T^{2} + \frac{3 ( v'-g'T )}{c} k_{a} g T^{2} \\ &\quad - \frac{2 v_{T}}{ c} k_{a} g T^{2} - \biggl( \frac{v_{\mathrm{atom}}}{c} a - \frac{v_{\mathrm{laser}}}{c} g \biggr) k_{a} T^{2} \end{aligned} $$
(17)

Microgravity is anticipated to play a pivotal role in advancing tests of the equivalence principle [40]. In our proposed scheme, we envision a single-photon-based ACI performing measurements in microgravity within the free fall facility of a drop tower. In this setup, a laser device is securely fixed within the free-fall chamber, descending simultaneously with the atoms. This configuration establishes relationships where the velocity of the atoms \(v_{\mathrm{atom}}\) equals the velocity of the laser device \(v_{\mathrm{laser}}\), and the gravitational acceleration g is equivalent to the acceleration a experienced by the laser device. Additionally, due to the consistent relative velocity between the laser and atoms throughout the interference, the frequency chirp of the laser becomes unnecessary. This unique approach simplifies the experimental conditions and enhances the precision of the GR test in the microgravity environment of a drop tower. Introducing a GR violation factor α, the effect of GR on internal atomic clocks in the Hamiltonian of Eq. (2) is rewritten as \((1+\alpha ) \frac{gz}{c^{2}}\). Consequently, we have the total phase shift deduced from Eq. (17) as

$$ \Delta \varphi _{\mathrm{total}} \approx - \frac{(2+ \alpha ) v_{T}}{c} k_{a} g T^{2} $$
(18)

Equation (18) reveals that, under microgravity conditions, single-photon-based ACIs are affected solely by the GR and special relativistic time dilation, with the non-relativistic phase shift and phase shifts induced by the FSL effect being eliminated. The equivalent altitude difference relies on the coherent clocks in free-fall is \(\Delta z= v_{T} T\). Considering typical parameters of \(v_{T}\) = 7 m/s, \(k_{a}\) = 107/m, and T = 0.7 s in a 10-meter drop tower, a GR test signal over 1 rad is obtained, corresponding to a relative test accuracy at the level of 10−5 corresponding to an equivalent altitude difference of two coherent atomic clocks is 4.9 meters.

4 Discussions and conclusions

The test of the GR effect necessitates information regarding the frequency difference between clocks situated at different heights. The GR test, when based on atomic clocks, can improve precision by augmenting the height difference between the two clocks. Although ACIs cannot achieve height differences comparable to those in atomic clock tests, the theoretical accuracy for the GR test in our proposed scheme is not inferior to experimental tests based on atomic clocks. This is primarily attributed to the distinct requirements these two approaches have for the precision of clock frequency measurements. In the GR tests based on atomic clocks, the initial step involves measuring the clock frequencies at different heights, and subsequently, the comparison of frequency differences between these clocks yields the corresponding signal for the GR test. This measurement process demands extremely high precision in clock frequency to achieve a meticulous test of the GR effect. On the contrary, ACIs can directly capture the phase information of the frequency difference accumulated over time due to the GR effect on clocks at different heights, which implying that the precision requirements for measuring clock frequencies using atomic interferometers are not particularly high. Furthermore, in our proposed scheme, we utilize the two atomic clocks arranged temporally in the ACI to test the GR effect. As atoms freely fall in a gravitational field, the clocks naturally assume different heights. In comparison to schemes [28, 30, 40] that involve the height difference induced by the momentum imparted by the absorbed photons, the height difference in our scheme can be increased by two orders of magnitude. This results in a corresponding two-order-of-magnitude enhancement in the GR test. The preceding discussion highlights the benefits of our GR test scheme and the considerable potential of single-photon-based ACIs. However, in comparison to two-photon-based ACIs, the construction of single-photon-based ACIs may necessitate more intricate experimental conditions. On one hand, for single-photon-based ACIs employing Sr atoms, the upper-level linewidth of Sr atoms is only on the order of mHz. This requires the laser involved in atomic interactions to achieve higher intensity for a substantial Rabi frequency. On the other hand, unlike two-photon-based ACIs that are sensitive to the phase difference between two lasers, controllable through methods like phase-locked loops, single-photon-based ACIs are more susceptible to laser phase noise. Consequently, the employed lasers must attain better phase noise control. Nevertheless, the internal clock with a substantial frequency inherent in single-photon-based ACIs presents a potential demonstrating significant promise for GR tests. Research groups are actively advancing experimental studies on single-photon-based ACIs [41–45], with the prospect of their application in fields such as low-frequency gravitational wave detection [46, 47].

In summary, our calculations involve determining the first-order relativistic effects in both single-photon-based and two-photon-based ACIs, employing the Hamiltonian rooted in mass-energy equivalence. By treating a Mach-Zehnder atom interferometer as two coherent atomic clocks, we focused on the calculation of the GR effect. Our findings suggest that the GR effect in a single-photon-based ACI is significantly magnified, rendering it comparable to the FSL effect. To further enhance accuracy, we considered the motion of the laser device in the Earth reference frame. Introducing the velocity of the laser device proves advantageous in suppressing phase shifts induced by non-GR terms, including the non-relativistic effect and the FSL effect. Proposing the implementation of a single-photon-based ACI within a 10-meter drop tower, we introduce a testing concept for GR expected to substantially improve testing accuracy in atom interferometers to the level of 10−5. Furthermore, we anticipate that this GR testing method will derive significant benefits from space experiments, utilizing the larger velocity of atoms in interferometers located in eccentric orbits around the Earth.