A Dual One-way Doppler Cancellation Scheme for Enhanced Gravitational Redshift Tests

TL;DR

The paper introduces a simplified dual one-way Doppler cancellation scheme for gravitational redshift tests that enhances sensitivity and reduces complexity in satellite-ground clock experiments, enabling more precise fundamental physics tests.

Contribution

It proposes a novel two-link frequency comparison method that doubles redshift sensitivity and halves engineering complexity compared to traditional triple-link schemes.

Findings

Reduces engineering complexity by 50%

Doubles gravitational redshift signal sensitivity

Residual systematic errors below 10^{-17}

Abstract

Gravitational redshift is an important prediction of General Relativity (GR). We propose a novel dual one-way frequency comparison scheme for gravitational redshift tests in satellite-ground clock experiments. Unlike conventional triple-link Doppler cancellation methods, our approach requires only two optical links, an uplink and a downlink, with Doppler data recorded independently at the satellite and ground station, respectively. Crucially, coherent combination of these measurements suppresses first-order Doppler frequency shift to second order while doubling the gravitational redshift signal. This scheme reduced engineering complexity (50$\%$ fewer links) and amplified redshift sensitivity with a factor of 2 for satellite-ground clock comparison. The scheme's viability hinges on high-precision clock synchronization ($\leq$ 0.1 ns for $10^{-18}$ clock comparison), which is potentially achievable via state-of-the-art time-transfer techniques. Systematic error analysis confirms residual Doppler, atmospheric, and Shapiro shifts remain below $10^{-17}$ for optimized orbits, establishing this scheme as a useful tool for probing fundamental physics beyond GR.

Tables

Table 1. Table 1: The performance for conventional triple-link DCS and dual one-way DCS.

Terms	Conventional triple-link DCS	Dual one-way DCS	Physical significance
Link architecture	3 optical links:	2 optical links:	50$\%$ link reduction
	1 plink and 2 down link	1 plink and 1 down link
Gravitational redshift signal	a factor of 1	a factor of 2	Doubled sensitivity
Doppler suppression	second-order residual Doppler	second-order residual Doppler	Equivalent suppression
Clock synchronization	No	Yes	-

Equations

$$\frac{\Delta f}{f}=(1+\alpha )\frac{\Delta U}{c^2}$$

$$\frac{f_2}{f_1}=\frac{d{\tau }_1}{d{t}_1}{\left(\frac{d{\tau }_2}{d{t}_2}\right)}^{-1}\frac{d{t}_1}{d{t}_2},$$

$$\frac{d{\tau }_1}{d{t}_1}{\left(\frac{d{\tau }_2}{d{t}_2}\right)}^{-1}=1-\frac{1}{c^2}\left(U_E({\mathbf{x}}_1)-U_E({\mathbf{x}}_2)+u_{\text{tid}}({\mathbf{x}}_1)-u_{\text{tid}}({\mathbf{x}}_2)+\frac{{\mathbf{v}}_g^2(t_1)}{2}-\frac{{\mathbf{v}}_s^2(t_2)}{2}\right),$$

$$t_1=t_2-T_{12}-{\Delta }_{12}^{\text{atm}}-{\Delta }_{12}^{\text{Shap}},$$

$$\frac{d{t}_1}{d{t}_2}=1-\frac{d{T}_{12}}{d{t}_2}-\frac{d{\Delta }_{12}^{\text{atm}}}{d{t}_2}-\frac{d{\Delta }_{12}^{\text{Shap}}}{d{t}_2},$$

$$y_{\text{up}}=\frac{f_2}{f_1}$$

$$y_{\text{do}}=\frac{f_4}{f_3}$$

$$y_{\text{dcs}}$$

$$U_E(\mathbf{x})=\frac{G{M}_E}{r}\sum _{l=0}^{\infty }\sum _{k=0}^l{\left(\frac{r_{0E}}{r}\right)}^l{P}_{lk}\left(\cos \theta \right)\left[C_{lk}\cos \left(k\phi \right)+S_{lk}\sin \left(k\phi \right)\right]$$

$$y_{\text{up}}^{\text{gr}}=\frac{G{M}_E}{c^2{r}_2}f(r_2,{\theta }_2,{\phi }_2)-\frac{G{M}_E}{c^2{r}_1}f(r_1,{\theta }_1,{\phi }_1)$$

$$y_{\text{dcs}}^{\text{gr}}=y_{\text{do}}^{\text{gr}}-y_{\text{up}}^{\text{gr}}\simeq 2\frac{G{M}_E}{c^2{r}_4}f(r_4,{\theta }_4,{\phi }_4)-2\frac{G{M}_E}{c^2{r}_3}f(r_3,{\theta }_3,{\phi }_3).$$

$$y_{\text{up}}^{\text{dop}}$$

$$y_{\text{dcs}}^{\text{dop}}=-\frac{2{\mathbf{D}}_{34}\cdot {\mathbf{a}}_g(t_4)}{c^2}+2\frac{({\mathbf{n}}_{34}\cdot {\mathbf{v}}_{34}(t_4){)}^2}{c^2}-2\frac{({\mathbf{v}}_g(t_4)\cdot {\mathbf{v}}_{34}(t_4))}{c^2}+y_{\text{dcs}}^{\text{dop3}},$$

$$y_{\text{up}}^{\text{atmo}}=\frac{d}{cdt}{\int }_{{\mathbf{x}}_1}^{{\mathbf{x}}_2}m({ϵ}_u)\left(\frac{{\rho }_a}{{\rho }_{as}}{N}_{ga}+\frac{{\rho }_w}{{\rho }_{ws}}{N}_{gw}\right)dz,$$

$$y_{\text{up}}^{\text{atmo}}=\frac{d}{cdt}\left[10^{-6}m({ϵ}_u)\left(4.16579\frac{f_h(\lambda )}{f(\phi ,H)}{P}_s+(5.316f_{nh}(\lambda )-3.759f_h(\lambda ))\frac{e_s}{f(\phi ,H)}\right)\right],$$

$$y_{\text{dcs}}^{\text{atmo}}=\frac{d}{cdt}\left[10^{-6}(m({ϵ}_d)-m({ϵ}_u))\left(4.16579\frac{f_h(\lambda )}{f(\phi ,H)}{P}_s+(5.316f_{nh}(\lambda )-3.759f_h(\lambda ))\frac{e_s}{f(\phi ,H)}\right)\right],$$

$$\begin{array}{r}{\Delta }_{\text{up}}^{\text{Shap}}=\frac{2G{M}_E}{c^3}\ln \frac{r_1+r_2+R_{12}}{r_1+r_2-R_{12}}.\end{array}$$

$$\begin{split}y^{\text{Shap}}_{\text{up}}=&-\frac{GM_{E}(r_{1}+r_{2})}{c^{3}r_{1}r_{2}}\bigg{[}\left(\frac{2}{1+\boldsymbol{{N}}_{1}\cdot\boldsymbol{{N}}_{2}}-\frac{r_{1}-r_{2}}{r_{1}+r_{2}}\right)\boldsymbol{{N}}_{12}\cdot(\boldsymbol{{v}}_{g}(t_{1})-\boldsymbol{{v}}_{s}(t_{2}))\\ &+\frac{2R_{12}}{r_{1}+r_{2}}\frac{\boldsymbol{{N}}_{1}\cdot\boldsymbol{{v}}_{g}(t_{1})+\boldsymbol{{N}}_{2}\cdot\boldsymbol{{v}}_{s}(t_{2})}{1+\boldsymbol{{N}}_{1}\cdot\boldsymbol{{N}}_{2}}\bigg{]},\end{split}$$

$$y_{\text{dcs}}^{\text{Shap}}=y_{\text{do}}^{\text{Shap}}-y_{\text{up}}^{\text{Shap}}$$

$$u_{\text{tid}}^{\text{ext}}(\mathbf{x})=\sum _{b\ne E}\left(U_b({\mathbf{r}}_{bE}+\mathbf{x})-U_b({\mathbf{r}}_{bE})-\mathbf{x}\cdot \nabla U_b({\mathbf{r}}_{bE})\right)$$

$$u_{\text{tid}}^{\text{int}}(\mathbf{x})=\frac{G{M}_E}{r}\sum _{l=0}^{\infty }\sum _{k=0}^l{\left(\frac{r_{0E}}{r}\right)}^l{P}_{lk}\left(\cos \theta \right)\left[\Delta C_{lk}\cos \left(k\phi \right)+\Delta S_{lk}\sin \left(k\phi \right)\right],$$

$$y_{\text{dcs}}^{\text{tid}}=y_{\text{do}}^{\text{tid}}-y_{\text{up}}^{\text{tid}}\simeq 2\left(\frac{u_{\text{tid}}({\mathbf{x}}_4)}{c^2}-\frac{u_{\text{tid}}({\mathbf{x}}_3)}{c^2}\right).$$

$$y_{\text{dcs}}^{\text{cs}}=-\frac{{\mathbf{n}}_{12}\cdot {\mathbf{a}}_{sg}(t_4)}{c}\delta t-\frac{{\mathbf{v}}_{12}^2(t_4)}{c{D}_{12}}\delta t+\frac{({\mathbf{n}}_{12}\cdot {\mathbf{v}}_{12}(t_4){)}^2}{c{D}_{12}}\delta t.$$

$$y_{\text{dcs}}^{\text{otd}}=-\frac{{\mathbf{n}}_{34}\cdot {\mathbf{a}}_{sg}(t_4)}{c}\Delta t.$$

$$S(f_F)=\sum _{n=-2}^2{h}_n{f}_F^n$$

$$y_{\text{up}}=\frac{f_2}{f_1}$$

$$\displaystyle\quad-\frac{\big{(}\boldsymbol{N}_{12}\cdot\boldsymbol{v}_{12}\big{)}}{c^{3}}\Bigg{[}\big{(}\boldsymbol{N}_{12}\cdot\boldsymbol{v}_{g}(t_{1})\big{)}^{2}+\frac{\boldsymbol{v}_{s}^{2}(t_{2})}{2}-\frac{\boldsymbol{v}_{g}^{2}(t_{1})}{2}\Bigg{]}+O(c^{-4}),$$

$$y_{\text{do}}=\frac{f_4}{f_3}$$

$$\displaystyle\quad-\frac{\big{(}\boldsymbol{N}_{34}\cdot\boldsymbol{v}_{34}\big{)}}{c^{3}}\Bigg{[}\big{(}\boldsymbol{N}_{34}\cdot\boldsymbol{v}_{s}(t_{3})\big{)}^{2}+\frac{\boldsymbol{v}_{g}^{2}(t_{4})}{2}-\frac{\boldsymbol{v}_{s}^{2}(t_{3})}{2}\Bigg{]}+O(c^{-4}),$$

$${\mathbf{R}}_{12}=-{\mathbf{R}}_{34}+{\mathbf{v}}_g\left(t_4\right)\cdot T_{14}-\frac{1}{2}{\mathbf{a}}_g\left(t_4\right)\cdot T_{14}^2-\Delta {\mathbf{r}}_s\left(\Delta t\right),$$

$${\mathbf{N}}_{12}$$

Full text

I introduction

General relativity (GR) is the cornerstone for describing the currently physical world at the macroscopic scale and has undergone countless experimental tests since its birth Will (2014). Although GR is currently regarded as the preeminent theory of gravity, it faces both theoretical and observational challenges. Theoretically, the pursuit of a Grand Unified Theory that can describe all fundamental phenomena persists Burgess (2004); Hossenfelder (2013); Langacker (1981). Observationally, the observational evidences accumulated over the past few decades, including intriguing puzzles of the dark matter and dark energy, pose potential challenges to GR Bertone and Hooper (2018); Peebles and Ratra (2003). Thus, the precise test of the validity of GR constitutes a critical step toward understanding fundamental physics.

Gravitational redshift is one of the most classical experiences of general GR, which predicts that a clock at a higher altitude on Earth run faster than one at a lower altitude. Experimental verification of gravitational redshift began with a series of landmark Pound-Rebka-Snider experiments in the 1960s, achieving a 1-10$\%$ precision Pound and Rebka (1959, 1960); Pound and Snider (1965). Gravity Probe A (GPA) improved precision to the level of $1.4\times 10^{-4}$ Vessot et al. (1980), and the space VLBI RadioAstron (RA) mission reported a test of the uncertainty $3.3\times 10^{-4}$ Nunes et al. (2023). Recently, the space-based experiment of Galileo satellites and ground-based experiment of Tokyo Skytree obtained uncertainties of $2.48\times 10^{-5}$ Delva et al. (2018); Herrmann et al. (2018) and $9.1\times 10^{-5}$ Takamoto et al. (2020), respectively. The remarkable advancements in atomic clocks and time-frequency transfer technologies over the past decade have reinvigorated gravitational redshift tests Giorgetta et al. (2013); Caldwell et al. (2024); Shen et al. (2022); Caldwell et al. (2025) and many space missions are ongoing or researching, such as Atomic Clock Ensemble in Space (ACES) Meynadier et al. (2018); Savalle et al. (2019), Space Optical Clock (SOC) Bongs et al. (2015), China Space Station (CSS) Shen et al. (2023), China’s Lunar exploration project CLEP Qin et al. (2024), NASA Discovery-class mission VERITAS De Marchi et al. (2023), and fundamental physics with a state-of-the-art optical clock in space (FOCOS) Derevianko et al. (2022). Furthermore,, there are also many attractive studies on spaceborne and labbased clocks for redshift Zhang et al. (2024); Litvinov (2024); Grotti et al. (2024); Zheng et al. (2023); Terno et al. (2023); Wei et al. (2025); Yu et al. (2025).

The space-based experiments have highly promising potential for improving tests of gravitational redshift. A paramount challenge among these is the first-order Doppler effect with the order of $10^{-6}$ to $10^{-5}$. For this challenge, the triple-link Doppler Cancellation Scheme (DCS) is typically employed in the experiments. By combining one-way and two-way Doppler information provided by three link signals, the first-order Doppler effect is suppressed to the second order while the gravitational redshift signal is retained. Then, combining the satellite’s orbit determination and speed measurement, a highly accurate test of gravitational redshift was conducted. The effectiveness of this method has been rigorously verified by experiments such as GPA and RA experiments Vessot et al. (1980); Nunes et al. (2023).

Here, we demonstrate a dual one-way frequency comparison scheme, named as dual one-way DCS, for testing gravitational redshift test in the space-based experiments. The scheme requires only two links: an uplink where Doppler information is recorded onboard the satellite, and a downlink where Doppler information is recorded at the ground station. By combining the Doppler information from both the uplink and downlink one-way frequency transfer during data processing, the first-order Doppler effect is effectively suppressed while the gravitational redshift signal is preserved and enhanced by a factor of 2. This scheme also enables precise gravitational redshift measurements. Compared to triple-link Doppler cancellation scheme, this scheme not only reduces the engineering complexity and cost by eliminating one signal link but also enhances the measurable gravitational redshift signal. By achieving an enhanced signal-to-noise ratio, this approach delivers a twofold improvement in gravitational redshift tests relative to conventional triple-link scheme. The trade-off for these advantages is the requirement for high-precision clock synchronization between the satellite and ground station. Fortunately, the clock synchronization has been demonstrated with high-precision accuracy Chen et al. (2024); Caldwell et al. (2023); Deschênes et al. (2016); Bergeron et al. (2019); Delva et al. (2012); Sinclair et al. (2019), which indicates the potential feasibility of this scheme. Beyond testing GR, enhanced redshift sensitivity could also constrain some new physics beyond standard paradigms, such as long-range fifth forces, dark matter, or Lorentz violation Brzeminski et al. (2022); Lévy and Uzan (2025); Derevianko and Pospelov (2014); Roberts et al. (2017); Qin et al. (2025); Castello et al. (2024).

This paper is structured as follows: Section II introduce the gravitational redshift and dual one-way Doppler Cancellation Scheme. Section III introduces the measurements of dual one-way DCS and discusses the various effects, such as Doppler effect, gravitational redshift, atmospheric frequency shift, clock synchronization etc. Section IV applied dual one-way DCS to analyse the test of gravitational redshift in some missions. Section V gives the conclusion.

II gravitational redshift and Doppler Cancellation Scheme

According to Einstein’s general relativity, a gravitational field can alter the rate of clocks. Near a massive body like Earth, a clock at higher altitude runs faster than one at lower altitude, a phenomenon known as gravitational redshift. A simple and convenient violation of gravitational redshift can be written as

[TABLE]

where $\Delta U$ is the difference of gravitational potential, $c$ is the vacuum speed of light , and $\alpha$ describes the departure from the predicted gravitational redshift that vanishes in the GR case. This effect can be tested by comparing the frequencies or times of two clocks with electromagnetic signal exchange. It can be seen that the greater the gravitational potential difference, the more sensitive it is to the gravitational redshift. Satellite-based clocks experience a significant gravitational potential difference relative to ground clocks. The configuration of the satellite-ground clock comparison is highly sensitive to the test of gravitational redshift. Utilizing state-of-the-art optical atomic clocks, tests of gravitational redshift in such configurations can achieve exceptional precision.

We consider the dual one-way DCS, as illustrated in Fig. 1. In this scheme, the satellite-ground clock comparison using dual one-way frequency transfer links comprises an uplink frequency comparison (red line) and a downlink frequency comparison (blue line). Ground clock $A$ and satellite clock $B$ perform frequency comparisons via optical signals. The process operates as follows:

Uplink clock comparison: At coordinate time $t_{1}$, the ground clock $A$ at position $\boldsymbol{{x}}_{1}$ emits an optical signal with frequency $f_{1}$. This signal propagates freely through space and is received at coordinate time $t_{2}$ by satellite clock $B$ at position $\boldsymbol{{x}}_{2}$. The satellite clock $B$ measures the received frequency as $f_{2}$, and records uplink data, $y_{\text{up}}$.

Downlink clock comparison: At coordinate time $t_{3}$, the satellite clock $B$ at position $\boldsymbol{{x}}_{3}$ sends a light signal towards the ground station with frequency $f_{3}$. After free-space propagation, ground clock $A$ at position $\boldsymbol{{x}}_{4}$ receives this signal at coordinate time $t_{4}$. The ground clock $A$ measures the received frequency as $f_{4}$, and record downlink data, $y_{\text{do}}$. Here, a time delay $\Delta t=t_{3}-t_{2}$ between coordinate times $t_{2}$ and $t_{3}$ is introduced. Considering clock synchronization, this time delay $\Delta t$ is very small. Ideally, this delay is 0.

This scheme thus generates two independent sets of frequency comparison data, one from the uplink and one from the downlink. A suitable combination of these measurements enables the determination of the gravitational redshift.

III The measurement of Doppler Cancellation Scheme

We consider a clock comparison experiment with two clocks in the Geocentric Celestial Reference System (GCRS). In order to study the measurement of dual one-way DCS, it is needed to give the clock comparisons of uplink and downlink. We take the uplink as an example for calculating the clock comparison, and the calculation for downlink is similar. Generally, the difference in frequencies between the two clocks can be characterized by the frequency shift $f_{2}/f_{1}$. The general form of the one-way frequency shift of uplink is given by

[TABLE]

where $\tau_{1}$ and $\tau_{2}$ are the corresponding proper times of coordinate times $t_{1}$ and $t_{2}$, respectively. On the right side of the equation, the first two terms are clock-dependent term containing gravitational redshift, time dilation and tidal effects; the third term is the path-dependent term, which includes Doppler effect, atmospheric effect, etc.

Firstly, we calculate the gravitational redshift term or clock-dependent term in the GCRS. The clock-dependent term can be calculated using the invariance of Riemannian space-time intervals. Considering the metric tensor of the GCRS, this term becomes

[TABLE]

where $U_{E}$ is the gravitational potential of the Earth, $u_{\text{tid}}$ is the tidal potential (the main contributions come from the Moon and the Sun), $\boldsymbol{{v}}$ is the velocity of the ground station or satellite, $\boldsymbol{{v}}_{g}(t_{1})$ is the velocity of ground station at time $t_{1}$, $\boldsymbol{{v}}_{s}(t_{2})$ is the velocity of satellite at time $t_{2}$.

Then, we consider the path-dependent term which is related to the transmission path of signal. This term can be calculated by the coordinate times between signal’s emission and reception. In the uplink, the signal’s emission time and reception time can be written as

[TABLE]

where $T_{12}=R_{12}/c=|\boldsymbol{{x}}_{2}-\boldsymbol{{x}}_{1}|/c$ is the travel time of the signal in Euclidean space, $\Delta_{12}^{\text{atm}}$ is the atmospheric delay, and $\Delta_{12}^{\text{Shap}}$ is the Shapiro gravitational delay. From the time interval Eq.(4) between the transmission and reception, the frequency shift of path-dependent term of uplink becomes

[TABLE]

where the second term represents the Dopper shift caused by the relative motion between ground station and satellite, the third term is the atmospheric frequency shift, and the fourth term is the gravitational frequency shift caused by Shapiro time delay.

From Eqs.(2) to (5), we can express the frequency comparison (or one-way frequency transfer) of the the uplink as

[TABLE]

In the general satellite-ground clock comparison, the velocity of satellites is about $3\sim 8$ km/s, and the first-order Doppler shift is the order of $10^{-5}$. For evaluating the frequency comparison to the level of $10^{-18}$, the accuracy requirement for speed measurement is about or better than $10^{-9}$ m/s. It is not achievable with the current technology. So, a Doppler cancellation scheme is applied to suppress Doppler shift to the second order. The atmospheric frequency shift can reach the order of $10^{-14}$, and the gravitational frequency shift also can reach the level of $10^{-14}$. The gravitational redshift term depends on the satellite’s orbit and the location of the ground station. For the ACES and CCS missions, the gravitational redshift is about $4\times 10^{-11}$. For the satellite-ground clock comparisons at higher orbits, the gravitational redshift can reach several $10^{-10}$. With the clock comparison at the level of $10^{-18}$, the test for gravitational redshift can reach $10^{-6}$ to $10^{-9}$. The two most significant challenges are: to achieve the $10^{-18}$ level satellite-ground clock comparisons in the engineering and to evaluate all the effects at the $10^{-18}$ level.

After a similar processing to uplink, we obtain the frequency comparison of the the downlink

[TABLE]

This equation can be used to estimate various effects of the downlink. Comparing to the frequency shift of uplink (6), the downlink contains almost the same first-order Doppler effect and the opposite gravitational redshift effect. Based on this characteristic, we can construct the measurement of dual one-way DCS, which can be written as follows

[TABLE]

On the right handle of Eq.(8), the first term represents the gravitational redshift effect. Within the Doppler cancellation scheme, the gravitational redshift is twice as large as that in a one-way frequency transfer. This enhancement is beneficial for improving the precision of gravitational redshift tests. The second term represents the residual Doppler effect with the $c^{-2}$ order and higher order. Due to the imperfect symmetry between the uplink and downlink paths, the first-order Doppler effect is not completely canceled after Doppler cancellation scheme, but is instead suppressed to a second order. The third and fourth terms represent the residual atmospheric delay effect and the gravitational time delay effect, respectively. Similar to the first-order Doppler effect, after Doppler cancellation scheme, their influences are significantly smaller than that in a one-way frequency transfer. The fifth term represents the tidal effect. The sixth term represents the influences of clock synchronization and onboard time delay. The last term is the clock frequency noise. In the following, we analyze and evaluate each term of these effects separately.

III.1 Gravitational redshift

The gravitational redshift is the primary effect targeted in the satellite-ground clock comparisons for testing general relativity’s gravitational redshift prediction, which is predominantly contributed by the Earth’s gravitational field. High-precision clock comparisons necessitate a high-accuracy evaluation of the Earth’s gravitational field. The current Earth gravitational field model is given by Earth Gravitational Model 2008 (EGM2008) Pavlis et al. (2012). The Earth’s gravitational potential is expanded in terms of spherical harmonic coefficients as follows

[TABLE]

where $GM_{E}$ is the Earth’s gravitational constant, $P_{lm}(\cos\theta)$ are the associated Legendre functions, $r_{0E}$ is the Earth’s equatorial radius, $\theta$ is latitude, $\varphi$ is longitude, $C_{lk}$ and $S_{lk}$ are the spherical harmonic coefficients determined through some gravity data. For convenience, we represent the gravitational potential $U_{E}(\boldsymbol{{x}})$ of EGM2008 in form of $GM_{E}f(r,\theta,\varphi)/r$. The uncertainty of EGM2008 in calculating the gravitational potential is about $1$ m2/s2, which corresponds to a gravitational redshift of $1\times 10^{-17}$. By combining the gravity data or more precise measurements, the uncertainty may reach the level of $0.1$ m2/s2.

By combining the EGM2008 model and one-way frequency transfer (6), the gravitational redshift for the uplink can be derived as

[TABLE]

where $f(r_{1},\theta_{1},\varphi_{1})$ and $f(r_{2},\theta_{2},\varphi_{2})$ represent the EGM2008’s expansions for positions $\boldsymbol{{x}}_{1}$ and $\boldsymbol{{x}}_{2}$, $r_{1}=|\boldsymbol{{x}}_{1}|$ and $r_{2}=|\boldsymbol{{x}}_{2}|$ are the positions of the ground station at coordinate time $t_{1}$ and satellite at coordinate time $t_{2}$, respectively. From the estimation of this equation, the gravitational redshift is the level of $10^{-11}$ to $10^{-10}$ for most satellite-ground clock comparison. For example, the gravitational redshift is about $-4\times 10^{-11}$ for ACES or CSS missions, and about $-5\times 10^{-10}$ for GPS case. With the state-of-the-art optical clocks, the test of gravitational redshift may reach $10^{-8}$.

Similar, the gravitational redshift of downlink $y^{\text{gr}}_{\text{do}}$ is given by $GM_{E}f(r_{4},\theta_{4},\varphi_{4})/(c^{2}r_{4})-GM_{E}f(r_{3},\theta_{3},\varphi_{3})/(c^{2}r_{3})$, which has same value as $-y^{\text{gr}}_{\text{up}}$. With this characteristic, the total gravitational redshift of Doppler cancellation scheme is given by

[TABLE]

In the measurement of dual one-way DCS, the magnitude of the gravitational redshift $y^{\text{gr}}_{\text{dcs}}$ is twice that of the downlink or uplink links. It implies that this scheme has an enhanced gravitational redshift, which is more sensitive to test the gravitational redshift. Considering satellite-ground clock comparisons with the optical clocks of identical accuracy, this scheme has the potential to enhance the testing precision by a factor of 2 compared to a conventional three-link DCS.

III.2 Doppler effect

The Doppler effect is the dominant effect in satellite-to-ground clock comparisons, arising from the relative velocity between the satellite and the ground station. This effect must be compensated for or suppressed, which is achieved by using a Doppler cancellation scheme. To study this effect and the efficacy of its suppression, we first consider the Doppler effect on the uplink. According to the Appendix.A, it can be expressed as:

[TABLE]

where $\boldsymbol{{N}}_{12}=\boldsymbol{{R}}_{12}/R_{12}$ is the unit vector of coordinate distance $\boldsymbol{{R}}_{12}$, and $\boldsymbol{{v}}_{12}=\boldsymbol{{v}}_{s}(t_{2})-\boldsymbol{{v}}_{g}(t_{1})$ is the relative velocity between satellite and ground station. Typically, the relative velocity between the satellite and ground station ranges from 3-8 km/s, corresponding to a Doppler shift on the order of $10^{-5}$. This is significantly larger than the other effects with several orders of magnitude. For the downlink, the corresponding Doppler frequency shift can be derived through an analogous process, achieved by substituting $1,2$ and $s,g$ in Eq. (12) with $3,4$ and $g,s$, respectively.

Combining first-order Doppler shifts of uplink and downlink, we can obtain the measurement of Doppler shift of the dual one-way DCS, which is the residual Doppler effect. However. in practical satellite-ground clock comparisons, experimental data or measurement quantities are typically recorded at the reception time of the light signal, such as times $t_{2}$ and $t_{4}$. This necessitates that all quantities within the Doppler shift should be expressed in terms of parameters referenced to the reception time. It is necessary to introduced some quantities at the reception time such as the instantaneous distance $\boldsymbol{{D}}_{34}=\boldsymbol{{x}}_{g}(t_{4})-\boldsymbol{{x}}_{s}(t_{4})$, satellite velocity $\boldsymbol{{v}}_{s}(t_{4})$, ground station velocity $\boldsymbol{{v}}_{g}(t_{4})$, and ground station acceleration $\boldsymbol{{a}}_{g}(t_{4})$ at coordinate time $t_{4}$. According to the Appendix.A, following appropriate mathematical treatment, the residual Doppler shift after Doppler cancellation scheme can be expressed as

[TABLE]

where $\boldsymbol{{n}}_{34}=\boldsymbol{{D}}_{34}/|\boldsymbol{{D}}_{34}|$ is the unit vector of the instantaneous distance $\boldsymbol{{D}}_{34}$, $\boldsymbol{{v}}_{34}(t_{4})=\boldsymbol{{v}}_{g}(t_{4})-\boldsymbol{{v}}_{s}(t_{4})$ represents the instantaneously relative velocity of satellite and ground station at coordinate time $t_{4}$. Note that all quantities in Eq.(13) are expressed as the values at time $t_{4}$. From Eq.(13), the residual Doppler shift is the order of $10^{-10}$, much smaller than first-order Doppler shift.

In addition, it can be seen that the difference between instantaneous distance $\boldsymbol{{D}}_{34}$ and coordinate distance $\boldsymbol{{R}}_{34}$ will lead to an error significantly exceeding the $10^{-18}$ level. In the experiment of clock comparison, the coordinate distance $\boldsymbol{{R}}_{34}$ and relativity velocity $\boldsymbol{{n}}_{34}$ depend on times $t_{3}$ and $t_{4}$, is in the real-time changing. It is very difficult to record the coordinate distance and corresponding unit vector $\boldsymbol{{n}}_{34}$ in the data and measurements. For example, in the clock comparison of ACES, the difference between $\boldsymbol{{D}}_{34}$ and $\boldsymbol{{R}}_{34}$ can lead to an estimated error of $5\times 10^{-18}$ in the residual Doppler effect. Therefore, in the equation for data processing, it is necessary to express all quantities as quantities related to the reception time, such as $\boldsymbol{{n}}_{34}$ and $\boldsymbol{{v}}_{34}(t_{4})$.

III.3 Atmospheric frequency shift

In satellite-ground clock comparisons, the light signals traversing the atmosphere experience an additional propagation delay, known as the atmospheric delay, which can be expressed as $\Delta_{\text{atmo}}=d_{\text{atmo}}/c$ with the atmospheric delay $d_{\text{atmo}}=\int Ndl$ where $N$ is the the group refractivity and the integration is calculated along the propagation path. The $d_{\text{atmo}}=m(\epsilon)d_{\text{atmo}}^{z}$ is typically modeled using the Marini-Murray model or the Mendes-Pavlis model Mendes and Pavlis (2004); Ciddor (1996), which incorporate a zenith propagation delay $d_{\text{atmo}}^{z}$ and a mapping function $m(\epsilon)$. Variations in the atmospheric delay induce an atmospheric frequency shift in the clock comparison. Considering the uplink, the atmospheric frequency shift can be expressed as

[TABLE]

where $\epsilon_{u}$ is the elevation angle of uplink signal, $\rho_{a}$ is the dry air density, $\rho_{as}$ is the standard dry air density at 15 ∘C and 101325 Pa, $N_{ga}$ is the dry air group refractivity, $\rho_{w}$ is the water vapor density, $\rho_{ws}$ is the pure water vapor density at 20 ∘C and 133 Pa, and $N_{gw}$ is the water vapor group refractivity. Combining the information of ground station and Eq.(14), the atmospheric frequency shift of the uplink can be expressed as

[TABLE]

where $\lambda$ is the vacuum wavelength ($\mu$m) of the signal, $\lambda$ is the latitude of ground station, $H$ is the height (km) of ground station, $P_{s}$ is the station’s surface barometric pressure (Pa), $e_{s}$ is the station’s surface water vapor pressure. The $f_{h}(\lambda)$, $f(\varphi,H)$ and $f_{nh}(\lambda)$ are the integration functions in Eq.(14), whose expressions can be found in Ref.Mendes and Pavlis (2004). For a simple estimation, the atmospheric frequency shift of the one-way link can reach the level of $10^{-14}$. From Eq.(15), the atmospheric frequency shifts have almost same value for the uplink and downlink signals.

Considering dual one-way DCS, the residual atmospheric frequency shift is given by

[TABLE]

where $\epsilon_{d}$ is the elevation angle of uplink signal. This equation can be used to estimate the atmospheric influences in the clock comparisons. Since the asymmetry of the uplink and downlink is small, the residual atmospheric frequency shift is much smaller than $10^{-14}$.

III.4 Gravitational frequency shift

From Einstein’s general relativity, the gravitational field can lead to an extra time delay in light travel time, known as Shapiro delay. In the clock comparison, the variations in the Shapiro delay induce a gravitational frequency shift. Assuming the metric is given, the standard methods to derive Shapiro delay are solving the null geodesic equation or the eikonal equation Zschocke (2016); Ashby and Bertotti (2010), and different approaches are alternative based on the Synge World function and time transfer function Le Poncin-Lafitte et al. (2004); Teyssandier and Le Poncin-Lafitte (2008); Hees et al. (2014, 2014). In the post-Newtonian approximation, the metric is expanded as $g^{\alpha\beta}=\eta^{\alpha\beta}+h^{\alpha\beta}$ with $\eta_{\alpha\beta}=$diag($-$1,+1,+1,+1), and the gravitational perturbation $h_{\alpha\beta}$. The Shapiro delay of light propagation in the gravitational field can be calculated by $\Delta^{\text{Shap}}=({R}/{2c})\int_{0}^{1}\left(h^{00}-2n^{i}h^{0i}+n^{i}n^{j}h^{ij}\right)_{x_{(l)}}dl$, where the integral is calculated along the straight line between the ground station and satellite. Considering the uplink signal, the Shapiro delay can be calculated by the uplink path as

[TABLE]

From this equation, the Shapiro time delay contributed by the Earth can reach the level of $10^{-10}$. From Eqs. (6) and (17), the gravitational frequency shift of the uplink is given by

[TABLE]

where $\boldsymbol{{N}}_{1}=\boldsymbol{{x}}_{1}/r_{1}$ and $\boldsymbol{{N}}_{2}=\boldsymbol{{x}}_{2}/r_{2}$. For a crude estimation, the gravitational frequency shift of the uplink can reach the level of $10^{-14}$. One can also derives the gravitational frequency shift $y^{\text{gr}}_{\text{do}}$ for the downlink by substituting 1,2 and $g,s$ in Eq.(18) with 3,4 and $s,g$, respectively. The values of gravitational frequency shift are almost same for the uplink and downlink signals.

In the measurement of dual one-way DCS, the residual gravitational frequency shift is given by

[TABLE]

This equation can be used to calculate the influences of Shapiro delay in the clock comparisons. Since the Shapiro time delays of the uplink and downlink seem to be the same, the residual gravitational frequency shift is much smaller than $10^{-14}$.

III.5 Tidal effects

Similar to the gravitational redshift, tidal potentials also induce slight variations in clock frequency. The tidal potential $u_{\text{tid}}(\boldsymbol{{x}})$ can be decomposed into two components, the external mass component $u^{\text{ext}}_{\text{tid}}(\boldsymbol{{x}})$ and the internal mass component $u^{\text{int}}_{\text{tid}}(\boldsymbol{{x}})$. The external part $u^{\text{ext}}_{\text{tid}}(\boldsymbol{{x}})$ is primarily contributed by the tidal potentials of the Sun and Moon, in the conventional Newtonian tidal potential form. The internal part $u^{\text{int}}_{\text{tid}}(\boldsymbol{{x}})$ is mainly from Earth’s tidal deformations, which comprises solid Earth tides, ocean tides, and pole tides. Solid Earth tides are the main influence in the internal mass part, which generates surface displacements of approximately 10-30 cm/day. The external and internal tidal potentials can be expressed as Petit and Luzum (2010)

[TABLE]

and

[TABLE]

where $U_{b}$ is the Newtonian gravitational potential of body $b$, $\boldsymbol{{r}}_{bE}$ is the vector connecting the center of mass of body $b$ to that of the Earth, $\nabla U$ is the gradient of potential, $\Delta C_{lk}$ and $\Delta D_{lk}$ are the changes of normalized potential coefficients due to Earth tides. From Eqs.(20) and (21), the magnitude of the external-mass tidal potentials increases with satellite orbital altitude, while the internal-mass tidal potential decreases correspondingly. The influence of tidal potentials can be characterized using Love numbers and more detailed computations can be found in Ref.Petit and Luzum (2010); Qin et al. (2020).

In the uplink signal, the tidal effect can be expressed as $y^{\text{tid}}_{\text{up}}=(u^{\text{int}}_{\text{tid}}(\boldsymbol{{x}}_{2})-u^{\text{int}}_{\text{tid}}(\boldsymbol{{x}}_{1}))/c^{2}$. From an estimation, the tidal frequency shift in uplink can reach the level of $10^{-17}$, and for medium/high Earth orbit (MEO/GEO) satellites, it may even reach the level of $10^{-16}$. For downlink paths, the tidal frequency shift has an opposite value as that of uplink.

In the measurement of dual one-way DCS, the total tidal frequency shift is expressed as

[TABLE]

The total tidal frequency shift is twice that of a one-way link. From Eqs.(20) and (21), external-mass tidal effects dominate for case of MEO/GEO satellites.

III.6 Clock synchronization and onboard time delay

In the dual one-way DCS, the frequency shift data of the uplink is recorded onboard the satellite, while downlink data is recorded at the ground station. Gravitational redshift tests based on the dual one-way DCS rely on combining uplink and downlink data. So clock desynchronization between ground station and satellite $\delta t$ will introduce errors in measurement of dual one-way DCS. Referenced to ground station time $t_{4}$, the clock desynchronization $\delta t$ induces an estimation error in the uplink Doppler shift as

[TABLE]

This error $y^{\text{cs}}_{\text{dcs}}$ propagates directly to a frequency shift error in the DCS gravitational redshift test. This equation quantify clock synchronization requirements for the dual one-way DCS. For example, for ACES/CCS missions, the clock comparison with the accuracy of $10^{-16}$ requires clock synchronization with the level 10 ns, and the clock comparison with the accuracy of $10^{-18}$ requires clock synchronization with the level $0.1$ ns. The remarkable advancements in time and frequency transfer implies that the clock synchronization with of femtosecond level has the potential to be realized in the near future Chen et al. (2024); Caldwell et al. (2023); Deschênes et al. (2016); Bergeron et al. (2019); Delva et al. (2012); Sinclair et al. (2019).

Additionally, onboard time delay $\Delta t=t_{3}-t_{2}$ introduces a DCS frequency shift as

[TABLE]

This frequency shift is dependent on the relative acceleration of the ground station and satellite. Under ideal conditions ($\Delta t=0$), this term vanishes. Typically, the accuracy of onboard time delay $\Delta t$ is better than the level of clock synchronization. For $0.1$ ns clock synchronization in the ACES/CCS missions, the corresponding frequency shift is the level of $10^{-18}$. For a rough estimation, the clock synchronization at the 10 ns nd 0.1 ns levels contribute uncertainties of the order of $10^{-6}$ and $10^{-8}$ for the gravitational redshift test.

III.7 Clock noise

The clock noise affects satellite-ground clock comparisons. The frequency output of an atomic clock can be modeled as superposition of statistically independent noises. Characterizing the stochastic noise properties, the one-sided power spectral density of fractional frequency fluctuations can be expressed as Lesage and Audoin (1979)

[TABLE]

where $f_{F}$ is the Fourier frequency variable, and $h_{n}$ is the noise intensity coefficient for each $f_{F}$. There are five fundamental noise types, white phase modulation noise ($f_{F}^{2}$), flicker phase modulation noise ($f_{F}^{1}$), white frequency modulation noise ($f_{F}^{0}$), flicker frequency modulation noise ($f_{F}^{-1}$), and random walk frequency modulation noise ($f_{F}^{-2}$). Based on the five types of clock noises, we can model and analyse the atomic clock frequency in the precision metrology application.

IV Application in the Testing Gravitational Redshift

The dual one-way DCS has an enhanced gravitational redshift. This scheme can be applied to the space missions of gravitational redshift test, such as ACES, CSS or other satellite-ground clock comparisons. Compared to conventional three-link DCS, our proposal scheme presents two advantages: a factor-of-2 enhancement in gravitational redshift and 50$\%$ reduction in the signal-link engineering requirement, as described in TABLE.1.

Presently, the accuracy and stability of the onboard satellite atomic clocks in space missions are approximately $10^{-16}$. Nevertheless, the current state-of-the-art optical atomic clocks on the ground can achieve an accuracy and stability level of $10^{-18}$. Therefore, we estimate the tests of gravitational redshift of clock comparisons with a level accuracy and stability of $1\times 10^{-16}$ and $1\times 10^{-18}$, and apply the dual one-way DCS in the ACES/CCS and Geosynchronous Earth Orbit (GEO) satellites. In addition, for realistic experimental implementations, it is set that the satellite position and velocity determination accuracies of 0.1 m and 0.1 mm/s respectively and the position and velocity determination of ground station exceeds these specifications. The dominant errors in relative position and velocity are satellite-dominated.

For satellite-ground clock comparisons at $1\times 10^{-16}$ accuracy and stability levels, we analyze ACES/CCS and GEO configurations. In the ACES/CCS mission (about 400 km altitude), the measurement of dual one-way DCS yields a gravitational redshift signal about $8\times 10^{-11}$ and residual Doppler frequency shift about $1\times 10^{-9}$. From the position and velocity error propagations, the uncertainty of residual Doppler frequency shift is smaller than $7\times 10^{-17}$. Systematic uncertainties from atmospheric delays, gravitational time delays, and tidal effects are much smaller than $1\times 10^{-16}$. These enable gravitational redshift tests at $1\times 10^{-6}$ level for the dual one-way DCS. In conventional triple-link DCS, the uncertainty of gravitational redshift is $(2\sim 3)\times 10^{-6}$. This demonstrates that under identical clock performance, dual one-way DCS yields a twofold improvement in gravitational redshift test relative to triple-link DCS. For geostationary orbits (about 36,000 km altitude), dual one-way DCS measurements exhibit a gravitational redshift about $1.2\times 10^{-9}$ and residual Doppler shift about $1\times 10^{-11}$. Considering the determination of position and velocity, the residual Doppler shift is smaller than $1\times 10^{-17}$, which significantly below clock noise. The other systematic uncertainties also are much smaller than $1\times 10^{-16}$. This GEO configuration demonstrates a potential for gravitational redshift tests at the $1\times 10^{-7}$ level, leveraging the enhanced signal-to-noise ratio at higher orbital altitude.

For satellite-ground clock comparisons at the $1\times 10^{-18}$ level, the error budget exhibits distinct characteristics. In the ACES/CCS configuration, both residual Doppler shift and gravitational redshift uncertainties are the level of $10^{-17}$ exceeding clock noise. Through improving satellite velocity measurements, the residual Doppler shift errors can be suppressed to $10^{-18}$, and other systematical uncertainties remain sub-$10^{-17}$. This enables gravitational redshift tests at the level of $10^{-7}$, mainly limited by Earth’s gravitational field model. Conversely, in the GEO case, residual Doppler shift uncertainty is about $10^{-18}$ level but gravitational redshift uncertainty persists at $10^{-17}$ level , yielding tests at $10^{-8}$ level, similarly constrained by Earth’s gravitational model. Consequently, advancing Earth gravitational field accuracy below $0.1$ m2/s2 is imperative for future $10^{-18}$ level clock comparisons, while such measurements reciprocally refine geopotential coefficients.

V conclusion

In conclusion, this work establishes a dual one-way DCS as a competitive alternative to triple-link DCS for gravitational redshift tests. It is composed of two signal links, an uplink and a downlink, with Doppler data recorded independently at the satellite and ground station, respectively. The Doppler cancellation measurement is established by coherent combination of these measurements. Based this dual one-way DCS, we analyse the influences of Doppler frequency shift, gravitational redshift, atmospheric shift, Shapiro shift, tidal shift, clock synchronization and clock noise. Our analysis demonstrates this dual one-way DCS can achieve three advantages: $i$. full suppression of first-order Doppler frequency shift to second order via coherent data combination; $ii$. a factor-of-2 enhancement in the measurable gravitational redshift signal, directly improving test sensitivity; $iii$. significant engineering simplification by eliminating one link signal, reducing mission cost and complexity.

To demonstrate this method, we study the gravitational redshift tests in the LEO (ACES/CSS) and GEO satellites. In the DCS measurement, the enhanced gravitational redshift signals can reach $8\times 10^{-11}$ and $1\times 10^{-9}$ for the LEO and GEO satellites, which potentially enables redshift tests at the $10^{-6}$ to $10^{-8}$ level. Systematic error budgets for LEO and GEO configurations confirm residual Doppler, atmospheric, Shapiro, and tidal shifts are controllable below $10^{-17}$-$10^{-18}$ levels with current technologies. The critical requirement is sub-nanosecond clock synchronization, which is feasible using high-precision time-transfer transfers, with 0.1 ns enabling $10^{-18}$ satellite-ground clock comparisons.

Future applications include next-generation missions targeting redshift tests at $10^{-8}$ level, where the amplified signal could probe non-metric gravity theories, fifth forces, or dark matter interactions. Achieving this potential necessitates improved Earth gravitational field models ($\delta U_{E}<0.1$ m2/s2) and picosecond-level synchronization schemes. This architecture has a potential to enhance reliability for space-based fundamental physics experiments.

VI Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grants No.12305062), Strategic Priority Research Program on Space Science, the Chinese Academy of Sciences (XDA30040400), and Fundamental Research Funds for the Central Universities, Sun Yat-sen University.

Appendix A Doppler frequency shift in the dual one-way DCS

This Appendix presents several relationships and results for the Doppler frequency shift in the dual one-way DCS. The one-way frequency transfer can be characterized as $f_{A}/f_{B}=d\tau_{B}/d\tau_{A}$, and the standard GR prediction have been studied some researches Blanchet et al. (2001); Linet and Teyssandier (2002). Ignoring the influences of atmospheric delay and Shapiro delay, the one-way frequency transfers of uplink and down link can be expanded to the order $c^{-3}$

[TABLE]

and

[TABLE]

where $w$ represent the scalar potential that is composed of the Earth gravitational potential $U_{E}$ and tidal potential $u_{\text{tid}}$, $\boldsymbol{{N}}=\boldsymbol{{R}}/R$ is the vector of coordinate distance $\boldsymbol{{R}}$, and $\boldsymbol{{v}}_{lm}$ is the relative velocity.

From two one-way frequency transfers, the frequency measurement of dual one-way DCS is given by $y_{\text{dcs}}=y_{\text{do}}-y_{\text{up}}$. Generally, all quantities in $y_{\text{dcs}}$ should be expressed in terms of parameters referenced to the reception time $t_{4}$. It is necessary to expressed the unit vector $\boldsymbol{N}_{12}$ in terms of the unit vector $\boldsymbol{N}_{34}$. From Fig.1, the relationship between the coordinate distances of uplink and downlink is given by

[TABLE]

where $\Delta\boldsymbol{r}_{s}\left(\Delta t\right)=\boldsymbol{v}_{s}(t_{2})\Delta t+(1/2)\boldsymbol{a}_{s}(t_{2})\Delta t^{2}$ is the satellite displacement in the onboard time delay $\Delta t$. From this equation, we obtain

[TABLE]

We also need reexpress the velocity of the satellite and ground station at emission in terms of the quantities at reception time, as follows

[TABLE]

[TABLE]

In addition, introducing the instantaneous distance $\boldsymbol{{D}}_{34}=\boldsymbol{{x}}_{g}(t_{4})-\boldsymbol{{x}}_{s}(t_{4})$ with its unit vector $\boldsymbol{{n}}_{34}$, one can express the term $\boldsymbol{{N}}_{34}$ in terms of the $\boldsymbol{{n}}_{34}$ as

[TABLE]

Taking these relativistic expansions into Eq.(8), the measurement of dual one-way DCS can be reexpressed in terms of the quantities at reception time $t_{4}$. After a long computation, we obtain

[TABLE]

where the first term is the gravitational redshift, the second term represents the tidal influence, the third term is the $c^{-2}$-order residual Doppler frequency shift, the fourth term is the $c^{-3}$-order residual Doppler frequency shift, and fifth term indicates the influences of the onboard time delay $\Delta t=t_{3}-t_{2}$ . The $y^{\text{gr}}_{\text{dcs}}$ is given by

[TABLE]

The is $y^{\text{tid}}_{\text{dcs}}$ given by

[TABLE]

The $y^{\text{dop2}}_{\text{dcs}}$ is given by

[TABLE]

where $\boldsymbol{{v}}_{34}(t_{4})=\boldsymbol{{v}}_{g}(t_{4})-\boldsymbol{{v}}_{s}(t_{4})$ is the relative velocity between satellite and ground station at time $t_{4}$. The $y^{\text{dop3}}_{\text{dcs}}$ is given by

[TABLE]

The $y^{\text{otd}}_{\text{dcs}}$ is given by

[TABLE]