Scalar gravitational Aharonov–Bohm effect: Generalization of the gravitational redshift

The original proposal for the electromagnetic Aharonov–Bohm (AB) effect¹ focused on the scalar and vector potentials of the electromagnetic interaction. The magnetic effect is a phenomenon where a charged particle's wave function is affected by the magnetic vector potential, $A→$⁠, even when both the electric and magnetic fields are zero.¹ Underlying this effect is the general concept of geometric phase,² which is apparent in many areas of physics,³ including condensed matter physics,^4,5 optics,^6,7 fluid mechanics,⁸ etc. On the other hand, the original work also proposed the scalar-electric AB effect, where the electric scalar potential, V, creates a geometric phase over time, with zero spatial variation so the electric field is zero. Since the experimental setup for the vector-magnetic AB effect is much easier to realize, there have been several tests of the vector-magnetic AB effect.^9,10 However, no clean scalar-electric AB effect has ever been implemented.¹¹ 

In the standard scalar-electric AB proposal, charges are sent along different paths with a potential difference between them but with no electric fields.¹ The observational signature is a shift in the quantum interference pattern of charges, which evolves with time as the two path lengths undergo different potentials.¹ In contrast, a more recent proposal¹² places a quantum system inside a Faraday cage with a time-varying scalar potential. This effectively compares the quantum system as the potential is turned on and off (a chopping experiment), which highlights the temporal nature of the scalar effect. In an analogous way to the AC Stark effect,¹³ this setup calculates that several energy sidebands in the spectrum of the quantum system will be produced, rather than a shift of interference fringes as observed in the originally proposed setup.¹ 

Since the gravitational potential is a Newtonian scalar potential, its analog in the gravitational sector closely resembles the scalar electromagnetic AB effect as discussed in Refs. 12 and 14. A recent experimental verification of the scalar gravitational AB effect¹⁵ employed a method similar to the original AB experiment.¹ In this setup, a matter beam was split into two paths, with one path experiencing a different gravitational potential than the other, resulting in a shift in the interference pattern when the beams were recombined.

The analog temporal setup to measure the scalar gravitational AB effect was recently proposed in Ref. 14, which considered a quantum system in orbit around a massive body. To be sensitive to the gravitational AB effect, the orbit must have non-zero eccentricity, so geometric phase effects related to the time-varying gravitation potential will be present. Since the quantum system will be in free fall, by the equivalence principle implies that the gravitational field is locally screened. However, the time-varying gravitational potential will change the energy levels of a quantum system, which should develop sidebands, which are harmonics of the orbit frequency, which are the signature of the scalar Aharonov–Bohm effect.

Experiments that have the potential to measure such geometric phase effects include precision atomic clocks in space, such as the Atomic Clock Ensemble in Space (ACES) mission¹⁶ on board the International Space Station (ISS), as well as other missions that propose optical clocks in space.^17–19 Also, data from Galileo clocks in orbits with non-zero eccentricity^20,21 could also exhibit a signal. This means that instead of measuring an interference pattern in an interferometric setup, the focus of this type of experiment is to measure the energy shifts in the quantum system caused by the time-varying gravitational potential. In this type of experiment, comparisons are made with atomic clock transition onboard a satellite with one on Earth, as illustrated in Fig. 1. The ground clocks experience a relatively constant gravitational potential, whereas the orbiting clocks are subject to a time-varying potential. Currently, most atomic clocks in space are microwave-based, typically using hydrogen, rubidium, or cesium atoms. These clocks rely on hyperfine transitions, where the ground state has an electron anti-aligned with the nucleus and the excited state has it aligned, as depicted in Fig. 1.

Assuming our atomic clock is in a satellite in an elliptical orbit so that the distance between the satellite and center of the Earth changes with time, the gravitational potential will be of the form $Φg(t)=−GM r(t)−1$⁠, where G is the Newton's constant, M is the mass of the Earth about which our quantum system will orbit, and $r(t)$ is the time-dependent distance between the satellite and one focus of the orbit, i.e., the center of the Earth. Since our quantum system is in the orbit (free fall) in a satellite, then by the equivalence principle, the quantum system will be locally in a zero gravitational field, i.e., the field has been transformed by going to a free falling frame. Assuming the satellite's trajectory as a Keplerian orbit (as shown in Fig. 1), $r(t)−1=r0−1+(A/r02)cos(Ωt)$⁠, where $r0=(ra+rp)/2$ is the average value of the apogee $ra$ and perigee $rp$⁠, and the value of $A=(ra−rp)/2$ is the average difference.

Note, if $α=0$ only, the $n=0$ part of the expansion in Eq. (17) survives and there are no sidebands, and if $α≪1$ only, the $n=±1$ sidebands are significant, and equivalent to a single tone frequency modulation, which will occur for extremely small eccentricities. When $α>1$⁠, overmodulation occurs, and extra sidebands are created, which may be searched for as a sign of this effect. In the case when $α≫1$⁠, one finds suppressed sidebands for $n≪α$⁠, until at $n∼nmax≈α$⁠, there is a sharp increase in the value of $Jn(α)$⁠, and hence the power in these sidebands. In this case, for $n>nmax$⁠, $Jn(α)$ exponentially decays to zero, so that sidebands beyond $nmax$ do not contribute.¹⁴ Thus, Eq. (18) may be thought of as a generalization of the gravitational redshift and predicts extra sidebands depending on the modulation parameter, $α$⁠, given in Eq. (16).

The Atomic Clock Ensemble in Space (ACES) mission¹⁶ places microwave atomic clocks on the International Space Station (ISS), which could measure this effect. The International Space Station is an almost circular, low Earth orbit with orbital parameters given by: (i) perigee and apogee radius from the center of the Earth being $rp=6.800×106$ and $ra=6.810×106$ m, respectively, which corresponds to a perigee altitude of 400 km and apogee altitude of 410 km, given that the Earth's radius is $rE≈6400$ km. These values give an eccentricity of $eISS=7.34×10−4$⁠. The period of a satellite with this apogee/perigee is about $TISS≈$ 90 min or 5400 s giving an angular frequency of $ΩISS=2πTISS=1.2×10−3rads$ (or frequency of $fISS=1.85×10−4 Hz$⁠). Substituting these values into Eq. (16) gives $αISS=2.6×10−9fph$⁠, where $fph$ is in Hz. Two types of atomic clocks are onboard the ISS, an active hydrogen maser of frequency 1.42 GHz and the PHARAO (Projet d'Horloge Atomique par Refroidissement d'Atomes en Orbite) cesium clock of frequency 9.192 631 77 GHz,²⁴ which will have modulation sidebands created through the gravitational modulation index of $αH=3.7$ and $αCs=23.8$⁠, respectively.

Data from Global Positioning Systems clocks have already been used to test fundamental physics,^25,26 such as tests on special relativity and the hunt for dark matter. The Galileo satellites, part of the European Space Agency's Galileo navigation system, mostly follow nearly perfect circular orbits.²⁷ However, two satellites were accidentally placed in elliptical orbits and provided an opportunity to achieve the best limits on measuring gravitational redshift by comparing the onboard hydrogen maser clocks with ground-based clocks.^20,21 It is also possible that the effects of the gravitational AB effect exists in these data, as the DC term in Eq. (18) is equivalent to the redshift, which was measured in Refs. 20 and 21 at a precision of parts in $105$⁠, while the Aharonov–Bohm phase predicts extra sidebands. The orbital period of the satellite is 12.94 h (21.5  $μ$ Hz) with an eccentricity of $eG=0.162$⁠, with a perigee and apogee radius from the center of the Earth of $rp=23,445$ and $ra=32,510 km$⁠, respectively, so $r0=27,977.5$ and $AG=453.2$ km; thus, the gravitational modulation parameter for the Galileo orbit is $αG=1.2×10−6fph$⁠, where $fph$ is in hertz. For the hydrogen maser clock transition at 1.42 GHz, $αg∼1699$⁠, with a predicted peak modulation sideband at around 36.5 mHz. Note, that this orbit causes a modulation, $αG$⁠, which is about 460 times greater than the modulation index induced by the ISS orbit. To be able to measure this peak frequency sideband and avoid aliasing, by the Nyquist–Shannon sampling theorem, a comparison measurement must be made at twice the rate of frequency (i.e., at 73 mHz) or less. This means the averaging time set for the comparison for a single measurement should be on the order of 14 s or less.

We have extended the original proposal for an experimental tests of the gravitational AB effect,¹⁴ which predicted that a time-varying gravitational potential will cause energy level shifts of an atomic clock transition. We have shown that the predicted phenomena can be viewed as a generalization of the gravitational redshift, which predicts the correct constant value of the redshift, plus additional sidebands described by the Jacobi-Anger expansion. The experimental verification of the redshift has been a focused test of general relativity over the years;^25,26,28,29 in such experiments, additional Doppler effects due to the velocity of the satellite are known to reduce the measurable signal but do not cancel it. Therefore, Doppler shifts must be properly accounted for to accurately determine the measurable gravitational effect for clock comparisons between one in an eccentric orbit and a stationary ground clock, and this remains an area for future research.

The authors would like to thank Andrei Derevianko, Doug Singleton, Nathan Inan, and Raymond Chiao for valuable discussions on this topic. This work was funded by the Australian Research Council Centre of Excellence for Engineered Quantum Systems, CE170100009, and the Australian Research Council Centre of Excellence for Dark Matter Particle Physics, CE200100008.