Quantum interferometry in external gravitational fields

TL;DR

This work develops a covariant framework for quantum interferometry in stationary space-times, unifying optical and matter-wave descriptions without assuming weak fields or invoking Einstein’s equations. By decomposing the space-time metric into lapse, shift, and a gauge-invariant spatial metric, it derives general phase-evolution formulas and ray-trajectory equations that depend on integrals of the lapse and shift along lightlike and timelike paths. The authors apply the framework to classic optical tests (Pound–Rebka, Tanaka, Stodolsky) and satellite proposals, as well as neutron and atomic fountain experiments, showing how gravity, redshift, and gravity gradients manifest as measurable phase shifts and how curvature enters only indirectly via gravity gradients or through Riemann tensor components. The framework provides exact, gauge-consistent predictions for phase shifts and detection probabilities, offering clear guidance for future high-precision tests of gravity with quantum probes and potential access to gravity-gradient–related curvature information. It also highlights practical challenges, such as isolating lapse-induced phases from Sagnac effects and managing optical losses, while outlining paths to extend the formalism to internal degrees of freedom and guided motion of massive particles.

Abstract

Current models of quantum interference experiments in external gravitational fields lack a common framework: while matter-wave interferometers are commonly described using the Schrödinger equation with a Newtonian potential, gravitational effects in quantum optics are modeled using either post-Newtonian metrics or highly symmetric exact solutions to Einstein's field equations such as those of Schwarzschild and Kerr. To coherently describe both kinds of experiments, this paper develops a unified framework for modeling quantum interferometers in general stationary space-times. This model provides a rigorous description and coherent interpretation of the effects of classical gravity on quantum probes.

Figures (7)

• Figure 1: Graphical illustration of different notions of isometric time-translations in flat space-time. The blue lines indicate the orbits of the group action $\Phi$. The central green surface is a plane of simultaneity for an inertial system; the surfaces above and below indicate how such a surface is deformed by different notions of time-translation: it is shifted in parallel under inertial motion (a), twisted under rotary motion (b), and tilted under hyperbolic motion (c).
• Figure 2: Simplified schematic of the fiber-optic interferometer proposed in Ref. 1983PhRvL..51..378T where a light ray is split into two paths that traverse fiber spools of equal length. The resulting phase shift at the output depends on the vertical separation between the two fiber spools via the redshift.
• Figure 3: Schematic representation of optical free-fall trajectories in a Mach–Zehnder interferometer, with dots marking which points along the two rays reach the point D simultaneously. If the system is inertial, interfering rays at D originate from the same point on the beam splitter A, which they pass at the same time (a). This no longer applies in uniformly accelerated interferometers, as is the case in gravimeters (b). Instead, the interfering rays must have a spatial and temporal offset at the first beam splitter A in order to reach the final beam splitter at the same space-time event. The gray area indicates the minimal beam width required to observe the gravitationally induced phase shift.
• Figure 4: Schematic representation of satellite experiments to measure the gravitationally induced phase shift on light. The setup \ref{['fig:satellite single Mach-Zehnder']} is a large-scale version of Tanaka’s setup (\ref{['s:Tanaka']}) and requires two uplinks. The setups \ref{['fig:satellite double Mach-Zehnder']} and \ref{['fig:satellite double Michelson']}, on the other hand, use a scheme similar to time-bin-encoding for which a single uplink suffices. All these setups, however, yield the same lapse-induced phase shift.
• Figure 5: Schematic representation of the neutron interferometer proposed by Overhauser and Colella in Ref. 1974PhRvL..33.1237O. In the absence of gravity the rays form a rhombus (a), but in the presence of gravity rays interfering at $D$ must have been injected into the interferometer at different heights (b). The dots indicate which points along the two rays reach the point D simultaneously and the gray region shows the minimal beam width required to observe phase shifts in the presence of gravity.