⚛️ quantum physics

Gravitational time dilation in quantum clock interferometry with entangled multi-photon states and quantum memories

This paper proposes a memory-assisted quantum clock interferometer using frequency-entangled multi-photon states to amplify gravitational time dilation effects, demonstrating that near-term laboratory experiments with existing quantum memories can observe entanglement dynamics driven by height differences as small as a few meters.

Original authors: Mustafa Gündoğan, Roy Barzel, Dennis Rätzel

Published 2026-09-22

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CC BY 4.0

Original authors: Mustafa Gündoğan, Roy Barzel, Dennis Rätzel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is often thought of as a force that pulls objects down, but in the realm of Einstein's general relativity, it is also a force that slows down time. The deeper you are in a gravitational field, the slower your clock ticks compared to someone higher up. This phenomenon, known as gravitational time dilation, has been confirmed many times with precise atomic clocks. However, a new question has emerged: does this slowing of time affect the strange, delicate rules of quantum mechanics? Specifically, if a quantum system acts like a clock and is placed in a superposition of two different heights, does the difference in time experienced at those heights cause the quantum system to lose its special properties? Answering this question is crucial because it probes the boundary between the physics of the very large, which governs gravity, and the physics of the very small, which governs atoms and light.

A team of researchers has proposed a way to test this using light and quantum memories. In their study, they describe a setup where a special kind of light, made of entangled photons, is split and stored in two different locations at different heights. One location is slightly higher than the other. Because of gravity, time passes at slightly different rates in these two spots. The researchers show that this difference in time creates a subtle shift in the quantum state of the light. When the light is retrieved and brought back together, this shift causes the light waves to interfere with each other in a specific pattern. The pattern changes depending on how long the light was stored and how far apart the two heights were. By measuring this pattern, scientists could detect the influence of gravity on quantum entanglement.

The core of the experiment involves a device called a quantum memory, which acts like a hard drive for light. Normally, light zips through space at incredible speeds, making it difficult to hold it in place long enough for gravity to make a noticeable difference. To solve this, the researchers use quantum memories to trap the light in the form of collective vibrations within atoms. They propose storing the light in two separate memories, one positioned above the other. The light is prepared in a state where it exists in a superposition of two different frequency colors. In one part of the superposition, the light of one color travels up to the higher memory, while the other color goes down to the lower memory. In the other part of the superposition, the paths are swapped. Because the memories are at different heights, the time experienced by the light in the upper memory is slightly different from the time in the lower memory.

When the light is released from the memories and recombined, the two paths interfere. If there were no difference in time, the light would interfere in a predictable way. However, the gravitational time dilation introduces a phase shift, a kind of lag, between the two paths. This lag causes the interference pattern to change. The researchers found that by using groups of entangled photons rather than just single pairs, they could make this effect much stronger. Specifically, if they use a group of N pairs of photons, the signal changes N times faster than it would with a single pair. This means that the experiment becomes much more sensitive, allowing them to detect the gravitational effect over much shorter distances or shorter storage times than previously thought possible.

The study calculates that with current technology, such as quantum memories based on rubidium or cesium atoms, this effect could be observed in a laboratory setting with a height difference of only 20 to 75 meters. If they use more advanced memories based on rare-earth ions, the required height difference could be reduced to just a few meters. The light would need to be stored for less than a second to a few seconds. This is a significant reduction from previous proposals that required interferometers stretching for tens of kilometers. The researchers also show that the signal remains visible even if the quantum memories are not perfectly efficient, which is a realistic constraint for current experiments.

This work suggests that we are on the verge of being able to observe how gravity directly influences quantum entanglement in a controlled laboratory environment. By using quantum memories to hold light in place and entangled multi-photon states to amplify the signal, the researchers have identified a practical path forward. The results indicate that the collapse and revival of the interference pattern, driven by the difference in proper time, can be measured with existing equipment. This would provide a direct operational signature of how time dilation affects quantum systems, offering a new way to explore the interface between gravity and quantum mechanics without needing to build massive, impractical structures. The study establishes that near-term laboratory conditions are sufficient to witness these dynamics, marking a step toward understanding how the fabric of spacetime interacts with the quantum world.

Technical Summary: Gravitational Time Dilation in Quantum Clock Interferometry with Entangled Multi-Photon States

Problem Statement
The paper addresses the challenge of observing gravitational time dilation effects on genuinely quantum degrees of freedom. While general relativistic phase shifts (e.g., the Colella-Overhauser-Werner effect) have been observed, a distinct phenomenon occurs when a quantum system with an internal "clock" structure (such as a two-level system) is placed in a superposition of different gravitational potentials. In this scenario, the internal state evolves according to local proper times, creating entanglement between internal and external degrees of freedom. This leads to a reduction and subsequent revival of interferometric visibility, serving as an operational signature of proper time.

Previous proposals for observing this effect using photons faced a significant scalability barrier: free-space propagation would require vertical interferometer separations of tens of kilometers to accumulate a measurable proper-time difference, rendering experiments impractical. While quantum memories were previously proposed to localize excitations and reduce the required spatial scale, the sensitivity remained limited to the two-photon regime.

Methodology
The authors propose a memory-assisted quantum clock interferometer utilizing frequency-bin entangled multi-photon states. The protocol involves the following steps:

State Preparation: A source generates a specific $2N$ -photon entangled state, $|\psi_{HOM}\rangle$ , which is a coherent superposition of two configurations: $N$ photons at frequency $\Omega_1$ in the upper arm and $N$ photons at frequency $\Omega_2$ in the lower arm, or vice versa. This state is distinct from standard $N00N$ states in the frequency domain, which do not exhibit the same phase sensitivity in this geometry.
Storage and Evolution: The photons are mapped into quantum memories (QM) located at two vertically separated heights ( $h_U$ and $h_L$ ). The memories store the excitations for a controllable local laboratory time $\tau_s$ . Due to the gravitational redshift, the local proper time differs between the two arms, causing the frequency components to accumulate a relative phase dependent on the proper-time difference.
Retrieval and Interference: After storage, the excitations are retrieved and converted back to optical photons. The two arms are recombined at a 50:50 beam splitter (BS) to form a Hong–Ou–Mandel (HOM) interferometer.
Detection: The authors analyze the detection statistics, specifically focusing on the parity of the photon number at the output ports. They derive analytic expressions for the probability of all photons exiting a single port and the expectation value of the parity operator.

Key Contributions and Results

$N$ -Fold Phase Amplification: The primary theoretical contribution is the demonstration that extending the scheme from entangled pairs ( $N=1$ ) to frequency-entangled $2N$ -photon inputs amplifies the proper-time dependent phase by a factor of $N$ . The interference phase is given by $\phi_{HOM} = N \Omega_- \Delta\Theta^{-1} \tau_s + \phi$ , where $\Omega_-$ is the frequency difference between the bins and $\Delta\Theta^{-1}$ is the differential inverse redshift.
Enhanced Sensitivity and Reduced Scale: This amplification leads to an $N$ -times faster collapse and revival of the interference signal compared to the two-photon case. Consequently, the required resource product of height ( $h$ ) and storage time ( $\tau_{ent}$ ) is reduced by a factor of $1/N$ . For a fixed height, the required storage time scales as $1/N$ ; for a fixed time, the required height scales as $1/N$ .
Parity Measurement vs. Post-Selection: The authors derive that while measuring the probability of all $2N$ photons bunching in one port ( $P_{\oplus\oplus}$ ) exhibits the correct phase dependence, its magnitude is penalized by a factor scaling with $N$ . In contrast, measuring the parity operator ( $\langle \hat{\Pi} \rangle$ ) accesses the full photon-number distribution, yielding a signal $\cos(\phi_{HOM})$ without the severe efficiency penalty, making it the preferred observable for larger $N$ .
Comparison with Mach-Zehnder Interferometry: The paper contrasts this HOM-based scheme with a Mach-Zehnder (MZ) interferometer using an $N$ -photon frequency- $N00N$ state. The MZ approach fails to provide $N$ -fold enhancement because the beam splitter randomizes the path information for frequency-superposed states. Furthermore, the MZ signal contains fast oscillations dependent on the sum frequency ( $\Omega_+$ ), whereas the HOM signal depends solely on the difference frequency ( $\Omega_-$ ), simplifying the signal analysis.
Feasibility Analysis: By incorporating finite memory efficiency and lifetime, the authors identify realistic experimental regimes. For parameters compatible with demonstrated Rubidium (Rb) and Cesium (Cs) memories, the first collapse of the interference signal occurs for height differences of 10–100 meters with sub-second to few-second storage times. Using specific combinations, such as Praseodymium-doped crystals and Rb, the required height could be reduced to the few-meter scale.

Significance and Claims
The paper claims to establish near-term laboratory conditions for observing entanglement dynamics driven by gravitational time dilation on a photonic platform. By integrating quantum memories with multi-photon entangled states, the work bridges the gap between theoretical proposals requiring kilometer-scale baselines and practical tabletop experiments.

The authors assert that their scheme provides a concrete instance of quantum metrological advantage in a relativistic setting: the $N$ -fold enhancement of the clock phase directly translates to reduced requirements for interferometer height, area, and storage time. This framework offers a pathway to probe the interface between gravity and quantum physics using experimentally accessible parameters, potentially extending to scenarios involving multiple spacetime branches or vertical scanning to extract information about spacetime curvature. The work builds upon previous foundations of memory-assisted interferometry but generalizes the approach to genuine multi-photon entangled states to achieve enhanced sensitivity.

Technical Summary: Gravitational Time Dilation in Quantum Clock Interferometry with Entangled Multi-Photon States

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