Measuring gravitational lensing time delays with quantum information processing

The gravitational fields of astrophysical bodies bend the light around them, creating multiple paths along which light from a distant source can arrive at Earth. Measuring the difference in photon arrival time along these different paths provides a means of determining the mass of the lensing system, which is otherwise difficult to constrain. This is particularly challenging in the case of microlensing, where the images produced by lensing cannot be individually resolved; existing proposals for detecting time delays in microlensed systems are significantly constrained due to the need for large photon flux and the loss of signal coherence when the angular diameter of the light source becomes too large. In this work, we propose a novel approach to measuring astrophysical time delays. Our method uses exponentially fewer photons than previous schemes, enabling observations that would otherwise be impossible. Our approach, which combines a quantum-inspired algorithm and quantum information processing technologies, saturates a provable lower bound on the number of photons required to find the time delay. Our scheme has multiple applications: we explore its use both in calibrating optical interferometric telescopes and in making direct mass measurements of ongoing microlensing events. To demonstrate the latter, we present a fiducial example of microlensed stellar flares sources in the Galactic Bulge. Though the number of photons produced by such events is small, we show that our photon-efficient scheme opens the possibility of directly measuring microlensing time delays using existing and near-future ground-based telescopes.

💡 Research Summary

The paper tackles the long‑standing challenge of measuring the tiny time delays (Δt) introduced by gravitational microlensing, where the multiple images produced by a lens cannot be spatially resolved. Traditional approaches rely on large photon fluxes and suffer from the finite‑source effect: if the source’s angular size is comparable to the Einstein radius, photons from different parts of the source experience different delays, washing out the interference pattern that encodes Δt. Moreover, the required number of photons scales linearly with the ratio of the maximum expected delay T to the photon coherence time t_c (i.e., O(T/t_c)), making many microlensing events inaccessible.

The authors propose a fundamentally different strategy that exploits quantum superposition and frequency‑domain interference. Each photon emitted by a point‑like source is in a superposition of the two lens‑induced paths; the resulting wavefunction is the sum of two wave packets separated by Δt. In the frequency domain this superposition creates an oscillatory modulation with a period of 1/Δt. By measuring the frequency of individual photons with a single‑photon spectrometer, one obtains samples from a distribution whose shape directly depends on Δt.

Algorithm 1 implements this idea: a broadband photon is frequency‑resolved, the measured frequency is fed into a maximum‑likelihood estimator, and Δt is recovered with precision ≈t_c. Crucially, the algorithm requires only O(log (T/t_c)) photons, an exponential improvement over classical methods. The authors prove that Ω(log (T/t_c)) is an information‑theoretic lower bound by modeling the lensing system as a communication channel and by reducing the problem to the dihedral hidden subgroup problem (DHSP), a well‑studied quantum‑algorithmic task. This reduction also shows that no algorithm can achieve a better sample complexity without violating known quantum lower bounds.

A second, time‑domain version (Algorithm 2) is introduced for scenarios where direct high‑resolution spectroscopy is impractical. Here, a non‑demolition measurement first narrows the photon’s frequency to a band of width 1/t′_c (t′_c ≥ t_c). The photon is then stored in a quantum memory that can only distinguish O(T/t′_c) temporal modes—far fewer than the Nyquist‑rate requirement Θ(T ω_0). A quantum Fourier transform (QFT) applied to this undersampled state yields an aliased frequency that still carries enough information to recover Δt via a DHSP‑style post‑processing. This approach can be realized with linear‑optics networks and, in principle, with digital quantum computers that encode arrival times in binary, achieving exponential compression of the required resources.

Implementation considerations focus on existing single‑photon spectrometers. Dual‑comb spectrometers achieve ~100 MHz resolution over a 10 GHz bandwidth, suitable for Δt down to ~10 ns (e.g., brown‑dwarf lenses). Time‑lens spectrometers reach ~20 kHz resolution but over a narrower bandwidth, allowing measurement of delays up to ~0.1 ms (e.g., solar‑mass primordial black holes). The main bottleneck is the broadband nature of thermal sources; many parallel spectrometers would be needed for very short Δt, motivating the development of next‑generation high‑resolution, wide‑band single‑photon detectors.

To demonstrate scientific impact, the authors analyze flares from M‑type red dwarfs as candidate sources. These flares are extremely compact (emitting regions of order kilometers) and can be as brief as seconds, ensuring that the finite‑source uncertainty δΔt is well below one carrier‑period (≈10⁻¹⁵ s). Consequently, the interference pattern survives, and the proposed quantum‑enhanced measurement can directly retrieve Δt. Measuring Δt for such events yields (i) the mass of the intervening microlens (including rogue planets, isolated black holes, or dark‑matter clumps) and (ii) an unprecedented constraint on the spatial size of stellar flare kernels, a quantity otherwise requiring kilometer‑scale optical baselines.

The paper also notes that the same technique can calibrate time delays in optical/IR interferometric telescope arrays, where precise path‑length matching is essential for coherent combination of light from distant telescopes. By using the quantum‑efficient algorithm, calibration can be performed rapidly with faint guide stars, reducing the need for bright artificial beacons.

In conclusion, the authors combine quantum information theory, optimal sampling bounds, and realistic photonic hardware to propose a provably optimal, photon‑efficient method for measuring gravitational‑lens time delays. Their analysis shows that with modest advances in single‑photon spectroscopy and quantum memory, direct measurement of microlensing delays—previously deemed infeasible—becomes attainable, opening new avenues for probing compact dark objects and stellar flare physics.