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arXiv:2505.02221

Fundamental Bounds of Wavefront Shaping of Spatially Entangled Photons

Ronen Shekel, Sébastien M. Popoff, Yaron Bromberg

Published May 4, 2025• 1 citation
arXivHTMLPDF

Abstract

Wavefront shaping enables control of classical light through scattering media. Extending these techniques to spatially entangled photons promises new quantum applications, but their fundamental limits, especially when both photons scatter, remain unclear. Here, we theoretically and numerically investigate the enhancement of two-photon correlations through thick scattering media. We analyze configurations where a spatial light modulator shapes one or both photons, either before or after the medium, and show that the optimal enhancement differs fundamentally from classical expectations. For a system with $N$ modes, we show that shaping one photon yields the classical enhancement $η\approx (π/4)N$, while shaping both photons before the medium reduces it to $η\approx (π/4)^2N$. However, in some symmetric detection schemes, when both photons are measured at the same mode, perfect correlations are restored with $η\approx N$, resembling digital optical phase conjugation. Conversely, shaping both photons after the medium leads to a complex, NP-hard-like optimization problem, yet achieves superior enhancements, up to $η\approx 4.6N$. These results reveal unique quantum effects in complex media and identify strategies for quantum imaging and communication through scattering environments.

DOI
10.48550/arxiv.2505.02221

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Wavefront shaping enables control of classical light through scattering media. Extending these techniques to spatially entangled photons promises new quantum applications, but their fundamental limits, especially when both photons scatter, remain unclear. Here, we theoretically and numerically investigate the enhancement of two-photon correlations through thick scattering media. We analyze configurations where a spatial light modulator shapes one or both photons, either before or after the medium, and show that the optimal enhancement differs fundamentally from classical expectations. For a system with NNN modes, we show that shaping one photon yields the classical enhancement η≈(π/4)Nη\approx (π/4)Nη≈(π/4)N, while shaping both photons before the medium reduces it to η≈(π/4)2Nη\approx (π/4)^2Nη≈(π/4)2N. However, in some symmetric detection schemes, when both photons are measured at the same mode, perfect correlations are restored with η≈Nη\approx Nη≈N, resembling digital optical phase conjugation. Conversely, shaping both photons after the medium leads to a complex, NP-hard-like optimization problem, yet achieves superior enhancements, up to η≈4.6Nη\approx 4.6Nη≈4.6N. These results reveal unique quantum effects in complex media and identify strategies for quantum imaging and communication through scattering environments.
Wavefront shaping enables control of classical light through scattering media. Extending these techniques to spatially entangled photons promises new quantum applications, but their fundamental limits, especially when both photons scatter, remain unclear. Here, we theoretically and numerically investigate the enhancement of two-photon correlations in two specific output modes through thick scattering media. We analyze three configurations: shaping one photon after the medium, shaping both photons before the medium, and shaping both photons after the medium. We show that each configuration yields fundamentally different enhancements compared to classical expectations. For a system with N modes, we show that shaping one photon yields the classical enhancement η ≈ (π/4)N, while shaping both photons before the medium reduces it to η ≈ (π/4)2N. However, in some symmetric detection schemes, when both photons are measured at the same mode, perfect correlations are restored with η ≈ N, resembling digital optical phase conjugation. Conversely, shaping both photons after the medium leads to a complex, NP-hard-like optimization problem, yet achieves superior enhancements, up to η ≈ 4.6N. These results reveal unique quantum effects in complex media and identify strategies for quantum imaging and communication through scattering environments.
Wavefront shaping is a key technique for mitigating scattering in complex media, enabling advanced imaging and optical communication. Yet existing approaches are inherently local, requiring active correction elements - such as spatial light modulators or deformable mirrors - to lie directly in the optical path of the scattered light, which limits their integration into compact imaging systems. Here, we experimentally demonstrate nonlocal wavefront shaping using spatially entangled photon pairs. By applying a phase correction to a photon that never interacts with the scattering medium, we compensate for the distortions experienced by its entangled partner and restore their initial spatial correlations. Our approach physically decouples the wavefront correction from the scattering medium, paving the way for imaging through complex media in compact and otherwise inaccessible systems.
This study falls within the emerging field of quantum optics in scattering media. By employing wavefront shaping, researchers successfully transmitted spatially entangled photon pairs through scattering media, mitigating optical disturbances by wavefront shaping. They maintained entanglement with a dimensionality of 17 after propagation, opening avenues for applications in quantum microscopy and quantum communication. Interestingly, this work also shows that in certain cases, quantum entanglement can be transmitted through a complex medium, while classical light remains scattered.
Quantum imaging enhances imaging systems performance, potentially surpassing fundamental limits such as noise and resolution. However, these schemes have limitations and are still a long way from replacing classical techniques. Therefore, there is a strong focus on improving the practicality of quantum imaging methods, with the goal of finding real-world applications. With this in mind, in this tutorial we describe how the concepts of classical light shaping can be applied to imaging schemes based on entangled photon pairs. We detail two basic experimental configurations in which a spatial light modulator is used to shape the spatial correlations of a photon pair state and highlight the key differences between this and classical shaping. We then showcase two recent examples that expand on these concepts to perform aberration and scattering correction with photon pairs. We include specific details on the key steps of these experiments, with the goal that this can be used as a guide for building photon-pair-based imaging and shaping experiments.
When light propagates through complex media, its output spatial distribution is highly sensitive to its wavelength. This fundamentally limits the bandwidth of applications ranging from imaging to communication. Here, we demonstrate analytically and numerically that the spatial correlations of hyper-entangled photon pairs, simultaneously entangled spatially and spectrally, remain stable across a broad bandwidth: The chromatic modal dispersion experienced by one photon is canceled to first order by its spectrally anti-correlated twin, defining a "two-photon bandwidth" that can far exceed its classical counterpart. We illustrate this modal dispersion cancellation in multimode fibers, thin diffusers and blazed gratings, and demonstrate its utility for broadband wavefront shaping of quantum states. These findings advance our fundamental understanding of quantum light in complex media with applications in quantum imaging, communication, and sensing.
Entangled photons provide transformative new paths in the fields of communication, sensing, and computing. However, when entangled photons propagate through a complex medium, their correlations are scrambled. Using wavefront shaping to compensate for the scattering and retrieve the two-photon correlations is challenging due to the low signal-to-noise ratio of the two-photon signal. While previous works partly addressed this challenge by using feedback from a strong classical laser beam that co-propagates with the entangled photons, such methods frequently depend on assumptions about the complex medium, limiting the applicability of quantum wavefront shaping. In this work, we propose and demonstrate a new feedback mechanism that is inspired by Klyshko’s advanced wave picture: the classical laser beam is emitted in one of the detection modes, counter-propagates with one of the entangled photons, reflects at the crystal plane, and co-propagates with the other. The new Klyshko feedback allows compensation of scattering in arbitrary samples and even in situations where each photon propagates through a different scattering medium. Since the advanced wave picture applies whenever optical reciprocity is valid, such Klyshko optimization can be used across a wide range of configurations, offering a robust and alignment-free setup. We therefore believe this protocol will open the door for real-world applications of quantum wavefront shaping.