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arXiv:2508.11418 · Physical Review A

Generation and certification of pure phase-entangled light

Rounak Chatterjee, Mayuresh S. Kanagal, Vikas S. Bhat, Kiran Bajar, Sushil A. Mujumdar

Published Dec 12, 2025• 1 citation
arXivHTMLPDF

Abstract

Biphoton systems exhibiting entanglement in position-momentum variables, known as spatial entanglement, are among the most intriguing and well-studied phenomena in quantum optics. A notable subset of these are phase entangled states, where entanglement manifests purely through correlations in the spatial phase of the wavefunction. While the generation of such states from biphotons via spontaneous parametric down-conversion has been explored, their physical implications and applications remain under-investigated. In this work, we theoretically and experimentally examine a unique form of phase entanglement known as `pure' phase entanglement. This state exhibits the unusual feature that the position of one photon is correlated with the momentum of the other. Unlike typical spatially entangled states, it shows no direct correlation in position or momentum between the two photons, underscoring that all correlations arise purely from the spatial phase of the wavefunction. We delve deeper into the theory of this state and experimentally construct it from known phase-entangled states. To certify its properties, we propose a setup that performs a "one-particle momentum measurement" and explore the various tunable parameters. We also highlight potential applications of this state in quantum optics and imaging experiments.

DOI
10.1103/jkvs-6v2y

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Biphoton systems exhibiting entanglement in position-momentum variables, known as spatial entanglement, are among the most intriguing and well-studied phenomena in quantum optics. A notable subset of these are phase entangled states, where entanglement manifests purely through correlations in the spatial phase of the wavefunction. While the generation of such states from biphotons via spontaneous parametric down-conversion has been explored, their physical implications and applications remain under-investigated. In this work, we theoretically and experimentally examine a unique form of phase entanglement known as `pure' phase entanglement. This state exhibits the unusual feature that the position of one photon is correlated with the momentum of the other. Unlike typical spatially entangled states, it shows no direct correlation in position or momentum between the two photons, underscoring that all correlations arise purely from the spatial phase of the wavefunction. We delve deeper into the theory of this state and experimentally construct it from known phase-entangled states. To certify its properties, we propose a setup that performs a "one-particle momentum measurement" and explore the various tunable parameters. We also highlight potential applications of this state in quantum optics and imaging experiments.
High-dimensional entanglement in the form of transverse spatial correlation between a pair of photons generated via spontaneous parametric downconversion is not only a valuable resource in many academic and real-life applications but also provides access to several intriguing quantum phenomena. One such non-intuitive phenomenon is phase entanglement, in which the biphoton state is correlated in the complex phase of its wavefunction. This state, which emerges during the propagation of the biphoton wavefunction, exhibits neither position nor momentum correlation, yet retains full entanglement. In this work, we experimentally explore this state in two distinct ways. The first is by tracking the vanishing spatial photon number correlation over propagation distances lying in [0,∞)\left[0,\infty\right)[0,∞), folded into a finite range using single-lens imaging. These observations show excellent agreement with our theoretical predictions based on the Double Gaussian (DG) approximation of the biphoton state. The second approach involves performing a two-photon interference experiment using a double slit and this state, which reveals the correlated phase front. We show, both theoretically and experimentally, that the observed two-photon interference structure is markedly different from that produced by position-correlated photons, as confirmed by computing the joint probability distribution of photons (JPD) and related metrics. Such interference using phase-entangled light has not been attempted before and opens avenues for advanced experiments and applications in the field of spatial entanglement.
Structured optical fields have led to several ground-breaking techniques in classical imaging and microscopy. At the same time, in the quantum domain, position-momentum entangled photon fields have been shown to have several unique features that can lead to beyond-classical imaging and microscopy capabilities. Therefore, it is natural to expect that position-momentum entangled two-photon fields that are structured can push the boundaries of quantum imaging and microscopy even further beyond. Nonetheless, the existing experimental schemes are able to produce either structured two-photon fields without position-momentum entanglement, or position-momentum entangled two-photon fields without structured correlations. In this article, by manipulating the phase-matching condition of the spontaneous parametric down-conversion process, we report experimental generation of two-photon fields with various structures in their spatial correlations. We experimentally measure the minimum bound on the entanglement of formation and thereby verify the position-momentum entanglement of the structured two-photon field. We expect this work to have important implications for quantum technologies related to imaging and sensing.
The complete measurement of the quantum state of two correlated photons requires reconstructing the amplitude and phase of the biphoton wavefunction. We show how, by means of spatially resolved single photon detection, one can infer the spatial structure of biphotons generated by spontaneous parametric down conversion. In particular, a spatially resolved analysis of the second-order correlations allows us to isolate the moduli of the pump and phase-matching contributions to the two-photon states. When carrying this analysis on different propagation planes, the free-space propagation of pump and phase-matching is observed. This result allows us, in principle, to gain enough information to also reconstruct the phase of the pump and the phase-matching and thus the full biphoton wavefunction. We show this in different examples where the pump is shaped as a superposition of orbital angular momentum modes or as a smooth amplitude with a phase structure with no singularities. The corresponding phase structure is retrieved employing maximum likelihood or genetic algorithms. These findings have potential applications in fast, efficient quantum state characterization that does not require phase locking of the unknown source with a reference biphoton.
In this thesis we investigate diverse aspects of spatial coherence of light. Non-classical fields containing two photons can be generated by a nonlinear optical process known as spontaneous parametric down conversion (SPDC). Among the questions we consider are: What is so special about spatial entanglement? How is it revealed in the fourth-order correlations? What are the differences between a highly entangled and a classically correlated state? How can the number of modes be manipulated and measured? For a two-photon system, we measure both intensities and two-photon correlations. To get deeper insights into how coherence affects interference, we also investigate completely classical sources.
Abstract We experimentally demonstrate the quantum entanglement in various spatial modes of the correlated photons generated via spontaneous parametric down-conversion in a type-II phase-matched periodically-poled KTP crystal. The two-fold coincidence measurement reveals the spatial entanglement between the photon pairs by projecting one of the correlated photons onto a Gaussian mode while scanning the spatial mode of its partner photon through selective projection (Gaussian, Laguerre–Gaussian (LG), Bessel-Gaussian (BG), and Airy–Gaussian (AG)) using a spatial light modulator (SLM). We also show the spatially resolved measurement of the non-conservation of orbital angular momentum via modal decomposition of the correlated photon in LG and BG bases.
We examine experimentally how the degree of position-momentum entanglement of photon pairs depends on the transverse coherence of the pump beam that excites them in a process of spontaneous parametric down-conversion. Using spatially incoherent light from a light-emitting diode, we obtain strong position correlation of the photons, but we find that transverse momentum correlation, and thus entanglement, is entirely absent. When we continuously vary the degree of spatial coherence on the pump beam, we observe the emergence of stronger momentum correlations and entanglement. We present theoretical arguments that explain our experimental results. Our results shed light on entanglement generation and can be applied to control entanglement for quantum information applications.