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arXiv:2505.17906 · Physical Review Applied

Tracking phase entanglement during propagation of down-converted photons

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

Published Jul 14, 2025• 3 citations
arXivHTMLPDF

Abstract

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 $\left[0,\infty\right)$, 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.

DOI
10.1103/qq7y-jdl1

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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.
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.
Control of spatial quantum correlations in bi-photons is one of the fundamental principles of Quantum Imaging. Up to now, experiments have been restricted to controlling the state of a single bi-photon, by using linear optical elements. In this work we demonstrate experimental control of quantum correlations in a four-photon state comprised of two pairs of photons. Our scheme is based on a high-efficiency parametric downconversion source coupled to a double slit by a variable linear optical setup, in order to obtain spatially encoded qubits. Both entangled and separable pairs have been obtained, by altering experimental parameters. We show how the correlations influence both the interference and diffraction on the double slit.
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.
Two-photon states generated through degenerate spontaneous parametric downconversion (SPDC) can exhibit sharp correlations in the transverse spatial coordinates. This property leads to unique free-space propagation features. Here, we show that a phase object placed in the image plane of the source affects the free space propagation of the SPDC in a way that is mathematically analogous to the Fresnel diffraction of a first-order coherent source. This effect can be observed via the extraction of correlation images. We demonstrate this prediction with an experiment where the diffraction of correlated bi-photons is detected using an event-based camera. The results allow us to reconstruct the phase structure of the sample via non-interferometric phase retrieval methods. We verify that the retrieved phase patterns exhibit a factor of two contrast enhancement due to the probe’s two-photon nature. Our findings offer a proof of principle application for quantitative phase imaging, particularly in regimes where low light conditions are required to prevent photo-damage and, at the same time, is desirable to preserve high resolution, contrast, and low noise.
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Entangled photons generated by spontaneous parametric downconversion are ubiquitous in quantum optics. In general, they exhibit a complex spatial photon count distribution. This spatial structure is responsible for seemingly surprising results concerning, e.g., complementarity such as the apparent simultaneous observation of interference fringes V and which-way information D at a double slit, as recently reported by Menzel et al. [Proc. Natl. Acad. Sci. USA109, 9314 (2012)10.1073/pnas.1201271109PNASA60027-8424]. We implement a complete quantitative model of the SPDC interaction that fully incorporates the effects of crystal anisotropies, phase matching, and the pump beam structure and allows for arbitrary manipulations of the SPDC light in the near and far fields. This enables us to establish an upper bound D2+V2≤1.47 for the experimental parameters reported by Menzel et al. We report new experimental results that agree excellently with these theoretical predictions. The new model enables a detailed quantitative analysis of this surprising result and the fair sampling interpretation of biphotons passing a double slit.