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Journal of Optics

Radial Schmidt mode detector of entangled photons

Radhika Prasad, Nilakshi Senapati, Abhinandan Bhattacharjee, Anand K. Jha

Published Jul 1, 2026• 1 citation
Publisher

Abstract

Abstract High-dimensional spatially entangled two-photon states generated by the spontaneous parametric down-conversion process (SPDC) have become a promising resource for several quantum information science applications. In order to harness high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is due to radial coordinates not having a Schmidt-decomposed form for the SPDC photons or a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that azimuthal averaging of SPDC two-photon states yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of the radial Schmidt spectrum of up to 50 radial Schmidt modes with about 98% fidelity.

DOI
10.1088/2040-8986/ae84e4

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Abstract High-dimensional spatially entangled two-photon states generated by the spontaneous parametric down-conversion process (SPDC) have become a promising resource for several quantum information science applications. In order to harness high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is due to radial coordinates not having a Schmidt-decomposed form for the SPDC photons or a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that azimuthal averaging of SPDC two-photon states yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of the radial Schmidt spectrum of up to 50 radial Schmidt modes with about 98% fidelity.
High-dimensional spatially entangled two-photon state generated by spontaneous parametric down-conversion process (SPDC) has become a promising resource for several quantum information science applications. For harnessing high-dimensional entanglement advantages, detection capability in the Schmidt basis is a necessity. Spatial entanglement has been explored in several modal bases, such as pixel, azimuthal, and radial modes. Among them, pixel and azimuthal entanglement have been widely utilized due to efficient access to their Schmidt modes, while radial-mode entanglement remains underexploited. This is because for radial coordinates, there is neither a Schmidt-decomposed form for the SPDC photons nor is there a technique for measuring high-dimensional radial Schmidt modes, which is a major roadblock in harnessing radial mode advantages. In this work, we first theoretically show that the azimuthal averaging of SPDC two-photon state yields a radial Schmidt-decomposed form under typical experimental situations. We then demonstrate an innovative approach for extracting the radial Schmidt modes and their spectrum by characterizing the density matrix in the radial basis of one of the SPDC photons. Finally, we report the first-ever measurement of radial Schmidt spectrum of upto 50 radial Schmidt modes with about 98\% fidelity.
Spontaneous parametric down-conversion (SPDC) is the most widely-used source of high-dimensional entangled two-photon states, and the entanglement in the spatial degree of freedom is considered best suited for harnessing high-dimensional advantages. Although the Schmidt basis provides a natural choice for state characterisation of entangled two-photon states in any degree of freedom, there is currently no technique that can measure the Schmidt basis of an entangled two-photon field. The existing techniques can only reconstruct the Schmidt spectrum when the Schmidt basis is known a priori. In contrast, we present a technique that measures the complete set of spatial Schmidt modes without any prior knowledge. Using this technique, we report measurement of states with over 3000 Schmidt modes -- highest reported yet -- with up to 98%\%% fidelity. We expect our work to significantly advance the harnessing of high-dimensional advantages in SPDC-based systems.
The full Schmidt decomposition of spatiotemporally entangled states generated from spontaneous parametric down-conversion (SPDC) has not been carried out until now due to the immense computational complexity arising from the large dimensionalities of the states. In this Letter, we utilize the rotational symmetry of the states to reduce the complexity by at least four orders of magnitude and carry out the decomposition to reveal the precise forms of the spatiotemporal Schmidt modes and the Schmidt spectrum spanning over 10^4 modes. We show that the Schmidt modes have a phase profile with a transverse spatial vortex structure that endows them with orbital angular momentum at all frequencies. In the high-gain regime, these Schmidt modes broaden and the Schmidt spectrum narrows with increasing pump strength. Our work can spur novel applications at the intersection of quantum imaging and spectroscopy that utilize entangled states produced from SPDC.
High-dimensional spatial entanglement generated via spontaneous parametric down-conversion (SPDC) provides a powerful resource for quantum information processing, yet its mode structure becomes increasingly complex when the pump occupies a higher-order Laguerre--Gaussian (LG) mode. Here, we develop an algebraic framework for characterizing biphoton spatial-mode entanglement by factorizing LG-pumped SPDC into a spatial-mode beam-splitter transformation followed by spatial two-mode squeezing. By choosing the biphoton LG basis radius as the geometric mean of the pump-beam radius and the crystal-induced correlation width, we obtain a natural modal basis in which the two operations can be treated separately. When the pump-beam radius matches the correlation width, spatial squeezing vanishes, and the two circular-mode components of the pump are independently conserved across the signal and idler photons, leading to conservation of both orbital angular momentum and the total spatial mode number. We further derive an analytical expression for the Schmidt number for arbitrary LG pump modes and squeezing strengths, revealing how the radial and azimuthal pump indices govern the dimensionality of the spatial entanglement. This algebraic characterization provides a unified description of the mode structure, conservation laws, and entanglement dimensionality of higher-order-LG-pumped SPDC.
Abstract It is well known that photons can carry a spatial structure akin to a ‘twisted’ or ‘rippled’ wavefront. Such structured light fields have sparked significant interest in both classical and quantum physics, with applications ranging from dense communications to light–matter interaction. Harnessing the full advantage of transverse spatial photonic encoding using the Laguerre–Gaussian (LG) basis in the quantum domain requires control over both the azimuthal (twisted) and radial (rippled) components of photons. However, precise measurement of the radial photonic degree-of-freedom has proven to be experimentally challenging primarily due to its transverse amplitude structure. Here we demonstrate the generation and certification of full-field LG entanglement between photons pairs generated by spontaneous parametric down conversion in the telecom regime. By precisely tuning the optical system parameters for state generation and collection, and adopting recently developed techniques for precise spatial mode measurement, we are able to certify fidelities up to 85% and entanglement dimensionalities up to 26 in a 43-dimensional radial and azimuthal LG mode space. Furthermore, we study two-photon quantum correlations between nine LG mode groups, demonstrating a correlation structure related to mode group order and inter-modal cross-talk. In addition, we show how the noise-robustness of high-dimensional entanglement certification can be significantly increased by using measurements in multiple LG mutually unbiased bases. Our work demonstrates the potential offered by the full spatial structure of the two-photon field for enhancing technologies for quantum information processing and communication.
The ability to efficiently characterize the spatial correlations of entangled states of light is critical for applications of many quantum technologies such as quantum imaging. Here, we demonstrate highly efficient theoretical and experimental characterization of the spatial Schmidt modes and the Schmidt spectrum of bright multiphoton entangled states of light produced from high-gain parametric down-conversion. In contrast to previous studies, we exploit the approximate quasihomogeneity and isotropy of the signal field and dramatically reduce the numerical computations involved in the experimental and theoretical characterization procedures. In our particular case where our experimental datasets consist of 5000 single-shot images of $256\ifmmode\times\else\texttimes\fi{}256$ pixels each, our method reduced the overall computation time by 2 orders of magnitude. This speedup would be even more dramatic for larger input sizes. Consequently, we are able to rapidly characterize the Schmidt modes and the Schmidt spectrum for a range of pump amplitudes and study their variation with increasing gain. Our results clearly reveal the broadening of the Schmidt modes and the narrowing of the Schmidt spectrum for increasing gain with agreement between theory and experiment.