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arXiv:2602.07949

Full Schmidt characterization of spatiotemporally entangled states produced from spontaneous parametric down-conversion

Rakesh Pradhan, Girish Kulkarni

quant-phphysics.opticsSubmitted Feb 8, 2026• Announced Feb 10, 2026• 0 citations
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Abstract

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.

Comments
6 pages, 3 figures, and supplemental material

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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.
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.
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.
Abstract In this work, we investigate the orbital angular momentum (OAM) spectrum of photons generated via spontaneous parametric down-conversion (SPDC) and study how shaping the transverse profile of the pump beam influences the resulting spectrum. Using a perfect optical vortex (POV) mode as the pump, we show that the OAM spectrum narrows significantly, leading to purer OAM states. We further examine the entanglement between the down-converted photons in the OAM basis, finding that Laguerre–Gaussian (LG) pumped photons exhibit increased higher-dimensional entanglement at the expense of mode purity.
Thin-film spontaneous parametric down-conversion, due to the relaxed phase-matching condition, generates photon pairs that are extremely broadband in frequency and angle and highly entangled in time/frequency and position/momentum. Here we show that it also provides a very broad orbital angular momentum (OAM) spectrum, which, in contrast to the case of bulk crystals, is decoupled from the frequency spectrum. To demonstrate it experimentally, we generate photon pairs in a 7 μm7~μ\mathrm{m}7 μm-thick lithium niobate film and compare them with those from a 500 μm500~μ\mathrm{m}500 μm-thick periodically poled lithium niobate crystal under identical pump conditions. The thin-film source exhibits a substantially broader OAM spectrum, with the azimuthal Schmidt number almost 9 times larger than that of the bulk source. On this highly entangled state, we perform quantum-state tomography in selected two- and four-dimensional OAM subspaces, demonstrating high-purity maximally entangled states. These results highlight thin nonlinear films as a promising platform for high-dimensional quantum-state generation with weak spatial--spectral coupling.
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.