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 article, 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 in quantum imaging and spectroscopy that utilize the spatiotemporal entanglement produced from SPDC.

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• Angular momentum of light
• Optical coherence
• Quantum entanglement
• Quantum optics