[8][9][10][11] . Here, we demonstrate on- as schematized in Fig. 1. In particular, we used a spectrally-filtered mode-locked laser to excite a single resonance of the microring at ~1550 nm wavelength, in turn producing pairs of correlated signal and idler photons spectrally-symmetric to the excitation field and which cover multiple resonances, see Fig. 1. The individual photons were intrinsically generated in a superposition of multiple frequency modes and owing the energy conservation of SFWM, this approach leads to the realization of a two-photon high-dimensional frequency-entangled state.We performed two experiments to characterize the dimensionality of the generated state. The large free spectral range (FSR) of the ring cavity (~200 GHz), i.e. the spectral separation between adjacent resonance modes, enabled us to use a commercially available telecommunications programmable filter (see Methods) for individually selecting and manipulating the states in these modes (given the filter's operational bandwidth of 1527.4 to 1567.5 nm, we were able to access 10 signal and 10 idler resonances). We measured the joint spectral intensity, describing the twophoton state's frequency distribution, see Methods. Specifically, we routed different frequency 4 modes of the signal and idler photons to two single photon detectors and counted photon coincidences for all sets of mode combinations. As shown in Fig. 2a, photon coincidences were measured only for mode combinations spectrally-symmetric to the excitation, a characteristic of frequency-entangled states. In addition, we evaluate the Schmidt number of our source. This parameter describes the lowest number of significant orthogonal modes in a bipartite system, and therefore describes its effective dimension. Through a Schmidt mode decomposition of the correlation matrix (see Methods), we extracted the lower bound for the Schmidt number to be 9.4, see Fig. 2b.Due to the narrow spectral linewidth of the photons (~800 MHz) and the related long coherence time (~0.6 ns), the effective time resolution of our full detection system (~100 ps) was sufficient to perform time-domain measurements and extract the maximal dimensionality of the state, seeMethods. Specifically, we measured the second-order coherence of the signal and idler fields using These measurements confirmed that one photon pair simultaneously spans multiple frequency modes, forming a high-dimensional entangled state of the form, with ∑| | 2 = 1 (Eq. 1).Here | ⟩ s and | ⟩ i are pure, single-frequency quantum states of the signal (s) and idler (i) photons, and k=1,2,…,D is the mode number, as indicated in Fig. 3 In general, the exploitation of quDit states for quantum information processing motivates the need for high-dimensional operations that enable access to multiple modes with a minimum number of components. While the individual elements (phase shifters and beam splitters) employed in the framework of spatial-mode quantum information processing usually operate on only one or two modes at a time 1 , the frequency...
