Envisioned large-scale quantum networks demand the ability to rapidly distribute and store entanglement between remote matter-based nodes using photons [1] . The minimum time required to distribute entanglement between two nodes is the time it takes light to travel between them, which over 100 km distance already 0.33 ms. The speed of light thus sets a fundamental limit on the achievable entanglement distribution rate: with a single quantum emitter in a node, one has to wait for the photon travel time before trying again with a new photon for any application that would require also classical communication (e.g. to realize a quantum repeater) [2] . Over 100 km of fiber, this limits the entanglement distribution attempt rate to 2 kHz and, given 1.5% transmission in the best telecom fibres, a success rate of 30 Hz. This is a general problem for entanglement distribution over long distances as far higher rates are critical for the practical realisation of quantum networks and their applications. To overcome this limit, several entanglement distribution attempts have to be done in parallel. This requires multiple modes of the node (memory qubits) to couple selectively to the multiple modes of the link.