Light states composed of multiple entangled photons - such as cluster states - are essential for developing and scaling-up quantum computing networks. Photonic cluster states with discrete variables can be obtained from single-photon sources and entangling gates, but so far this has only been done with probabilistic sources constrained to intrinsically-low efficiencies, and an increasing hardware overhead. Here, we report the resource-efficient generation of polarization-encoded, individually-addressable, photons in linear cluster states occupying a single spatial mode. We employ a single entangling-gate in a fiber loop configuration to sequentially entangle an ever-growing stream of photons originating from the currently most efficient single-photon source technology - a semiconductor quantum dot. With this apparatus, we demonstrate the generation of linear cluster states up to four photons in a single-mode fiber. The reported architecture can be programmed to generate linear-cluster states of any number of photons with record scaling ratios, potentially enabling practical implementation of photonic quantum computing schemes.

The ability to generate light in a pure quantum state is essential for advances in optical quantum technologies. However, obtaining quantum states with control in the photon-number has remained elusive. Optical light fields with zero and one photon can be produced by single atoms, but so far it has been limited to generating incoherent mixtures, or coherent superpositions with a very small one-photon term. Here, we report on the on-demand generation of quantum superpositions of zero, one, and even two photons, via pulsed coherent control of a single artificial atom. Driving the system up to full atomic inversion leads to the generation of quantum superpositions of vacuum and one photon, with their relative populations controlled by the driving laser intensity. A stronger driving of the system, with $2\pi$-pulses, results in a coherent superposition of vacuum, one and two photons, with the two-photon term exceeding the one-photon component, a state allowing phase super-resolving interferometry. Our results open new paths for optical quantum technologies with access to the photon-number degree-of-freedom.