  Common methods to achieve photon number resolution rely on fast on-off
single-photon detectors in conjunction with temporal or spatial mode
multiplexing. Yet, these methods suffer from an inherent trade-off between the
efficiency of photon number discrimination and photon detection rate. Here, we
introduce a method of photon number resolving detection that overcomes these
limitations by replacing mode multiplexing with coherent absorption of a single
optical mode in a distributed detector array. Distributed coherent absorption
ensures complete and uniform absorption of light among the constituent
detectors, enabling fast and efficient photon number resolution. As a
proof-of-concept, we consider the case of a distributed array of
superconducting nanowire single-photon detectors with realistic parameters and
show that deterministic absorption and arbitrarily high photon number
discrimination efficiency can be achieved by increasing the number of detectors
in the array. Photon number resolution without optical mode multiplication
provides a simple yet effective method to discriminate an arbitrary number of
photons in large arrays of on-off detectors or in smaller arrays of mode
multiplexed detectors.
