  We investigate scaling of work and efficiency of a photonic Carnot engine
with the number of quantum coherent resources. Specifically, we consider a
generalization of the "phaseonium fuel" for the photonic Carnot engine, which
was first introduced as a three-level atom with two lower states in a quantum
coherent superposition by [M. O. Scully, M. Suhail Zubairy, G. S. Agarwal, and
H. Walther, Science {\bf 299}, 862 (2003)], to the case of $N+1$ level atoms
with $N$ coherent lower levels. We take into account atomic relaxation and
dephasing as well as the cavity loss and derive a coarse grained master
equation to evaluate the work and efficiency, analytically. Analytical results
are verified by microscopic numerical examination of the thermalization
dynamics. We find that efficiency and work scale quadratically with the number
of quantum coherent levels. Quantum coherence boost to the specific energy
(work output per unit mass of the resource) is a profound fundamental
difference of quantum fuel from classical resources. We consider typical modern
resonator set ups and conclude that multilevel phaseonium fuel can be utilized
to overcome the decoherence in available systems. Preparation of the atomic
coherences and the associated cost of coherence are analyzed and the engine
operation within the bounds of the second law is verified. Our results bring
the photonic Carnot engines much closer to the capabilities of current
resonator technologies.
