  Starting from a quantum description of multiple Lambda-type 3-level atoms
driven with a coherent microwave field and incoherent optical pumping, we
derive a microscopic model of lasing from which we move towards a consistent
macroscopic picture. Our analysis applies across the range of system sizes from
nanolasers to the thermodynamic limit of conventional lasing. We explore the
necessary conditions to achieve lasing without inversion by calculating the
non-equilibrium steady state solutions of the model at, and between, its
microscopic and macroscopic limits. For the macroscopic picture, we use
mean-field theory to present a thorough analysis of the lasing phase
transition. In the microscopic case, we exploit the underlying permutation
symmetry of the density matrix to calculate exact solutions for N 3-level
systems. This allows us to show that the steady state solutions approach the
thermodynamic limit as N increases, restoring the sharp non-equilibrium phase
transition in this limit. We demonstrate how the lasing phase transition and
degree of population inversion can be adjusted by simply varying the phase of
the coherent driving field. The high level of quantum control presented by this
microscopic model and the framework outlined here have applications to further
understanding and developing nanophotonic technology.
