Stochastic correction to the Maxwell-Bloch equations via the positive $P$ representation

Focusing on two-level atoms, we apply the positive $P$ representation to a full-wave mixed bosonic and fermionic system of Jaynes-Cummings type and identify an advantageous degree of freedom in the choice of the involved nonorthogonal fermionic basis states. On this basis, we propose a stochastic correction to the Maxwell-Bloch equations by relating them to a stochastic differential equation on a nonclassical phase space, which captures the full second quantization dynamics of the system. This approach explores the connection between semiclassical and field-quantized treatments of light-matter interaction and can potentially be used for the simulation of nonclassical light sources while retaining the main advantages of a semiclassical model.