becomes continuous with a divergent photon number expectation above it.

The exact solutions were obtained by mapping the driven Jaynes-Cummings model onto the evolution of a Dirac particle in an electromagnetic field. The relativistic covariance of the Dirac equation was then exploited and the solutions mapped back to the quantum optical system. Under this correspondence, dipole coupling and drive amplitude are related to magnetic and electric fields. When the former dominates, the interaction between spin and magnetic field creates a harmonic potential and discrete states naturally arise. When the latter dominates, the effective potential is unbounded from below and continuous solutions are encountered.

Aside from providing a theoretical method for finding solutions to the driven Jaynes-Cummings Rabi model, the introduced correspondence opens up a further connection between quantum phase transitions in quantum optical and condensed matter systems.

This research was supported by the Marsden fund of the Royal Society of New Zealand. R.G.J was supported on a scholarship funded by the New Zealand Tertiary Education Committee through the Dodd-Walls Centre for Photonic and Quantum Technologies. R.G.J. thanks R. Jáuregui for insightful discussions.

References {#references .unnumbered}
==========

[1]{}

Jaynes E T and Cummings F W 1963, *Proc. IEEE* **51**, 89 Meekhof 