consider dissipation through two decay rates: one, $\kappa$, is related to damping of the field mode, and the other, $\gamma/2$, to spontaneous emission of the two-level system. It is found that even in the presence of these rates, the critical point identified in the Dirac-particle analogue \[equation (\[back\_2\])\] remains as the organizing center of the transition. For non-vanishing $\kappa$ the system reaches a steady-state in the long-time limit with zero photon number expectation below critical drive and a finite value above it. The interplay between $\kappa, \lambda$ and $\epsilon$ determine the maximum photon value, allowing for the definition of a thermodynamical limit in this system where threshold behaviour and its relation to a phase transition can be studied. For the generalized model ($\eta\neq0$), damping of the cavity mode is considered in Ref. [@Gutierrez_2017b].

![Bloch sphere representation of the polarization $\langle \hat{\mathbf{\sigma}} \rangle$ below (orange) and above (blue) the critical point $\epsilon_{0} = \lambda /2$. Two branches arise above the critical point since $\phi^{+}_{D}$ is also a solution of the JC Hamiltonian for $\varepsilon = 0$ under $E \rightarrow - E$ \[see equations (\[corr\_6a\]) and (\[corr\_6b\])\].[]{data-label="fig:dipole"}](tikz_bloch_5){width=".5\linewidth"}

Squeezing and Lorentz transformations
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A word is in order regarding the apparent relationship between 