Lorentz transformations and squeezing, as found in the solutions below critical drive. The eigenstates found in the privileged frame, equation (\[eige\_8b\]), correspond to the dressed states of the Jaynes-Cummings model. When the Lorentz transformation was applied to reach the laboratory frame, the position and accompanying momentum operator appeared rescaled. This would translate to a squeezing transformation of these states. In reality, the natural length in the privileged frame is a relativistic invariant, $$l = \sqrt{ \frac{\hbar c }{ e ( F_{\mu \nu}F^{\mu \nu} )^{1/2}}} \, ,$$ which rules the behaviour of the system, and in this sense the transformation does not generate squeezing. The value of this quantity also helps to explain how the behaviour of the particle reacts towards the flipping of the potential introduced between magnetic and electric field-dominated cases. A real (complex) value determines discrete (continuous) solutions.

The creation and annihilation operators given in equation (\[corr\_7\]), however, are defined in terms of $l_{\mathcal{B}}$, not the relativistic invariant. They are affected by the Lorentz transformation. In the optical analogue these operators act upon the photon number of the field mode and the squeezing refers to photons injected into the mode through the coherent drive [@Alsing_1992_2]. In the mechanical 