Relativistic Maxwell-Bloch Equations with Applications to Astrophysics

Abstract

We derive relativistic Maxwell-Bloch equations for potential applications in astronomical environments, where various radiative processes are known to occur, including the maser action and Dicke's superradiance. We show that for both phenomena a radiating system's response is preserved at different relative velocities between the system's rest frame and the observer, while the relevant timescales and the radiation intensity transform as expected from relativistic considerations. We verify that the level of coherence between groups of emitters travelling at different speeds is unchanged in all reference frames. We also derive relativistic versions of the maser equations applicable in the steady-state regime.

Figures (5)

• Figure 1: Superradiance (SR) and maser emission at $z=L'/\gamma$ for relative velocities $\beta=0$ (solid blue), 0.5 (dashed orange), and $-0.5$ (dash-dotted green) with $n_t'=2\times10^{4}$ m$^{-3}$. The shape of the curves is preserved regardless of the velocity but squeezed/stretched depending on the direction of motion relative to the observer.
• Figure 2: Same as Fig. \ref{['fig:1']} with $n_t'=6\times10^{3}$ m$^{-3}$. Similar observations can be made for the intensities and timescales as for Fig. \ref{['fig:1']}.
• Figure 3: Endfire intensities of systems in the rest frame with one or two velocity channels, each channel of population density difference $n'_t=6\times10^{3}$ m$^{-3}$. From top to bottom, the panels correspond to systems of non-superradiant single channel, superradiant two-channel with $\Delta v'=0$, and non-superradiant two-channel with $\Delta v'=40\,dv'$, respectively.
• Figure 4: Endfire intensities of a two-channel system with $\Delta v'=40\,dv'$ moving at centre velocities $\beta=0$ (blue), 0.5 (orange) and $-0.5$ (green) relative to the observer. Both channels possess a rest frame population density difference of $n'_t=6\times10^{3}$ m$^{-3}$. As expected, the same level of interaction is maintained, while the intensities scale with the factor $(1+\beta)/(1-\beta)$.
• Figure 5: Same as Figure \ref{['fig:4']} but with the population density difference of each channel increased to $n_t'=1.2\times10^4\,\text{m}^{-3}$. A superradiance response is observed at all three velocities, with the same intensity scaling factor described above.