  The $5d^1$ ordered double perovskites present an exotic playground for
studying novel multi-polar physics due to large spin-orbit coupling. We present
Re L3 edge resonant inelastic X-ray scattering (RIXS) results that reveal the
presence of the dynamic Jahn-Teller effect in the A$_2$MgReO$_6$ (A = Ca, Sr,
Ba) family of $5d^1$ double perovskites. The spin-orbit excitations in these
materials show a strongly asymmetric lineshape and exhibit substantial
temperature dependence, indicating that they are dressed with lattice
vibrations. Our experimental results are explained quantitatively through a
RIXS calculation based on a spin-orbit-lattice entangled electronic ground
state with the dynamic Jahn-Teller effect taken into consideration. We find
that the spin-orbit-lattice entangled state is robust against magnetic and
structural phase transitions as well as against significant static Jahn-Teller
distortions. Our results illustrate the importance of including vibronic
coupling for a complete description of the ground state physics of $5d^1$
double perovskites. Usage: Secondary publications and information retrieval
purposes.
