Magic Conditions for Multiple Rotational States of Bialkali Molecules in Optical Lattices

We investigate magic-wavelength trapping of ultracold bialkali molecules in the vicinity of weak optical transitions from the vibrational ground state of the X$^1\Sigma^+$ potential to low-lying rovibrational states of the b$^3\Pi_0$ potential, focussing our discussion on the $^{87}$Rb$^{133}$Cs molecule in a magnetic field of $B=181\,$G. We show that a frequency window exists between two nearest neighbor vibrational poles in the dynamic polarizability where the trapping potential is "near magic" for multiple rotational states simultaneously. We show that the addition of a modest DC electric field of $E=0.13\,\text{kV}/\text{cm}$ leads to an exact magic-wavelength trap for the lowest three rotational states at a angular-frequency detuning of $\Delta_{v'=0} = 2\pi\times 218.22$\,GHz from the X$^1\Sigma^+ (v=0, J=0)\rightarrow$ b$^3\Pi_0 (v'=0, J=1)$ transition. We derive a set of analytical criteria that must be fulfilled to ensure the existence of such magic frequency windows and present an analytic expression for the position of the frequency window in terms of a set of experimentally measurable parameters. These results should inform future experiments requiring long coherence times on multiple rotational transitions in ultracold polar molecules.