Getting coherent state superpositions to stay put in phase space: $Q$ functions and one dimensional integral representations of generator eigenstates

Abstract: We study quantum mechanics in the phase space associated with the coherent state (CS) manifold of Lie groups. Eigenstates of generators of the group are constructed as one dimensional integral superpositions of CS along their orbits. We distinguish certain privileged orbits where the superposition is in phase. Interestingly, for closed in phase orbits, the geometric phase must be quantized to $2\pi\mathbb{Z}$, else the superposition vanishes. This corresponds to exact Bohr-Sommerfeld quantization. The maximum of the Husimi-Kano $Q$ quasiprobability distribution is used to diagnose where in phase space the eigenstates of the generators lie. The $Q$ function of a generator eigenstate is constant along each orbit. We conjecture that the maximum of the $Q$ function corresponds to these privileged in phase orbits. We provide some intuition for this proposition using interference in phase space, and then demonstrate it for canonical CS ($H_4$ oscillator group), spin CS ($SU(2)$) and $SU(1,1)$ CS, relevant to squeezing.