Velocity correlations of vortices and rarefaction pulses in compressible planar quantum fluids

We present a quantitative analytical framework for calculating two-point velocity correlations in compressible quantum fluids, focusing on two key classes of superfluid excitations: vortices and rarefaction pulses. We employ two complementary approaches. First, we introduce a new Ansatz for vortex cores in planar quantum fluids that enhances analytic integrability. This Ansatz yields closed-form expressions for power spectra and velocity correlations for general vortex distributions. Using it, we identify distinct signatures of short- and long-range velocity correlations corresponding to vortex dipoles and vortex pairs, respectively. Second, we analyze the fast rarefaction pulse regime of the Jones-Roberts soliton. By applying the asymptotic high-velocity wave function, we derive analytic expressions for the velocity power spectrum and correlation function, capturing the soliton characteristic length scale. We validate our analytical results for the homogeneous system against numerical treatment of a large trapped system, finding close quantitative agreement. Our findings provide the first analytic treatment of velocity correlations for Jones-Roberts solitons in quantum fluids of light [M. Baker-Rasooli et al., Phys. Rev. Lett. 134, 233401 (2025)] and establish a foundation for characterizing vortices and solitons in compressible quantum fluids.