Active feedback cooling of levitated dielectric particles is a pivotal technique for creating ultrasensitive sensors and probing fundamental physics. Here we demonstrate phase-adaptive feedback cooling of silica nanoparticles optically trapped in standing-wave potential formed by two co-linearly polarized counterpropagating diffraction-free guided modes in a hollow-core photonic crystal fiber at room temperature. Unlike standard laser intensity- or Coulomb force-based feedback, our approach modulates the relative optical phase between the counterpropagating fundamental modes proportionally to the particle's axial momentum. This generates a Stokes-like dissipative force which effectively damps the center-of-mass motion without introducing excess heating and can also work with uncharged particles. At 2 mbar air pressure, the axial center-of-mass temperature of a 195 nm silica particle is reduced by half upon application of the feedback and to 58.6 K at 0.5 mbar. The measured mechanical spectra agree well with our analytical model, validating the cooling mechanism. We envision this approach will open up pathways towards long-range, coherent control of mesoscopic particles inside hollow-core fibers, offering a fiber-integrated versatile platform for future quantum manipulation.

We present several analytical approaches to the Dicke superradiance problem, which involves determining the time evolution of the density operator for an initially inverted ensemble of $N$ identical two-level systems undergoing collective spontaneous emission. This serves as one of the simplest cases of open quantum system dynamics that allows for a fully analytical solution. We explore multiple methods to tackle this problem, yielding a solution valid for any time and any number of spins. These approaches range from solving coupled rate equations and identifying exceptional points in non-Hermitian evolution to employing combinatorial and probabilistic techniques, as well as utilizing a quantum jump unraveling of the master equation. The analytical solution is expressed as a residue sum obtained from a contour integral in the complex plane, suggesting the possibility of fully analytical solutions for a broader class of open quantum system dynamics problems.