Quantum scattering of droplets by wells and barriers in one-dimensional Bose-Bose mixtures
We investigate, both analytically and numerically, the scattering of quasi-one-dimensional quantum droplets from Pöschl-Teller potential wells and barriers. For attractive wells, we find a sharp transition between complete reflection and transmission at a critical incident velocity for both small and large flat-top droplets. The scattering interactions differ: small, soliton-like droplets form a spatially symmetric trapped mode at the critical velocity, showing their compressibility and coherence characteristics, while large droplets develop a spatially asymmetric trapped state, revealing incompressibility and internal structure. The critical velocity depends non-monotonically on atom number: it rises in the small, compressible-droplet regime, falls in the incompressible, flat-top regime, and turns at the crossover point. We also show that the reflectionless well generates a $π$-phase shift, strongly altering droplet-droplet collisions relative to free space. The persistence of a confined mode after collisions between trapped and incident droplets depends sensitively on their relative phase. For the repulsive barrier, we identify regimes of complete reflection, partial return, and full transmission, depending on incident velocity, barrier height, and particle number. Our predictions match direct numerical simulations in all cases.