Incommensurate dynamic correlations and continuum scattering in BiCu$_2$PO$_6$

We report comprehensive inelastic neutron scattering measurements on single crystals of the frustrated two-leg ladder BiCu$_2$PO$_6$, whose ground state is a spin liquid phase with no static magnetic correlations down to 5 K. A combination of triple-axis and time-of-flight experiments were performed to explore magnetic excitations over a broad range of phase space. Operation of the instruments in a high resolution configuration enabled a detailed measurement of the dynamical structure factor over many Brillouin zones; revealing an extremely rich and highly unusual magnetic excitation spectrum. Two branches of steeply dispersing long-lived spin excitations are observed with gaps of 1.90(9) meV and 3.95(8) meV. Significant frustrating next-nearest-neighbor interactions along the ladder-leg drive the minimum of each excitation branch to incommensurate wavevectors 0.574$\pi$ and 0.553$\pi$ for the lower and upper energy branches respectively. Intriguingly, the spin excitations merge into a broad continuum near the top of each excitation band which persists to an upper boundary of 40 meV.