We study the escape rate of flux quanta in a long Josephson junction having an asymmetric spatial inhomogeneous critical current density. We show that such a junction can behave as a quantum ratchet when it is driven by an ac current in the presence of a magnetic field. The rectification gives rise to an onset of the dc voltage across the junction. The usual approach of particle-like tunneling cannot describe this rectification, and a quantum field theory description is required. We also show that under definite conditions the rectification direction and, consequently the dc voltage, can change its sign when varying the temperature near the crossover temperature between the quantum and classical regimes. ; Comment: 4 pages, 3 figures