  In this paper we define a quantization of the Double Ramification Hierarchies
of [Bur15b] and [BR14], using intersection numbers of the double ramification
cycle, the full Chern class of the Hodge bundle and psi-classes with a given
cohomological field theory. We provide effective recursion formulae which
determine the full quantum hierarchy starting from just one Hamiltonian, the
one associated with the first descendant of the unit of the cohomological field
theory only. We study various examples which provide, in very explicit form,
new $(1+1)$-dimensional integrable quantum field theories whose classical
limits are well-known integrable hierarchies such as KdV, Intermediate Long
Wave, Extended Toda, etc. Finally we prove polynomiality in the ramification
multiplicities of the integral of any tautological class over the double
ramification cycle.
