  The complex dynamical phases of quantum systems are dictated by atomic
interactions that usually evoke an emergent periodic order. Here, we study a
quantum many-body system with two competing and substantially different
long-range interaction potentials where the dynamical instability towards
density order can give way to a superglass phase, i. e., a superfluid
disordered amorphous solid, which exhibits local density modulations but no
long-range periodic order. We consider a two-dimensional BEC in the
Rydberg-dressing regime coupled to an optical standing wave resonator. The
dynamic pattern formation in this system is governed by the competition between
the two involved interaction potentials: repulsive soft-core interactions
arising due to Rydberg dressing and infinite-range sign changing interactions
induced by the cavity photons. The superglass phase is found when the two
interaction potentials introduce incommensurate length scales. The dynamic
formation of this peculiar phase without any externally added disorder is
driven by quantum fluctuations and can be attributed to frustration induced by
the two competing interaction energies and length scales.
