Fig. 1.

Metastability and system overview. (A) Mean-field results from the toy model. In the presence of short-range interactions $U_{\text{𝗌}}$ and global-range interactions $U_{\text{𝗅}}$ atoms placed in a lattice potential can show metastable behavior. States (indicated by circles) can be protected by an energy barrier and the present state of the system depends on its history, leading to hysteresis. The Mott insulator (orange line) and the charge density wave (green lines) are stable (solid), metastable (dashed), or unstable. (B) Our system consists of a Bose–Einstein condensate coupled to a single mode of an optical resonator in the presence of 3D optical lattices. The atoms can create a particle imbalance $\mathrm{\Theta }$ by arranging in a checkerboard pattern which maximizes scattering of photons from a z lattice (not shown) into the resonator mode. (C) Schematic phase diagram of the system with a superfluid (SF, gray), a lattice supersolid (SS, blue), a Mott insulator (MI, orange), and a charge-density wave (CDW, green) phase. The shaded region between the MI and CDW indicates a region of hysteresis between the phases. The black arrow illustrates the experimental sequence: We prepare the atoms in the SF phase and ramp up the 3D optical lattices to increase $U_{\text{𝗌}}$, which brings the system into the MI phase. Subsequently, we carry out a detuning ramp toward cavity resonance which increases $U_{\text{𝗅}}$.