Bose-Einstein condensation and many-body localization of rotational excitations of polar molecules following a microwave pulse

We study theoretically the collective dynamics of rotational excitations of polar molecules loaded into an optical lattice in two dimensions. We explore the collective many-body phases that form following a microwave pulse. We show that, owing to the long-range interactions between molecules and energy conservation in this isolated system, the rotational excitations can form a Bose-Einstein condensate with long-range order, even for the natural (undressed) dipole interactions. This manifests itself as a divergent $T_2$ coherence time of the rotational transition even in the presence of inhomogeneous broadening. The dynamical evolution of a dense gas of rotational excitations shows regimes of nonergodicity, characteristic of many-body localization and localization protected quantum order.

Figure 1

Phase diagram of the disorder-free system. The prepared state is always close to the $T=0^{-}$ ground state because of small spin-wave corrections and hence thermalizes in the BEC phase for all $\rho$. The BEC transition line represents the energy of the system at $T=T_{\mathrm{C}}$ when the condensate density vanishes.

Figure 2

Phase diagram of the disordered system for $\rho =\frac{1}{2}$. The BEC transition line is the energy of the system corresponding to the transition temperature. Isolated systems below this energy thermalize in the uncondensed phase. The dashed line is the energy of the prepared state with $\rho =\frac{1}{2}$ which thermalizes in the BEC phase for $J/W>j_{\mathrm{Th}}$. The mean-field description breaks down for $J/W<j_{\mathrm{MF}}\sim 0.24$. In Fig. 3 we compare the dynamics at points A and B.

Figure 3

Dynamical evolution of the condensate fraction, computed via ${\rho }_0=\frac{(B_x^2+B_y^2)}{4J^2}$ from mean-field theory. For strong disorder the condensate fraction decays to zero at long times, so the initial state completely decoheres.

Figure 4

Comparison of the condensate fraction for the thermalized state with that found from the long-time dynamics of mean-field theory (for increasing system sizes $N$) for a system prepared with a coherent microwave pulse. The prepared state retains coherence in a regime $j_1<J/W<j_{\mathrm{Th}}$ for which the thermalized state would be completely incoherent, showing nonergodic dynamics in the disordered system. The dotted-dashed line is the long-time behavior for a system prepared in a state of minimal coherence, described in the text, with the same mean energy.