Nonequilibrium Phase Transition of Interacting Bosons in an Intra-Cavity Optical Lattice

We investigate the nonlinear light-matter interaction of a Bose-Einstein condensate trapped in an external periodic potential inside an optical cavity which is weakly coupled to vacuum radiation modes and driven by a transverse pump field. Based on a generalized Bose-Hubbard model which incorporates a single cavity mode, we include the collective backaction of the atoms on the cavity light field and determine the nonequilibrium quantum phases within the nonperturbative bosonic dynamical mean-field theory. With the system parameters adapted to recent experiments, we find a quantum phase transition from a normal phase to a self-organized superfluid phase, which is related to the Hepp-Lieb-Dicke superradiance phase transition. For even stronger pumping, a self-organized Mott insulator phase arises.

Figure 1

Experimental setup. The external laser beams (blue) form a square lattice in the $xz$ plane with a lattice constant ${\lambda }_p/2$ (blue pattern). The linear polarization of the pump beams (red) along the $y$ direction allows scattering of photons into the cavity. This leads to a square lattice in the $xz$ plane rotated with respect to the external lattice by 45° with the lattice constant ${\lambda }_p/\sqrt{(}2)$ (red pattern). Each minimum of the pump-induced lattice potential coincides with a minimum of the external lattice potential. The wavelength of all beams is ${\lambda }_p$.

Figure 2

Top panel: Order parameter $\mathrm{\Phi }$ of the self-organized phase and two exemplary density distributions. Middle panel: Effective potential $V_{\mathrm{eff}}\equiv V_x=V_{\mathrm{cl}}+U_0|\alpha {|}^2$ along the cavity axis (left ordinate) and mean photon number $|\alpha {|}^2$ in the cavity (right ordinate). Bottom panel: Resulting hopping amplitude ${\tilde{J}}_x$ along the cavity ($x$) axis (left ordinate) and calculated on-site interaction $U$ (right ordinate). All panels are vs the transverse pump laser strength $V_p$ (in units of $E_{\mathrm{rec}}$). The cavity decay rate, detuning, and external classical potential are $\kappa =50E_{\mathrm{rec}}$, ${\mathrm{\Delta }}_c=-2.8E_{\mathrm{rec}}$, and $V_{\mathrm{cl}}=-5E_{\mathrm{rec}}$.

Figure 3

(a) Phase boundary between the superfluid and the self-organized phase for varying cavity decay rates $\kappa$ and critical pump laser strengths $V_p^{\text{crit}}$ (both in units of $E_{\mathrm{rec}}$). The dashed line represents a square-root fit. (b) Superfluid order parameter ${\phi }_{\mathrm{SF}}$ (left ordinate) and SO order parameter $\mathrm{\Phi }$ (right ordinate) plotted against the pump strength $V_p$ (in units of $E_{\mathrm{rec}}$). The cavity decay rate and the detuning are set to $\kappa =50E_{\mathrm{rec}}$ and ${\mathrm{\Delta }}_c=-2.8E_{\mathrm{rec}}$, respectively.

Figure 4

Phase diagram with respect to ${\mathrm{\Delta }}_c$ and $V_p$ (both in units of $E_{\mathrm{rec}})$. N/SF refers to the normal superfluid phase, SO/SF to the self-organized superfluid phase, and SO/MI to the self-organized Mott insulator. The inset shows a zoom of the main plot for small values of $|{\mathrm{\Delta }}_c|$ ($\kappa =50E_{\mathrm{rec}}$).