Dicke-Model Phase Transition in the Quantum Motion of a Bose-Einstein Condensate in an Optical Cavity

We show that the motion of a laser-driven Bose-Einstein condensate in a high-finesse optical cavity realizes the spin-boson Dicke model. The quantum phase transition of the Dicke model from the normal to the superradiant phase corresponds to the self-organization of atoms from the homogeneous into a periodically patterned distribution above a critical driving strength. The fragility of the ground state due to photon measurement induced backaction is calculated.

Figure 1

Photon (dashed red lines) and motionally excited atom (solid blue lines) numbers in the ground state. Thick lines represent the contributions form the mean fields (${\alpha }_0^2$ and ${\beta }_0^2$, these are the photon and atom excitation numbers divided by $N$, respectively), which can be the order parameters of the phase transition. Thin lines represent the incoherent excitations due to the squeezing, given by the quantum averages $⟨a^{†}a⟩$ and $⟨b^{†}b⟩$ taken in the ground state. Parameters are ${\delta }_C=-100{\omega }_R$, $u=-0.1{\omega }_R$.

Figure 2

Diffusion out from the ground state. The rate of increase of normal mode excitations (dashed line) and that of photons and motionally excited atoms (solid line) with coarse graining $|{\delta }_C{|}^{-1}\ll \delta t\ll {\omega }_R^{-1}$. The almost overlapping dash-dotted line is derived from an adiabatic elimination method and is given by the analytic result of Eq. (13). Parameters are ${\delta }_C=-100{\omega }_R$, $u=-0.1{\omega }_R$.