Nonequilibrium Quantum Phase Transition in a Hybrid Atom-Optomechanical System

We consider a hybrid quantum many-body system formed by a vibrational mode of a nanomembrane, which interacts optomechanically with light in a cavity, and an ultracold atom gas in the optical lattice of the out-coupled light. The adiabatic elimination of the light field yields an effective Hamiltonian which reveals a competition between the force localizing the atoms and the membrane displacement. At a critical atom-membrane interaction, we find a nonequilibrium quantum phase transition from a localized symmetric state of the atom cloud to a shifted symmetry-broken state, the energy of the lowest collective excitation vanishes, and a strong atom-membrane entanglement arises. The effect occurs when the atoms and the membrane are nonresonantly coupled.

Figure 1

Sketch of the hybrid system. A nanomechanical membrane in an optical cavity is optically coupled to the vibrational motion of a distant atom gas.

Figure 2

(a) Normalized potential energy surface $\epsilon (\mathcal{S},\lambda )$ as a function of $\mathcal{S}$ and $\lambda$ for $V=200{\omega }_R$. The red line indicates the minimum $\epsilon ({\mathcal{S}}_0,\lambda )=0$. (b) Positive value of ${\mathcal{S}}_0$ and (c) condensate width ${\sigma }_0$ as a function of $\lambda$ for different values $V$ as indicated in (c). The dashed curves show the GPE results for comparison. The inset in (b) shows the order parameter ${\mathcal{S}}_0$ as a function of $\lambda /{\lambda }_c$ for which all data points collapse to a single curve. For all panels, we have used $g=0$, ${\mathrm{\Omega }}_{\mathrm{m}}=100{\omega }_R$, and $\gamma =20{\omega }_R$.

Figure 3

(a) Atom-membrane entanglement by the logarithmic negativity of the ground state ($\gamma =0$) between membrane and either atom displacement (purple) or condensate width (pink). (b) Frequencies of collective excitations. (c) Decay rates of the collective eigenmodes (some curves are scaled by the factors indicated). Curves in blue, green, and red correspond to the membrane mode, the condensate width mode, and the atomic displacement excitation, respectively. The dashed curves show the GPE results. The parameters are $V=200{\omega }_R$, ${\mathrm{\Omega }}_{\mathrm{m}}=100{\omega }_R$, $\gamma =20{\omega }_R$, and $g=0$.

Figure 4

(a) Membrane excitation frequency and (b) momentum distribution $|n(k)|$ of the atoms in a single well. (c) Width of $|n(k)|$ in dependence of $\lambda$. The dotted vertical line indicates ${\lambda }_c$. The parameters are $V=200{\omega }_R$, ${\mathrm{\Omega }}_{\mathrm{m}}=100{\omega }_R$, $\gamma =20{\omega }_R$, and $g=0$.