Cavity-enhanced long-distance coupling of an atomic ensemble to a micromechanical membrane

We discuss a hybrid quantum system where a dielectric membrane situated inside an optical cavity is coupled to a distant atomic ensemble trapped in an optical lattice. The coupling is mediated by the exchange of sideband photons of the lattice laser, and is enhanced by the cavity finesse as well as the square root of the number of atoms. In addition to observing coherent dynamics between the two systems, one can also switch on a tailored dissipation by laser cooling the atoms, thereby allowing for sympathetic cooling of the membrane. The resulting cooling scheme does not require resolved sideband conditions for the cavity, which relaxes a constraint present in standard optomechanical cavity cooling. We present a quantum mechanical treatment of this modular open system which takes into account the dominant imperfections, and identify optimal operation points for both coherent dynamics and sympathetic cooling. In particular, we find that ground state cooling of a cryogenically precooled membrane is possible for realistic parameters.

Figure 1

(a) Micromechanical membrane in a cavity coupled to a distant atomic ensemble by means of a laser (red, frequency ${\omega }_{\mathrm{L}}$). The motion of the membrane shifts the optical lattice dipole trap of the atoms, while the motion of the atoms changes the radiation pressure on the membrane. Inset: The atoms are modeled as two-level systems, which are far off resonant with respect to the laser. (b) Refractive index profile used in the one-dimensional model [Eq. (6)].

Figure 2

Light fields involved in mediating the effective coupling. The laser-enhanced interactions (coupling constants $g_{\mathrm{m}}$ and $g_{\mathrm{at}}$) correspond to a conversion between laser photons ${\omega }_{\mathrm{L}}$ (black) and sideband photons ${\omega }_{\mathrm{L}}\pm {\omega }_{\text{at},\mathrm{m}}$ [blue (dark) and red (light)]. The response profile of the cavity is indicated by the green dashed line and easily accommodates both sidebands in the bad cavity regime discussed in the text.

Figure 3

Coherent dynamics: We plot the ratios ${\gamma }_{\mathrm{at}}^{\mathrm{diff}}/g$, ${\gamma }_{\mathrm{m}}^{\mathrm{diff}}/g$, and ${\gamma }_{\mathrm{m}}^{\mathrm{th}}/g$ as a function of finesse to illustrate the strong-coupling conditions in Eq. (34). One clearly observes a tradeoff between mechanical heating (${\gamma }_{\mathrm{m}}^{\mathrm{th}}$) and radiation pressure noise (${\gamma }_{\mathrm{m}}^{\mathrm{diff}}$). Parameters other than $\mathcal{F}$ are taken from Table .

Figure 4

Sympathetic cooling of the membrane: Mechanical steady-state occupation number. (a) Approximate total occupation number $n_{\mathrm{ss}}$ with its individual contributions in the adiabatic limit [Eq. (38)] as a function of $\mathcal{F}$ and for ${\gamma }_{\mathrm{at}}^{\mathrm{cool}}=240$ kHz. For comparison, the exact mechanical occupation $n_{\mathrm{ss}}^{\mathrm{exact}}$, which results from solving the QLEs (31) and (35), is depicted as a black line. (b) Exact steady-state occupation number $n_{\mathrm{ss}}^{\mathrm{exact}}$ as a function of $\mathcal{F}$ and ${\gamma }_{\mathrm{at}}^{\mathrm{cool}}$. The top axis displays the corresponding coherent coupling $g$ and the contours indicate the associated occupation numbers. For both plots the parameters from Table have been used.