Optical lattices with micromechanical mirrors

We investigate a setup where a cloud of atoms is trapped in an optical lattice potential of a standing-wave laser field which is created by retroreflection on a micromembrane. The membrane vibrations itself realize a quantum mechanical degree of freedom. We show that the center-of-mass mode of atoms can be coupled to the vibrational mode of the membrane in free space. Via laser cooling of atoms a significant sympathetic cooling effect on the membrane vibrations can be achieved. Switching off laser cooling brings the system close to a regime of strong coherent coupling. This setup provides a controllable segregation between the cooling and coherent dynamics regimes, and allows one to keep the membrane in a cryogenic environment and atoms at a distance in a vacuum chamber.

Figure 1

A laser field impinging from the right is partially reflected off a dielectric membrane and forms a standing-wave optical potential for an atomic ensemble. Vibrations of the membrane’s fundamental mode will shift the standing-wave field, shaking atoms in the optical lattice. Conversely, oscillations of the atomic cloud (center of mass motion) will change the intensity of left- (right-) propagating field components, thus shaking the membrane via changing the radiation pressure on it. The membrane can be kept in a cryogenic environment, and the atoms at a distance in a vacuum chamber.

Figure 2

(a) Schematic illustration of mode functions $A_{\sigma }(k,z)$ of $R$ and $L$ modes for fields impinging from the right and left, respectively. All modes are in vacuum but a single one in the $R$ field at frequency ${\omega }_l$ which is driven by a laser field. (b) Mediated action of atomic COM fluctuations on the membrane: At the advanced time $t^{+}$ atoms absorb a photon from the right-propagating component of the laser field (with amplitude $\sqrt{\mathfrak{r}}\alpha$, straight arrow) and reemit a sideband photon to the $R$ field [$b_R^{†}(t^{+})$, wavy arrow], receiving a momentum kick of $2\hslash k_l$ [cf. the term in Eq. (2a)]. At a later time $t$ the sideband photon is annihilated [] and transferred back to the laser field, giving a momentum kick to the membrane [cf. first term in Eq. (2b)]. (c) Reverse action: Via corresponding emission and reabsorption processes as given in Eqs. (2b) and (2c), membrane vibrations affect atomic COM fluctuations. In these processes both the $R$ (upper panel) and the $L$ field (lower panel) contribute, effectively mediating a stronger action than the process shown in (b).

Figure 3

Cooling factor $f=\mathrm{n̄}/{\mathrm{n̄}}_{\mathrm{SS}}$ versus effective coupling $g$ between the atomic COM motion and the membrane vibrations and rate of Raman sideband laser cooling ${\gamma }_{\mathrm{at}}^{\mathrm{cool}}$, as determined from an exact numerical solution of the effective master equation (3) for parameters as given in the case study. A cooling factor of ${10}^4$ yields a steady state occupation ${\mathrm{n̄}}_{\mathrm{SS}}$ below 1 if the system is precooled to below 500 mK. At room temperature the sympathetic cooling effect will be still clearly observable.