Laser Cooling of a Nanomechanical Resonator Mode to its Quantum Ground State

We show that it is possible to cool a nanomechanical resonator mode to its ground state. The proposed technique is based on resonant laser excitation of a phonon sideband of an embedded quantum dot. The strength of the sideband coupling is determined directly by the difference between the electron-phonon couplings of the initial and final states of the quantum dot optical transition. Possible applications of this scheme include generation of nonclassical states of mechanical motion.

Figure 1

Top: Schematic diagram of a GaAs bridge of length $L$, width $b$ and thickness $d$ (see $\mathbf{s}$), with an embedded InAs quantum dot (D); W denotes the wetting layer. Bottom: Vertical cross sections through the axis of the bridge illustrating the deformations suffered by the QD, when the beam is bent slightly in the vertical direction ($x$). $\mathbf{r}$ and $\mathbf{t}$ correspond, respectively, to positive and negative deflection, and $\mathbf{s}$ shows the equilibrium configuration. Each volume element parallel to the axis undergoes a simple extension or compression.

Figure 2

Level diagram of the QD coupled to the resonator for perturbative $\Omega$. The cooling (heating) cycle consists of the excitation of the QD and subsequent spontaneous photon emission, and is denoted by solid (dashed) lines. The black (gray) path corresponds to the case where phonon number is changed during the laser absorption (spontaneous emission).

Figure 3

Optimal final number of phonons ${\tilde{n}}_f$ vs initial number $n_i$ (solid line) for $\Gamma /{\omega }_0=1/4$, ${\eta }^{2}Q=100$. The optimum results from searching for the laser parameters (${\delta }_{\mathrm{o}\mathrm{p}\mathrm{t}}$, ${\Omega }_{\mathrm{o}\mathrm{p}\mathrm{t}}$) that minimize $n_f$. There are three regimes (dotted lines): (I) for $n_i\le (\Gamma /4{\omega }_0{)}^2$ laser addressing of the structure is not useful (i.e., $n_f\ge n_i$), (II) for small $n_i$ [$n_i\ll {\eta }^{2}Q(\Gamma /4{\omega }_0{)}^2$], ${\tilde{n}}_f$ is constant and ${\delta }_{\mathrm{o}\mathrm{p}\mathrm{t}}\approx {\omega }_0$, ${\Omega }_{\mathrm{o}\mathrm{p}\mathrm{t}}^2\ll {\omega }_0^2$, (III) for large $n_i$ [$n_i\gg {\eta }^{2}Q(\Gamma /4{\omega }_0{)}^2$] the optimum is proportional to $n_i$ with the ratio ${\tilde{n}}_f/n_i$ bounded by $1/{\eta }^{2}Q$ and ${\delta }_{\mathrm{o}\mathrm{p}\mathrm{t}}\sim {\omega }_0$, ${\Omega }_{\mathrm{o}\mathrm{p}\mathrm{t}}\sim {\omega }_0$. The dashed line corresponds to $n_i/{\eta }^{2}Q+(\Gamma /4{\omega }_0{)}^2$.