Ground State Cooling of a Nanomechanical Resonator in the Nonresolved Regime via Quantum Interference

Ground state cooling of a nanomechanical resonator coupled to a superconducting flux qubit is discussed. By inducing quantum interference to cancel unwanted heating excitations, ground state cooling becomes possible in the nonresolved regime. The qubit is modeled as a three-level system in $\Lambda$ configuration, and the driving fluxes are applied such that the qubit absorption spectrum exhibits electromagnetically induced transparency, thereby canceling the unwanted excitations. As our scheme allows the application of strong cooling fields, fast and efficient cooling can be achieved.

Figure 1

(a) A nanomechanical resonator (red curve) coupling to a superconducting flux qubit. (b) The level diagram of the flux qubit. (c) Effective level scheme in the Lamb-Dicke (LD) limit. (d) Example absorption spectrum of the cooling field. Indicated are the transition frequencies for the different absorption channels in the LD limit.

Figure 2

(a) Cooling limit $n_{ss}$ against initial phonon number $N_i$. Dashed lines show Eq. (3), and solid lines show full numerical results with $\Gamma =0$, ${\Gamma }_{\varphi }=0$. The dash-dotted lines show numerical results including decoherence $\Gamma =0.02\gamma$, ${\Gamma }_{\varphi }=0.04\gamma$ as measured in [26]. Results are shown for two different cooling field strenghts. Parameters are ${\Omega }_e=\sqrt{\nu (\nu -\Delta )}$ [$\sqrt{\nu (\nu -\Delta )/2}$] for weak [strong] cooling fields and $\Delta =-3\gamma$, except for the dash-dotted curve with ${\Omega }_e={\Omega }_g$ which shows ${\Delta }_g=-2.85\gamma$, ${\Delta }_e=-3\gamma$, and ${\Omega }_e=0.53\gamma$. (b) Cooling limit $n_{ss}$ as a function of the detuning ${\Delta }_e$. Parameters are $N_i=16$ and ${\Omega }_e={\Omega }_g$. Shown are (i) Eq. (3), and (ii) numerical results for $\Gamma ={\Gamma }_{\varphi }=0$, both with ${\Omega }_e={\Omega }_e^{\mathrm{opt}}≔\sqrt{\nu (\nu -{\Delta }_e)/2}$ and ${\Delta }_e={\Delta }_g$. The other curves are numerical results for ${\Delta }_g=0.99{\Delta }_e$, $\Gamma =0.02\gamma$, and (iii) ${\Gamma }_{\varphi }=0$, ${\Omega }_e=0.58{\Omega }_e^{\mathrm{opt}}$, (iv) ${\Gamma }_{\varphi }=0.04\gamma$, ${\Omega }_e=0.64{\Omega }_e^{\mathrm{opt}}$ and (v) ${\Gamma }_{\varphi }=0.08\gamma$, ${\Omega }_e=0.64{\Omega }_e^{\mathrm{opt}}$.