Cooling a micromechanical resonator by quantum back-action from a noisy qubit

We study the role of qubit dephasing in cooling a mechanical resonator by quantum back-action. With a superconducting flux qubit as a specific example, we show that ground-state cooling of a mechanical resonator can only be realized if the qubit dephasing rate is sufficiently low.

Figure 1

(Color online) Schematic of our setup. A doubly clamped mechanical beam is incorporated in a superconducting flux qubit (Josephson junctions are indicated by crosses). The initial bias in the loop is controlled by a magnetic flux ${\Phi }_{z0}$ in the $z$ direction. A coupling magnetic field $B_0$ is applied in the $y$ direction, leading to a coupling of the motion of the beam and the qubit because the supercurrent leads to a Lorentz force on the beam. A microwave line introduces a microwave bias to the qubit loop.

Figure 2

(Color online) Inset: dependence of the final phonon number $n$ on the scaled detuning $\delta \omega /{\omega }_{\text{b}}$ and the scaled drive amplitude $A/{\omega }_{\text{b}}$ (with ${\epsilon }_0=800\text{MHz}$). The plot range is limited from $n=0$ to $n=1$ (i.e., the ground-state cooling regime). The exterior region (shown in blue) corresponds to $n>1$. Main panel: cooling power $\eta \equiv N/n-1$ as a function of the scaled detuning $\delta \omega /{\omega }_{\text{b}}$ between the microwave drive and qubit energy splitting along the dashed line in the inset. The other parameters are the same as those used to estimate the cooling limit in the text.

Figure 3

(Color online) Final phonon number of the MR versus scaled qubit bias. The three curves correspond to qubits with the same relaxation rate $\left({\Gamma }_{\downarrow }^{\left(0\right)}\approx 4\text{MHz}\right)$ but different dephasing rates [from bottom to top $\alpha =0$ (red), $\alpha =0.008$ (blue), and $\alpha =0.08$ (black)].