Cooling a micromechanical beam by coupling it to a transmission line

We study a method to cool down the vibration mode of a micromechanical beam using a capacitively coupled superconducting transmission line. The Coulomb force between the transmission line and the beam is determined by the driving microwave on the transmission line and the displacement of the beam. When the frequency of the driving microwave is smaller than that of the transmission line resonator, the Coulomb force can oppose the velocity of the beam. Thus, the beam can be cooled. This mechanism, which may enable us to prepare the beam in its quantum ground state of vibration, is feasible under current experimental conditions.

Figure 1

(Color online) (a) Schematic layout for the proposed device and (b) its equivalent circuit. A superconducting coplanar stripline, forming a 1D TLR, provides a cavity. A doubly clamped beam (central small dark red rectangle) is placed between two horizontal superconducting lines. This red beam is capacitively coupled to the central (hatched) superconducting line at a maximum of the voltage standing wave in the 1D TLR. The capacitances $C_0$ allow the input and output signals to be coupled to the central (hatched) stripline. This allows us to measure the amplitude and phase of the 1D TLR and apply dc and rf pulses to the 1D TLR.

Figure 2

(Color online) (a) Schematics of the beam (mass and spring) capacitively coupled to a conductor and (b) its equivalent circuit. The charging energy of the beam is determined by the voltage $V_a$ and variable capacitor $C_b$ between the beam and the 1D transmission line.

Figure 3

(Color online) The effective elastic constant $K^{\prime }$ (arbitrary units) versus the detuning $\Delta ={\omega }_{\mathrm{rf}}-{\omega }_{a0}$ (scaled by $\gamma$). When ${\omega }_{\mathrm{rf}}<{\omega }_{a0}$, the beam can be cooled. Here, the region $\Delta >0$ $(\Delta <0)$ is shaded by the red (blue) color, representing the heating (cooling) of the beam, respectively.

Figure 4

(Color online) Cooling effect $T_{\mathrm{eff}}∕T_0$ versus the oscillating frequency of the beam ${\omega }_b$ and the detuning $\Delta ={\omega }_{\mathrm{rf}}-{\omega }_{a0}$, both rescaled by $\gamma$, for the typical parameters listed in the text. There is a cooling window for ${\omega }_b$ and $\Delta$. The cooling effect is normalized to unity. The vertical bar at the right refers to cooling effects (in blue) and not heating (in red). The darker blue corresponds to a greater cooling effect $T_{\mathrm{eff}}∕T_0$. The beam is heated in the red region and becomes unstable in the green region.