Quantum theory of transmission line resonator-assisted cooling of a micromechanical resonator

We propose a quantum description of the cooling of a micromechanical flexural oscillator by a one-dimensional transmission line resonator via a force that resembles cavity radiation pressure. The mechanical oscillator is capacitively coupled to the central conductor of the transmission line resonator. At the optimal point, the micromechanical oscillator can be cooled close to the ground state, and the cooling can be measured by homodyne detection of the output microwave signal.

Figure 1

(Color online) Schematic of a MR located at the center of a one-dimensional TLR. The external microwave drive field enters from the left and drives the TLR. The signal at the output on the right end can be used to measure the motion of the MR via homodyne detection (Refs. 32, 44, 47, 49).

Figure 2

(Color online) Variance of position $⟨\delta x^2⟩$ in units of $\hslash /m{\omega }_b$ as a function of effective detuning $\Delta$. The dashed lines are obtained by numerically evaluating the integral in Eq. (8). The solid lines are obtained by using the approximate expression (10). Here, $T=6\times {10}^3\hslash {\omega }_b/k_B$, $g_0=3\times {10}^{-5}{\omega }_b\sqrt{m{\omega }_b/\hslash }$, ${\omega }_a=2\times {10}^4{\omega }_b$, $\epsilon =2.5\times {10}^3{\omega }_b$, ${\gamma }_b=0.25\times {10}^{-4}{\omega }_b$, and $\kappa ={\omega }_b$ (upper lines) or $\kappa =0.2{\omega }_b$ (lower lines).

Figure 3

(Color online) Final mean phonon number in the steady state $n_b^f$ vs effective detuning $\Delta$ and decay rate $\kappa$ of the TLR ($T=3\times {10}^4\hslash {\omega }_b/k_B$; for the other parameters, see Fig. 2).

Figure 4

(Color online) (a) Optimal final mean phonon number obtained by numerically evaluating the integral in Eq. (8) (dashed line) or using the approximate expression in Eqs. (10, 13) (solid line) as a function of $\kappa$ at the optimal effective detuning $\Delta ={\omega }_b$. (b) Logarithm of the ratio of the corresponding effective damping rate ${\gamma }_b^{\text{eff}}\left({\omega }_b\right)$ to $\kappa$ as a function of $\kappa$. Here, $T=3\times {10}^3\hslash {\omega }_b/k_B$. For the other parameters, see Fig. 2.

Figure 5

(Color online) The final mean phonon number vs effective detuning $\Delta$ for three initial temperatures: $T=10\text{mK}$ (dotted lines), $T=30\text{mK}$ (solid lines), and $T=100\text{mK}$ (dot-dashed lines). Here, $\kappa =0.1{\omega }_b$, $m=1.5\times {10}^{-13}\text{kg}$, ${\omega }_b=4\text{MHz}$, and ${\gamma }_b=0.25\times {10}^{-4}{\omega }_b$ (equivalently, $Q_b\equiv {\omega }_b/{\gamma }_b=4\times {10}^4$). For the other parameters, see Fig. 2.

Figure 6

(Color online) The final effective mean phonon number $n_b^f$ is plotted as a function of $\Delta$ for different quality factors of the MR: $Q_b=4\times {10}^4$, ${10}^5$, $4\times {10}^5$, and ${10}^6$ (from up to down). The corresponding damping rates are ${\gamma }_b=100$, 40, 10, and 4 Hz. Here, $T=10\text{mK}$. For the other parameters see, Fig. 5.