Figure 1

(Color online) (a) Schematic diagram of a transmission line resonator (TLR), in blue, and a SQUID-nanomechanical beam system, in dark yellow and dark blue. Here, “JJ” represents a Josephson junction and $E_{\text{in}}$, $E_{\text{out}}$ are, respectively, the input and output electromagnetic fields of the transmission line resonator. (b) Diagram of the quantum circuit.Reuse & Permissions

Figure 2

(Color online) Schematic diagram of the feedback control circuit. The “DSP” denotes a digital signal processor which works as the integral estimator of the system state by solving Eqs. (B3, B4). The “A/D” and “D/A” represent the analog-to-digital and digital-to-analog signal converters. The output of the estimator is fed into a linear amplifier circuit to obtain a control signal which is further used to drive the input electromagnetic field of the transmission line resonator and the input of the estimator.Reuse & Permissions

Figure 3

(Color online) Squeezing the position fluctuations of the beam with the parameters given in Eqs. (29, 30, 31). (a) Variances of the position and momentum of the beam (in units of $\hslash$) versus the normalized control parameter ${\tilde{\upsilon }}_p={\upsilon }_p/{\omega }_T$. Recall that ${\upsilon }_p$ is the linear feedback gain [see Eq. (19)]. The green line with triangles and the blue dashed line represent, respectively, the controlled variances of the position $V_{x_M}^c/\hslash$ and momentum $V_{p_M}^c/\hslash$ of the beam. The red line with asterisks and the blue solid line represent the uncontrolled variances $V_{x_M}^{uc}/\hslash$ and $V_{p_M}^{uc}/\hslash$ of the beam, which coincide because the beam without control is in a coherent thermal state with equal variances of position and momentum. (b) Products of the variances (in units of ${\hslash }^2$) versus the normalized control parameter ${\tilde{\upsilon }}_p={\upsilon }_p/{\omega }_T$. The green line with asterisks and the blue solid line denote the uncontrolled trajectory $V_{x_M}^{uc}{V}_{p_M}^{uc}/{\hslash }^2$ and the controlled trajectory $V_{x_M}^c{V}_{p_M}^c/{\hslash }^2$, respectively. The red dashed line at 1/4 represents the Heisenberg uncertainty limit ${\hslash }^2/4$.Reuse & Permissions

Figure 4

(Color online) Squeezing the momentum fluctuations of the beam with the parameters given in Eqs. (29, 30, 32). (a) Variances of the position and momentum of the beam (in units of $\hslash$) versus the normalized control parameter ${\tilde{\upsilon }}_p={\upsilon }_p/{\omega }_T$. The green line with triangles and the blue dashed line represent, respectively, the controlled variances of the position $V_{x_M}^c/\hslash$ and momentum $V_{p_M}^c/\hslash$ of the beam. The red line with asterisks and the blue solid line represent the uncontrolled variances $V_{x_M}^{uc}/\hslash$ and $V_{p_M}^{uc}/\hslash$ of the beam, which coincide because the beam without control is in a coherent thermal state with equal variances of position and momentum. (b) Products of the variances (in units of ${\hslash }^2$) versus the normalized control parameter ${\tilde{\upsilon }}_p={\upsilon }_p/{\omega }_T$. The green line with asterisks and the blue solid line denote the uncontrolled trajectory $V_{x_M}^{uc}{V}_{p_M}^{uc}/{\hslash }^2$ and the controlled trajectory $V_{x_M}^c{V}_{p_M}^c/{\hslash }^2$, respectively. The red dashed line at 1/4 represents the Heisenberg uncertainty limit ${\hslash }^2/4$.Reuse & Permissions

Figure 5

(Color online) Cooling the nanomechanical beam for (a) the average photon number $\overline{n}$, and (b) the effective temperature $T_{\text{eff}}$ versus the normalized control parameter ${\tilde{\upsilon }}_p={\upsilon }_p/{\omega }_T$. The blue solid lines representing the average photon number $n^c$ and effective temperatures $T_{\text{eff}}^c$ under control are below the green lines with asterisks representing the corresponding average photon number $n^{uc}$ and effective temperature $T_{\text{eff}}^{uc}$ without control, which means that the designed feedback control could effectively cool the fluctuations of the beam induced by thermal noises.Reuse & Permissions