Cavity optomechanics with feedback-controlled in-loop light

It has recently been shown [Rossi et al., Phys. Rev. Lett. 119, 123603 (2017); Phys. Rev. Lett. 120, 073601 (2018)] that feedback-controlled in-loop light can be used to enhance the efficiency of optomechanical systems. We analyze the theoretical ground at the basis of this approach and explore its potentialities and limitations. We discuss the validity of the model, analyze the properties of in-loop cavities, and show how they can be used to observe coherent optomechanical oscillations also with a weakly coupled system, improve the sideband cooling performance, and increase ponderomotive squeezing.

Figure 1

The feedback loop: A field quadrature at phase ${\theta }_{\mathrm{fb}}$ is detected, and the corresponding photocurrent is used to modulate the amplitude $X_{\mathrm{in}}$ of the field itself, while the field phase $Y_{\mathrm{in}}$ remains unaffected.

Figure 2

Photocurrent power spectrum normalized such that the power spectrum for a coherent field is equal to 1 and evaluated for the filter function defined in Eq. (14) (with ${\varphi }_{\mathrm{fb}}=0$). Lines from dark to light red correspond to values of ${\overline{g}}_{\mathrm{fb}}$ which range from ${\overline{g}}_{\mathrm{fb}}=-1/(2\sqrt{\eta }cos{\theta }_{\mathrm{fb}})$ to ${\overline{g}}_{\mathrm{fb}}=1/(2\sqrt{\eta }cos{\theta }_{\mathrm{fb}})$, and the horizontal red solid line corresponds to ${\overline{g}}_{\mathrm{fb}}=0$. The maxima of both the lightest and darkest curves diverge. The minima are indicated by the horizontal dashed line at the value $1/4$.

Figure 3

Power spectrum of the in-loop field $S_{X_{\mathrm{in}}^{\left(\varphi \right)}\left(\omega \right)}$ evaluated for the filter function defined in Eq. (14) (with ${\varphi }_{\mathrm{fb}}=0$) and for (a) $\varphi ={\theta }_{\mathrm{fb}}$ and (b) $\varphi ={\theta }_{\mathrm{fb}}-\pi /3$. Lines from dark to light red correspond to values of ${\overline{g}}_{\mathrm{fb}}$ which range from ${\overline{g}}_{\mathrm{fb}}=-1/(2\sqrt{\eta }cos{\theta }_{\mathrm{fb}})$ to ${\overline{g}}_{\mathrm{fb}}=1/(2\sqrt{\eta }cos{\theta }_{\mathrm{fb}})$, and the horizontal red solid line corresponds to ${\overline{g}}_{\mathrm{fb}}=0$. Both spectra are found under the same condition of Fig. 2. In panel (a), we observe zeros of the power spectrum in correspondence with the minima of the photocurrent. In panel (b), the minima are shifted and found for a value of ${\overline{g}}_{\mathrm{fb}}$ different from that corresponding to the minima of the photocurrent.

Figure 4

The feedback loop: a quadrature at phase ${\theta }_{\mathrm{fb}}$ of the field transmitted through a Fabry-Pérot optical cavity (detuned by $\mathrm{\Delta }$ from the input field and with dissipation rates ${\kappa }_1$ and ${\kappa }_2$) is detected, and the corresponding photocurrent is used to modulate the input amplitude $X_{\mathrm{in}}$. In this case, the feedback can be closed also by measuring the reflected field.

Figure 5

Photocurrent power spectrum normalized such that the power spectrum for a coherent field is equal to one, and evaluated for the flat filter function ${\tilde{g}}_{\mathrm{fb}}\left(\omega \right)$ defined in Eq. (14) (with ${\varphi }_{\mathrm{fb}}=0$), with perfect detection efficiency ($\eta =1$), and including a symmetric cavity (${\kappa }_1={\kappa }_2$). Different colors correspond to different values of ${\overline{g}}_{\mathrm{fb}}$ with dark to light red corresponding to increasing values in the range of feedback stability as defined in Sec. 3b, and the horizontal solid red line corresponds to ${\overline{g}}_{\mathrm{fb}}=0$. The thick gray line indicates the position of the cavity with $\kappa =1/{\tau }_{\mathrm{fb}}$ and $\mathrm{\Delta }=10/{\tau }_{\mathrm{fb}}$. Feedback closed (a) in transmission and (b) in reflection.

Figure 6

Total feedback transfer function $G_{\mathrm{fb}}\left(\omega \right)$ (see Sec. 3b) evaluated for the same parameters and feedback configurations of Fig. 5 [(a) in transmission and (b) in reflection], but with ${\overline{g}}_{\mathrm{fb}}=1$. The horizontal lines indicate the values of $G_{\mathrm{fb}}^{>}$ and $G_{\mathrm{fb}}^{<}$ introduced in Sec. 3b, such that the feedback is stable when $1/G_{\mathrm{fb}}^{>}<{\overline{g}}_{\mathrm{fb}}<1/G_{\mathrm{fb}}^{<}$. The blue solid lines are the real part $\mathrm{Re}\left[G_{\mathrm{fb}}\left(\omega \right)\right]$, the red dashed lines are the imaginary part $\mathrm{Im}\left[G_{\mathrm{fb}}\left(\omega \right)\right]$, and the yellow dotted lines are the absolute value $|G_{\mathrm{fb}}\left(\omega \right)|$. The thick gray line indicates the position of the cavity with $\kappa =1/{\tau }_{\mathrm{fb}}$ and $\mathrm{\Delta }=10/{\tau }_{\mathrm{fb}}$.

Figure 7

Power spectrum of the quadrature at phase ${\theta }_{\mathrm{un}}={\theta }_{\mathrm{fb}}$ of the field lost by the cavity from the unused output port. It is evaluated for the same parameters and feedback configurations of Fig. 5. In panel (a), the feedback is closed in transmission so this plot corresponds to the reflected field. In panel (b), instead the feedback is closed in reflection and this plot shows the fluctuations of the transmitted field. The unused (out-of-loop) output field always exhibits fluctuations enhanced with respect to the vacuum noise level (which is here set to 1).

Figure 8

The feedback loop: a quadrature at phase ${\theta }_{\mathrm{fb}}$ of the field transmitted through a cavity which contains a mechanical element (at frequency ${\omega }_{\mathrm{m}}$ and which interacts with the cavity field with strength $G$) is detected, and the corresponding photocurrent is used to modulate the input amplitude $X_{\mathrm{in}}$. The feedback can be closed also by measuring the reflected field.

Figure 9

Power spectrum of the cavity quadrature $F$ evaluated for a single-sided cavity (${\kappa }_{\mathrm{fb}}=\kappa$ and feedback closed in reflection). These plots are evaluated for the flat feedback filter function defined in Eq. (14), for the feedback parameters (${\overline{g}}_{\mathrm{fb}}$ and ${\varphi }_{\mathrm{fb}}$) that suppress anti-Stokes scattering according to Eq. (103), and for the value of ${\theta }_{\mathrm{fb}}$ that maximizes the corresponding Stokes scattering according to Eq. (104). The corresponding total feedback transfer function $G_{\mathrm{fb}}\left(\omega \right)$ is reported in the insets (the blue solid lines correspond to the real part and the red dashed ones to the imaginary part), and they show that the feedback is stable (namely $\mathrm{Re}\left\{G_{\mathrm{fb}}\right\}<1$ at the frequencies at which $\mathrm{Im}\left\{G_{\mathrm{fb}}\right\}=0$). Plot (a) refers to a short feedback delay time ${\tau }_{\mathrm{fb}}=0.1/{\omega }_{\mathrm{m}}$, and plot (b) corresponds to a larger value ${\tau }_{\mathrm{fb}}=5/{\omega }_{\mathrm{m}}$. The values of the curves at the frequencies $\pm {\omega }_{\mathrm{m}}$, indicated by the vertical lines, determine the values of the Stokes and anti-Stokes scattering rates, and they are equal for both cases (this indicates that the optimized values of the scattering rates are independent of the specific delay time). The solid dark lines correspond to perfect detection efficiency $\eta =1$, and the dashed lines correspond to $\eta =0.7$. The light solid lines are the corresponding result without feedback (namely the result valid for standard sideband cooling). The other parameters are $G=0.1{\omega }_{\mathrm{m}}$, $\mathrm{\Delta }={\omega }_{\mathrm{m}}$, and $\kappa ={\omega }_{\mathrm{m}}$.

Figure 10

[(a), (b)] Steady-state number of mechanical excitations as a function of the cavity detuning $\mathrm{\Delta }$ and the cavity decay rate $\kappa$ evaluated [using the perturbative result of Eq. (98) with the rates defined in Eq. (101)], for a single-sided cavity (feedback closed in reflection), with the flat feedback filter function (14), and with (a) perfect detection efficiency $\eta =1$ and (b) $\eta =0.7$. (c) Corresponding result for standard sideband cooling. The feedback parameters (${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$ and ${\varphi }_{\mathrm{fb}}$) are optimized at each point in order to achieve the optimal result of Eq. (106). The other parameters are ${\omega }_{\mathrm{m}}=10\mathrm{MHz}$, $\gamma ={10}^{-4}{\omega }_{\mathrm{m}}$, $n_{\mathrm{th}}=131$ (corresponding to a temperature of $100\mathrm{mK}$), and $G=0.2{\omega }_{\mathrm{m}}$.

Figure 11

Ponderomotive squeezing with feedback-controlled light: The squeezing of a quadrature of the reflected field at phase ${\theta }_{\mathrm{un}}$ is enhanced when the quadrature of the transmitted field at phase ${\theta }_{\mathrm{fb}}$ is detected, and the corresponding photocurrent is used to modulate the input amplitude $X_{\mathrm{in}}$. The feedback can be also closed in reflection and in this case one would enhance the ponderomotive squeezing of the transmitted field, and the role of ${\kappa }_{\mathrm{fb}}$ and ${\kappa }_{\mathrm{un}}$ would be exchanged (i.e., ${\kappa }_{\mathrm{un}}={\kappa }_2$ and ${\kappa }_{\mathrm{fb}}={\kappa }_1$).

Figure 12

Squeezing spectrum $S_{\mathrm{out},\mathrm{un}}^{\left({\theta }_{\mathrm{un}}\right)}\left(\omega \right)$, of the field reflected by a symmetric cavity (${\kappa }_1={\kappa }_2$) with the feedback closed in transmission, evaluated for ${\omega }_{\mathrm{m}}=10$ MHz, $\gamma ={10}^{-4}{\omega }_{\mathrm{m}}$, $T=100$ mK, $\mathrm{\Delta }=0$, $\kappa ={\kappa }_1+{\kappa }_2={\omega }_{\mathrm{m}}$, ${\tau }_{\mathrm{fb}}=0.1{\omega }_{\mathrm{m}}$, $G=0.5{\omega }_{\mathrm{m}}$, $\eta =1$, and the flat feedback filter of Eq. (14). The dark lines are with feedback, the thin light lines are without feedback, and the thick light lines correspond to a single-sided cavity with equal total decay rate and no feedback. The solid lines are found for the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$, which optimize the squeezing for the values given above and at the frequency corresponding to the vertical line. The dashed lines are found by optimizing the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$ for all the frequencies.

Figure 13

Squeezing spectrum $S_{\mathrm{out},\mathrm{un}}^{\left({\theta }_{\mathrm{un}}\right)}\left(\omega \right)$, of the reflected field with the feedback closed in transmission, at the frequency indicated by the vertical line in Fig. 12, as a function of the detection efficiency $\eta$. The solid lines are found for the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$, which optimize the squeezing at the value of $\eta$ indicated by the vertical line. The dashed lines are found by optimizing the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$ for all the values of $\eta$. The other parameters and line styles are as in Fig. 12.

Figure 14

Squeezing spectrum $S_{\mathrm{out},\mathrm{un}}^{\left({\theta }_{\mathrm{un}}\right)}\left(\omega \right)$, of the reflected field with the feedback closed in transmission, at the frequency indicated by the vertical line in Fig. 12, as a function of the optomechanical coupling strength $G$. The solid lines are found for the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$, which optimize the squeezing at the value of $G$ indicated by the vertical line. The dashed lines are found by optimizing the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$ for all the values of $G$. The other parameters and line styles are as in Fig. 12.

Figure 15

Squeezing spectrum $S_{\mathrm{out},\mathrm{un}}^{\left({\theta }_{\mathrm{un}}\right)}\left(\omega \right)$, of the reflected field with the feedback closed in transmission, at the frequency indicated by the vertical line in Fig. 12, as a function of the decay rate of the first mirror ${\kappa }_1$ (with constant $\kappa$). The solid lines are found for the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$, which optimize the squeezing at the value of ${\kappa }_1$ indicated by the vertical line. The dashed lines are found by optimizing the values of ${\theta }_{\mathrm{un}}$, ${\theta }_{\mathrm{fb}}$, ${\overline{g}}_{\mathrm{fb}}$, and ${\varphi }_{\mathrm{fb}}$ for all the values of ${\kappa }_1$. The other parameters and line styles are as in Fig. 12.