Dissipative and dispersive cavity optomechanics with a frequency-dependent mirror

An optomechanical microcavity can considerably enhance the interaction between light and mechanical motion by confining light to a subwavelength volume. However, this comes at the cost of an increased optical loss rate. Therefore, microcavity-based optomechanical systems are placed in the unresolved-sideband regime, preventing sideband-based ground-state cooling. A pathway to reduce optical loss in such systems is to engineer the cavity mirrors, i.e., the optical modes that interact with the mechanical resonator. In our work, we analyze such an optomechanical system, whereby one of the mirrors is strongly frequency dependent, i.e., a suspended Fano mirror. This optomechanical system consists of two optical modes that couple to the motion of the suspended Fano mirror. We formulate a quantum-coupled-mode description that includes both the standard dispersive optomechanical coupling as well as dissipative coupling. We solve the Langevin equations of the system dynamics in the linear regime showing that ground-state cooling from room temperature can be achieved even if the cavity is per se not in the resolved-sideband regime, but achieves effective sideband resolution through strong-optical-mode coupling. Importantly, we find that the cavity output spectrum needs to be properly analyzed with respect to the effective laser detuning to infer the phonon occupation of the mechanical resonator. Our work also predicts how to reach the regime of nonlinear quantum optomechanics in a Fano-based microcavity by engineering the properties of the Fano mirror.

Figure 1

(a) Sketch of the optomechanical setup, consisting of a double-sided optical cavity with one movable frequency-dependent mirror, and (b) the setup's coupled-mode picture. The cavity mode ($\widehat{a}$) is coupled to electromagnetic environments on each side of the double-sided cavity (blue circles). The left mirror is frequency dependent and its internal mode ($\widehat{d}$) is also coupled to the left electromagnetic environment. The left mirror has a mechanical degree of freedom ($\widehat{q}$), which is coupled to a phonon environment (red circle). The cavity is driven with a laser at frequency ${\omega }_{\text{las}}$. See Secs. 2a1 and 2b1 for definitions of all indicated variables.

Figure 2

Illustration of the relevant parameter regimes. Each box details one aspect of the different regimes for a specific set of parameters. We have indicated in green in which regime the studied devices (see Sec. 2d and Table 1) are operating. In this article, where we focus on strong optical coupling, the effective “$-$” mode (highlighted in red) has a small linewidth and therefore characterizes the physics of the device, in contrast to the highly damped (and hence irrelevant) “$+$” mode. The optomechanical-coupling and sideband-resolution regimes in the lower two boxes therefore refer to the “$-$” mode.

Figure 3

Effective couplings [Eq. (15)] as functions of detuning for (a) device (i), (b) device (ii), and (c) device (iii). The couplings are complex numbers and the top panels show the absolute values of the couplings while the bottom panels show the phase ${\tilde{g}}_{}=\left|{\tilde{g}}_{}\right|{\text{e}}^{i\varphi }$. The dotted black line corresponds to the effective coupling ${\tilde{g}}_{}^{\text{std}}=g_a^{\omega }\overline{a}$ of a standard optomechanical device with the same sideband resolution as devices (ii) and (iii). The parameters are given in Table 1.

Figure 4

Optical response to the variation of different detunings and bath couplings as a function of the “$-$”-mode detuning for (a) device (i), (b) device (ii), and (c) device (iii). We have plotted each partial derivative from Eq. (26). The hatched areas indicate detunings for which the respective devices are unstable due to heating [see Fig. 5 and Appendix pp1-s2a]. The parameters are given in Table 1.

Figure 5

(a) Mechanical frequency shift $\delta {\mathrm{\Omega }}_{\text{mec}}$ (top panel) and optical contribution to the mechanical damping rate ${\mathrm{\Gamma }}_{\text{opt}}$ (bottom panel) as a function of the detuning ${\tilde{\mathrm{\Delta }}}_{-}$ for devices (ii) and (iii). The plots were obtained by evaluating Eqs. (27) at $\omega ={\mathrm{\Omega }}_{\text{mec}}$. The hatched areas indicate detunings for which the respective devices are unstable due to the heating. The dotted black lines represent a standard resolved-sideband optomechanical device where the laser power has been adjusted to get a similar cooling on the red sideband. (b), (c) Components of the mechanical frequency shift $\delta {\mathrm{\Omega }}_{\text{mec}}$ (top panels) and optical contribution to the mechanical damping rate ${\mathrm{\Gamma }}_{\text{opt}}$ (bottom panels) as a function of the detuning ${\tilde{\mathrm{\Delta }}}_{-}$ around the red sideband [gray dotted box in (a)] for device (ii) and device (iii) at $\omega ={\mathrm{\Omega }}_{\text{mec}}$ [see Eq. (28)]. The parameters are given in Table 1.

Figure 6

Noise power spectra of (a) device (i), (b) device (ii), and (c) device (iii). Upper row of each panel: Noise power spectrum of the mechanical position $S_q\left[\omega \right]$ and light output quadratures on the left side $S_{X_{\text{out},\text{L}}}\left[\omega \right]$ and $S_{P_{\text{out},\text{L}}}\left[\omega \right]$ as functions of the effective detuning ${\tilde{\mathrm{\Delta }}}_{-}$ and of the frequency $\omega$. The dotted red line shows the mechanical frequency shift $\delta {\mathrm{\Omega }}_{\text{mec}}\left[{\mathrm{\Omega }}_{\text{mec}}\right]$. Lower row of each panel: Area corresponding to the integral over $\omega$ of the corresponding spectrum in the upper row. All the plotted quantities have been normalized by their maximum values; all parameters are given in Table 1.

Figure 7

Optomechanical cooling in the weak-coupling limit and resolved-sideband regime ${\tilde{\kappa }}_{-}\ll {\mathrm{\Omega }}_{\text{mec}}$. We show results for (a) device (ii) and (b) device (iii) (see parameters in Table 1). Top panels: average photon numbers in the cavity ($|\overline{a}{|}^2$, solid blue) and in the Fano mirror ($|\overline{d}{|}^2$, dashed orange); center panels: anti-Stokes (solid green) and Stokes (dashed red) rates; bottom panels: obtained average phonon number in the mechanical fluctuations (solid purple), the gray line indicates the thermal phonon number ${\overline{n}}_{\text{mec}}$. All functions are plotted in dependence of the detuning between the laser and the effective “$-$” mode. The dotted black lines correspond to the result for a standard optomechanical device with the same sideband resolution (we have only plotted $A_{-}$ in the middle panel), and the hatched areas indicate detunings for which the system is unstable due to heating.

Figure 8

Steady-state phonon number in the mechanical fluctuations [Eq. (35)]. Results as function of the laser power at ${\tilde{\mathrm{\Delta }}}_{-}={\mathrm{\Omega }}_{\text{mec}}$ are shown for (a) device (ii) and (b) device (iii) at room temperature, $T_{\text{mec}}=300$ K (solid blue line), and at cryogenic temperature, $T_{\text{mec}}=4$ K (dashed-dotted green line). For the room-temperature case, the dotted gray line indicates the thermal phonon number, the dashed orange line the weak-coupling approximation [Eq. (38)], and the solid gray line the minimum reachable phonon number. The purple diamonds indicate the laser powers used in the other figures as indicated in Table 1. (c) Minimum steady-state phonon number in the mechanical fluctuations (minimized on ${\tilde{\mathrm{\Delta }}}_{-}$ and $P_{\text{las}}$) as function of the sideband resolution, where ${\tilde{\kappa }}_{-}$ is tuned via ${\tilde{\kappa }}_d$ and ${\tilde{\mathrm{\Delta }}}_d$. The solid blue line was obtained with a device differing from device (iii) only by the laser power with ${\delta }_{\mathrm{\Delta }}={\delta }_{\kappa }\lambda /\sqrt{{\tilde{\kappa }}_a{\tilde{\kappa }}_d}$. The purple star indicates devices (ii) and (iii) and the dotted black line corresponds to the minimum ${\overline{n}}_{\text{fin}}$ for a standard optomechanical device, with the ideal parameters ${\gamma }_a={\kappa }_a={\tilde{\kappa }}_{-}/2$. The dashed gray line indicated the ground-state cooling threshold. All other parameters are given in Table 1.

Figure 9

Steady-state phonon number in the mechanical fluctuations [Eq. (35)] as function of the laser power at ${\tilde{\mathrm{\Delta }}}_{-}={\mathrm{\Omega }}_{\text{mec}}$ for device (iv). The curves have been plotted on the laser power range where the device's dynamics can be linearized (Sec. 2c).

Figure 10

Steady-state phonon number ${\overline{n}}_{\text{fin}}$ (in blue, left axis) and relevant stability condition (in orange, right axis) as a function of (a) the detuning ${\tilde{\mathrm{\Delta }}}_{-}$ and (b) the laser power $P_{\text{las}}$ for devices (ii) and (iii). See Table 1 for the other parameters.

Figure 11

Steady-state photon numbers, for the cavity mode in green and the mirror mode in red, in the classical steady state $|\overline{c}{|}^2$ (solid lines) and in the fluctuations $\langle \delta {\widehat{c}}^{†}\delta {\widehat{c}}^{}\rangle$ (dashed lines at $T_{\text{mec}}=300 \mathrm{K}$ and dotted lines at $T_{\text{mec}}=4 \mathrm{K}$) as a function of (a), (b) the detuning ${\tilde{\mathrm{\Delta }}}_{-}$ and (c)–(f) the laser power $P_{\text{las}}$ (at ${\tilde{\mathrm{\Delta }}}_{-}={\mathrm{\Omega }}_{\text{mec}}$) for the device indicated on top of each panel. The dotted vertical black lines in (c) and (d) indicate the laser powers used in all the figures where $P_{\text{las}}$ is not on the $x$ axis. See Table 1 for the other parameters of devices (ii) and (iii) and Table 2 for the parameters of devices (iv) and (v).

Figure 12

Factor $D_{Q_{\text{out},\text{L}}}\left[{\mathrm{\Omega }}_{\text{mec}}\right]$ for $Q=X$ (solid orange) and $Q=P$ (dashed blue) for devices (i)–(iii) as a function of the detuning ${\tilde{\mathrm{\Delta }}}_{-}$. See Table 1 for the parameters.

Figure 13

Mechanical position and momentum fluctuations $\langle \delta {\widehat{q}}^2\rangle$ and $\langle \delta {\widehat{p}}^2\rangle$ as a function of laser power for devices (ii), (iii), and (iv). See Tables 1 and 2 for the parameters.