Multimode optomechanical cooling via general dark-mode control

The dark-mode effect is a stubborn obstacle for ground-state cooling of multiple degenerate mechanical modes optomechanically coupled to a common cavity-field mode. Here we propose an auxiliary-cavity-mode method for simultaneous ground-state cooling of two degenerate or near-degenerate mechanical modes by breaking the dark mode. We find that the introduction of the auxiliary cavity mode not only breaks the dark-mode effect, but also provides a new cooling channel to extract the thermal excitations stored in the dark mode. Moreover, we study the general physical-coupling configurations for breaking the dark mode in a generalized network-coupled four-mode optomechanical system consisting of two cavity modes and two mechanical modes. We find the analytical dark-mode-breaking condition in this system. This method is general and it can be generalized to break the dark-mode effect and to realize the simultaneous ground-state cooling in a multiple-mechanical-mode optomechanical system. We also demonstrate the physical mechanism behind the dark-mode breaking by studying the breaking of dark-state effect in the $N\text{−}\mathrm{type}$ four-level atomic system. Our results not only provide a general method to control various dark-mode and dark-state effects in physics, but also present an opportunity to the study of macroscopic quantum phenomena and applications in multiple-mechanical-resonator systems.

Figure 1

(a) Schematic of the $N\text{−}\mathrm{type}$ four-mode optomechanical system. An intermediate-coupling cavity mode $a$ with the resonance frequency ${\omega }_c$ is optomechanically coupled to two mechanical modes $b_1$ and $b_2$, with the corresponding resonance frequencies ${\omega }_1$ and ${\omega }_2$. An auxiliary cavity mode $a_s$ with the resonance frequency ${\omega }_s$ is optomechanically coupled to the mechanical mode $b_1$. The coupling strength between the cavity mode $a$ $\left(a_s\right)$ and the mechanical mode $b_{l=1,2}$ $\left(b_1\right)$ is denoted by $g_{l=1,2}$ $\left(g_{s1}\right)$. The cavity mode $a$ $\left(a_s\right)$ is driven with the driving frequency ${\omega }_L$ $\left({\omega }_d\right)$ and the driving amplitude $\mathrm{\Omega }$ $\left({\mathrm{\Omega }}_s\right)$. The decay rates of the intermediate cavity mode, the auxiliary cavity mode, and the two mechanical modes are denoted by $\kappa$, ${\kappa }_s$, and ${\gamma }_{l=1,2}$, respectively. (b) Coupling configuration associated with the approximate linearized Hamiltonian (11). The auxiliary cavity mode $\delta a_s$ (with driving detuning ${\mathrm{\Delta }}_s^{\prime })$ is coupled to the two hybrid mechanical modes $B_{\pm }$ (with resonance frequency ${\omega }_{\pm })$ via the effective coupling strengths $G_{s\pm }$. The cavity mode $\delta a$ (with driving detuning ${\mathrm{\Delta }}_c^{\prime })$ is coupled to the hybrid mechanical mode $B_{+}$ via the effective coupling strength $G_{+}$. The phonon-hopping interaction between the two hybrid mechanical modes is denoted by $\zeta$. In the degenerate-mechanical-mode case and in the absence of auxiliary cavity, i.e., $\zeta =0$ and $G_{s\pm }=0$, the hybrid mechanical mode $B_{-}$ is decoupled from both the cavity mode $\delta a$ and the hybrid mechanical mode $B_{+}$

Figure 2

Final mean phonon numbers (a) and (c) $n_1^f$ and (b) and (d) $n_2^f$ of the two mechanical modes versus the scaled driving detuning (a) and (b) ${\mathrm{\Delta }}_c^{\prime }/{\omega }_1$ and (c) and (d) ${\mathrm{\Delta }}_s^{\prime }/{\omega }_1$ and the scaled cavity-field decay rate (a) and (b) $\kappa /{\omega }_1$ and (c) and (d) ${\kappa }_s/{\omega }_1$. The parameters are (a) and (b) ${\mathrm{\Delta }}_s^{\prime }/{\omega }_1=1$ and ${\kappa }_s/{\omega }_1=0.1$ and (c) and (d) ${\mathrm{\Delta }}_c^{\prime }/{\omega }_1=1$ and $\kappa /{\omega }_1=0.1$. The other parameters are ${\omega }_2/{\omega }_1=1$, ${\gamma }_1/{\omega }_1={\gamma }_2/{\omega }_1={10}^{-5}$, $G_1/{\omega }_1=G_2/{\omega }_1=0.05$, $G_{s1}/{\omega }_1=0.08$, and ${\overline{n}}_1={\overline{n}}_2=1000$.

Figure 3

Final mean phonon numbers $n_1^f$ and $n_2^f$ versus the frequency ratio ${\omega }_2/{\omega }_1$ and the scaled cavity-field decay rate $\kappa /{\omega }_1$ in the cases of (a) and (b) dark-mode unbreaking $(G_{s1}/{\omega }_1=0)$ and (c) and (d) dark-mode breaking $(G_{s1}/{\omega }_1\ne 0)$. The other parameters are ${\mathrm{\Delta }}_c^{\prime }={\mathrm{\Delta }}_s^{\prime }={\omega }_1$, ${\kappa }_s/{\omega }_1=0.1$, $G_1/{\omega }_1=G_2/{\omega }_1=0.05$, $G_{s1}/{\omega }_1=0.08$, ${\gamma }_1/{\omega }_1={\gamma }_2/{\omega }_1={10}^{-5}$, and ${\overline{n}}_1={\overline{n}}_2=1000$.

Figure 4

Final mean phonon numbers (a) $n_1^f$ and (b) $n_2^f$ as functions of $G_{s1}/{\omega }_1$ when the scaled cavity-field decay rate takes various values ${\kappa }_s/{\omega }_1=0.4$, 0.8, and 1.2. The insets show close-ups of the final mean phonon numbers as functions of $G_{s1}/{\omega }_1$, with the horizontal axis ranging from 0.05 to 0.3. The other parameters are ${\omega }_2/{\omega }_1=1$, ${\mathrm{\Delta }}_c^{\prime }/{\omega }_1={\mathrm{\Delta }}_s^{\prime }/{\omega }_1=1$, $\kappa /{\omega }_1=0.1$, ${\gamma }_1/{\omega }_1={\gamma }_2/{\omega }_1={10}^{-5}$, $G_1/{\omega }_1=G_2/{\omega }_1=0.05$, and ${\overline{n}}_1={\overline{n}}_2=1000$.

Figure 5

(a) Schematic of the network-coupled four-mode optomechanical system consisting of two cavity modes and two mechanical modes. In addition to the couplings and notation introduced in Fig. 1, here we introduce three new couplings marked by dashed lines: the phonon- and photon-hopping interactions with the coupling strengths $\eta$ and $J$, respectively, and the optomechanical-coupling strength $g_{s2}$ between the auxiliary cavity mode $a_s$ and the second mechanical mode $b_2$. (b) Coupling configuration associated with the approximate linearized Hamiltonian (26). Here we add a tilde to the couplings and notation introduced in Fig. 1. We also introduce the photon-hopping strength $J$ between the two cavity modes.

Figure 6

Fourteen coupling configurations of the network-coupled four-mode optomechanical system, where $J$, $g_l$ $\left(g_{sl}\right)$ (for $l=1,2)$, and $\eta$ are the photon-hopping coupling strength, the optomechanical-coupling strength between the intermediate (auxiliary) cavity mode and the $l\mathrm{th}$ mechanical mode, and the phonon-hopping coupling strength, respectively. Here the two optomechanical couplings between the cavity mode $a$ and the two mechanical modes $b_1$ and $b_2$ are kept and the other couplings can be closed on demand. Then there are 14 different coupling configurations. Concretely, the one-coupling-closed cases include the following: (a) The coupling channel $J$ is closed $(J=0)$, (b) the coupling channel $g_{s1}$ is closed $(g_{s1}=0)$, (c) the coupling channel $g_{s2}$ is closed $(g_{s2}=0)$, and (d) the coupling channel $\eta$ is closed $(\eta =0)$. The two-coupling-closed cases include the following: (e) The coupling channels $J$ and $g_{s1}$ are closed $(J=g_{s1}=0)$, (f) the coupling channels $J$ and $g_{s2}$ are closed $(J=g_{s2}=0)$, (g) the coupling channels $J$ and $\eta$ are closed $(J=\eta =0)$, (h) the coupling channels $g_{s1}$ and $g_{s2}$ are closed $(g_{s1}=g_{s2}=0)$, (i) the coupling channels $g_{s1}$ and $\eta$ are closed $(g_{s1}=\eta =0)$, and (j) the coupling channels $g_{s2}$ and $\eta$ are closed $(g_{s2}=\eta =0)$. In addition, the cases in which three couplings are closed include the following: (k) The coupling channels $J$, $g_{s1}$, and $g_{s2}$ are closed $(J=g_{s1}=g_{s2}=0)$, (l) the coupling channels $J$, $g_{s1}$, and $\eta$ are closed $(J=g_{s1}=\eta =0)$, (m) the coupling channels $J$, $g_{s2}$, and $\eta$ are closed $(J=g_{s2}=\eta =0)$, and (n) the coupling channels $g_{s1}$, $g_{s2}$, and $\eta$ are closed $(g_{s1}=g_{s2}=\eta =0)$. We point out that the single-photon optomechanical-coupling strengths $g_1$, $g_2$, $g_{s1}$, and $g_{s2}$ are related to the linearized optomechanical-coupling strengths $G_1$, $G_2$, $G_{s1}$, and $G_{s2}$ in the four-mode optomechanical system.

Figure 7

Final mean phonon numbers $n_1^f$ and $n_2^f$ versus the scaled decay rate $\kappa /{\omega }_1$ in various coupling configurations: (a) and (b) cases where one coupling is closed, i.e., $J=0$ (blue solid curves), $\eta =0$ (red circles), $G_{s1}=0$ (cyan dash-dotted curves), and $G_{s2}=0$ (green dashed curves); (c) and (d) two-coupling-closed cases, i.e., $J=\eta =0$ (blue solid curves), $G_{s1}=G_{s2}=0$ (red circles), $\eta =G_{s1}=0$ (green dashed curves), $J=G_{s1}=0$ (cyan dash-dotted curves), $J=G_{s2}=0$ (brown squares), and $\eta =G_{s2}=0$ (purple triangles); and (e) and (f) three-coupling-closed cases, i.e., $J=G_{s1}=G_{s2}=0$ (blue solid curves), $\eta =G_{s1}=G_{s2}=0$ (red circles), $J=\eta =G_{s1}=0$ (brown dash-dot curves), and $J=\eta =G_{s2}=0$ (purple dashed curves). The other parameters are ${\omega }_2/{\omega }_1=1$, ${\gamma }_1/{\omega }_1={\gamma }_2/{\omega }_1={10}^{-5}$, $\kappa /{\omega }_1={\kappa }_s/{\omega }_1=0.1$, $J/{\omega }_1=\eta /{\omega }_1=0.03$, ${\mathrm{\Delta }}_c^{\prime }/{\omega }_1={\mathrm{\Delta }}_s^{″}/{\omega }_1=1$, $G_1/{\omega }_1=G_2/{\omega }_1=0.05$, $G_{s1}/{\omega }_1=G_{s2}/{\omega }_1=0.08$, and ${\overline{n}}_1={\overline{n}}_2=1000$. Note that the values of $J$, $\eta$, $G_{s1}$, and $G_{s2}$ presented here work when the corresponding coupling channels are open.

Figure 8

(a) Final mean phonon numbers $n_1^f$ and $n_2^f$ versus the scaled decay rate $\kappa /{\omega }_1$ in the cases of $G_{s2}/{\omega }_1=4G_{s1}/{\omega }_1=0.08$ (blue dashed curve and red solid curve, respectively) and $G_{s1}/{\omega }_1=4G_{s2}/{\omega }_1=0.08$ (red circles and blue squares, respectively). (b) Final mean phonon numbers $n_1^f$ (blue solid curves) and $n_2^f$ (red dashed curves) versus the ratio $G_{s2}/G_{s1}$ when the linearized optomechanical-coupling strength $G_{s1}/{\omega }_1=0.08$. The other parameters used in this system are ${\omega }_2/{\omega }_1=1$, ${\gamma }_1/{\omega }_1={\gamma }_2/{\omega }_1={10}^{-5}$, $\kappa /{\omega }_1={\kappa }_s/{\omega }_1=0.1$, $J/{\omega }_1=\eta /{\omega }_1=0.03$, ${\mathrm{\Delta }}_c^{\prime }/{\omega }_1={\mathrm{\Delta }}_s^{″}/{\omega }_1=1$, $G_1/{\omega }_1=G_2/{\omega }_1=0.05$, and ${\overline{n}}_1={\overline{n}}_2=1000$.

Figure 9

Part of the energy levels and transitions of the system in either the (a) absence or (b) presence of the auxiliary cavity mode $a_s$. Here the states $|m,j,k\rangle$ and $|m,n,j,k\rangle$ correspond to the states ${|m,j,k\rangle }_{a,b_1,b_2}$ and ${|m,n,j,k\rangle }_{a,a_s,b_1,b_2}$, respectively. The states ${|m\rangle }_a$, ${|n\rangle }_{a_s}$, ${|j\rangle }_{b_1}$, and ${|k\rangle }_{b_2}$ denote the number states of the cavity mode $a$, auxiliary cavity mode $a_s$, mechanical mode $b_1$, and mechanical mode $b_2$, respectively.

Figure 10

(a) $N$-mechanical-mode optomechanical system consisting of an intermediate cavity mode optomechanically coupled to $N$ mechanical modes. (b) Both the phonon-hopping interactions between two neighboring mechanical modes and the optomechanical interaction between the auxiliary cavity mode and the mechanical mode $b_1$ are introduced into the $N$-mechanical-mode optomechanical system.

Figure 11

(a) and (b) Final mean phonon numbers $n_l^f$ as functions of the scaled driving detuning ${\tilde{\mathrm{\Delta }}}_c/{\omega }_1$ in the dark-mode-unbreaking $(G_s/{\omega }_1=0$ and ${\eta }_l/{\omega }_1=0)$ or -breaking $(G_s/{\omega }_1=0.1$ and ${\eta }_l/{\omega }_1=0.06)$ cases for (a) $N=3$ and (b) $N=4$. (c) and (d) Final mean phonon numbers $n_l^f$ as functions of the scaled cavity-field decay rate $\kappa /{\omega }_1$ for (c) $N=3$ and (d) $N=4$. The other parameters are ${\omega }_l/{\omega }_1=1$, ${\mathrm{\Delta }}_s^{\prime }/{\omega }_1=1$, ${\kappa }_s/{\omega }_1=0.1$, ${\eta }_l/{\omega }_1=0.06$, ${\gamma }_l/{\omega }_1={10}^{-5}$, $G_l/{\omega }_1=0.05$, $G_s/{\omega }_1=0.1$, ${\overline{n}}_l=1000$, and (a) and (b) $\kappa /{\omega }_1=0.1$ and (c) and (d) ${\tilde{\mathrm{\Delta }}}_c/{\omega }_1=1$.

Figure 12

(a) Schematic of the three-level system, which consists of three states $|e\rangle$, $|f\rangle$, and $|g\rangle$, with the corresponding energies $E_e$, $E_f$, and $E_g$. The coupling strengthes of the transition processes $|g\rangle \leftrightarrow |e\rangle$ and $|f\rangle \leftrightarrow |e\rangle$ are denoted by ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ with the corresponding detunings ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$, respectively. (b) Schematic of the $N\text{−}\mathrm{type}$ four-level system. In addition to the couplings and states introduced in (a), we introduce an auxiliary state $|d\rangle$ coupled to the lower state $|f\rangle$ with the coupling strength ${\mathrm{\Omega }}_3$ and the detuning ${\mathrm{\Delta }}_3$.

Figure 13

(a) Probability $P_e^{\left[s\right]}$ of the excited state $|e\rangle$ in three eigenstates versus the amplitude ratio ${\mathrm{\Omega }}_1/{\mathrm{\Omega }}_2$ in the three-level system. (b) Probability $p_e^{\left[s\right]}$ of four eigenstates versus the amplitude ratio ${\mathrm{\Omega }}_3/{\mathrm{\Omega }}_2$ in the four-level system.