Nonreciprocal optomechanically induced transparency and enhanced ground-state cooling in a reversed-dissipation cavity system

Jiaming Zhang; Yueqian Li; Yan Zhang

doi:10.1364/OE.505747

1. Introduction

Cavity optomechanics [1–3], an exploratory field investigating the interaction between light fields and mechanical motion, has been successfully demonstrated in diverse systems including the Fabry-Perot cavity [4,5], whispering-gallery microcavity [6,7] and photonic crystal mirror [8] to yield various quantum phenomena, such as the cooling of mechanical modes [9–11], quantum entanglement [12–14], optical nonreciprocity [15,16], slow/fast light [17], photon blockade [18,19] mechanical state estimation [20], and optomechanically induced transparency (OMIT) [21–24]. As its prominent application, OMIT is the result of destructive interference between the anti-Stokes field and the probe field [25]. The efficacy of OMIT is primarily characterized by the optical transmission and delay properties, directly impacting information transfer efficiency and storage duration, respectively. The optomechanical interaction has also been harnessed to realize the optical nonreciprocity [26–30], which has garnered substantial attention due to its important applications with the suppression of spurious modes and unwanted signals in quantum signal processing and communication. Numerous OMIT-based nonreciprocal devices, such as the optomechanical isolator [31], circulator [32,33] and router [34] have been designed for optical communication [35]. Recently, the concept of dissipation engineering [36,37] has been extensively developed within domains such as superconducting qubits [38,39], superconducting resonators [40], and Rydberg atoms [41]. Therefore, the extension of this concept to the optomechanical system holds significant promise. Most optomechanical experiments conducted to date are within the framework of the so-called normal dissipation regime, where the linewidth of the mechanical resonator resonance is typically significantly narrower than that of the corresponding cavity mode. However, recent reports have introduced a novel perspective of the reversed-dissipation regime in optomechanics. This intriguing configuration is characterized by a mechanical dissipation rate that notably surpasses the cavity linewidth [42–46]. The successful demonstration of such a reversed-dissipation regime highlights its ability to tailor microwave fields for amplification or to enhance optomechanical entanglement [47,48]. The mechanical resonator in the reversed-dissipation regime can serve as a dissipative quantum reservoir relative to the cavities for the optomechanical system. When coupled to two cavities, it can undergo an adiabatic elimination process, transforming into an effectively complex coupling with a phase factor between two cavities [49]. Differing from the coherent coupling process, the dissipative process is recognized as anti-PT-symmetric, frequently harnessed in the design of non-Hermitian systems [50–52] and the attainment of exceptional points (EPs) [53]. At these EPs, the photon transmission can exhibit nonreciprocity and chirality, which may have important implications for quantum information, such as building directional quantum interfaces, cascaded quantum networks, and topological-protected state transfer and storage mechanisms. In addition, the EPs of the reversed-dissipation system may also open doors to the creation of novel optomechanical sensors and actuators with the enhancement of the sensitivity [54] without the requirement of gain. Recent advancements in experimental implementations facilitate the realization of the reversed dissipation, such as the ultra-high-Q microwave cavity [46], mechanical resonators damped by an auxiliary electromagnetic mode [55] or embedded by erbium ions [45]. The engineering of this dissipative process has been realized in various experimental setups, including superconducting circuits [56,57] and microcavities [58]. In addition, the complex dissipative coupling in diverse systems has been used to facilitate the nonreciprocal transmission and amplification [59,60], implement unitary operations [61], trap light [62], and so on. Nevertheless, a comprehensive exploration is also needed to discover distinct characteristics and potential applications of the optomechanical system with dissipative couplings, especially the quantum coherent effects. For instance, the distinct OMIT observed within the reversed-dissipation regime may introduce unique features to some phenomena and applications, such as optomechanical nonlinearity, photon blockade, slow and fast light, precision measurement, and force sensors. It may be beneficial to the enhancement of optomechanical entanglement and ground-state cooling and has the potential to enable novel functionalities, such as a unique type of parametric instability.

Here, we delve into the exploration of OMIT and ground-state cooling for a reversed-dissipation multimode cavity optomechanical system. This setup contains two directly coupled cavities via the coherent coupling, a low-Q mechanical resonator, and a high-Q mechanical resonator. The adiabatic elimination of the highly dissipative mechanical mode can result in a complex dissipative coupling mechanism for the reversed-dissipation regime, while the lowly dissipative one is harnessed to support OMIT. Our results find OMIT characterized by asymmetric spectra with the absence of a phase-matching condition. Notably, the degree of asymmetry is subject to modulation by both the phase and strength of the dissipative coupling. This phase-dependent OMIT introduces nonreciprocity, allowing for unidirectional transmission with the presence of the phase-matching condition and the balance between dissipative and coherent couplings, i.e., at EPs. Here, a series of EPs could appear due to the destructive interference between the coherent and dissipative coupling processes with the dissipative coupling phase [53]. Note that, in this reversed-dissipation regime, the unidirectional bandwidth is no longer limited by the linewidth of the mechanical mode but is improved to the linewidth of the cavity mode. Furthermore, we have introduced an innovative approach to enhance ground-state cooling by harnessing the reversed-dissipation technique, operating under the same conditions that enable unidirectional OMIT. Our findings shed light on the intriguing possibilities the reversed-dissipation method offers in advancing the fields of optical nonreciprocity and quantum state engineering.

2. Model and effective Hamiltonian of the system

As shown in Fig. 1, the system consists of two cavity modes characterized by annihilation operators $a$ and $b$, along with a low-Q mechanical resonator by $m$ and a high-Q resonator by $d$, with resonant frequencies $\omega _{a,b,m,d}$, respectively. $g_{a,b,0}$ describes the relevant optomechanical coupling strengths while $G$ the direct coherent coupling strength between two cavities. $\kappa _{a}=\kappa _{a, i}+\kappa _{a, e}$ ($\kappa _{b}=\kappa _{b, i}+\kappa _{b, e}$) denotes the loss for modes $a$ ($b$), where $\kappa _{a, i}\left (\kappa _{a, e}\right )$ and $\kappa _{b, i}\left (\kappa _{b, e}\right )$ are the internal (external) loss. $\gamma _{m,d}$ are the dissipation of the mechanical modes $m$ and $d$, respectively. Considering fields with frequency $\omega _{1,2}$ and amplitude $\varepsilon _{1,2}$, respectively, driving on cavities $a$ and $b$, and in the rotating frame at the frequency $\omega _{r}=\left (\omega _{1}+\omega _{2}\right )/ 2$, and in the rotating frame at the frequency, the Hamiltonian is

(1)$$\begin{aligned} & H=\Delta_{a} a^{{\dagger}} a+\Delta_{b} b^{{\dagger}} b+\omega_{m} m^{{\dagger}} m+\omega_{d} d^{{\dagger}} d+G\left(a^{{\dagger}} b+b^{{\dagger}} a\right) -g_{a} a^{{\dagger}} a\left(m^{{\dagger}}+m\right) \\ &-g_{b} b^{{\dagger}} b\left(m^{{\dagger}}+m\right)-g_{0} a^{{\dagger}} a\left(d^{{\dagger}}+d\right) +i\varepsilon_{1} \left(a^{{\dagger}} e^{{-}i \delta_{r} t}-a e^{i \delta_{r} t}\right) +i\varepsilon_{2} \left(b^{{\dagger}} e^{i \delta_{r} t}- b e^{{-}i \delta_{r} t}\right), \end{aligned}$$

where $\Delta _{a,b}=\omega _{a,b}-\omega _{r}$ and $\delta _{r}=\left (\omega _{1}-\omega _{2}\right ) / 2$. When we assume $\omega _{a}=\omega _{b}$ and $\omega _{1}=\omega _{2}$ hereafter, $\Delta _{a}=\Delta _{b}=\Delta$ and $\delta _{r}=0$. When the input quantum noises are neglected, the system’s dynamics are governed by the Langevin equations (LEs) as

(2)$$\begin{aligned} \dot{a} &={-}\left(\kappa_{a}+i \Delta\right) a-i G b+i g_{a}\left(m^{{\dagger}}+m\right) a+i g_{0}\left(d^{{\dagger}}+d\right) a+\varepsilon_{1}, \\ \dot{b}&={-}\left(\kappa_{b}+i \Delta\right) b-i G a+i g_{b}\left(m^{{\dagger}}+m\right) b+\varepsilon_{2}, \\ \dot{m}&={-}\left(\gamma_{m}+i \omega_{m}\right) m+i g_{a} a^{{\dagger}} a+i g_{b} b^{{\dagger}} b, \\ \dot{d}&={-}\left(\gamma_{d}+i \omega_{d}\right) d+i g_{0} a^{{\dagger}} a. \end{aligned}$$

Fig. 1. Schematic diagram of the cavity optomechanical system consisting of two cavity modes ($a$ and $b$) and two mechanical modes ($d$ and $m$). $E_{I, in}$ ($E_{I, out}$) denotes the input (output) signal amplitude from port I (I=A, B, C, D). Cavity $a$ ($b$) is coupled to the upper (lower) waveguide for probe signal propagation.

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Under the strongly driving, we use displacement transformation to linearize Eq. (2), i.e., $\tau =\tau _{s}+\delta \tau$ ($\tau =a,b,m,d$) with $\tau _{s}$ ($\delta \tau$) being the steady-state (quantum fluctuation) of operators. Then, the linearized LEs for the quantum fluctuations are expressed as

(3)$$\begin{aligned} \delta \dot{a}&={-}\left(\kappa_{a}+i \bar{\Delta}_{a}\right) \delta a-i G \delta b+i G_{a}\left(\delta m^{{\dagger}}+\delta m\right)+i G_{0}\left(\delta d^{{\dagger}}+\delta d\right), \\ \delta \dot{b}&={-}\left(\kappa_{b}+i \bar{\Delta}_{b}\right) \delta b-i G \delta a+i G_{b}\left(\delta m^{{\dagger}}+\delta m\right), \\ \delta \dot{m}&={-}\left(\gamma_{m}+i \omega_{m}\right) \delta m+i\left(G_{a}^{*} \delta a+G_{a} \delta a^{{\dagger}}\right)+i\left(G_{b}^{*} \delta b+G_{b} \delta b^{{\dagger}}\right), \\ \delta \dot{d}&={-}\left(\gamma_{d}+i \omega_{d}\right) \delta d+i\left(G_{0}^{*} \delta a+G_{0} \delta a^{{\dagger}}\right), \end{aligned}$$

where $\bar {\Delta }_{a}= \Delta -g_{a}\left (m_{s}^{*}+m_{s}\right )-g_{0}\left (d_{s}^{*}+d_{s}\right )$ and $\bar {\Delta }_{b}= \Delta -g_{b}\left (m_{s}^{*}+m_{s}\right )$ are the effective cavity detuning, while $G_{a}= g_{a} a_{s}, G_{b}= g_{b} b_{s},G_{0}= g_{0} a_{s}$ the linearized optomechanical coupling strengths. We move Eq. (3) into another interaction picture by introducing the slowly moving operators, i.e., $\delta a=\delta \tilde {a} e^{-i \bar {\Delta }_{a} t}$, $\delta b=\delta \tilde {b} e^{-i \bar {\Delta }_{b} t}$, $\delta m=\delta \tilde {m} e^{-i \omega _{m} t}$, and $\delta d=\delta \tilde {d} e^{-i \omega _{d} t}$. The mechanical mode $m$ can be adiabatically eliminated when operating within the reversed-dissipation regime (rendering it nearly decoupled from the system). Then, under the rotating wave approximation with $\bar {\Delta }_{a}\left (\bar {\Delta }_{b}\right )+\omega _{m}>G_{a}, G_{b}, \kappa _{a}, \kappa _{b}, \gamma _{m},\gamma _{d}$, we can get

(4)$$\begin{aligned} \delta \dot{\tilde{a}} & ={-}\left(\kappa_{a}+\frac{\left|G_{a}\right|^{2}}{\gamma_{m}}\right) \delta \tilde{a}-\left(\frac{G_{a} G_{b}^{*}}{\gamma_{m}}+i G\right) \delta \tilde{b}+i G_{0} \delta\tilde d e^{i\left(\bar\Delta_{a}-\omega_{d}\right) t}, \\ \delta \dot{\tilde{b}} & ={-}\left(\kappa_{b}+\frac{\left|G_{b}\right|^{2}}{\gamma_{m}}\right) \delta \tilde{b}-\left(\frac{G_{a}^{*} G_{b}}{\gamma_{m}}+i G\right) \delta \tilde{a}, \\ \delta \dot{\tilde{d}} & ={-}\gamma_{d} \delta \tilde{d}+i G_{0}^{*} \delta\tilde ae^{{-}i\left(\bar\Delta_{a}-\omega_{d}\right) t}, \end{aligned}$$

by setting $\bar {\Delta }_{a} = \bar {\Delta }_{b} = \omega _{m} = \omega _{d}= \Delta$ and $\kappa _{a} = \kappa _{b}= \kappa$. Then, by transforming back to the original picture, we can achieve

(5)$$\begin{aligned} \delta \dot{a} & ={-}\left(\kappa+\frac{|G_{a}|^{2}}{\gamma_{m}}+i \Delta\right) \delta a-\left(\frac{G_{a} G_{b}^{*}}{\gamma_{m}}+i G\right) \delta b+i G_{0} \delta d, \\ \delta \dot{b} &={-}\left(\kappa+\frac{|G_{b}|^{2}}{\gamma_{m}}+i \Delta\right) \delta b-\left(\frac{G_{a}^{*} G_{b}}{\gamma_{m}}+i G\right) \delta a, \\ \delta \dot{d} & ={-}\left(\gamma_{d}+i \omega_{d}\right) \delta d+i G_{0}^{*} \delta a. \end{aligned}$$

Therefore, the effective Hamiltonian for the reversed-dissipation system can be expressed as

(6)$$\begin{aligned} H_{eff}& = {[\Delta-i\left(\kappa+J\right)] a^{{\dagger}} a+[\Delta-i\left(\kappa+J\right)] b^{{\dagger}} b} +\left(\omega_{d}-i \gamma_{d}\right) d^{{\dagger}} d \\ & +G\left(a^{{\dagger}} b+b^{{\dagger}} a\right)-i J\left(a^{{\dagger}} b e^{{-}i \theta}+a b^{{\dagger}} e^{i \theta}\right)-G_{0} a^{{\dagger}} d-G_{0}^{*} a d^{{\dagger}}, \end{aligned}$$

where we assume $\left |G_{a}\right |=\left |G_{b}\right |$ and $\kappa =\kappa _{i}+\kappa _{e}+J$, while ensuring that the cavities consistently satisfy the critical-coupling condition $\kappa =2(J+\kappa _{i})$. Here $J=\left |G_{a}\right |^{2} / \gamma _{m}$ is the effective dissipative coupling strength. Note that both $G_{a,b}$ are complex with phase factors depending on $\varepsilon _{\mathrm {1,2}}$, respectively. Consequently, the dissipative coupling phase $\theta =\arg \left (G_{a}^{*} G_{b} / \gamma _{m} \right )$ can be meticulously controlled by the two driving fields. Such an anti-PT-symmetric dissipative coupling with phase factors can exert a pronounced influence on the response of this non-Hermitian system. This influence is particularly prominent at the exceptional points, arising when the dissipative and coherent couplings are balanced ($J=G$) [53].

3. Phase-dependent optical response properties

Considering a probe signal with frequency $\omega _{p}$ incident from port A, the quantum LEs are

(7)$$\begin{aligned} \dot{a} & ={-}(i \delta+\kappa) a-\left(J \mathrm{e}^{{-}i \theta}+i G\right) b+iG_{0}d+\sqrt{\kappa} E_{A,\mathrm{in}}, \\ \dot{b} & ={-}(i \delta+\kappa) b-\left(J \mathrm{e}^{i \theta}+i G\right) a, \\ \dot{d} & ={-}(i \delta+\gamma_{d}) d+i G_{0}^{*}a, \end{aligned}$$

where $\delta =\Delta -\omega _{p}=\omega _{d}-\omega _{p}$ is the detuning between the cavity mode and probe signal. Here, we set $\sqrt {2\kappa _e}= \sqrt {\kappa }$ as mentioned above. With the steady-state solution $\langle a\rangle$ of Eq. (7), we use the standard input-output relation $E_{B,\mathrm {out}} = E_{A,\mathrm {in}}-\sqrt {\kappa }\langle a\rangle$ to deduce the transmission coefficient $T_{BA}$ for energy transfer from port $A$ to $B$ as

(8)$$T_{BA}=\frac{E_{B,\mathrm{out}}}{E_{A,\mathrm{in}}}=1-\frac{\kappa\left(i \delta+\kappa\right)\left(i \delta+\gamma_{d}\right)}{M},$$

with $M=(i \delta +\kappa )^{2}\left (i \delta +\gamma _{d}\right )-\left (J \mathrm {e}^{-i \theta }+i G\right )\left (J \mathrm {e}^{i \theta }+i G\right )\left (i \delta +\gamma _{d}\right )+|G_{0}|^{2}(i \delta +\kappa )$.

Here, we investigate the phase-dependent optical responses of the reversed-dissipation system. We set the dissipative coupling phase to $\theta =\pi$ in Fig. 2(a) and observe how the transmission spectra $\left | T_{BA} \right |$ vary with the changes in the dissipative coupling strength $J$. Starting with $J=0$, corresponding to the absence of dissipative coupling between the two cavities, $\left | T_{BA} \right |$ exhibits typical OMIT spectra characterized by a symmetrical transparency window centered around the resonance $\delta =0$. With increasing $J$ to enough strong value, we realize the OMIT effect, which may be vanished by sufficiently strong $J$ in the reversed-dissipation regime due to the adiabatic elimination of the mechanical mode $m$. Absorption near the transparency window intensifies, influenced by the introduction of external loss by the dissipation coupling. It’s worth noting that in the reversed-dissipation regime, the two absorption valleys of OMIT are no longer symmetric. In particular, they lose their symmetry about $\delta =0$ with different depths due to the frequency shift effect. Such an effect arises from the dissipative coupling strength $J$ when all of the time operators are transferred back to the frequency domain during the Fourier transform unless the system adheres to the phase-matching condition $\theta /\frac {\pi }{2}=n, (n=1,3,5\cdots )$. The phase-matching condition can eliminate the phase-dependent term [the second term of $M$ in Eq. (8)] with the balance between the dissipative and coherent couplings. This observation is exemplified in Fig. 2(c) with $\theta /\frac {\pi }{2}=1$, where a symmetric OMIT window is evident, albeit with deeper valleys compared with the yellow curve in Fig. 2(a). The symmetry of the two valleys stems from the normal-mode splitting due to the phonon-photon coupling. However, when the phase-matching condition is broken, as illustrated in Fig. 2(b) with $\theta / \frac {\pi }{2}=0$, spectral asymmetry appears due to the phase-dependent frequency shift. Moreover, we can achieve a reversal profile by turning the phase to $\pi$ as shown in Fig. 2(d).

Fig. 2. (a) Transmission amplitude $\left | T_{BA} \right |$ as a function of detuning $\delta$ with $\theta =\pi$ for coupling strength $J = 0$ (yellow solid curve), $J = 0.5G$ (green dashed curve), $J = G$ (blue solid curve), and $J = 2G$ (red dashed curve). Transmission amplitude $\left | T_{BA} \right |$ as a function of detuning $\delta$ with $J = G$ for $\theta =0$ in (b), $\theta =\frac {\pi }{2}$ in (c), and $\theta =\pi$ in (d). The internal decay rate $\kappa _{i}$ could be $1$ MHz ($13$ kHz) for the optical (microwave) cavity. Here $G= 10\kappa _{i}$, $G_{0}= 9\kappa _{i}$, and $\gamma _{d}=1 \times 10^{-4}\kappa _{i}$.

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Then, we explore the nonreciprocity of the light across the cavities $a$ and $b$ by examining the transmission spectra. The corresponding transmission coefficients are

(9)$$\begin{aligned} T_{DA} & = \frac{E_{D, \mathrm{out}}}{E_{A, \mathrm{in}}}={-}\frac{\kappa\left(i G+J \mathrm{e}^{i \theta}\right)\left(i \delta+\gamma_{d}\right)}{M}, \\ T_{AD} & = \frac{E_{A, \mathrm{out}}}{E_{D, \mathrm{in}}}={-}\frac{\kappa\left(i G+J \mathrm{e}^{{-}i \theta}\right)\left(i \delta+\gamma_{d}\right)}{M}. \end{aligned}$$

In Figs. 3(a) and (b), we plot transmission amplitudes $\left | T_{DA} \right |$ and $\left | T_{AD} \right |$ as functions of the dissipative coupling strength $J$ and detuning $\delta$, respectively. Fixing the system in the phase-matching condition $\theta /\frac {\pi }{2}=1$, the spectra in Figs. 3(a) and (b) exhibit notable distinctions. These distinctions suggest the effective realization of optical nonreciprocity within the system. This is due to the combination of the interference between different transmission pathways in this closed-loop configuration and the external losses including mode dissipations of the modes and the energy out of the other ports [63]. Around $J=G$, $\left | T_{AD} \right |$ becomes negligible. This feature can be clearly demonstrated in Fig. 3(c), where $\left | T_{AD} \right | =0$ and $\left | T_{DA} \right |$ shows OMIT spectra, with the balanced dissipative and coherent couplings ($J=G$). This implies that photons can propagate solely from port A to D without encountering losses within the OMIT window, while transmission in the opposite direction is forbidden. This characteristic highlights our ability to achieve unidirectional OMIT with the phase-matching condition $\theta / \frac {\pi }{2}=n, (n=1,3,5\cdots )$ when $J=G$. This phenomenon benefits from the external high-Q mechanical mode $d$, as only the reversed-dissipative mechanical mode $m$ fails to induce the OMIT window in the regime of relatively strong coupling, such as $J=G$. Furthermore, we demonstrate the capability to control the direction of unidirectional transmission by adjusting the phase to $\theta /\frac {\pi }{2}=3$, as shown in Fig. 3(d). The transmission spectra can revert to a reciprocal mode (not shown here) with the broken phase-matching condition $\theta / \frac {\pi }{2}=n, (n=0,2,4\cdots )$ with $J=G$, leading to a situation where $\left | T_{DA} \right |$ and $\left | T_{AD} \right |$ in Eq. (9) become equal. Such a unidirectional transmission behavior can be referred to as parity-dependent unidirectional transport. Thus, such a tunable nonreciprocal scheme holds promise as a phase-modulation isolator and a parity-dependent circulator.

Fig. 3. (a) Transmission amplitudes $\left | T_{DA} \right |$ and (b) $\left | T_{AD} \right |$ against coupling strength $J$ and detuning $\delta$ with phase-matching condition $\theta =\frac {\pi }{2}$. Transmission amplitudes $T_{DA}$ and $T_{AD}$ as functions of detuning $\delta$ for $\theta =\frac {\pi }{2}$ in (c) and $\theta =\frac {3\pi }{2}$ in (d). Here $J = G$ and other parameters are the same as in Fig. 2.

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4. Enhanced ground-state cooling of the mechanical resonator

We study the cooling dynamics of the mechanical resonator in the reversed-dissipation system. To achieve a more precise exploration of ground-state cooling, accounting for all damping and noise effects, we examine the evolution of the system described by the nonlinear quantum LEs as

(10)$$\begin{aligned} \delta \dot{a}&={-}\left(\kappa_{a}+i \bar{\Delta}_{a}\right) \delta a-i G \delta b+i G_{a}\delta m+i G_{0}\delta d+\sqrt{2\kappa_{a}} a_{\mathrm{in}}, \\ \delta \dot{b}&={-}\left(\kappa_{b}+i \bar{\Delta}_{b}\right) \delta b-i G \delta a+i G_{b}\delta m+\sqrt{2\kappa_{b}} b_{\mathrm{in}}, \\ \delta \dot{m}&={-}\left(\gamma_{m}+i \omega_{m}\right) \delta m+iG_{a}^{*} \delta a+iG_{b}^{*} \delta b\sqrt{2\gamma_{m}} m_{\mathrm{in}}, \\ \delta \dot{d}&={-}\left(\gamma_{d}+i \omega_{d}\right) \delta d+iG_{0}^{*} \delta a+\sqrt{2\gamma_{d}} d_{\mathrm{in}}, \end{aligned}$$

where the subscripts "in" stands for the corresponding noise operators satisfying the correlation relations $\left \langle j_{\mathrm {in}}(t) j_{\mathrm {in}}^{\dagger }\left (t^{\prime }\right )\right \rangle = \left (N_{j}+1\right ) \delta \left (t-t^{\prime }\right )$ where $j=a,b,m,d$. Here $N_{a,b}$ represent the equilibrium mean thermal photon number of the microwave bath for cavities $a$ and $b$, respectively, while $N_{m,d}$ the thermal phonon number at the frequency of mode $m$ and $d$. As set above, $\bar {\Delta }_{a} = \bar {\Delta }_{b}=\Delta$ and $\kappa _{a} = \kappa _{b}= \kappa$. The quadrature operators for all modes and their respective input noises are

(11)$$\begin{aligned} \delta \alpha &= \left(\delta j+\delta j^{{\dagger}}\right) / \sqrt{2}, \quad \delta \beta = \left(\delta j-\delta j^{{\dagger}}\right) / \sqrt{2} i , \\ \delta \alpha _{\mathrm{in}} &= \left(j_{\mathrm{in}} +j_{\mathrm{in}}^{{\dagger}}\right) / \sqrt{2}, \quad \delta \beta_{\mathrm{in}}= \left(j_{\mathrm{in}}-j_{\mathrm{in}}^{{\dagger}}\right) / \sqrt{2} i , \end{aligned}$$

where $\alpha =q_1$ and $\beta =p_1$ for $j=m$, $\alpha =q_2$ and $\beta =p_2$ for $j=d$, $\alpha =x_1$ and $\beta =y_1$ for $j=a$, and $\alpha =x_2$ and $\beta =y_2$ for $j=b$. The dynamics of quantum fluctuation operators can be written in matrix form as

(12)$$\dot{\mathbf{U}}(t) = \mathbf{A}(t)\mathbf {U}(t)+\mathbf{N}(t),$$

where $\mathbf {U}(t)=\left [\delta q_{1}, \delta p_{1},\delta q_{2}, \delta p_{2}, \delta x_{1}, \delta y_{1}, \delta x_{2}, \delta y_{2}\right ]^{T}$ is the vector of quadrature fluctuation operators, $\mathbf {N}(t)=\left [\sqrt { 2\gamma _{m}} \delta q_{1,\mathrm {in}}, \sqrt { 2\gamma _{m}} \delta p_{1,\mathrm {in}},\sqrt { 2,\gamma _{d}} \delta q_{2,\mathrm {in}}, \sqrt { 2\gamma _{d}} \delta p_{2,\mathrm {in}}, \sqrt { 2\kappa } \delta x_{1,\mathrm {in}}, \sqrt { 2\kappa } \delta y_{1,\mathrm {in}},\sqrt { 2\kappa }\right.\\\left. \delta x_{2,\mathrm {in}},\sqrt { 2\kappa } \delta y_{2,\mathrm {in}}\right ]^{T}$ is the vector of noises, and $\mathbf {A}(t)$ is the drift matrix as

(13)$$\mathbf{A}(t) = \left(\begin{array}{cccccccc}-\gamma_{m} & \omega_{m} & 0 & 0 & 0 & -G_{a} & 0 & -G_{b} \\ -\omega_{m}\ & -\gamma_{m} & 0 & 0 & G_{a} & 0 & G_{b} & 0\\ 0 & 0 & -\gamma_{d} & \omega_{d} & 0 & -G_{0} & 0 & 0\\ 0 & 0 & -\omega_{d} & -\gamma_{d} & G_{0} & 0 & 0 & 0 \\ 0 & -G_{a} & 0 & -G_{0} & -\kappa & \Delta & 0 & G \\ G_{a} & 0 & G_{0} & 0 & -\Delta & -\kappa & -G & 0 \\ 0 & -G_{b} & 0 & 0 & 0 & G & -\kappa & \Delta \\ G_{b} & 0 & 0 & 0 & -G & 0 & -\Delta & -\kappa\end{array}\right).$$

In the stable regime, quantum fluctuations will evolve to an asymptotic Gaussian state. However, with the above linearized dynamics of the quantum fluctuations and considering the zero-mean Gaussian nature of the quantum noises, the evolution of quantum fluctuations can be described by a 8 $\times$ 8 covariance matrix $\mathbf {V}(t)$ with the elements $\mathbf {V}_{k, l}=\left \langle \mathbf {U}_{k}(t) \mathbf {U}_{l}(t)+\mathbf {U}_{l}(t) \mathbf {U}_{k}(t)\right \rangle / 2$. Then, we can derive the dynamics equation governing the evolution of the covariance matrix as

(14)$$\dot{\mathbf{V}}(t)=\mathbf{A}(t) \mathbf{V}(t)+\mathbf{V}(t) \mathbf{A}^{\mathrm{T}}(t)+\mathbf{D},$$

where $\mathbf {A}^{\mathrm {T}}(t)$ denotes the transpose of $\mathbf {A}(t)$ and $\mathbf {D}=\operatorname {Diag}[\gamma _{m}(2 N_{m}+1),\gamma _{m}\left (2 N_{m}+1\right ), \gamma _{d}(2 N_{d}+1), \gamma _{d}(2 N_{d}+1),\kappa (2 N_{a}+1),\kappa (2 N_{a}+1),\kappa (2 N_{b}+1),\kappa (2 N_{b}+1)]$ is the matrix of noise correlation.

For numerically solving Eq. (14) as an inhomogeneous first-order differential equation, we use the initial value $\mathbf {V}(0)=\operatorname {Diag} \left [\frac {1}{2}, \frac {1}{2},N_{d}+\frac {1}{2}, N_{d}+\frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {1}{2}, \frac {1}{2}\right ]$. Then, we explore the dynamics of the mechanical resonator $d$ to explore the enhancement of cooling behaviors by dissipation engineering using the final steady thermal occupation of $d$ as

(15)$$\bar{n}=\left\langle d^{{\dagger}}d\right\rangle= \frac{1}{2}\left(\mathbf{V}_{3,3}+\mathbf{V}_{4,4}-1\right).$$

Figure 4(a) depicts the time evolution of mean phonon number $\bar {n}$. Notably, compared to the case of $J=0$ [see the green dotted curve in Fig. 4 (a)], representing a typical optomechanical system, the presence of the mechanical resonator $m$ related to the dissipative coupling process may weaken the ground-state cooling. This can be observed according to the yellow dash-dotted curve and the red dashed curve in Fig. 4(a) and occurs due to the unavoidable noise introduced by large dissipation $\gamma _m$, which tends to diminish the effectiveness of the cooling process. However, the ground-state cooling can be enhanced with the proper parameter condition. When setting the balance between dissipative and coherent couplings ($J=G$) corresponding to the Eps, we emphasize the critical role of the phase. Remarkably, the final $\bar {n}$ can be significantly reduced to a much lower value when the phase-matching condition ($\theta =\pi /2$) is satisfied. For a more thorough investigation of this effect, we plot Fig. 4(b) to show the steady-state $\bar {n}$ versus the varying $J$ with $\theta =\pi /2$. This figure reveals a sharp decrease in the final $\bar {n}$ as the system approaches $J=G$. Conversely, the final $\bar {n}$ exhibits a rapid increase when the system deviates from the EPs. This observation implies that the optimal cooling conditions, achievable through the modulation of both $\theta$ and $J$, are specifically tailored to the regime of the unidirectional OMIT. The underlying mechanism is underpinned by the effective suppression of Stokes heating processes through the cooperation of the dissipative coupling process and the phase modulations.

Fig. 4. (a) Time evolution of mean phonon number $\bar {n}$ for $J = G$ and $\theta =\pi /2$ (blue, solid curve), $J = G$ and $\theta =\pi$ (yellow, dash-dotted curve), $J =2G$ and $\theta =\pi /2$ (red, dashed curve) and $J = 0$ (green, dotted curve). (b) The steady-state mean phonon number $\bar {n}$ versus dissipative coupling strength $J$. Here $G=k_i$ and other parameters are the same as in Fig. 2.

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The optomechanical systems for nonreciprocal transmission have been experimentally demonstrated in several multimode cavity optomechanical setups, including silica microtoroid [31,64,65], and optomechanical crystal [66]. These experimental setups provide an ideal foundation for realizing our proposal. In our system, the coherent coupling can be effectively tuned by changing the distance between the fiber waveguide and the optical microcavity [31,64,65] or by modifying the coupling capacitance (mutual inductance) of superconducting circuits [67,68]. Concerning the reversed-dissipation coupling, recent experimental studies have shown that the decay rates of aluminum drums [55], silica microspheres [42], and mechanical resonators can be made substantially larger than the decay rate of the corresponding cavity resonator operating in microwave or optical frequency bands. These advancements in engineering decay rates of mechanical modes facilitate the implementation of dissipative coupling in our proposal in microwave and optical domains.

5. Conclusion

To conclude, we investigate the nonreciprocal OMIT spectra and the enhanced cooling capabilities within a cavity optomechanical system in the reversed-dissipation regime. In the system, two cavities are coupled with a direct coherent coupling and an indirect dissipative coupling with phase factors arising from the adiabatic elimination of a low-Q mechanical resonator. We can achieve the phase-dependent unidirectional OMIT spectra, whose direction and profiles can be controlled via phase modulation, even with the strong dissipative coupling. Notably, the OMIT window exhibits an asymmetric profile due to the frequency shift effect inherent to the reversed-dissipation regime. For nonreciprocity with the presence of a balance between the dissipative coupling and the coherent coupling, the parity-dependent unidirectional transmission can appear at the phase-matching condition. Furthermore, we explore the potential for enhancing the ground-state cooling of a mechanical mode in the reversed-dissipation system. The optimization of cooling is achieved by fine-tuning both the dissipative coupling and phase. Particularly noteworthy is that the cooling effect can be significantly enhanced in the conditions of balanced dissipative and coherent couplings and phase-matching, specifically tailored to the regime of the unidirectional OMIT. Our proposed scheme has broader applicability and can be extended to other bosonic systems, including magnons, atomic spin excitation, and photonic crystals.

Funding

National Natural Science Foundation of China (12074061).

Acknowledgement

The authors thank Lei Du for fruitful discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Nonreciprocal optomechanically induced transparency and enhanced ground-state cooling in a reversed-dissipation cavity system

Abstract

1. Introduction

2. Model and effective Hamiltonian of the system

3. Phase-dependent optical response properties

4. Enhanced ground-state cooling of the mechanical resonator

5. Conclusion

Funding

Acknowledgement

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (15)

Optics Express