Abstract

We study the optical transmission characteristics of coupled spinning optomechanical resonators with pump-probe driven lasers. Under the steady-state conditions, we focus on how changing the optical Sagnac effect due to same or opposite spinning directions of the resonators can give rise to non-reciprocal and delayed probe light transmission. We find that coupled resonators can exhibit distinct transmission features, can generate negative group delays (slow as well as fast light) and offer additional control of the probe light transmission as compared to the case of a single spinning resonator. Our results can be useful in achieving chiral light propagation in quantum communication technologies without using traditional magneto-optical means.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Nonreciprocal optical devices have gained a lot of attraction due to their ability to break the symmetry of an experiment under the source and detector exchange even in the presence of noise [13]. Primary examples of such devices are optical circulators [4] and isolators [5] which have found applications in quantum networking, one-way optical communication protocols and topological photonics [68]. For a recent experiment on optomechanical circulators see [9]. Some of the underlying physical effects utilized in these devices include optical nonlinearity [10], magneto-optical crystal based Faraday rotation [11,12], reservoir engineering [13] and photonic Aharonov-Bohm effect [14].

In addition to nonreciprocity, tunable slow and fast light propagation [15,16] is sometimes another requirement for on-demand photon transmission in chip devices. Typically, in quantum optical settings, the slow and fast light is achieved through the phenomenon of electromagnetically induced transparency (EIT) [17] in which a strong pump field driven three-level atomic medium becomes transparent to a weak probe field due to quantum coherence. More recently EIT has been studied in disparate setups, for instance in metamaterials [18], circuit quantum electrodynamics [19], waveguide quantum electrodynamics [20,21] and optomechanics [22]. Analogous to EIT in other physical setups, optomechanics also manifests EIT due to interference among different decay channels incorporating mechanical side-bands. This phenomenon in optomechanics is commonly referred to as optomechanically induced transparency (OMIT) [2325].

In this context, cavity quantum optomechanics [26] is particularly a noticeable example due to its ability to host both nonreciprocity [27] and slow and fast light propagation [28] in the same setup. More recently, hybrid atom-optomechanical systems (with single atoms, Bose-Einstein condensates and Kerr-type nonlinear medium coupled to optomechanical systems) [2932] have made these studies more interesting due to coherent coupling among different degrees of freedom. The presence of atom-like system can lead to for instance improved ground state cooling of a mechanical oscillator [33], steady-state mechanical squeezing [34], the appearance of novel correlations [35] and qubit-assisted enhancement of optomechanical interaction [36].

However, besides coupling emitters to an optomechanical cavity, there are other fascinating possibilities of introducing hybrid degrees of freedom in quantum optomechanics [3739]. In particular, recently Lü et al. have investigated a fiber coupled cavity optomechanical system where the cavity (ring-resonator in their study) is capable of spinning/rotating as well [40]. They have shown that spin degrees of freedom can considerably modify the probe light transmission and even can introduce nonreciprocity with slow light propagation. In this same regard, Maayani et al. have demonstrated that optical non-reciprocal transmission can be achieved by coupling spinning resonators with flying couplers [41]. For further very recent developments (such as nanoparticle sensing and nonreciprocal photon blockade) with setups involving spinning resonators we direct reader to the Refs. [42,43].

Motivated by Lü and Maayani et. al studies, in this work we consider two spinning coupled resonators. We concentrate on the question that how the presence of different spinning directions can impact our ability to alter the probe light transmission. We find that different spin directions can considerably change the transport of probe photons. Our study has three central findings: (1) For a set of experimentally feasible parameters, we notice by selecting clockwise spinning of both resonators one can observe the emergence of a “W” like transmission pattern for negative detunings, which cannot be achieved by using a single spinning resonator. (2) We find that by fixing the detuning at $10MHz$ and varying spinning rates up to $120$kHz, the group delay of probe light remains positive (slow light) for a single spinning resonator but can take negative values (fast light) for two counterclockwise spinning resonators. We observed that this fast light feature can only be achieved with two spinning coupled resonators. (3) Moreover, we also notice that by choosing the same spinning directions in both resonators one can achieve both higher transmission as well as fast light propagation without requiring any optical non-linearity or magneto-optical effect. These unique features confirm that the compound spinning device can provide a more flexible and powerful way to steer or enhance the nonreciprocal optical propagation in practice.

The paper is organized as follows: In Sec. 2. we outline the theoretical model of the system by presenting the Hamiltonian. Followed by this, in Sec. 3. we present the steady-state analysis of probe transmission. Sec. 4. is devoted to the discussion of results. In Sec. 5., we summarize our results and present a glimpse of possible future research directions. Finally, in appendices, we report the remaining transmission spectra covering all different cases of spinning.

2. Physical setup and Hamiltonian

We consider a vertical array of two coupled optomechanical spinning microresonators as depicted in Fig. 1. The Hamiltonian of the problem can be decomposed into three pieces: $\hat {H}_{0}$, $\hat {H}_{dr}$, and $\hat {H}_{int}$ which, respectively, represent the free, drive, and interaction part. Transforming into a frame rotating with pump frequency $\omega _{l_{j}}$, with $\hbar =1$ and under the rotating wave approximation the Hamiltonian is expressed as

$$\begin{aligned} & \hat{H}=\hat{H}_{0}+\hat{H}_{int}+\hat{H}_{dr},\\ & \hat{H}_{0}=\sum^{2}_{j=1}(\Delta_{c_j}\hat{a}^{{\dagger}}_{j}\hat{a}_{j}+\frac{\hat{p}^{2}_{j}}{2m_{j}}+\frac{1}{2}m_{j}\omega^{2}_{m_{j}}\hat{x}^{2}_{j}+\frac{\hat{p}^{2}_{j\theta}}{2m_{j}r^{2}_{j}}),\\ & \hat{H}_{dr}=\sum^{2}_{j=1}i\sqrt{\kappa_{ex}}(\varepsilon_{l_{j}}\hat{a}^{{\dagger}}_{j}+\varepsilon_{p_{1}}\hat{a}^{{\dagger}}_{1}e^{{-}i(\omega_{p}-\omega_{l_{j}})t}-H.c.),\\ & \hat{H}_{int}={-}\sum^{2}_{j=1}\xi_{j}\hat{x}_{j}\hat{a}^{{\dagger}}_{j}\hat{a}_{j}+J_{1,2}(a^{{\dagger}}_{1}\hat{a}_{2}+\hat{a}^{{\dagger}}_{2}\hat{a}_{1}). \end{aligned}$$
The first term in $\hat {H}_0$ describes the free Hamiltonian of two isolated resonant optical modes (each belonging to its own resonator) with frequency $\omega _{c_{j}} (j=1,2)$. In our model, both resonators are driven by a strong pump/control field (pump frequency for first (second) resonator $\omega _{l_{1}}(\omega _{l_{2}})$). $\Delta _{c_{j}}=\omega _{c_{j}}-\omega _{l_{j}}$ is the detuning. $\hat {a}_{j}(\hat {a}^{\dagger }_{j})$ is the photonic destruction (creation) operator in the $j$th resonator. $\hat {p}_{j} (\hat {p}_{j_{\theta }})$ appearing in the second (fourth) term in $\hat {H}_0$ describes the linear (angular) momentum operator while $m_j$ is the effective mass and $r_j$ is the radius of the $j$th resonator. The strong pump fields also excite breathing mechanical-modes with frequency $\omega _{m_{j}}$. The third term in $\hat {H}_0$ represents the free evolution of such mechanical mode with displacement operator $\hat {x}_j$. The presence of the angular momentum terms opens up the possibility of spinning of resonators. We represent the spinning/rotating frequency of the first (second) resonator by $\Omega _{1}(\Omega _{2})$, where conventionally we take $\Omega _{j}>0$ for clockwise rotary direction.

 

Fig. 1. Scheme of the system consisting of two series-coupled spinning optomechanical resonators.

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In our model, the bottom resonator is coupled to an optical (typically tapered) fiber which guides two incoming fields namely a weak probe field with amplitude $\varepsilon _{p_{1}}$ and a strong pump field with amplitude $\varepsilon _{l_{1}}$. The respective powers of the incoming probe and pump fields $P_{in}$ and $P_{l_{1}}$ are related to these amplitudes through : $\varepsilon _{p_{1}}=\sqrt {\frac {P_{in}}{\hbar \omega _{p}}}\hspace {4mm} \textit {and}\hspace {4mm}\varepsilon _{l_{1}}=\sqrt {\frac {P_{l_{1}}}{\hbar \omega _{l_{1}}}},$ where $\omega _{p}$ and $\omega _{l_{1}}$ are the frequencies of the probe and pump light, respectively. Terms in $H_{dr}$ show these drives. Close to the bottom resonator the fields in the fiber evanescently couple to the resonator with an external cavity loss rate given by $\kappa ^{(1)}_{ex}$ (or simply $\kappa _{ex}$) and start circulating in a counterclockwise direction due to the process of total internal reflection. Under realistic conditions, resonators suffer from the internal cavity loss which is represented by $\kappa ^{(1)}_{in}$ and $\kappa ^{(2)}_{in}$ for first and second resonators, respectively.

The pump and probe laser fields entering the bottom resonator can couple to the upper resonator through the inter-resonator coupling mechanism (characterized by the parameter $J_{1,2}$). This is described by the last two terms in $\hat {H}_{int}$. The geometry of our setup is such that a field entering in the upper resonator from the bottom resonator circulates in the clockwise direction. The optical mode in each resonator is additionally coupled with the mechanical mode of the same resonator through the standard non-linear optomechanical interaction [26] with strength parameter $\xi _{j}$ (see first term in $\hat {H}_{int}$).

It is well-known that due to the rotation, optical mode frequency experiences Sagnac-Fizeau shift [44] which transforms

$$\begin{aligned} & \omega_{c_{j}}\longrightarrow\omega_{c_{j}}+\Delta_{sag_{j}},\\ & \textrm{where}\hspace{2mm}\Delta_{sag_{j}}:=\frac{n_{j}r_{j}\Omega_{j}\omega_{c_{j}}}{c}\Bigg(1-\frac{1}{n^{2}_{j}}-\frac{\lambda_{j}}{n_{j}}\frac{dn_{j}}{d\lambda_{j}}\Bigg). \end{aligned}$$
$\Delta _{sag_{j}}$, $n_{j}$, $r_{j}$ are the Sagnac-Fizeau shift, refractive index, and radius of the $j$th resonator. $c$ is the speed of light and $\Omega =\frac {d\theta }{dt}$ is the spinning rate. $dn_{j}/d\lambda _{j}$ term represents negligibly small relativistic (dispersion) correction in the Sagnac-Fizeau shift. In Eq.(2), the first term in the parenthesis shows Sagnac contribution which arises from the rotation of the resonators. While the two last terms with negative signs take into account the Fizeau drag due to the light propagation through a moving resonator medium (see Eq. (11) and the corresponding text in [44] for further details). Finally, we would like to mention that the series configuration of resonators considered in this work is similar to the setup mentioned in [45] where two coupled optical modes with asymmetric waveguide interaction were studied.

3. Output probe light transmission

3.1 Heisenberg-Langevin equations of motion

Relevant operators’ equations of motion in the Heisenberg picture can be worked out using Eq. (1). We obtain a set of coupled equations as

$$\begin{aligned} & \frac{d\hat{a}_{1}(t)}{dt}={-}i(\Delta_{c_{1}}-\xi_{1}\hat{x}_{1}-i\beta_{1})\hat{a}_{1}+\sqrt{\kappa_{ex}}(\varepsilon_{l_{1}}+\varepsilon_{p_{1}}e^{{-}i\eta_{1}t})-iJ_{1,2}\hat{a}_{2},\\ & \frac{d\hat{a}_{2}(t)}{dt}={-}i(\Delta_{c_{2}}-\xi_{2}\hat{x}_{2}-i\beta_{2})\hat{a}_{2}+\sqrt{\kappa_{ex}}\varepsilon_{l_{2}}-iJ_{1,2}\hat{a}_{1},\\ & \frac{d^{2}\hat{x}_{1}(t)}{dt^{2}}=\frac{\xi_{1}}{m_{1}}\hat{a}^{{\dagger}}_{1}\hat{a}_{1}-\omega^{2}_{m_{1}}\hat{x}_{1}+\frac{\hat{p}^{2}_{1\theta}}{m^{2}_{1}r^{3}_{1}}-\gamma_{m_{1}}\frac{d\hat{x}_{1}}{dt},\\ & \frac{d^{2}\hat{x}_{2}(t)}{dt^{2}}=\frac{\xi_{2}}{m_{2}}\hat{a}^{{\dagger}}_{2}\hat{a}_{2}-\omega^{2}_{m_{2}}\hat{x}_{2}+\frac{\hat{p}^{2}_{2\theta}}{m^{2}_{2}r^{3}_{2}}-\gamma_{m_{2}}\frac{d\hat{x}_{2}}{dt},\\ & \frac{d\hat{\theta}_{1}(t)}{dt}=\frac{\hat{p}_{1\theta}}{m_{1}r^{2}_{1}}, \hspace{10mm}\frac{d\hat{\theta}_{2}(t)}{dt}=\frac{\hat{p}_{2\theta}}{m_{2}r^{2}_{2}},\\ & \frac{d\hat{p}_{j\theta}(t)}{dt}=0,\hspace{10mm} \frac{d\hat{p}_{2\theta}(t)}{dt}=0.\\ \end{aligned}$$
Note that in the above set of equations we have phenomenologically added the optical and mechanical decay rates $\beta _{j}$ and $\gamma _{m_{j}}$, respectively. From the last equation, equations of motion for the expectation values of these operators can be readily obtained as
$$\begin{aligned} & \frac{d\langle\hat{a}_{1}(t)\rangle}{dt}={-}i(\Delta_{c_{1}}-i\beta_{1})\langle\hat{a}_{1}\rangle+i\xi_{1}\langle\hat{x}_{1}\hat{a}_{1}\rangle+\sqrt{\kappa_{ex}}(\varepsilon_{l_{1}}+\varepsilon_{p_{1}}e^{{-}i\eta_{1}t})-iJ_{1,2}\langle\hat{a}_{2}\rangle,\\ & \frac{d\langle\hat{a}_{2}(t)\rangle}{dt}={-}i(\Delta_{c_{2}}-i\beta_{2})\langle\hat{a}_{2}\rangle+i\xi_{2}\langle\hat{x}_{1}\hat{a}_{2}\rangle+\sqrt{\kappa_{ex}}\varepsilon_{l_{2}}-iJ_{1,2}\langle\hat{a}_{1}\rangle,\\ & \frac{d^{2}\langle\hat{x}_{1}(t)\rangle}{dt^{2}}={-}(\omega^{2}_{m_{1}}+\gamma_{m_{1}}\frac{d}{dt})\langle\hat{x}_{1}\rangle+\frac{\xi_{1}}{m_{1}}\langle\hat{a}^{{\dagger}}_{1}\hat{a}_{1}\rangle+\frac{\langle\hat{p}^{2}_{1\theta}\rangle}{m^{2}_{1}r^{3}_{1}},\\ & \frac{d^{2}\langle\hat{x}_{2}(t)\rangle}{dt^{2}}={-}(\omega^{2}_{m_{2}}+\gamma_{m_{2}}\frac{d}{dt})\langle\hat{x}_{2}\rangle+\frac{\xi_{2}}{m_{2}}\langle\hat{a}^{{\dagger}}_{2}\hat{a}_{2}\rangle+\frac{\langle\hat{p}^{2}_{2\theta}\rangle}{m^{2}_{2}r^{3}_{2}},\\ & \frac{d\langle\hat{\theta}_{1}(t)\rangle}{dt}=\frac{\langle\hat{p}_{1\theta}\rangle}{m_{1}r^{2}_{1}},\hspace{10mm}\frac{d\langle\hat{\theta}_{2}(t)\rangle}{dt}=\frac{\langle\hat{p}_{2\theta}\rangle}{m_{2}r^{2}_{2}},\\ & \frac{d\langle\hat{p}_{1\theta}(t)\rangle}{dt}=0, \hspace{10mm}\frac{d\langle\hat{p}_{2\theta}(t)\rangle}{dt}=0. \end{aligned}$$
$\eta _{j}\equiv \omega _{p}-\omega _{l_{j}}$ is the pump-probe detuning and $\beta _{j}=1/2(\kappa _{ex}\delta _{j1}+\kappa ^{(j)}_{in})$ is the net photon leakage rate from the $j$th cavity. Note that the Kronecker delta $\delta _{j1}$ is used to represent that the $\kappa _{ex}$ term only contributes for the bottom resonator (i.e. when $j=1$).

3.2 Steady-state analysis

Next, in order to proceed with the analytic solution, we apply the so-called mean-field approximation [25]. To this end, we express all correlations as the product of the average value of operators. For instance, we express $\langle \hat {x}_{1}\hat {a}_{1}\rangle = \langle \hat {x}_{1}\rangle \langle \hat {a}_{1}\rangle$, $\langle \hat {x}_{2}\hat {a}_{2}\rangle = \langle \hat {x}_{2}\rangle \langle \hat {a}_{2}\rangle$, $\langle \hat {a}^{\dagger }_{1}\hat {a}_{1}\rangle =\langle \hat {a}^{\dagger }_{1}\rangle \langle \hat {a}_{1}\rangle$ and $\langle \hat {a}^{\dagger }_{2}\hat {a}_{2}\rangle =\langle \hat {a}^{\dagger }_{2}\rangle \langle \hat {a}_{2}\rangle$. We then follow the standard procedure (see for instance [46,47]) and decompose expectation value of all operators as a sum of their steady-state value and small fluctuations around the steady-state value in the following form

$$\begin{aligned} & \langle\hat{a}_{1}\rangle\longrightarrow a_{1}+\delta a_{{-}1}e^{{-}i\eta_{1}t}+\delta a_{{+}1}e^{i\eta_{1}t},\\ & \langle\hat{a}_{2}\rangle\longrightarrow a_{2}+\delta a_{{-}2}e^{{-}i\eta_{2}t}+\delta a_{{+}2}e^{i\eta_{2}t},\\ & \langle\hat{x}_{1}\rangle\longrightarrow x_{1}+\delta x_{1}e^{{-}i\eta_{1}t}+\delta x^{{\ast}}_{1}e^{i\eta_{1}t},\\ & \langle\hat{x}_{2}\rangle\longrightarrow x_{2}+\delta x_{2}e^{{-}i\eta_{2}t}+\delta x^{{\ast}}_{2}e^{i\eta_{2}t}. \end{aligned}$$
By inserting Eq. (5) into (4) we easily drive the steady-state values as
$$\begin{aligned} & a_{1}=\frac{(\sqrt{\kappa_{ex}}\varepsilon_{l}-iJ_{1,2} a_{2})}{\beta_{1}+i\Delta_{c_{1}}-i\xi_{1} x_{1}},\hspace{4mm}a_{2}=\frac{(\sqrt{\kappa_{ex}}\varepsilon_{l}-iJ_{2,1} a_{1})}{\beta_{2}+i\Delta_{c_{2}}-i\xi_{2} x_{2}},\\ & x_{1}=\frac{(\xi_{1}|a_{1}|^{2}+m_{1}r_{1}\Omega^{2}_{1})}{m_{1}\omega^{2}_{m_{1}}},\hspace{4mm} x_{2}=\frac{(\xi_{2}|a_{2}|^{2}+m_{2}r_{2}\Omega^{2}_{2})}{m_{2}\omega^{2}_{m_{2}}}. \end{aligned}$$
$|\Omega _{j}|=\frac {d\theta _{j}}{dt}$ is the magnitude of the spinning rate. Likewise, the fluctuating part of the expectation values of the operators can be worked out as
$$\begin{aligned} & \delta a_{{-}1}(\beta_{1}+i\Delta_{c_{1}}-i\xi_{1}x_{1}-i\eta_{1})-i\xi_{1}a_{1}\delta x_{1}=\sqrt{\kappa_{ex}}\varepsilon_{p_{1}}-iJ_{1,2}\delta a_{2},\\ & \delta a_{{-}2}(\beta_{2}+i\Delta_{c_{2}}-i\xi_{2}x_{2}-i\eta_{2})-i\xi_{2}a_{2}\delta x_{2}={-}iJ_{1,2}\delta a_{1},\\ & \delta a^{{\ast}}_{{+}1}(\beta_{1}-i\Delta_{c_{1}}+i\xi_{1}x_{1}-i\eta_{1})+i\xi_{1}a^{{\ast}}_{1}\delta x_{1}=iJ_{1,2}\delta a^{{\ast}}_{2},\\ & \delta a^{{\ast}}_{{+}2}(\beta_{2}-i\Delta_{c_{2}}+i\xi_{2}x_{2}-i\eta_{2})+i\xi_{2}a^{{\ast}}_{2}\delta x_{2}=iJ_{1,2}\delta a^{{\ast}}_{1},\\ & m_1(\omega^{2}_{m_{1}}-\eta_{1}-i\eta_{1}\gamma_{m_{1}})\delta x_{1}=\xi_{1}(a^{{\ast}}_{1}\delta a_{{-}1}+a_{1}\delta a_{{+}1}),\\ & m_2(\omega^{2}_{m_{2}}-\eta_{2}-i\eta_{2}\gamma_{m_{2}})\delta x_{2}=\xi_{2}(a^{{\ast}}_{2}\delta a_{{-}2}+a_{2}\delta a_{{+}2}). \end{aligned}$$
In the derivation of Eq. (6) and (7) we have adopted a perturbation approach where in all decompositions we have assumed the steady-state mean values to be much larger than the fluctuations i.e. $|a_{1}|>>|\delta a_{\pm 1}|$, $|a_{2}|>>|\delta a_{\pm 2}|$, $|x_{1}|>>\lbrace |\delta x_{1}|,|\delta x^{\ast }_{1}|\rbrace$ and $|x_{2}|>>\lbrace |\delta x_{2}|,|\delta x^{\ast }_{2}|\rbrace$.

3.3 Probe field transmission rate

The transmission rate $T$ of the probe field is defined as

$$T\equiv{\mid} t_{p}\mid^{2}=\frac{<\hat{a}^{{\dagger}}_{out}\hat{a}_{out}>}{<\hat{a}^{{\dagger}}_{in}\hat{a}_{in}>}.$$
The probe field travels through the fiber in the right direction. Therefore, even though the probe field input ($\hat {a}_{in}$) and output ($\hat {a}_{out}$) operators act at different ports of the fiber, they can still be related through the standard Collett and Gardiner input-output relationship [48] as both fields travel in the same (rightward) channel. Therefore, we use
$$\hat{a}_{out}=\hat{a}_{in}-\sqrt{\kappa_{ex}}\delta a_{{-}1}=\varepsilon_{p_{1}}-\sqrt{\kappa_{ex}}\delta a_{{-}1}.$$
Since the probe is a classical field, therefore, we have replaced the input operator by the probe field amplitude $\varepsilon _{p_{1}}$ in Eq. (9). Using Eq. (9) in (8), the probe transmission rate can be expressed in terms of $\delta a_{-1}$ as
$$T=\Bigg| 1-\frac{\sqrt{\kappa_{ex}}}{\varepsilon_{p_{1}}}\delta a_{{-}1}\Bigg|^{2}.$$
Hence, to find out the net transmission rate $T$, we’ll follow the recipe of solving equation sets Eq. (6) and (7) simultaneously to obtain the required $\delta a_{-1}$.

4. Results and discussion

4.1 Parameters and experimental feasibility

We begin by mentioning the set of experimentally feasible parameters [40,49,50] considered in this work. Apart from spinning directions, we take all resonators to be identical with $m= 2$ng, $\omega _{m}=200$MHz, $\gamma _{m}=0.2$MHz, optical wavelength $\lambda = 1.55\mu$m, refractive index $n =1.44$, speed of light in the optical medium $v =3\times 10^{8}$/n, $\omega _{c}=193.5$ THz, quality factor of the optical resonator $Q =3\times 10^{7}$, $\kappa _{ex}=\omega _{c}/Q$, $\kappa ^{(j)}_{in} =\kappa _{ex}$, $P_{l} =10$W, $r =0.25$mm and $\xi =\omega _{c}/r$. The frequency of the driving field for both resonators is also assumed to be the same. Here we take resonator-resonator coupling $J=J_{1,2}=\kappa _{ex}/2, \kappa _{ex}$ and $2\kappa _{ex}$. The strength of parameter $J$ can be controlled by altering the separation between the resonators. For instance, Peng et al. [51] have experimentally shown that by reducing the gap between two resonators to $\sim 5\mu$m a strong inter-resonator coupling can be established. This strong coupling leaves its signature on the transmission spectrum in the form of the splitting of two resonances. For a relevant discussion on the coupling between a whispering-gallery microdisk resonator with a tapered fiber see [52].

Throughout this work, we have treated fiber-resonator coupling $\kappa _{ex}$ and resonator-resonator coupling $J$ independent of spinning rate $\Omega$. Our reason for keeping $\kappa _{ex}$ constant relies on a recent spinning resonator experiment [41] where authors have found a perfect match between experiment and theory while keeping $\kappa _{ex}$ constant. Since till date to best of our knowledge, there is no experiment performed with two coupled spinning microresonators, therefore to check how valid it is to treat $J$ constant, we performed a straightforward calculation. The main aim of our calculation was to see how much $\Omega$ can change $\xi x$ value which can then alter the inter-resonator separation (which defines the $J$ value). Using Eq. (6), we found at $\Omega =100kHz$, $x$ turns out to be $\sim 0.063nm$. Comparing this with a typical static inter-resonator separation value $\sim 5\mu m$ [51], we conclude that spinning will have almost five orders of magnitude smaller effect on the inter-resonator coupling. Hence it is fully justified to treat $J$ constant for the problem of coupled spinning resonators.

Furthermore, in this work, we consider rotation speeds up to $100$kHz which seems challenging from the point of view of keeping motion stability as well as efficient optical coupling. However, we would like to point out that this value is not entirely far from implementation if we take into account the continued improvement in the optical resonator technology. For instance, in a recent study [41] for a $1.1$mm radius resonator, a spinning rate of $3kHz$ has already been experimentally achieved to perform non-reciprocal light transmission.

4.2 Probe transmission: non-reciprocal transport

We now present the discussion of our results. The main focus of these results is to analyze how different spinning directions in the ring resonators can impact the probe transmission rate $T$. To this end, we concentrate on the following cases

  • • both resonators are non-spinning,
  • • both resonators have clockwise spin,
  • $1^{st}$ has clockwise, $2^{nd}$ has counterclockwise spin,
  • $1^{st}$ has counterclockwise, $2^{nd}$ has clockwise spin,
  • • only one of the two resonators is spinning,
  • • and both resonators have counterclockwise spin.

4.2.1 Impact of resonator-resonator coupling: W-like spectral pattern

We start with Fig. 2(a) where we compare probe transmission of a single- and a double-resonator problem without the involvement of any spin (i.e. $\Omega _{1}=\Omega _{2}=0$). For a single resonator case ($N=1$ gray dotted curve in Fig. 2(a)) one finds

$$\delta a_{{-}1}={-}\frac{ \sqrt{\kappa_{ex}}\varepsilon_{p_{1}}\lbrace i\xi^{2}\vert a\vert^{2}+m(\widetilde{\beta}_{-})\Gamma_{m} \rbrace }{ i\xi^{2}\vert a\vert^{2}(\widetilde{\beta}_{-})-(\widetilde{\beta}^{{\ast}}_{-})\lbrace i\xi^{2}\vert a\vert^{2}+m(\widetilde{\beta}_{-})\Gamma_{m}\rbrace },$$
where $\widetilde {\beta }_{-}+i\eta =(\beta -i\Delta _{c}+ix\xi )$, $\widetilde {\beta }^{\ast }_{-}+i\eta =(\beta +i\Delta _{c}-ix\xi )$ and $\Gamma _{m}=\omega ^{2}_{m}-i\gamma _{m}\eta -\eta ^{2}$. From the plot we notice the appearance of the standard OMIT transmission with peak residing at the resonance point ($\Delta _{p}\equiv \omega _{l}-\omega _{p}=0$). As reported in [23] the transparency window linewidth is given by $\gamma _{m}+\xi ^{2}|a|^{2}/(m^{2}\omega ^{2}_{m}\beta )$ which for our case takes the numerical value of $\sim 2$MHz. For some related studies on non-reciprocal light propagation in non-spinning single optomechanical and optical resonators we direct reader to [5355].

 

Fig. 2. Probe transmission rate as a function of detuning $\Delta _{p}$ for a single resonator and a double coupled resonator system. (a) Absence of spin in all cases and increasing resonator-resonator coupling $J$ for the two-resonator case. (b) Bottom resonator spinning in the clockwise direction (with rate $\Omega _1=40$kHz) while upper resonator may or may not be spinning in the same direction. We have also incorporated the scenario when the upper resonator is not spinning. Note that in this and later plots, for $N=2$ we have assumed $J/\kappa _{ex}=1$ and $\vert \Omega _{1}\vert =\vert \Omega _{2}\vert =\vert \Omega \vert$ unless stated otherwise.

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For a double-resonator setup without spin, the values of $a_1, a_2, x_1$ and $x_2$ are given in Eq. (6). We find that as we increase the resonator-resonator coupling strength $J$ from $0.5\kappa _{ex}$ to $2\kappa _{ex}$ the whole transmission spectrum lifts upwards. However, the location of the OMIT peak stays unaffected. Additionally, for $J>\kappa _{ex}$ on each side of the OMIT peak, two points of suppressed transmission are observed due to mode splitting. For instance for $\Delta _{p}<0$ and $J=2\kappa _{ex}$ (red solid curve in Fig. 2(a)) two lowest $T$ values appear at $\Delta _{p}\sim -1$MHz and $\sim -13$MHz. Moreover, we find that the separation between the two lowest transmission points on positive or negative $\Delta _{p}$-axis can be enhanced by increasing the $J$ value. For instance, for $J=2\kappa _{ex}$ ($\kappa _{ex}=6.5$ MHz) the separation between the lowest transmission points either on the positive or negative $\Delta _{p}$-axis turns out to be almost $2\kappa _{ex}$. Notice that the analytic expression of $\delta a_{-1}$ for two-resonators is mathematically involved and therefore not reported here.

Next, in Fig. 2(b) we introduce the spin degree of freedom. For a single resonator case (gray dotted curve) it is known that the steady-state values of $\langle \hat {a}\rangle$ and $\langle \hat {x} \rangle$ follow [40]

$$a=\frac{\sqrt{\kappa_{ex}}\varepsilon_{l}}{\beta+i\Delta_{c}-i\xi x}, \hspace{2mm} x=\frac{(\xi|a|^{2}+mr\Omega^{2})}{m\omega^{2}_{m}}.$$
Clearly when the resonator is capable of spinning ($\Omega \neq 0$), the value of $\xi x$ as well as $\Delta _{sag}$ modify which influence the full spectrum. In particular, we note that in both $\Omega >0$ and $\Omega <0$ situations the OMIT peaks are slightly red-shifted (move towards $\Delta _{p}<0$) as discussed in [40].

From plots in Fig. 2(b) we find that either the spin directions are the same or opposite, the transmission shows a considerably different profile as compared to the single-resonator transmission. When both resonators spin in the same direction, for example clockwise, an important feature of the coupled resonators is the splitting of the spectrum due to resonant mode-coupling between the two resonators. As shown in Fig. 2(b), this leads to a unique “W”-like transmission pattern with two symmetric dips for negative values of probe detuning $\Delta _p$ (see solid red curve). When the bottom resonator spins and upper resonator is kept static (blue dotted-dashed curve), we observe asymmetric splitting of the spectrum due to off-resonant coupling between two cavity modes. We noticed that when both resonator spin directions are opposite, i.e. $\Omega _1>0$ and $\Omega _2<0$ (curve not shown here) the transmission follows a profile similar to blue dotted dashed curve. For remaining cases of spinning directions (not addressed in Fig. 2(b)), we direct reader to Appendix A.

From Fig. 2(b) we also notice that the one-spinning case can be quite different from the two-spinning case even if one of the two resonators in both situations have the same spinning direction (e.g., $\Omega _1$ has the same sign in blue dotted-dashed and solid red curves in Fig. 2(b)). These features extend down to higher spinning speeds as well (see Fig. 7 in Appendix). Overall these trends clearly demonstrate that the spin degree of freedom in two resonators provide additional control as compared to the single spinning resonator and thus can be used for the probe transmission alteration.

4.2.2 Non-reciprocal probe light transmission

In our setup, there are two ports of the fiber from which the probe light can be launched. Optical nonreciprocity in this context means passing of light if probe light enters from one port and blocking if the probe enters from the opposite port. If we take into account the spinning of resonators then we notice that for a fixed spinning direction, say clockwise, probe entering from the right (left) port will propagate in a counterclockwise (clockwise) direction in the bottom resonator. This establishes a connection between the launching direction of the probe light and spinning direction of the resonators. Thus, in the following, we use this connection and discuss nonreciprocity by focusing on different spinning directions.

As discussed in [40] for a single resonator problem with $|\Omega |=100$kHz and $\xi x=48.47$ MHz, at $\Delta _{p}$ value of $40kHz$ we can achieve $T> 0.9$ (pass) and $T\sim 0$ (block) by adjusting the spinning directions in the clockwise and counterclockwise directions, respectively for a left incoming probe field. We find a second coupled resonator can introduce new features in $T$ versus $\Delta _{p}$ profile at $|\Omega |=100$kHz. For example, when both resonators spin in the same direction (solid red curves in Fig. (3)), a “W” like transmission appears which can be controlled by changing the sign of the spinning directions and value of $J$. In this situation with $\xi x=48.47$ MHz, when both resonators spin in a clockwise direction $T$ takes $90$% maximum value (solid thick red curve). But when the spinning direction is changed to counterclockwise direction the same probe field incoming from left port suffers minimum transmission (see thin red solid curve). It is worthwhile to mention here that we also find that these non-reciprocal probe transmission patterns require higher spinning rates (close to $100$kHz).

 

Fig. 3. Probe transmission for larger spinning rate $|\Omega |=100$kHz. We have focused on a frequency region where non-reciprocal light transmission is evident. Rest of the parameters are the same as used in Fig. 2.

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Finally, in the scenario in which both resonators spin in different directions the transmission achieves zero value when the bottom resonator spin in the counterclockwise direction (see Appendix B for the plot). Whereas, if the spin of the resonators is reverted then the transmission becomes maximum with $T>80$% around $\Delta _{p}=-35$MHz. Thus we conclude, if $\Omega _{1}<0, \Omega _{2}<0$ specifies the case when pump and probe are applied from the right direction then flipping the applied probe direction to left can lead to the non-reciprocal probe transmission.

4.3 Group delay: slow and fast light control

In order to characterize slow and fast light propagation [22], we use probe field group delay $\tau _{g}$ as a quantifying parameter. It is defined as

$$\tau_{g}=\frac{d\textrm{arg}(t_{p})}{d\Delta_{p}}$$
Consequently one can also define the group delay enhancement factor $G.D.$ for two-resonators as
$$G.D.=\frac{\tau_{g}(\Omega_{1}\neq 0,\Omega_{2}\neq 0)}{\tau_{g}(\Omega_{1}=0,\Omega_{2}=0)}\Bigg\vert_{\Delta_{p}}-1.$$
Here we focus on the group delay itself and plot $\tau _{g}$ in Fig. 4 as we vary the spinning rate $|\Omega |$ at a fixed value of $\Delta _{p}=10$MHz. We notice that the behavior of $\tau _{g}$ is sensitive to $\Delta _{p}$ and $|\Omega |$ values. For single counterclockwise spinning resonator (thin dotted gray curve in Fig. 4(a)), $\tau _{g}$ gradually increases and reaches a maximum around $15$kHz and then decays to almost zero value as we increase the spinning rate to $120$kHz. On the other hand, for double resonators (Fig. 4(b)), $\Omega _{1}<0$, $\Omega _{2}<0$ situation leads to negative $\tau _{g}$ values for $|\Omega |\lesssim 25$kHz. We would like to emphasize that this feature i.e. $\tau _g<0$ (which allows the possibility of fast light) is unique to double resonator setup and cannot be achieved with a single spinning resonator. It is interesting to note that for all other possible spinning combinations with double resonators, we always obtain $\tau _g>0$ i.e. slow light (see Appendix C).

 

Fig. 4. The group delay of probe light as a function of spinning rate magnitude for (a) a single and (b) a double spinning resonator. In the double resonator case, we have only plotted the unique situation where for a range of $|\Omega |$ fast light can be achieved. For comparison, we have plotted the single spinning resonator case in (a) where one can only achieve slow light for all $|\Omega |$ value. Parameters are the same as used in Fig. 2.

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4.4 Transmission rate enhancement factor (E.F.)

In order to quantify how much transmission is augmented when two resonators take different spinning directions at a fixed probe detuning $\Delta _{p}$ value, we define the transmission rate enhancement factor (E.F.) as

$$E.F.=\frac{T(\Omega_{1}\neq 0, \Omega_{2}\neq 0)}{T(\Omega_{1}=0,\Omega_{2}=0)}\Bigg\vert_{\Delta_{p}}-1.$$
For a single spinning resonator the expression of the E.F. simplifies to
$$E.F.=\frac{T(\Omega\neq 0)}{T(\Omega=0)}\Bigg\vert_{\Delta_{p}}-1.$$
From the inspection of Fig. 2 we notice that under all spinning directions $T$ takes almost the same value at the OMIT point ($\Delta _{p}=0$). Therefore, we choose an off-resonant value of $\Delta _{p}=10MHz$ to clearly show considerable change in the maximum value of the probe transmission through the E.F.

For a single resonator, we notice as we increase $|\Omega |$ value for $\Omega >0$ scenario (thick dotted gray curve in Fig. 5), E.F. shows growth such that for $|\Omega |\geq 80$kHz E.F. takes $\sim 45$% value. On the contrary, the corresponding $\Omega <0$ scenario (thin dotted gray curve in Fig. 5) shows a declining trend (negative E.F. value) up to $|\Omega |\approx 25$kHz. Crossing this point, E.F. becomes positive and approaches the E.F. asymptotic value around $|\Omega |=100kHz$ as achieved in $\Omega >0$ case. For two-resonator setup, we only focus on the particular case in which we can achieve the fast light i.e. when $\Delta _{p}=10$MHz (see Fig. 4). Very interestingly, we point out for the same case ($\Omega _{1}<0$, $\Omega _{2}<0$) in addition to fast light almost double enhancement in the probe light transmission can be simultaneously realized. This shows that the availability of a second spinning resonator can be utilized for a fast and an enhanced retrieval of output probe light transmission which may find applications in quantum communication protocols.

 

Fig. 5. Off-resonance transmission enhancement factor as a function of spinning rate for the double resonator case in which one can achieve the fast light. For comparison we have plotted the corresponding single resonator enhancement factor as well. Parameters are the same as used in Fig. 2.

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5. Summary, remarks and outlook

To recapitulate, we investigated the probe transmission properties in two coupled spinning optomechanical resonators. For $\Omega =0$ case, single resonator leads to OMIT. The presence of a second coupled resonator lifts the transmission upwards as well as enhances the separation between the absorption peaks as resonator-resonator coupling ($J$) value is elevated. When $\Omega \neq 0$ was considered we found the value of $\xi x$ changes for both single and double resonator cases. This, for instance, led to spin direction-dependent OMIT peak shift for single resonator case and the emergence of a “W” like pattern for $\Delta _{p}<0$ values for double clockwise direction spinning resonator scenario. The case in which we set $\Omega _{2}=0$ and $\Omega _{1}\neq 0$ the transmission and $E.F.$ retain an asymmetric profile. Therein, we find that by switching on-and-off the spinning of the second resonator provides an additional tunability in the transmission profile.

When we selected a higher spinning rate for both resonators ($|\Omega _{1}|=|\Omega _{2}|=100$kHz) and focused around $\Delta _{p}=40$MHz, we noticed that the same and opposite rotary directions produced oscillations in the curves corresponding to the minimum and maximum transmission regions. This indicated the possibility of nonreciprocal probe light transmission by altering the probe launching direction. For the double resonator case, when both resonators spin in the counterclockwise direction between $5\lesssim |\Omega |\lesssim 20$kHz, we found a high transmission ($E.F.\sim 60$%) and fast light ($\tau _{g}\sim -10$ns) can be simultaneously realized. We found that this fast light feature is unique to double spinning resonators and may provide novel applications which are not possible to attain by a single spinning resonator.

As the final remarks, we would like to highlight two points. Firstly, in all of this work, we have assumed both microresonators to be identical. However, in reality, it is very hard to manufacture two resonators with exactly the same properties. Therefore, we consider one commonly encountered scenario where there is a mismatch between optical mode frequencies [56,57]. By taking $\omega _{c_2}=\omega _{c_1}/2$ (plots not shown here but results are summarized), we notice that even in the absence of spinning, the frequency mismatch causes a red shift in the lowest transmission points. When we introduce spinning of the two resonators in the same direction, we observe the symmetry between the two dips in the “W” feature of the transmission is lifted. In the opposite spinning case, we point out that the shoulder and the dip feature near the resonance point are elevated. Albeit, we point out that in all cases the location and the value of OMIT peak remain unaffected. However, overall we conclude that an optical mode frequency mismatch greatly changes the probe light transmission.

Secondly, besides considering both resonators to be optomechanical (as studied in [58,59]) there are other interesting variations considerable in this setup. For instance, one can study the case where one resonator is purely optical spinning resonator, while the other is a non-spinning optomechanical cavity. With such a system, one can, for example, investigate the nonreciprocal amplification of phonons [60]. Another interesting extension of this work can be the study of many-body physics in 1D or 2D lattices of coupled spinning optomechanical resonators. It is also possible to investigate a situation when there exists a relative phase between the optical pumps driving the resonators which can lead to interesting coherent optical effects. We leave these as the possible future directions of this work.

Appendix A: Transmission in the scenario opposite to Fig. 2(b)

When both spinning directions are inverted compared to Fig. 2(b), we observe similar features but with different spectra due to the nonreciprocity of the spinning resonators (see Fig. 6). When we consider only one of the two resonators to be spinning (say $\Omega _{1}\lessgtr 0$ but $\Omega _{2}=0$) we find $T$ remains asymmetric.

 

Fig. 6. Probe transmission rate as a function of detuning $\Delta _{p}$ for a double coupled resonator system. Bottom resonator spinning in the counterclockwise direction (with rate $\Omega _1=40$kHz) while upper resonator may or may not be spinning in the same direction. We have also incorporated the scenario when the upper resonator is not spinning. We have set $J/\kappa _{ex}=1$.

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Appendix B: Non-reciprocal probe light transmission for a wider ranger of $\Delta _p$ values

In Fig. 7 we have plotted the probe transmission spectra for a wider range of frequencies ($-125\leq \Delta _p\leq 0$) and at a higher spinning rate ($|\Omega |=100kHz$). Various scenarios of spinning directions have been considered. The main focus of this plot is the yellow highlighted region which shows the nonreciprocal probe light transmission. The magnified version of this figure has been presented in Fig. 3 and discussed there.

 

Fig. 7. Probe transmission for larger spinning rate $|\Omega |=100$kHz. The yellow highlighted regions show the possibility of optical nonreciprocal transmission with different combinations of spinning directions. Rest of the parameters are the same as used in Fig. 2.

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Appendix C: Slow light propagation for different spinning directions

In Fig. 8 we have plotted the group delay as a function of $\Delta _p$. Several different cases of spinning directions have been considered for double resonator problem. In all of these cases we note that different degrees of slow light can be acquired. As an example, we observe in the case when $\Omega _{1}<0$ and $\Omega _{2}=0$ group delay reaches almost zero value around $12$kHz and then shows oscillations on the positive vertical axis.

 

Fig. 8. The group delay of probe light as a function of spinning rate magnitude for various spin direction options. Parameters are the same as used in Fig. 2.

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Funding

Miami University College of Arts and Science start up funding; National Natural Science Foundation of China (11474087, 11774086).

References

1. L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327(4), 1050–1077 (2012). [CrossRef]  

2. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011). [CrossRef]  

3. A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011). [CrossRef]  

4. M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016). [CrossRef]  

5. C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015). [CrossRef]  

6. L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017). [CrossRef]  

7. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017). [CrossRef]  

8. A. Metelmann and A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95(1), 013837 (2017). [CrossRef]  

9. Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018). [CrossRef]  

10. X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014). [CrossRef]  

11. J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013). [CrossRef]  

12. D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$,” Phys. Rev. X 7(2), 021048 (2017). [CrossRef]  

13. A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015). [CrossRef]  

14. L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic aharonov–bohm effect,” Opt. Lett. 40(22), 5140–5143 (2015). [CrossRef]  

15. R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009). [CrossRef]  

16. R. Pant, A. Byrnes, C. G. Poulton, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic-chip-based tunable slow and fast light via stimulated brillouin scattering,” Opt. Lett. 37(5), 969–971 (2012). [CrossRef]  

17. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]  

18. N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef]  

19. A. Abdumalikov Jr, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010). [CrossRef]  

20. D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010). [CrossRef]  

21. I. M. Mirza and J. C. Schotland, “Influence of disorder on electromagnetically induced transparency in chiral waveguide quantum electrodynamics,” J. Opt. Soc. Am. B 35(5), 1149–1158 (2018). [CrossRef]  

22. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011). [CrossRef]  

23. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010). [CrossRef]  

24. A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013). [CrossRef]  

25. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010). [CrossRef]  

26. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics: nano-and micromechanical resonators interacting with light (Springer, 2014).

27. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009). [CrossRef]  

28. C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013). [CrossRef]  

29. B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014). [CrossRef]  

30. Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018). [CrossRef]  

31. I. M. Mirza, “Real-time emission spectrum of a hybrid atom-optomechanical cavity,” J. Opt. Soc. Am. B 32(8), 1604–1614 (2015). [CrossRef]  

32. I. M. Mirza, “Strong coupling optical spectra in dipole–dipole interacting optomechanical tavis- cummings models,” Opt. Lett. 41(11), 2422–2425 (2016). [CrossRef]  

33. W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017). [CrossRef]  

34. D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016). [CrossRef]  

35. J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017). [CrossRef]  

36. J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015). [CrossRef]  

37. T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-hermitian time-floquet systems,” Phys. Rev. Lett. 120(8), 087401 (2018). [CrossRef]  

38. R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013). [CrossRef]  

39. B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019). [CrossRef]  

40. H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017). [CrossRef]  

41. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018). [CrossRef]  

42. R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018). [CrossRef]  

43. H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018). [CrossRef]  

44. G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000). [CrossRef]  

45. Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010). [CrossRef]  

46. H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015). [CrossRef]  

47. R. W. Boyd, Nonlinear optics (Academic press, 2003).

48. C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985). [CrossRef]  

49. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010). [CrossRef]  

50. Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015). [CrossRef]  

51. B. Peng, Ş. K. Özdemir, J. Zhu, and L. Yang, “Photonic molecules formed by coupled hybrid resonators,” Opt. Lett. 37(16), 3435–3437 (2012). [CrossRef]  

52. F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017). [CrossRef]  

53. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012). [CrossRef]  

54. K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014). [CrossRef]  

55. F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016). [CrossRef]  

56. J. Li and Y. Wu, “Quality of photon antibunching in two cavity-waveguide arrangements on a chip,” Phys. Rev. A 98(5), 053801 (2018). [CrossRef]  

57. Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016). [CrossRef]  

58. S. Davuluri and S. Zhu, “Controlling optomechanically induced transparency through rotation,” Europhys. Lett. 112(6), 64002 (2015). [CrossRef]  

59. S. Davuluri, “Optomechanics for absolute rotation detection,” Phys. Rev. A 94(1), 013808 (2016). [CrossRef]  

60. Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018). [CrossRef]  

References

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  1. L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327(4), 1050–1077 (2012).
    [Crossref]
  2. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
    [Crossref]
  3. A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011).
    [Crossref]
  4. M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
    [Crossref]
  5. C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
    [Crossref]
  6. L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
    [Crossref]
  7. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
    [Crossref]
  8. A. Metelmann and A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95(1), 013837 (2017).
    [Crossref]
  9. Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
    [Crossref]
  10. X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
    [Crossref]
  11. J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
    [Crossref]
  12. D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
    [Crossref]
  13. A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
    [Crossref]
  14. L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic aharonov–bohm effect,” Opt. Lett. 40(22), 5140–5143 (2015).
    [Crossref]
  15. R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
    [Crossref]
  16. R. Pant, A. Byrnes, C. G. Poulton, E. Li, D.-Y. Choi, S. Madden, B. Luther-Davies, and B. J. Eggleton, “Photonic-chip-based tunable slow and fast light via stimulated brillouin scattering,” Opt. Lett. 37(5), 969–971 (2012).
    [Crossref]
  17. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
    [Crossref]
  18. N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
    [Crossref]
  19. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
    [Crossref]
  20. D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010).
    [Crossref]
  21. I. M. Mirza and J. C. Schotland, “Influence of disorder on electromagnetically induced transparency in chiral waveguide quantum electrodynamics,” J. Opt. Soc. Am. B 35(5), 1149–1158 (2018).
    [Crossref]
  22. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
    [Crossref]
  23. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
    [Crossref]
  24. A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
    [Crossref]
  25. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
    [Crossref]
  26. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics: nano-and micromechanical resonators interacting with light (Springer, 2014).
  27. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
    [Crossref]
  28. C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
    [Crossref]
  29. B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
    [Crossref]
  30. Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
    [Crossref]
  31. I. M. Mirza, “Real-time emission spectrum of a hybrid atom-optomechanical cavity,” J. Opt. Soc. Am. B 32(8), 1604–1614 (2015).
    [Crossref]
  32. I. M. Mirza, “Strong coupling optical spectra in dipole–dipole interacting optomechanical tavis- cummings models,” Opt. Lett. 41(11), 2422–2425 (2016).
    [Crossref]
  33. W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
    [Crossref]
  34. D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
    [Crossref]
  35. J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
    [Crossref]
  36. J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
    [Crossref]
  37. T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-hermitian time-floquet systems,” Phys. Rev. Lett. 120(8), 087401 (2018).
    [Crossref]
  38. R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
    [Crossref]
  39. B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
    [Crossref]
  40. H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
    [Crossref]
  41. S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
    [Crossref]
  42. R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
    [Crossref]
  43. H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018).
    [Crossref]
  44. G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000).
    [Crossref]
  45. Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010).
    [Crossref]
  46. H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
    [Crossref]
  47. R. W. Boyd, Nonlinear optics (Academic press, 2003).
  48. C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
    [Crossref]
  49. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
    [Crossref]
  50. Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
    [Crossref]
  51. B. Peng, Ş. K. Özdemir, J. Zhu, and L. Yang, “Photonic molecules formed by coupled hybrid resonators,” Opt. Lett. 37(16), 3435–3437 (2012).
    [Crossref]
  52. F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
    [Crossref]
  53. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20(7), 7672–7684 (2012).
    [Crossref]
  54. K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
    [Crossref]
  55. F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
    [Crossref]
  56. J. Li and Y. Wu, “Quality of photon antibunching in two cavity-waveguide arrangements on a chip,” Phys. Rev. A 98(5), 053801 (2018).
    [Crossref]
  57. Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
    [Crossref]
  58. S. Davuluri and S. Zhu, “Controlling optomechanically induced transparency through rotation,” Europhys. Lett. 112(6), 64002 (2015).
    [Crossref]
  59. S. Davuluri, “Optomechanics for absolute rotation detection,” Phys. Rev. A 94(1), 013808 (2016).
    [Crossref]
  60. Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
    [Crossref]

2019 (1)

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

2018 (9)

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018).
[Crossref]

J. Li and Y. Wu, “Quality of photon antibunching in two cavity-waveguide arrangements on a chip,” Phys. Rev. A 98(5), 053801 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

I. M. Mirza and J. C. Schotland, “Influence of disorder on electromagnetically induced transparency in chiral waveguide quantum electrodynamics,” J. Opt. Soc. Am. B 35(5), 1149–1158 (2018).
[Crossref]

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-hermitian time-floquet systems,” Phys. Rev. Lett. 120(8), 087401 (2018).
[Crossref]

2017 (8)

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
[Crossref]

W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
[Crossref]

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

A. Metelmann and A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95(1), 013837 (2017).
[Crossref]

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

2016 (6)

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

S. Davuluri, “Optomechanics for absolute rotation detection,” Phys. Rev. A 94(1), 013808 (2016).
[Crossref]

Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

I. M. Mirza, “Strong coupling optical spectra in dipole–dipole interacting optomechanical tavis- cummings models,” Opt. Lett. 41(11), 2422–2425 (2016).
[Crossref]

2015 (8)

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

I. M. Mirza, “Real-time emission spectrum of a hybrid atom-optomechanical cavity,” J. Opt. Soc. Am. B 32(8), 1604–1614 (2015).
[Crossref]

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
[Crossref]

L. Yuan, S. Xu, and S. Fan, “Achieving nonreciprocal unidirectional single-photon quantum transport using the photonic aharonov–bohm effect,” Opt. Lett. 40(22), 5140–5143 (2015).
[Crossref]

S. Davuluri and S. Zhu, “Controlling optomechanically induced transparency through rotation,” Europhys. Lett. 112(6), 64002 (2015).
[Crossref]

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

2014 (3)

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
[Crossref]

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

2013 (4)

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
[Crossref]

C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
[Crossref]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref]

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

2012 (4)

2011 (3)

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

2010 (6)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010).
[Crossref]

Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express 18(8), 8367–8382 (2010).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
[Crossref]

2009 (3)

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
[Crossref]

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref]

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

2005 (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

2000 (1)

G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000).
[Crossref]

1985 (1)

C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
[Crossref]

Abdumalikov, A.

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Agarwal, G. S.

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Albrecht, B.

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Alu, A.

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics: nano-and micromechanical resonators interacting with light (Springer, 2014).

Astafiev, O.

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Bai, C.-H.

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

Belotelov, V. I.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

Bo, F.

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

Boyd, R. W.

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
[Crossref]

R. W. Boyd, Nonlinear optics (Academic press, 2003).

Byrnes, A.

Carmon, T.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Chen, A.

W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
[Crossref]

Chen, B.

Chen, G.

Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Chen, Y.-F.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Cheng, Y.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Chin, J. Y.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

Cho, S. U.

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Choi, D.-Y.

Chong, Y.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

Christodoulides, D. N.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
[Crossref]

Ciuti, C.

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
[Crossref]

Clarke, J.

A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011).
[Crossref]

Clerk, A.

A. Metelmann and A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95(1), 013837 (2017).
[Crossref]

Clerk, A. A.

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
[Crossref]

Collett, M.

C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
[Crossref]

Cui, Y.

Dahan, R.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

Davuluri, S.

S. Davuluri, “Optomechanics for absolute rotation detection,” Phys. Rev. A 94(1), 013808 (2016).
[Crossref]

S. Davuluri and S. Zhu, “Controlling optomechanically induced transparency through rotation,” Europhys. Lett. 112(6), 64002 (2015).
[Crossref]

De Chiara, G.

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

Deák, L.

L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327(4), 1050–1077 (2012).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Devoret, M. H.

A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011).
[Crossref]

Dong, C.-H.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Dregely, D.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

Eggleton, B. J.

Eichenfield, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Eisfeld, A.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
[Crossref]

El-Ganainy, R.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
[Crossref]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Fan, S.

Favero, I.

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
[Crossref]

Feng, L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Fleury, R.

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-hermitian time-floquet systems,” Phys. Rev. Lett. 120(8), 087401 (2018).
[Crossref]

Floess, D.

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

Fülöp, T.

L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327(4), 1050–1077 (2012).
[Crossref]

Gao, F.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

Gao, Z.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

Gardiner, C.

C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
[Crossref]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Geng, Z.

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

Giessen, H.

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Gong, S.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Grudinin, I. S.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
[Crossref]

Gullo, N. L.

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

Guo, G.-C.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Gupta, S. D.

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
[Crossref]

Habermeier, H.-U.

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

Hafezi, M.

Hakonen, P.

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Hassan, A. U.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

Heikkilä, T. T.

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Hentschel, M.

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

Hilico, A.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

Hill, J. T.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Huang, J.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Huang, R.

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

Huang, S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Jiang, C.

Jiang, Y.

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

Jiao, J.

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

Jiao, Y.-F.

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

Jing, H.

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

Junge, C.

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Kamal, A.

A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011).
[Crossref]

Kástel, J.

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Kippenberg, T. J.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics: nano-and micromechanical resonators interacting with light (Springer, 2014).

Kligerman, Y.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

Koutserimpas, T. T.

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-hermitian time-floquet systems,” Phys. Rev. Lett. 120(8), 087401 (2018).
[Crossref]

Kronwald, A.

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref]

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Lee, H.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
[Crossref]

Levy, M.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
[Crossref]

Li, B.

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

Li, E.

Li, J.

J. Li and Y. Wu, “Quality of photon antibunching in two cavity-waveguide arrangements on a chip,” Phys. Rev. A 98(5), 053801 (2018).
[Crossref]

Li, L.

W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
[Crossref]

Li, Q.

Li, X.

Li, Z.

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

Liao, J.-Q.

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

Lin, G.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Lin, Q.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Lin, X.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

Lipson, M.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref]

Liu, H.

Liu, N.

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Liu, X.

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
[Crossref]

Liu, Y.-L.

Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
[Crossref]

Liu, Y.-x.

Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
[Crossref]

Lodahl, P.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

Lu, G.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Lu, M.-H.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Lu, T.-X.

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

Lü, H.

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018).
[Crossref]

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

Lü, X.-Y.

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

Luther-Davies, B.

Maayani, S.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

Madden, S.

Mahmoodian, S.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

Malykin, G. B.

G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000).
[Crossref]

Manipatruni, S.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Marquardt, F.

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics: nano-and micromechanical resonators interacting with light (Springer, 2014).

Massel, F.

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Metelmann, A.

A. Metelmann and A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95(1), 013837 (2017).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
[Crossref]

Miranowicz, A.

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

Miri, M.-A.

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

Mirza, I. M.

Mitsch, R.

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Monifi, F.

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

Moses, E.

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

Nakamura, Y.

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Nie, W.

W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
[Crossref]

Niu, Y.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Nori, F.

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

H. Jing, H. Lü, S. Özdemir, T. Carmon, and F. Nori, “Nanoparticle sensing with a spinning resonator,” Optica 5(11), 1424–1430 (2018).
[Crossref]

Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

O’Shea, D.

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Özdemir, S.

Özdemir, S. K.

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

B. Peng, Ş. K. Özdemir, J. Zhu, and L. Yang, “Photonic molecules formed by coupled hybrid resonators,” Opt. Lett. 37(16), 3435–3437 (2012).
[Crossref]

Painter, O.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
[Crossref]

Palma, G. M.

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

Pant, R.

Pashkin, Y. A.

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Paternostro, M.

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

Peng, B.

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

B. Peng, Ş. K. Özdemir, J. Zhu, and L. Yang, “Photonic molecules formed by coupled hybrid resonators,” Opt. Lett. 37(16), 3435–3437 (2012).
[Crossref]

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Pichler, H.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

Pirkkalainen, J.-M.

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Poulton, C. G.

Qiu, M.

Rabl, P.

Rauschenbeutel, A.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Restrepo, J.

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
[Crossref]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Robinson, J. T.

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref]

Rogers, B.

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

Ruesink, F.

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Sayrin, C.

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Scherer, A.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Scheucher, M.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Schneeweiss, P.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Schotland, J. C.

Shen, Z.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Shi, X.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

Sillanpää, M.

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Sørensen, A. S.

D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010).
[Crossref]

Steinle, T.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

Stobbe, S.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

Stritzker, B.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

Su, Y.

Sun, F.-W.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Tian, L.

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

Tikhodeev, S. G.

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

Tsai, J.

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Tuorila, J.

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Twamley, J.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Vahala, K. J.

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
[Crossref]

Verhagen, E.

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

Volz, J.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Wang, D.-Y.

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

Wang, G.-Z.

Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
[Crossref]

Wang, H.-F.

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

Wang, T.

Wang, Y.-Z.

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

Wehlus, T.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Weiss, T.

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Will, E.

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

Winger, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

Witthaut, D.

D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010).
[Crossref]

Wu, Y.

J. Li and Y. Wu, “Quality of photon antibunching in two cavity-waveguide arrangements on a chip,” Phys. Rev. A 98(5), 053801 (2018).
[Crossref]

Xia, K.

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

Xu, J.

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

Xu, S.

Xu, X.

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

Xu, Y.-L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Yan, M.

Yang, L.

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

B. Peng, Ş. K. Özdemir, J. Zhu, and L. Yang, “Photonic molecules formed by coupled hybrid resonators,” Opt. Lett. 37(16), 3435–3437 (2012).
[Crossref]

Yang, Z.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

Yuan, L.

Zagoskin, A. M.

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Zeng, W.

W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
[Crossref]

Zhang, B.

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

Zhang, G.

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

Zhang, J.

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

Zhang, S.

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

Zhang, Y.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Zhu, A.-D.

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

Zhu, J.

Zhu, S.

S. Davuluri and S. Zhu, “Controlling optomechanically induced transparency through rotation,” Europhys. Lett. 112(6), 64002 (2015).
[Crossref]

Zoller, P.

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

Zou, C.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Zou, X.-B.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

Ann. Phys. (1)

L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327(4), 1050–1077 (2012).
[Crossref]

Appl. Phys. Lett. (1)

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103(16), 161105 (2013).
[Crossref]

Europhys. Lett. (1)

S. Davuluri and S. Zhu, “Controlling optomechanically induced transparency through rotation,” Europhys. Lett. 112(6), 64002 (2015).
[Crossref]

J. Mod. Opt. (1)

R. W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56(18-19), 1908–1915 (2009).
[Crossref]

J. Opt. Soc. Am. B (2)

Nat. Commun. (4)

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9(1), 1797 (2018).
[Crossref]

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film faraday rotation,” Nat. Commun. 4(1), 1599 (2013).
[Crossref]

F. Ruesink, M.-A. Miri, A. Alu, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7(1), 13662 (2016).
[Crossref]

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. Hakonen, and M. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

Nat. Mater. (1)

N. Liu, L. Langguth, T. Weiss, J. Kástel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the drude damping limit,” Nat. Mater. 8(9), 758–762 (2009).
[Crossref]

Nat. Phys. (1)

A. Kamal, J. Clarke, and M. H. Devoret, “Noiseless non-reciprocity in a parametric active device,” Nat. Phys. 7(4), 311–315 (2011).
[Crossref]

Nature (3)

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472(7341), 69–73 (2011).
[Crossref]

S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, “Flying couplers above spinning resonators generate irreversible refraction,” Nature 558(7711), 569–572 (2018).
[Crossref]

New J. Phys. (1)

D. Witthaut and A. S. Sørensen, “Photon scattering by a three-level emitter in a one-dimensional waveguide,” New J. Phys. 12(4), 043052 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Optica (1)

Photonics Res. (2)

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367–371 (2017).
[Crossref]

Phys. Rev. A (11)

K. Xia, G. Lu, G. Lin, Y. Cheng, Y. Niu, S. Gong, and J. Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Phys. Rev. A 90(4), 043802 (2014).
[Crossref]

C. Gardiner and M. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
[Crossref]

J. Li and Y. Wu, “Quality of photon antibunching in two cavity-waveguide arrangements on a chip,” Phys. Rev. A 98(5), 053801 (2018).
[Crossref]

Y.-L. Liu, G.-Z. Wang, Y.-x. Liu, and F. Nori, “Mode coupling and photon antibunching in a bimodal cavity containing a dipole quantum emitter,” Phys. Rev. A 93(1), 013856 (2016).
[Crossref]

S. Davuluri, “Optomechanics for absolute rotation detection,” Phys. Rev. A 94(1), 013808 (2016).
[Crossref]

X. Liu, S. D. Gupta, and G. S. Agarwal, “Regularization of the spectral singularity in $\mathcal {PT}$PT-symmetric systems by all-order nonlinearities: Nonreciprocity and optical isolation,” Phys. Rev. A 89(1), 013824 (2014).
[Crossref]

A. Metelmann and A. Clerk, “Nonreciprocal quantum interactions and devices via autonomous feedforward,” Phys. Rev. A 95(1), 013837 (2017).
[Crossref]

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96(1), 013808 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
[Crossref]

Phys. Rev. Appl. (1)

Y. Jiang, S. Maayani, T. Carmon, F. Nori, and H. Jing, “Nonreciprocal phonon laser,” Phys. Rev. Appl. 10(6), 064037 (2018).
[Crossref]

Phys. Rev. Lett. (7)

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104(8), 083901 (2010).
[Crossref]

Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, “Topological acoustics,” Phys. Rev. Lett. 114(11), 114301 (2015).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-hermitian time-floquet systems,” Phys. Rev. Lett. 120(8), 087401 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref]

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102(21), 213903 (2009).
[Crossref]

A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Y. A. Pashkin, Y. Nakamura, and J. Tsai, “Electromagnetically induced transparency on a single artificial atom,” Phys. Rev. Lett. 104(19), 193601 (2010).
[Crossref]

Phys. Rev. X (3)

D. Floess, M. Hentschel, T. Weiss, H.-U. Habermeier, J. Jiao, S. G. Tikhodeev, and H. Giessen, “Plasmonic analog of electromagnetically induced absorption leads to giant thin film faraday rotation of 14$^0$0,” Phys. Rev. X 7(2), 021048 (2017).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5(2), 021025 (2015).
[Crossref]

C. Sayrin, C. Junge, R. Mitsch, B. Albrecht, D. O’Shea, P. Schneeweiss, J. Volz, and A. Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Phys. Rev. X 5(4), 041036 (2015).
[Crossref]

Phys.-Usp. (1)

G. B. Malykin, “The sagnac effect: correct and incorrect explanations,” Phys.-Usp. 43(12), 1229–1252 (2000).
[Crossref]

Quantum Meas. Quantum Metrol. (1)

B. Rogers, N. L. Gullo, G. De Chiara, G. M. Palma, and M. Paternostro, “Hybrid optomechanics for quantum technologies,” Quantum Meas. Quantum Metrol. 2(1), 11–43 (2014).
[Crossref]

Rev. Mod. Phys. (1)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Sci. Rep. (4)

W. Zeng, W. Nie, L. Li, and A. Chen, “Ground-state cooling of a mechanical oscillator in a hybrid optomechanical system including an atomic ensemble,” Sci. Rep. 7(1), 17258 (2017).
[Crossref]

D.-Y. Wang, C.-H. Bai, H.-F. Wang, A.-D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. 6(1), 24421 (2016).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
[Crossref]

F. Bo, Ş. K. Özdemir, F. Monifi, J. Zhang, G. Zhang, J. Xu, and L. Yang, “Controllable oscillatory lateral coupling in a waveguide-microdisk-resonator system,” Sci. Rep. 7(1), 8045 (2017).
[Crossref]

Science (3)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

M. Scheucher, A. Hilico, E. Will, J. Volz, and A. Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354(6319), 1577–1580 (2016).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref]

Other (2)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics: nano-and micromechanical resonators interacting with light (Springer, 2014).

R. W. Boyd, Nonlinear optics (Academic press, 2003).

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Figures (8)

Fig. 1.
Fig. 1. Scheme of the system consisting of two series-coupled spinning optomechanical resonators.
Fig. 2.
Fig. 2. Probe transmission rate as a function of detuning $\Delta _{p}$ for a single resonator and a double coupled resonator system. (a) Absence of spin in all cases and increasing resonator-resonator coupling $J$ for the two-resonator case. (b) Bottom resonator spinning in the clockwise direction (with rate $\Omega _1=40$kHz) while upper resonator may or may not be spinning in the same direction. We have also incorporated the scenario when the upper resonator is not spinning. Note that in this and later plots, for $N=2$ we have assumed $J/\kappa _{ex}=1$ and $\vert \Omega _{1}\vert =\vert \Omega _{2}\vert =\vert \Omega \vert$ unless stated otherwise.
Fig. 3.
Fig. 3. Probe transmission for larger spinning rate $|\Omega |=100$kHz. We have focused on a frequency region where non-reciprocal light transmission is evident. Rest of the parameters are the same as used in Fig. 2.
Fig. 4.
Fig. 4. The group delay of probe light as a function of spinning rate magnitude for (a) a single and (b) a double spinning resonator. In the double resonator case, we have only plotted the unique situation where for a range of $|\Omega |$ fast light can be achieved. For comparison, we have plotted the single spinning resonator case in (a) where one can only achieve slow light for all $|\Omega |$ value. Parameters are the same as used in Fig. 2.
Fig. 5.
Fig. 5. Off-resonance transmission enhancement factor as a function of spinning rate for the double resonator case in which one can achieve the fast light. For comparison we have plotted the corresponding single resonator enhancement factor as well. Parameters are the same as used in Fig. 2.
Fig. 6.
Fig. 6. Probe transmission rate as a function of detuning $\Delta _{p}$ for a double coupled resonator system. Bottom resonator spinning in the counterclockwise direction (with rate $\Omega _1=40$kHz) while upper resonator may or may not be spinning in the same direction. We have also incorporated the scenario when the upper resonator is not spinning. We have set $J/\kappa _{ex}=1$.
Fig. 7.
Fig. 7. Probe transmission for larger spinning rate $|\Omega |=100$kHz. The yellow highlighted regions show the possibility of optical nonreciprocal transmission with different combinations of spinning directions. Rest of the parameters are the same as used in Fig. 2.
Fig. 8.
Fig. 8. The group delay of probe light as a function of spinning rate magnitude for various spin direction options. Parameters are the same as used in Fig. 2.

Equations (16)

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H ^ = H ^ 0 + H ^ i n t + H ^ d r , H ^ 0 = j = 1 2 ( Δ c j a ^ j a ^ j + p ^ j 2 2 m j + 1 2 m j ω m j 2 x ^ j 2 + p ^ j θ 2 2 m j r j 2 ) , H ^ d r = j = 1 2 i κ e x ( ε l j a ^ j + ε p 1 a ^ 1 e i ( ω p ω l j ) t H . c . ) , H ^ i n t = j = 1 2 ξ j x ^ j a ^ j a ^ j + J 1 , 2 ( a 1 a ^ 2 + a ^ 2 a ^ 1 ) .
ω c j ω c j + Δ s a g j , where Δ s a g j := n j r j Ω j ω c j c ( 1 1 n j 2 λ j n j d n j d λ j ) .
d a ^ 1 ( t ) d t = i ( Δ c 1 ξ 1 x ^ 1 i β 1 ) a ^ 1 + κ e x ( ε l 1 + ε p 1 e i η 1 t ) i J 1 , 2 a ^ 2 , d a ^ 2 ( t ) d t = i ( Δ c 2 ξ 2 x ^ 2 i β 2 ) a ^ 2 + κ e x ε l 2 i J 1 , 2 a ^ 1 , d 2 x ^ 1 ( t ) d t 2 = ξ 1 m 1 a ^ 1 a ^ 1 ω m 1 2 x ^ 1 + p ^ 1 θ 2 m 1 2 r 1 3 γ m 1 d x ^ 1 d t , d 2 x ^ 2 ( t ) d t 2 = ξ 2 m 2 a ^ 2 a ^ 2 ω m 2 2 x ^ 2 + p ^ 2 θ 2 m 2 2 r 2 3 γ m 2 d x ^ 2 d t , d θ ^ 1 ( t ) d t = p ^ 1 θ m 1 r 1 2 , d θ ^ 2 ( t ) d t = p ^ 2 θ m 2 r 2 2 , d p ^ j θ ( t ) d t = 0 , d p ^ 2 θ ( t ) d t = 0.
d a ^ 1 ( t ) d t = i ( Δ c 1 i β 1 ) a ^ 1 + i ξ 1 x ^ 1 a ^ 1 + κ e x ( ε l 1 + ε p 1 e i η 1 t ) i J 1 , 2 a ^ 2 , d a ^ 2 ( t ) d t = i ( Δ c 2 i β 2 ) a ^ 2 + i ξ 2 x ^ 1 a ^ 2 + κ e x ε l 2 i J 1 , 2 a ^ 1 , d 2 x ^ 1 ( t ) d t 2 = ( ω m 1 2 + γ m 1 d d t ) x ^ 1 + ξ 1 m 1 a ^ 1 a ^ 1 + p ^ 1 θ 2 m 1 2 r 1 3 , d 2 x ^ 2 ( t ) d t 2 = ( ω m 2 2 + γ m 2 d d t ) x ^ 2 + ξ 2 m 2 a ^ 2 a ^ 2 + p ^ 2 θ 2 m 2 2 r 2 3 , d θ ^ 1 ( t ) d t = p ^ 1 θ m 1 r 1 2 , d θ ^ 2 ( t ) d t = p ^ 2 θ m 2 r 2 2 , d p ^ 1 θ ( t ) d t = 0 , d p ^ 2 θ ( t ) d t = 0.
a ^ 1 a 1 + δ a 1 e i η 1 t + δ a + 1 e i η 1 t , a ^ 2 a 2 + δ a 2 e i η 2 t + δ a + 2 e i η 2 t , x ^ 1 x 1 + δ x 1 e i η 1 t + δ x 1 e i η 1 t , x ^ 2 x 2 + δ x 2 e i η 2 t + δ x 2 e i η 2 t .
a 1 = ( κ e x ε l i J 1 , 2 a 2 ) β 1 + i Δ c 1 i ξ 1 x 1 , a 2 = ( κ e x ε l i J 2 , 1 a 1 ) β 2 + i Δ c 2 i ξ 2 x 2 , x 1 = ( ξ 1 | a 1 | 2 + m 1 r 1 Ω 1 2 ) m 1 ω m 1 2 , x 2 = ( ξ 2 | a 2 | 2 + m 2 r 2 Ω 2 2 ) m 2 ω m 2 2 .
δ a 1 ( β 1 + i Δ c 1 i ξ 1 x 1 i η 1 ) i ξ 1 a 1 δ x 1 = κ e x ε p 1 i J 1 , 2 δ a 2 , δ a 2 ( β 2 + i Δ c 2 i ξ 2 x 2 i η 2 ) i ξ 2 a 2 δ x 2 = i J 1 , 2 δ a 1 , δ a + 1 ( β 1 i Δ c 1 + i ξ 1 x 1 i η 1 ) + i ξ 1 a 1 δ x 1 = i J 1 , 2 δ a 2 , δ a + 2 ( β 2 i Δ c 2 + i ξ 2 x 2 i η 2 ) + i ξ 2 a 2 δ x 2 = i J 1 , 2 δ a 1 , m 1 ( ω m 1 2 η 1 i η 1 γ m 1 ) δ x 1 = ξ 1 ( a 1 δ a 1 + a 1 δ a + 1 ) , m 2 ( ω m 2 2 η 2 i η 2 γ m 2 ) δ x 2 = ξ 2 ( a 2 δ a 2 + a 2 δ a + 2 ) .
T t p 2 = < a ^ o u t a ^ o u t > < a ^ i n a ^ i n > .
a ^ o u t = a ^ i n κ e x δ a 1 = ε p 1 κ e x δ a 1 .
T = | 1 κ e x ε p 1 δ a 1 | 2 .
δ a 1 = κ e x ε p 1 { i ξ 2 | a | 2 + m ( β ~ ) Γ m } i ξ 2 | a | 2 ( β ~ ) ( β ~ ) { i ξ 2 | a | 2 + m ( β ~ ) Γ m } ,
a = κ e x ε l β + i Δ c i ξ x , x = ( ξ | a | 2 + m r Ω 2 ) m ω m 2 .
τ g = d arg ( t p ) d Δ p
G . D . = τ g ( Ω 1 0 , Ω 2 0 ) τ g ( Ω 1 = 0 , Ω 2 = 0 ) | Δ p 1.
E . F . = T ( Ω 1 0 , Ω 2 0 ) T ( Ω 1 = 0 , Ω 2 = 0 ) | Δ p 1.
E . F . = T ( Ω 0 ) T ( Ω = 0 ) | Δ p 1.

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