## 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 [1–3]. 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 [6–8]. 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) [23–25].

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) [29–32] 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 [37–39]. 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

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

## 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

#### 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

#### 3.3 Probe field transmission rate

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

## 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

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]

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).

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

Consequently one can also define the group delay enhancement factor $G.D.$ for two-resonators as#### 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

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.

## 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.

## 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.

## 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.

## Funding

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

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