1. Introduction

Quantum coherence, a main characteristic of quantum physics distinguishing from classical physics [1], promotes the deep understanding of various quantum phenomena and it has important practical applications in many fields such as quantum information processing [2,3], quantum thermodynamics [4,5] and biological science [6]. In quantum optics, the correlation functions were used to characterize quantum coherence [7,8] and achieved great success in understanding it, but they did not rigorously quantify quantum coherence itself. In recent years, the quantification and measurement of quantum coherence aroused more and more interest. In 2014, Baumgratz et al. presented a seminal work to quantify quantum coherence in a rigorous framework by adopting the viewpoint of coherence as a physical resource [9]. Then, Napoli et al. introduced an operational and observable measure by defining the robustness of coherence [10]. Later, Streltsov et al. proposed to quantify the quantum coherence by using entanglement via incoherent operations [11]. In 2016, Xu established a theory for quantifying the continuous variable quantum coherence based on the relative entropy [12]. Using such a method, the macroscopic quantum coherence was widely studied in recent years. As a prominent example, Zheng et al. investigated the macroscopic quantum coherence of a driven cavity optomechanical system with the optical and the mechanical dissipations, they also gave the relationship between coherence and quantum mutual information [13]. Very recently, Li et al. studied quantum coherence in a cavity optomechanical system with two mechanical oscillators and analyzed coherence transfer between the optical cavity and oscillators [14]. In addition, quantum coherence in other macroscopic systems has also been studied, such as quadratically coupled optomechanical system [15], Bose-Einstein condensates [16,17], graphene sheet [18] and Josephson junction [19], etc.

On the other hand, the cavity optomechanics with whispering-gallery-mode (WGM) resonators has been well developed and many remarkable research results have been achieved [20,21]. In particular, cavity optomechanics with spinning resonators has attracted much attention recently. In such a spinning device, many novel physical phenomena induced by the Sagnac-Fizeau shifts are experimentally verified or theoretically predicted. For instance, optomechanically induced transparency (OMIT) was studied in a spinning microresonator, the results indicate the rotation-induced Sagnac frequency shift can strongly affect the transmission rate and the group delay of the signal, leading to the emergence of the nonreciprocal optical sidebands [22]. The second-order OMIT sidebands [23] were also theoretically investigated in such a spinning system. In a recent experiment, Carmon et al. realized optical nonreciprocity through a spinning resonator with two optical modes propagating in opposite directions [24]. The resonance frequencies of the two optical modes are different because they experience the opposite Sagnac-Fizeau shift. The optical nonreciprocity, attributed to the Sagnac effect in this spinning system, was used to realize the optical isolator with up to 99.6% isolation. Jiao et al. studied nonreciprocal entanglement and found that the quantum nonreciprocity can protect quantum entanglement against backscattering [25]. Subsequently, the WGM cavity optomechanics with spinning resonator has been suggested to realize the nonreciprocal phonon lasers [26,27], nonreciprocal photon blockades [28,29], anti-PT symmetry and its spontaneous breaking without resorting to the nonlinearity [30], nanoparticle sensing [31] and nonreciprocal chaos [32]. Nonreciprocal optical solitons in a spinning nonlinear resonator [33] and the optical nonreciprocity and slow light in coupled spinning optomechanical resonators [34] were also studied.

In recent years, based on the reciprocity breaking caused by time-reversal asymmetry, the optical nonreciprocal physics has made great progress in both its realization and applications. In the early stage, the conventional method to achieve the optical nonreciprocity usually relies on the magneto-optical effect of bias magnetic field [35,36]. But such a method has the disadvantages of bulkiness and high loss of magnetic devices, which is not beneficial to integrate on the circuit. To develop magnet-free optical nonreciprocity, many different physical effects such as optical nonlinearity [37–40], optomechanics [41–45], refractive index modulation [46–48], thermal motion of hot atoms [49,50] and Sagnac effect [23–25] have been utilized in recent years.

In this paper, we investigate a spinning optomechanical resonator system which supports a mechanical mode and two optical modes with opposite directions: clockwise (CW) and counterclockwise (CCW), and one of the optical modes is driven by an external laser field. The one-, two- and three-mode quantum coherences are calculated by considering the influence of the various parameters including the driven laser power, the coupling strength of the optical modes, the rotating angular velocity, the optical and the mechanical decay rates. In particular, we study the nonreciprocity of the quantum coherences from the Sagnac effect of the rotating resonator.

This paper is organized as follows. In Sec. 2 the theoretical model for a spinning optomechanical resonator system is introduced and the Hamiltonian of the system is given. In Sec. 3, the Heisenberg-Langevin equations and their steady-state solutions are derived, then the linearization of the Heisenberg-Langevin equations by using the standard method yields the drift matrix $A$. Based on the drift matrix $A$, we derive the one-, two- and three-mode quantum coherences. In Sec. 4, the one-, two- and three-mode quantum coherences are numerically calculated under the influence of various physical parameters. And the nonreciprocal quantum coherences are investigated finally. The section 5 gives an explanation of the experimental feasibility and a brief conclusion.

2. Model and Hamiltonian

To set the stage, we consider a spinning whispering-gallery-mode resonator with both CW and CCW optical modes. The resonator is supposed to support a mechanical radial breathing mode which is induced by radiation pressure, it is driven by a strong pump laser field with frequency $\omega _{l}$ via the evanescent field of a nearby tapered fiber [51], as shown in Fig. 1. Here we first assume that the resonator rotates counterclockwise with an angular velocity $\Omega$, then the correction frequencies of the CW and the CCW modes under Sagnac effect are $\omega _{\textrm{cw} }=\omega _{\textrm{0}}+|\Delta _{\textrm{s}}|$ and $\omega _{\textrm{ccw} }=\omega _{\textrm{0}}-|\Delta _{\textrm{s}}|$, respectively, here $\omega _{ \textrm{0}}$ is the resonance frequency of the resonator and $\Delta _{\textrm{s}}=\pm \Omega \frac {nR\omega _{\textrm{0}}}{c}(1-\frac {1}{n^{2}}-\frac {\lambda }{n}\frac {dn}{ d\lambda })\cong \pm \Omega \frac {nR\omega _{\textrm{0}}}{c}(1-\frac {1}{n^{2}} )$ is Sagnac-Fizeau shift, in which $n$ is the refractive index of the material, $R$ is the radius of the resonator, $c$ is the speed of light in vacuum and $\lambda$ is the wavelength of light in vacuum. The dispersion term $dn/d\lambda$ describes the relativistic origin of Sagnac effect which is relatively small and therefore can be ignored [24,52].

 

Fig. 1. Schematic of the quantum coherence for a spinning WGM resonator system: (a) The resonator spins at an angular velocity of $\Omega$ in a fixed CCW direction. By driving the system from the left-hand (right-hand) side, the light from the tapered fiber will experience a Sagnac-Fizeau shift in the resonator due to rotation. (b) Interactions among mechanical mode $b$ and optical modes with loss, one of the optical modes being driven by laser field.

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The Hamiltonian of the system in a rotating frame with respect to the frequency $\omega _{l}$ of driving laser is of the form

3. Dynamical analysis and macroscopic quantum coherence

The dynamics of the system can be described by a set of nonlinear Langevin equations, which are the Heisenberg equations derived from Eq. (1) with the corresponding dissipation term and fluctuation term added [53]:

The annihilation operators $\hat {a}_{\textrm{cw}}^{\textrm{in}}$ and $\hat {a}_{ \textrm{ccw}}^{\textrm{in}}$ comply with the time-domain correlation functions [54,55]

We can get the steady-state value of the coupled optomechanical system from Eq. (2) as follows

Now we can linearize the dynamics of coupled optomechanical system by using a standard linearization method, in which each Heisenberg operator is expanded into its steady-state value plus a small fluctuation with zero-mean value [55], i.e., $\hat {q}=\bar {q}+\delta \hat {q}$, $\hat {p}=\bar {p }+\delta \hat {p}$, $\hat {a}_{j}=\bar {a}_{j}+\delta \hat {a}_{j}$ $(j=$cw$,$ccw $)$, under the action of strong driving, we can safely ignore the higher-order nonlinear terms and obtain the following linearized Langevin equations by defining the optical quadrature operators $\hat {X} _{j}=(\hat {a}_{j}+\hat {a}_{j}^{\dagger })/\sqrt {2}$, $\hat {Y}_{j}=(\hat {a}_{j}- \hat {a}_{j}^{\dagger })/\sqrt {2}i$ $(j=$cw$,$ccw$)$, the optical quadrature fluctuation operators $\delta \hat {X}_{j}=(\delta \hat {a}_{j}+\delta \hat {a} _{j}^{\dagger })/\sqrt {2}$, $\delta \hat {Y}_{j}=(\delta \hat {a} _{j}-\delta \hat {a}_{j}^{\dagger })/\sqrt {2}i$ and their corresponding noise operators $\delta \hat {X}_{j}^{i\textrm{n}}=(\delta \hat {a}_{j}^{ \textrm{in}}+\delta \hat {a}_{j}^{\textrm{in}^{\dagger }})/\sqrt {2}$, $\delta \hat {Y}_{j}^{ \textrm{in}}=(\delta \hat {a}_{j}^{\textrm{in}}-\delta \hat {a}_{j}^{\textrm{in}^{\dagger }})/\sqrt {2} i$.

The linearized Langevin Eqs. (7) can be rewritten in a compact matrix form as follows

According to the Routh-Hurwitz criterion [57], only when the real parts of all the eigenvalues of the drift matrix $A$ are negative can the coupled system reach its steady state. Adopted the parameters in this paper, numerical calculation is carried out to verify that the system meets the stability requirements. More details are provided in the Appendix. On the other hand, the maximal value of $\Omega =30$ kHz used in this paper is largely less than the mechanical limit for the angular velocity posed, $2.8$ MHz, in the supplementary material of Ref. [25].

Since the dynamics of the optomechanical system is approximately linear, the system with an initial Gaussian state always remains in Gaussian state [13]. According to Ref. [12], a Gaussian state $\rho$ can be fully described by the first and the second moments where the first moment is defined as $\vec {d}=(d_{x},d_{y})$ with $d_{x}$ and $d_{y}$ corresponding, respectively, to the steady-state values of the optical quadrature operators $\hat {X}_{j}$ and $\hat {Y}_{j}$ for the optical mode ${j}$ and the dimensionless position and momentum for the mechanical mode, which can be obtained from Eq. (6), and the second moment is characterized by a $6\times 6$ covariance matrix $V$ with its elements defined as $V_{\alpha \beta }=\frac {1}{2}\langle u_{\alpha }(t)u_{\beta }(t^{\prime })+u_{\beta }(t^{\prime })u_{\alpha }(t)\rangle \; (\alpha,\beta =1,2,\ldots,6)$ .

When the system satisfies the stability condition, the steady-state covariance matrix $V$ can be achieved by solving the Lyapunov equation [58]

We solve Eq. (10) and write the covariance matrix $V$ in the form of block matrix

Now we reduce the covariance matrix $V$ to a $2\times 2$ submatrix $V_{jk}$ $(j,k=$cw$,$ccw$,$m$)$

Following Ref. [13], the quantum coherence of one-mode Gaussian state $\rho (L_{j},\vec {d})$ $(j=$cw$,$ccw$,$m$)$ can be calculated as

Similarly, the quantum coherence of a two-mode Gaussian state $\rho (V_{jk},\vec {d})$ $(j,k=$cw$,$ccw$,$m$)$ can be calculated as

By using Eq. (13) and Eq. (15) we can calculate various one- and two-mode quantum coherences which will be shown in the next section.

Actually, by using the covariance matrix $V$ defined in Eq. (11) we can also calculate the three-mode quantum coherence as follows

4. Numerical results of quantum coherences

In this section we numerically calculate various quantum coherences defined above and discuss the influence of several physical quantities, including the input laser power $P_{l}$, the coupling strength of the optical modes $J$, the decay rate of optical fields $\gamma _{\textrm{a}}$, the decay rate of the mecanicial mode $\gamma _{\textrm{m}}$ and the angular velocity of the mechanical resonator $\Omega$, on them, and we discuss the nonreciprocity of the various quantum coherences.

4.1 Quantum coherence of the optomechanical system

For the further numerical calculation, the parameters adopted in this work are based on the state-of-art experiments reported in Ref. [24,60] in which the power and the wavelength in vacuum of input light are, respectively, $P_{l}=0.1$ W, $\lambda =1.55\; \mu$m, the resonance frequency of the cavity $\omega _{\textrm{0}}=3.87\pi \times 10^{14}$ Hz, the cavity quality factor $Q=3.2\times 10^{7}$, the optical decay rate $\gamma _{ \textrm{a}}=\omega _{\textrm{0}}/Q$, the mass, the radius, the refractive index of the material, the frequency of the mechanical resonator and the decay rate of the mechanical mode are respectively $m=20\; \mu$g, $R=0.22$ mm, $n=1.48$, $\omega _{m}=63$ MHz and $\gamma _{\textrm{m}}=5.2$ kHz, the optical coupling strength $J/\gamma _{\textrm{a}}=0.5$, the angular velocity of the mechanical resonator rotating around its axis $\Omega =10$ kHz and the temperature $T=10$ K. Unless otherwise specified, the above parameters remain unchanged.

Let us now explore the one-mode quantum coherences $C_{\textrm{cw}},C_{\textrm{ccw}},C_{\textrm{m}}$, the two-mode quantum coherences $C_{\textrm{cw,ccw}},C_{ \textrm{cw,m}},C_{\textrm{ccw,m}}$ and the three-mode quantum coherence $C_{ \textrm{tot}}$ as a function of the input laser power $P_{l}$, which are shown in Fig. 2. It is apparent that all the quantum coherences become larger with the increase of input laser power.

 

Fig. 2. The quantum coherences versus the input laser power $P_{l}$, (a) the one-mode quantum coherences and (b) the two- and the three-mode quantum coherences.

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For $P_{l}=0$, we have all the quantum coherences equal to 0, indicating that the building of each quantum coherence depends on the input laser power. In fact, from $\varepsilon _{l}=\sqrt {2\gamma _{\textrm{a} }P_{l}/\hbar \omega _{l}}$ and Eq. (6), we can find that when $P_{l}=0$, $\bar {a}_{\textrm{cw}}$, $\bar {a}_{\textrm{ccw}}$ and $\bar {q}$ are all equal to 0, therefore all the quantum coherences are equal to zero. For $P_{l}\neq 0$, Fig. 2(a) shows that the two optical quantum coherences $C_{ \textrm{cw}}$ and $C_{\textrm{ccw}}$ increase sharply to a certain larger value, then they increase gently as $P_{l}$ increases, while the mechanical quantum coherence $C_{\textrm{m}}$ increases slower than $C_{\textrm{cw}}$ and $C_{\textrm{ccw}}$. Besides, $C_{\textrm{m}}$ is always smaller than $C_{\textrm{cw}}$ and $C_{\textrm{ccw}}$. This can be understood if one considering the environment of the optical fields is supposed to be a vacuum noise, while the environment of the mechanical resonator is a thermal noise. The thermal noise inhibits the increase of the mechanical quantum coherence $C_{\textrm{m} }$ more seriously, therefore the value of $C_{\textrm{m}}$ increases slower and it is always smaller than that of $C_{\textrm{cw}}$ and $C_{\textrm{ccw}}$ [13].

In Fig. 2(b), the numerical results indicate that the three two-mode quantum coherences meet a relation $C_{\textrm{cw,ccw}}>$ $C_{\textrm{cw,m}}$ > $C_{\textrm{ccw,m}}$. In the following we would like to give our explanation about this relation. First of all, one should bear in mind that the driving laser directly pumps the CW mode, thus $C_{\textrm{cw,ccw}}$ and $C_{\textrm{cw,m}}$ are directly related to the CW mode and increase more significantly than $C_{\textrm{ccw,m}}$. Secondly, we compare $C_{\textrm{cw,ccw} }$ with $C_{\textrm{cw,m}}$. As well known, the two-mode quantum coherence contains the one-mode quantum coherences and the coherence difference [13] as defined $C_{\textrm{cw,ccw}}=\Delta C_{\textrm{cw,ccw}}+C_{\textrm{cw}}+C_{ \textrm{ccw}}$ and $C_{\textrm{cw,m}}=\Delta C_{\textrm{cw,m}}+C_{\textrm{cw}}+C_{ \textrm{m}}$, where $\Delta C_{j,k}$ is the corresponding coherence difference. Note that among all the components of the two-mode quantum coherence the coherence difference $\Delta C_{j,k}$ is very small compared to the one-mode quantum coherences, for instance, when $P_{l}=0.1$ W, $C_{\textrm{cw,ccw}}\approx 63.1$, $C_{\textrm{cw}}\approx 32.8$, $C_{\textrm{ccw}}\approx 29.7$, then we can obtain $\Delta C_{\textrm{cw,ccw}}$ $\approx 0.6$ which is much smaller than both $C_{\textrm{cw}}$ and $C_{\textrm{ccw}}$. So the value of $C_{\textrm{cw,ccw} }$ is close to $C_{\textrm{cw}}+C_{\textrm{ccw}}$. Similarly, the value of $C_{ \textrm{cw,m}}$ is close to $C_{\textrm{cw}}+C_{\textrm{m}}$. Considering $C_{ \textrm{ccw}}>C_{\textrm{m}}$, thus we have $C_{\textrm{cw,ccw}}$ > $C_{\textrm{cw,m}}$. Moreover, Fig. 2(b) indicates that the three-mode coherence $C_{\textrm{tot}}$ is larger than each of the two-mode coherences $C_{\textrm{cw,ccw}},C_{\textrm{cw,m}},C_{\textrm{ccw,m}}$, which can be understood due to the $C_{\textrm{tot}}$ containing three one-mode quantum coherences and coherence difference [61]. Finally, we have $C_{\textrm{tot}}>C_{\textrm{cw,ccw}}$ > $C_{\textrm{cw,m} }$ > $C_{\textrm{ccw,m}}$ for a certain value of $P_{l}$.

In the following, we discuss the quantum coherences as a function of the coupling strength between the optical modes, $J$. It can be found from Fig. 3(a) that, with the increase of $J$, $C_{\textrm{cw}}$ and $C_{\textrm{m}}$ keep almost unchanged, while $C_{\textrm{ccw}}$ increases rapidly to a certain value at first, then its increasing trend becomes flat. The reason is that, with the increase of $J$, more photons of CW mode transfer into CCW mode, yielding the increase of photon number in CCW mode. In particular, even if $J$ increases from 0 to a small value, Eq. (6b) indicates $\bar {a}_{\textrm{ccw}}$ can reach a large value because $\varepsilon _{l}=\sqrt {2\gamma _{\textrm{a}}P_{l}/\hbar \omega _{l}}$ is actually very large, so, even very small, as long as $J\neq 0$, $C_{\textrm{ccw}}$ suddenly increases as illustrated at the beginning of Fig. 3(a). However, due to $J$ is the optical mode coupling strength, it has little influence on the mechanical resonator, so $C_{\textrm{m}}$ changes little with $J$. As for $C_{\textrm{cw}}$, it can be understood that $C_{\textrm{cw}}$ is almost unaffected with the change of $J$, if one taking into account the CW mode is driven directly by the laser field and system is treated as a steady state.

 

Fig. 3. The quantum coherences versus the coupling strength between the optical modes $J$, (a) the one-mode quantum coherences and (b) the two- and the three-mode quantum coherences.

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In Fig. 3(b), $C_{\textrm{cw,ccw}},C_{\textrm{ccw,m}}$ and $C_{\textrm{tot}}$ increase rapidly to a certain value at first, then they have a flat increasing trend, while $C_{\textrm{cw,ccw}}$ keeps almost unchanged. The reason for this is similar to the explanation about the one-mode quantum coherences. Obviously $C_{\textrm{cw,ccw}},C_{\textrm{ccw,m}}$ and $C_{\textrm{tot} }$ which are related to the CCW mode have the similar changing trend as $C_{ \textrm{ccw}}$. But $C_{\textrm{cw,m}}$, determined mainly by $C_{\textrm{cw}}$ and $C_{\textrm{m}}$ as mentioned above, keeps unchanged with the increase of $J$. Moreover, we find that the values of $C_{\textrm{ccw,m}}$ and $C_{\textrm{tot }}$ are the same at $J=0$, this is because there is no photon in the CCW mode in such a case.

Now let us investigate the influence of optical decay rate $\gamma _{\textrm{a} }$ on the quantum coherences. It can be seen from Fig. 4(a) and (b) that with the increase of $\gamma _{\textrm{a}}$ all the quantum coherences increase sharply at first, and quickly reach their maximal values, and then decrease gradually. In general, the optical decay generally destroys cavity fields and reduces the quantum coherences, but on the other hand it is helpful to establish the cavity field through the relation $\varepsilon _{l}= \sqrt {2\gamma _{\textrm{a}}P_{l}/\hbar \omega _{l}}$, which is proportional to $\sqrt {\gamma _{\textrm{a}}}$. When $\gamma _{\textrm{a}}$ is small, the latter plays a main role, while the former plays a less important role. With $\gamma _{\textrm{a}}$ gradually increasing, the decay effect and the pump effect reach a balance and the corresponding quantum coherences achieve their maximal values. When $\gamma _{\textrm{a}}$ is beyond the balance point, the decay effect surpasses pump effect, and the corresponding quantum coherences reduce.

 

Fig. 4. The quantum coherences versus the ratio $\gamma _{\textrm{a} }/\gamma _{\textrm{a}_{\textrm{0}}}$ ($\gamma _{\textrm{a}_{\textrm{0}}}$ is fixed and it is determined by the parameters $\omega _{\textrm{0}}$ and $Q$ listed in subsection 4.1 through the relation $\omega _{\textrm{0}}/Q$ ), (a) the one-mode quantum coherences and (b) the two- and the three-mode quantum coherences.

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Finally, we investigate the influence of the decay rate of the mechanical mode $\gamma _{\textrm{m}}$ on the quantum coherences as shown in Fig. 5. We find from Fig. 5(a) that all the three one-mode quantum coherences decrease monotonically with the increase of $\gamma _{\textrm{m}}$. The physical mechanism behind this can be explained as follows. The influence of $\gamma _{\textrm{m}}$ is embodied in the diffusion matrix $D$ of the Lyapunov Eq. (10) through the relation $\gamma _{\textrm{m}}(2n_{\textrm{th}}+1)$, we can find that the increase of $\gamma _{\textrm{m}}$ means the increase of effective thermal noise, yielding the decrease of the three one-mode quantum coherences. However, with the increase of $\gamma _{\textrm{m}}$ the decrease of the two optical quantum coherences $C_{ \textrm{cw}}$ and $C_{\textrm{ccw}}$ are much smaller than that of mechanical quantum coherence $C_{\textrm{m}}$. This can be understood because $\gamma _{\textrm{m}}$ has a direct restraining influence on the mechanical mode, while influence of $\gamma _{\textrm{m}}$ on $C_{\textrm{cw}}$ and $C_{\textrm{ccw}}$ is indirect through the coupling of the optical modes to the mechanical mode.

 

Fig. 5. The quantum coherences versus the ratio $\gamma _{\textrm{m} }/\gamma _{\textrm{m}_{\textrm{0}}}$ (here $\gamma _{\textrm{m}_{\textrm{0}}}$ is fixed as listed in first paragraph of subsection 4.1), (a) the one-mode quantum coherences and (b) the two- and the three-mode quantum coherences.

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The situation for $C_{\textrm{cw,ccw}},\; C_{\textrm{cw,m}},\; C_{\textrm{ccw,m}}$ and $C_{\textrm{tot}}$ in Fig. 5(b) is similar to the one-mode quantum coherences, decreasing monotonically with the increase of $\gamma _{\textrm{m} }$. As $\gamma _{\textrm{m}}$ increases, $C_{\textrm{cw,m}},\; C_{\textrm{ccw,m}}$ and $C_{\textrm{tot}}$ decrease faster than $C_{\textrm{cw,ccw}}$ because they are related directly to the mechanical mode.

4.2 Nonreciprocal quantum coherence

In the previous subsection, we have discussed the quantum coherences in the situation that the input pump always drives the CW optical mode from left as illustrated in Fig. 1(a), left-driving for short, and the resonator rotates counterclockwise. Next we shall explore in this subsection the situation that the input pump drives the CCW optical mode from right, right-driving for short, while the resonator still rotates counterclockwise. Then we compare quantum coherences in both left- and right-driving cases to investigate their nonreciprocity. Note that to investigate quantum coherences for right-driving one should change the driving term of the CW mode, that is the last term of Eq. (1), into $i\hbar \varepsilon _{l}(\hat {a}_{\textrm{ccw} }^{\dagger }-\hat {a}_{\textrm{ccw}})$, a driving term of the CCW mode.

In the following, we label superscripts “L” and “R” to stand respectively for left- and right-driving. For example, regarding the one-mode mechanical quantum coherences, $C_{\textrm{m}}^{\textrm{L}}\;(C_{\textrm{m}}^{ \textrm{R}})$ stands for the mechanical quantum coherence with left-driving (right-driving). Considering in the situation of left-driving, $C_{\textrm{cw}}^{ \textrm{L}}$ is the quantum coherence of the driven optical mode and $C_{ \textrm{ccw}}^{\textrm{L}}$ is the quantum coherence of the optical mode that is not driven, while in the situation of right-driving, $C_{\textrm{ccw}}^{\textrm{R}}$ is the quantum coherence of the driven optical mode and $C_{\textrm{cw} }^{\textrm{R}}$ is the quantum coherence of the optical mode that is not driven, so we shall illustrate $C_{\textrm{cw}}^{\textrm{L}}$ and $C_{\textrm{ccw }}^{\textrm{R}}$ in the same plot in order to compare the driven optical modes, while $C_{\textrm{ccw}}^{\textrm{L}}$ and $C_{\textrm{cw}}^{\textrm{R}}$ in the other plot to compare the not driven optical modes. Regarding the two- and three-mode quantum coherences, the driven optical mode and not driven optical mode are treated in a similar way. For instance, $C_{\textrm{tot}}^{ \textrm{L}}$ $(C_{\textrm{tot}}^{\textrm{R}})$ is the three-mode quantum coherence with left-driving (right-driving). Considering in the situation of left-driving, $C_{\textrm{cw,m}}^{\textrm{L}}$ is the quantum coherence of the driven optical mode and the mechanical mode, and $C_{\textrm{ccw,m}}^{\textrm{L} }$ is the quantum coherence of the not driven optical mode and the mechanical mode, while in the situation of right-driving, $C_{\textrm{ccw,m}}^{\textrm{R}}$ is the quantum coherence of the driven optical mode and the mechanical mode and $C_{\textrm{cw,m}}^{\textrm{R}}$ is the quantum coherence of the not driven optical mode and the mechanical mode, so we illustrate $C_{ \textrm{cw,m}}^{\textrm{L}}$ and $C_{\textrm{ccw,m}}^{\textrm{R}}$ in the same plot for comparison, while $C_{\textrm{ccw,m}}^{\textrm{L}}$ and $C_{\textrm{cw,m} }^{\textrm{R}}$ in the other plot.

In Fig. 6, the quantum coherences are plotted as a function of the rotating velocity $\Omega$ with left- and right-driving. We can see that the two quantum coherences in each plot are the same when $\Omega =0$, meaning that the quantum coherences are independent of the driving directions and the system is apparently reciprocal. However, when $\Omega \neq 0$, the two quantum coherences in each plot gradually separate as $\Omega$ increases, one monotonically increasing in the case of right-driving, the other monotonically decreasing in the case of the left-driving, embodying the nonreciprocity of the system in the various quantum coherences.

 

Fig. 6. Nonreciprocal quantum coherence: the quantum coherence versus the rotating velocity $\Omega$ with CW- and CCW-driving, (a1), (b1), (c1) the one-mode nonreciprocal quantum coherences and (a2), (b2), (c2) the two- and the three-mode nonreciprocal quantum coherences.

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The physical mechanism for the nonreciprocity can be explained by the time-reversal asymmetry of the system [24,62,63]. If the resonator does not rotate, no matter which mode, CW or CCW, is driven, their resonance frequencies are always $\omega _{\textrm{0}}$. But if the resonator rotates counterclockwise, the situation will be very different. As mentioned above, the resonance frequencies of the CW and CCW modes under Sagnac effect are respectively $\omega _{\textrm{cw} }=\omega _{\textrm{0}}+|\Delta _{\textrm{s}}|$ and $\omega _{\textrm{ccw} }=\omega _{\textrm{0}}-|\Delta _{\textrm{s}}|$. In the left-driving case, the driven mode’s resonance frequency is obviously $\omega _{\textrm{cw} }=\omega _{\textrm{0}}+|\Delta _{\textrm{s}}|$, while in the right-driving case, the driven mode’s resonance frequency is $\omega _{\textrm{ccw} }=\omega _{\textrm{0}}-|\Delta _{\textrm{s}}|$, thus the rotation of the resonator results in different frequency shifts of the driven optical mode for the left- and right-driving, this breaks the time-reversal symmetry of the system and leads to the nonreciprocity in the quantum coherences. In fact, such nonreciprocity also exists in many physical quantities and the quantum processes [22,23,25–34].

5. Discussion and conclusion

5.1 Experimental feasibility of the quantum coherence

In this subsection, we give a brief discussion about the experimental feasibility for achieving quantum coherence in such a system. As well known, the manufacturing process of WGM micro-resonator is very mature and high-quality WGM resonators have been created in many research groups [20,21]. The effective temperature $T=10$ K can be achieved by cooling the WGM resonator. In fact, a lower temperature such as $T=1.65$ K has been realized experimentally [64]. As mentioned above, the nonreciprocity of quantum coherences are resulted from rotation of the resonator, to achieve apparent nonreciprocity the extreme large rotation angular velocity is required. Fortunately, several experiments have verified the feasibility recently [24,65,66]. For example, in a recent experiment, Carmon et al. fabricated a WGM silicon sphere with the radius $r=1.1$ mm and they rotated the spherical resonator by fixing it to a turbine, achieving the angular velocity of resonator $\Omega =6.6$ kHz [24]. After that, faster angular velocities of resonators were realized experimentally by using the levitated nanomechanical rotors with smaller radiuses [67,68]. Moreover, a photonic coupler technique has been developed by making the tapered fibers fly aerodynamically close to the rapidly rotating spherical resonators with a separation of only a few nanometers [24]. We have shown in the Appendix that the stability conditions can be met in the case $T=10$ K. In addition, the quantum coherence can be measured by the method of homodyne detection [53]. Obviously, this provides a solid basis to verify the conclusion obtained in this work.

5.2 Conclusion

In summary, we theoretically investigate the various macroscopic quantum coherences including one-, two- and three-mode ones in a spinning WGM resonator system. It is found that in such a system (a) the quantum coherences increase from zero to a certain value as $P_{l}$ increases, and in two- and three-mode quantum coherences the one-mode quantum coherences play a main role; (b) the quantum coherences related directly to the optical mode not driven by the external laser field are more dependent on the optical coupling strength $J$ because the photons in such a mode are transferred from the driven optical mode; (c) the quantum coherences are influenced by both optical and mechanical decay rates, with the positive and negative influences of the former resulted respectively from the driving and decay of the optical modes, while the latter has only the negative influence; (d) the quantum coherences in the spinning system are nonreciprocal, they monotonically increase as $\Omega$ increases when the driving direction of the laser field is consistent with the rotation direction of the resonator, otherwise they decrease monotonically. Our results are meaningful to the understanding of the macroscopic quantum coherences and their nonreciprocity in the spinning optomechanical system.

Appendix: stability conditions of the system

According to Routh-Hurwitz criterion [57], the stability of the linearized Langevin Eqs. (8) in the main text should meet the following conditions:

(19)$$\mathcal{C}_{1}>0,\;\mathcal{C}_{2}>0,\;\mathcal{C}_{3}>0,\;\mathcal{C} _{4}>0,$$

(20)$$4\gamma _{\textrm{a}}+\gamma _{\textrm{m}}>0,$$

(21)$$\mathcal{C}_{1}\left( 2\gamma _{\textrm{a}}+\gamma _{\textrm{m}}\right) >8\gamma _{\textrm{a}}^{2}\left( \gamma _{\textrm{a}}+\gamma _{\textrm{m}}\right) +\gamma _{\textrm{m}}\omega _{\textrm{m}}^{2},$$

(22)$$J^{4}+5\gamma _{\textrm{a}}^{2}(\omega _{\textrm{m}}^{2}-\gamma _{\textrm{a} }^{2})+\tilde{\Delta}_{\textrm{ccw}}^{2}+\tilde{\Delta}_{\textrm{cw}}^{2}+\gamma _{\textrm{a}}^{2}\mathcal{C}_{1}>\mathcal{C}_{3},$$

(23)$$\gamma _{\textrm{m}}\mathcal{C}_{2}+2\gamma _{\textrm{a}}\left( 2J^{2}+2\gamma _{\textrm{a}}^{2}+\tilde{\Delta}_{\textrm{ccw}}^{2}+\tilde{\Delta}_{\textrm{cw} }^{2}\right) >2\gamma _\textrm{a}\mathcal{C}_{3},$$

(24)$$\omega _{\textrm{m}}^{2}\mathcal{C}_{2}+\left( \tilde{\Delta}_{\textrm{ccw}} \tilde{\Delta}_{\textrm{cw}}-J^{2}\right) \left( \tilde{\Delta}_{\textrm{ccw}}+ \tilde{\Delta}_{\textrm{cw}}\right) \mathcal{C}_{4}>\left( J^{2}+\gamma _{ \textrm{a}}^{2}+\tilde{\Delta}_{\textrm{ccw}}\tilde{\Delta}_{\textrm{cw}}\right) \mathcal{C}_{3} ,$$

where

$$\begin{aligned}&\mathcal{C}_{1}=2J^{2}+6\gamma _{\textrm{a}}^{2}+4\gamma _{\textrm{a}}\gamma _{ \textrm{m}}+\tilde{\Delta}_{\textrm{ccw}}^{2}+\tilde{\Delta}_{\textrm{cw} }^{2}+\omega _{\textrm{m}}^{2},\\ &\mathcal{C}_{2}=\left[ \left( J^{2}+\gamma _{\textrm{a}}^{2}\right) ^{2}+\left( \tilde{\Delta}_{\textrm{ccw}}^{2}+\tilde{\Delta}_{\textrm{cw} }^{2}\right) \gamma _{\textrm{a}}^{2}+\left( \tilde{\Delta}_{\textrm{ccw}}\tilde{ \Delta}_{\textrm{cw}}-2J^{2}\right) \tilde{\Delta}_{\textrm{ccw}}\tilde{\Delta}_{ \textrm{cw}}\right] ,\\ &\mathcal{C}_{3}=\frac{\hslash }{m}[2JG_{\textrm{ccw}}^{x}G_{\textrm{cw} }^{x}+2JG_{\textrm{ccw}}^{y}G_{\textrm{cw}}^{y}+(G_{\textrm{ccw}}^{x})^{2}\tilde{ \Delta}_{\textrm{ccw}}+(G_{\textrm{ccw}}^{y})^{2}\tilde{\Delta}_{\textrm{ccw}}\\ &+\left( (G_{\textrm{cw}}^{x})^{2}+(G_{\textrm{cw}}^{y})^{2}\right) \tilde{ \Delta}_{\textrm{cw}}],\\ &\mathcal{C}_{4}=\frac{\hslash }{m}\left[ (G_{\textrm{ccw}}^{x})^{2}+(G_{ \textrm{cw}}^{x})^{2}+(G_{\textrm{ccw}}^{y})^{2}+(G_{\textrm{cw}}^{y})^{2}\right] . \end{aligned}$$

Based on the above conditions Eqs. (19)–(24), we plot Fig. 7 to show the stability region for the angular velocity $\Omega$ and the optical coupling strength $J$ with the parameters provided in the main text. Obviously, the maximal values of $\Omega =30$ kHz and coupling strength $J/\gamma _{\textrm{a}}=0.5$ selected in the paper meet the stability conditions.

 

Fig. 7. The stability conditions. The green part represents stability region and blue part represents instability region.

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Funding

Natural Science Foundation of Shanghai (20ZR1441600); National Natural Science Foundation of China (11804225, 12074261).

Acknowledgments

The author thank Akash Kundu for his insightful discussions.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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