Cavity optomechanics (COM) research has progressed rapidly in recent years ^[1,2], leading to prospects and applications of nano-COM technologies, such as quantum state transducers ^[3–6], highly sensitive motion sensors ^[7,8], and coherent phonon lasing ^[9–11]. Few-photon-based COM has also been explored and is particularly useful for applications in COM-based quantum phononic engineering or quantum information processing ^[12–15]. Another example closely related to this work is the experimental demonstrations of optomechanically induced transparency (OMIT) ^[16–20]. Nonlinear OMIT effects and OMIT-based switching of slow to fast light have also been shown ^[19–25]. These advances indicate that versatile coherent control of light is achievable with solid-state COM devices.

In OMIT, radiation-pressure-induced breathing-mode oscillations of the boundary of the resonator play a key role, as in most COM devices. The possible role of the rotation of the resonator itself, however, has not been explored. In a rotating device, an additional phase is accumulated for the propagating light, known as the optical Sagnac effect ^[26,27]. Recently, exotic rotation-induced effects have been revealed in purely optical systems ^[28–33]. For example, the rotation-induced mode splitting in an optical microresonator, i.e., the frequency difference between the clockwise (CW) and the counterclockwise (CCW) modes, was revealed to provide a method for sensitive measurements of the angular velocity of the medium ^[28]; in addition, optical chiral symmetry breaking and other relevant effects have also been investigated ^[29–33]. In acoustics, we note that sound isolation was achieved in a recent experiment by using a circulating fluid medium ^[34].

Here we study the OMIT in a rotating microresonator. We show that both the transmission rate and the group delay of the probe can be flexibly tuned by steering the Sagnac shift. We find that the optical sidebands strongly depend on the rotary direction of the resonator, indicating a new scheme to achieve optical nonreciprocality, which is distinct from nonlinearity-based schemes as already demonstrated in recent experiments ^[35–39]. We stress that our work, using only a spinning resonator, is clearly different from a recent proposal: in Refs. ^[40,41], a COM system placed on a rotating table, including the cavity, the driving source, and the detectors, were considered to detect the Coriolis force or to turn off/on the OMIT, with a fixed rotary direction and also a cavity restricted along the diameter of the table. The main features of our work here, i.e., the optical nonreciprocality (transmitted or blocked) and the tunable group delay, by steering rotary directions of the resonator itself, were not revealed in all previous works. In addition, our work holds the promise to be extended to, e.g., a spinning nonlinear resonator, coupled spinning resonators, or a lattice of spinning resonators, to further steer the optical nonreciprocality, the slow light, and even topological COM effects ^[42,43].

As shown in Fig. 1, we consider a rotating whispering-gallery-mode (WGM) microresonator coupled to a stationary tapered fiber. This resonator, with optical resonance frequency ωa and intrinsic loss γa=ωa/Q (Q is the optical quality factor), is driven by a strong pump field at frequency ωl and a weak probe at frequency ωp. The amplitudes of the pump and probe fields are, respectively, ϵl=Pl/ℏωl, and ϵp=Pin/ℏωp, where Pl or Pin denotes the pump or probe power, respectively. The resonator also contains a mechanical mode at frequency ωm, driven by the pump field, and the frequency of rotation is denoted as Ω. Because the Sagnac effect, the frequency of the optical mode experiences a Sagnac–Fizeau shift ^[44], i.e., where n or r is the refractive index or the radius of the resonator, respectively, c is the light speed in vacuum, and the dispersion term dn/dλ, characterizing the relativistic origin of the Sagnac effect, is relatively small in typical materials (∼1%).

The Hamiltonian of the system, in a frame rotating at the pump frequency ωl, can be written at the simplest level as (ℏ=1) where Δ+=Δa+Δsag, Δa=ωa−ωl, ξ=ωa/r is the COM coupling, γex is the loss due to the resonator–fiber coupling, a is the optical annihilation operator, and x, p, θ and pθ denote the displacement, momentum, rotation angle, and angular momentum operators ^[41], respectively, with the commutation relations [x,p]=[θ,pθ]=i. To compare with the familiar OMIT spectrum ^[16], here we ignore the Kerr effect of the materials, which, as demonstrated in a recent transient-OMIT experiment, can be compensated for by shifting the driving frequency, even for a strong driving ^[18]. Thermal effects can be further weakened in a similar way, besides using a cryostat device ^[16].

The Heisenberg equations of motion of the system are then written as where γm is the mechanical damping rate, η=ωp−ωl is the frequency detuning between the probe and pump lights, and, for simplicity, the total optical loss is denoted as β=(γa+γex)/2.

The steady-state values of the dynamical variables can be obtained as where Ω=θ˙ is the speed of rotation. Clearly, besides the optical pump, the rotation also affects the values of both the mechanical displacement x¯ and the intracavity optical amplitude a¯. This means that the effective COM coupling or the breathing mode oscillations can be tuned by the rotation, which in turn results in modified OMIT properties of the system.

In order to see this, we use the standard procedure by expanding every operator as the sum of its steady value and a small fluctuation, i.e.,

By substituting Eq. (6) into Eq. (4), we have

Solving these equations leads to

The expectation value of the output field can then be obtained by using the input–output relation, i.e., where ain and aout are the input and output field operators, respectively, and the transmission rate of the probe field turns out to be

This sets the stage for our discussion of the impact of rotation on the transmission rate and the group delay of the probe light.

The transmission T is shown in Fig. 2 as a function of the detuning Δp≡ωp−ωa and the rotation speed, with experimentally feasible values ^[45], i.e., n=1.44, Q=3×107, r=0.25mm, λ=1.55μm, γex=γa=ωa/Q, Pl=10W, Δa=ωm, m=2μg, ωm=200MHz, and γm=0.2MHz. For the familiar stationary case, the OMIT window appears around the resonance, Δp=0, due to destructive interference of the probe absorption. As shown in Ref. ^[16], the OMIT linewidth is approximately γm+ξ2|a¯|2/βm2ωm2; in the presence of rotation, the modified values of x¯ and a¯ change the OMIT spectrum; see the red and blue curves in Fig. 2(a). Note that the rotation leads to a strong enhancement of ξx¯, resulting in redshifts of the OMIT peaks, i.e., from Δp=0 to Δp<0. Even at the resonance, significant differences emerge for different rotary directions. To clearly see this, we define the enhancement factor of transmission at the resonance

Interestingly, as shown in Fig. 2(b), f increases with the rotation speed, but with different ratios for Ω>0 or Ω<0. For Ω=120kHz, the transmission at Δp=0 is enhanced by ∼11%. This holds the promise to be useful for practical applications of OMIT in coherent control of light.

For the stationary case, the OMIT peak appears at the resonance Δp=0, and for the off-resonance cases, a symmetric sideband dip appears on each side of the peak, indicating strong absorption of the off-resonance probe. Our scheme provides a new way to tune the positions of the OMIT peak and dips. In particular, for the off-resonance cases, the transmission rates of the probe depend on the rotary directions of the resonator (even transmitted or blocked for the input probe). This can be referred to as nonreciprocal optical sidebands (for other ways of tuning OMIT sidebands, see, e.g., Ref. ^[24]). For example, the frequency shift of the lower sideband is approximately −ξx¯∓|Δsag|, where ∓ corresponds to Ω>0 or Ω<0. In Figs. 2(c) and 2(d), we choose Ω=100kHz and thus ξx¯≈49MHz, and |Δsag|≈12MHz. In particular, T(Ω>0)>0.9, while T(Ω<0)≈0 at Δp≈−37MHz, i.e., featuring nonreciprocality in this situation. This indicates that, by tuning Ω and Δp, the probe light can be transmitted or blocked. Note that for a fixed rotary direction, Ω<0 corresponds to the case with the pump and probe lights applied from the right of the fiber. In this case, the probe coming from the left or the right can be transmitted or blocked. A similar feature was also observed in a purely optical experiment with a spinning resonator ^[46].

Accompanied with the OMIT, the slow light can also be observed ^[17]. This can be characterized by the group delay of the probe light:

As shown in Fig. 3, τg now depends on both Ω and Δp. For Δp=0, the probe light tends to pass with a higher transmission [Fig. 2(b)] but also with a higher velocity [Fig. 3(a)], highlighting different roles of OMIT and the Sagnac effect; for Δp≠0, we define the enhancement factor of group delay:

Figure 3(b) shows D as a function of Ω and Δp. Clearly, to achieve both a higher transmission and a higher group delay, one needs to carefully choose the values of Ω and Δp. For example, for Ω=100kHz and Δp=−75MHz, we have D≈22 and T≈0.85, indicating the possibility of achieving enhanced slow light with a still considerable transmission rate. In comparison with conventional COM devices, our spinning-OMIT scheme provides a new degree of freedom to adjust the properties of optical propagations.

In summary, we have studied OMIT in a spinning resonator. We find that the mechanical oscillation and the intracavity photons are strongly affected by the rotation-induced Sagnac effect, which in turn leads to significantly modified OMIT properties, including the transmission rate and the group delay of the probe. In particular, the optical sidebands, depending on the directions of rotation, exhibit a highly asymmetric feature (i.e., transmitted or blocked for incoming signals from left or right). We note that the Sagnac effect can be applied to induce different shifts for the forward and backward signals in Brillouin-scattering-induced transparency ^[47], in which the nonreciprocal light propagation may be more easily observed with a lower rotation speed.

The rotation of the resonator is now up to ∼100kHz, i.e., still far less than the mechanical frequency. We expect that by further enhancing the rotation speed, new effects such as optical chaotic dynamics or other interesting effects (even rotational–vibrational coupling) ^[48,49] may be detected, which, however, need to be explored in future work. We believe that spinning devices hold the promise to find applications in a wide range of COM-based applications such as optical isolating ^[35] or chiral sensing ^[50].

Acknowledgment. Insightful discussions with Shai Maayani, Tal Carmon, and Chun-Hua Dong are acknowledged.