Loss-induced transparency in optomechanics

We study optomechanically induced transparency (OMIT) in a compound system consisting of coupled optical resonators and a mechanical mode, focusing on the unconventional role of loss. We find that optical transparency can emerge at the otherwise strongly absorptive regime in the OMIT spectrum, by using an external nanotip to enhance the optical loss. In particular, loss-induced revival of optical transparency and the associated slow-to-fast light switch can be identified in the vicinity of an exceptional point. These results open up a counterintuitive way to engineer micro-mechanical devices with tunable losses for e.g., coherent optical switch and communications.

Here we probe the EP features in OMIT, without using any active gain, but increasing the optical loss [55,56]. We note that in a recent experiment [55], by placing an external nanotip near a microresonator and thus increasing the optical loss, counterintuitive EP features, i.e., suppression and revival of lasing were demonstrated [55]. Similar EP features in optical transmissions, i.e., loss-induced transparency (LIT) were reported previously in a purely optical experiment [56]. Our purpose here is to show the LIT features in OMIT devices, i.e., loss-induced suppression and revival of optical transparency at the EP. In addition, we find that by increasing the optical loss, strong absorption regimes in conventional OMIT can become transparent, accompanying by a slow-to-fast light switch in the vicinity of the EP (for similar reversed-OMIT features, see also Ref. [34] in an active COM system). The unconventional role of loss on the higher-order OMIT sidebands [28-30] is also probed. These results indicate a counterintuitive way to achieve optical switch and communications with OMIT devices, without the need of any active gain or complicated materials.

Tapered fiber
Schematic diagram of the compound COM system, with an additional optical loss γ tip induced by a Cr-coated nanofiber tip on the right (i.e., purely optical) resonator [55,56]. (b) The frequency spectrum of the compound COM system, with the red line or the blue lines denoting the red sideband (Stokes process) or the blue sidebands (anti-Stokes process), respectively [18,19].
As shown in Fig. 1, we consider two whispering-gallery-mode (WGM) microtoroid resonators coupled through evanescent fields, with the tunable coupling strength J and the intrinsic optical loss γ 1 or γ 2 , respectively [44,55]. The external lights are input and output via tapered-fiber waveguides. As in Refs. [55,56], an additional optical loss γ tip is induced on the right (i.e., purely optical) resonator by a chromium (Cr)-coated silica nanofiber tip [57], in order to see the loss effects on such a compound COM system. The left resonator, supporting also a mechanical mode of frequency ω m and an effective mass m, is driven by a strong red-detuned pump laser at frequency ω L and a weak probe laser at frequency ω P [18,19], with the optical field amplitudes respectively, where for simplicity we take γ 1 = γ 2 = γ c and P L or P in is the power of the pump or the probe light. In a frame rotating at frequency ω L , the Hamiltonian of this compound COM system can be written at the simplest level as where ω c is the resonant frequency of the optical mode, a 1 (a † 1 ) and a 2 (a † 2 ) are the optical bosonic annihilation (creation) operators, g denotes the COM coupling strength, x or p is the mechanical displacement or momentum operator, and the optical detunings are The Heisenberg equations of motion of this compound system are where Γ m is the mechanical loss rate. For ε P ε L , we can take the probe light as a perturbation, the dynamical variables can be expressed as a i = a i,s + δa i (i =1,2) and x = x s + δx, where the steady-state solutions of the system, by setting all the derivatives of the variables as zero, are easily obtained as For comparisons, we first consider the purely optical case [56] by ignoring the COM coupling. In this special case, by using the input-output relation [58] a out 1 = a in 1 − √ 2γ 1 a 1 , we can derive the optical transmission rate as where ∆ i = ω P − ω i (i = 1, 2) is the detuning between the probe and the cavity mode. For simplicity, here we take ∆ 1 = ∆ 2 = ∆ P . As shown in Fig. 2(a), the LIT feature can be clearly seen at the resonance (i.e., ∆ P = 0) in the transmission spectrum, that is, the transmission rate firstly decreases and then increases by increasing the tip loss γ tip [56]. The turning point (TP) position turns out to be which, for the parameter values chosen here, corresponds to γ TP tip /γ c = 3 (illustrated in Fig. 2(b)). Interestingly, we also note that by increasing γ tip , the strong-absorption regimes in the conventional transmission spectrum (at ∆ P = ±11 MHz) become transparent, see Fig. 2(b), which is not reported in Ref. [56]. This phenomenon is induced by the reduction of interference caused by the tip loss. And from our numerical estimation, γ TP tip is ∼ 0 for ∆ P = ±11 MHz, i.e., by reversing the tip loss to an active gain, it is possible to reverse the dip in the EIT spectrum to a peak as already observed in the recent reversed EIT experiment performed by T. Oishi and M. Tomita [59]. LIT is generally viewed as the evidence of the EP emergence in this lossy system [56], or the existence of hidden parity-time symmetry (under a suitable mathematical transformation) [60]. The eigenfrequencies of this coupled optical system are For ω 1 = ω 2 = ω c , the EP condition is simplified as γ EP tip = γ 1 − γ 2 + 2J, or for the parameter values chosen here, γ EP tip /γ c = 4, see Fig. 2(c-d). Clearly, γ TP tip can be close to but not exactly coincides with γ EP tip , due to the fact that the TP depends on the detuning ∆ 1,2 while the EP does not (for similar features, see also Ref. [55]). Now we consider the role of COM coupling in LIT. For this aim, we express the dynamical variables as the sum of their steady-state values and small fluctuations to the first order, i.e., x = x s + δx (1) + · · ·, a i = a i,s + δa (1) i + · · · (i = 1, 2), with which we can rewrite the equations of motion as Here the higher-order terms such as δx (1) δa (1) 1 will be neglected since they only contribute to the higher-order sidebands [28].
Then by using the ansatz: we obtain the solutions for the fluctuation operators as where With these results at hand, by using the standard input-output relation [58], we can obtain the transmission rate of the probe light which describes the relation of the output field amplitude and the input field amplitude at the probe frequency. Figure 3 shows the transmission rate T P of the probe as a function of ∆ P and γ tip . We see that (i) the strong-absorption regimes at ∆ p = ±11 MHz become transparent by increasing the tip loss, e.g., T P is increased from zero to ∼ 0.35 or ∼ 0.6 for γ tip /γ c ∼ 3 or 8, see Fig. 3(b) which is same as the purely optical case. In contrast, (ii) for the resonant case (∆ P = 0), the OMIT peak tends to be lowered, i.e., T P decreases for more tip loss, with its linewidth firstly decreased but then increased again, see Fig. 3(a,c). More interestingly, (iii) for the intermediate regime (∆ P = ±3 MHz), the feature as LIT in purely optical systems [55,56] in the resonant case can be clearly seen, i.e., T P firstly drops down to zero but then increases again for more tip loss, with the turning point γ TP tip /γ c = 3, see Fig. 3(d). A more intuitive analysis of these phenomena in the various parametric regimes is shown in Fig. 3(e) and the TP and EP are illustrated in the figure. In Fig. 3(f), the transmission rate is plotted as a function of γ tip and ∆ P which are both continuously varying to give a comprehensive view.
Based on the analyses above, we find that the LIT emerging at the resonance in the purely optical system now moves to the off-resonance regime of a specific detuning (∆ P = ±3 MHz) in the COM system. We will give an analysis of this difference in the following section. By comparing the linearized equations of δa (1) 1+ corresponding to the optical case and the COM case where and With chosen parameters above, we can numerically estimate a frequency shift of gx s + Re(C 1 ) to be ∼ 3 MHz in the steady-state case. For comparison with purely optical system, we choose ∆ L = 0 and ∆ P = 0. This frequency shift caused by the COM interaction is matched well with the parametric regimes where we find the LIT at the detuning of ∆ P = ±3 MHz in the transmission spectrum of OMIT. Now we turn to the slow-to-fast-light switch at the EP. Slowing or advancing of light can be associated with the OMIT process due to the abnormal dispersion [19]. This feature can be characterized by the group delay of the probe light Figure 4 shows the group delay as a function of γ tip at different values of ∆ P . We find that at ∆ P = −3 MHz, the probe light experiences a fast-to-slow switch in the vicinity of the EP, a feature which is similar to the reverted OMIT reported previously in an active COM system [34]. This provides a new method to achieve coherent optical group-velocity switch by tuning the optical loss, which as far as we know, has not been demonstrated previously in purely optical systems. In view of the sensitive change of τ g at the EP, this also could be used for e.g., EP-enhanced sensing of external particles entering into the mode volume of the resonator [61][62][63].

Second-order LIT in COM
In contrast to the linear systems [55,56], the tip loss can also affect the higher-order process originating from intrinsic nonlinear COM interactions. In order to see this, we use the following x = x s + δx (1) + δx (2) + · · ·, a i = a i,s + δa (1) i + δa (2) i + · · · (i = 1, 2), with Then by solving the Eq. (4), with the aid of Eqs. (11,22,23) and neglecting the higher-order terms more than second order, we get the second-order solutions with K (2) = m(−4 2 − 2i Γ m + ω 2 m ) and As defined in Ref.
[28], the efficiency of the second-order sideband process is Figure 5(a) shows the impact of the tip loss on the second-order sideband of OMIT. We find that, in contrast to the linear cases, the efficiency η increases by enhancing the loss γ tip . A comparison between T P and η is shown in Fig. 5(b-d). Figure 5(b) shows that T P decreases by increasing γ tip at the resonance while η is enhanced, a feature which was firstly revealed in Ref.
[28]. However, for non-resonance cases, e.g., ∆ P = ±3 MHz or ∆ P = ±11 MHz, η increases by increasing γ tip , which is evidently different from that for T P , see Fig. 5(c-d). Clearly, these results on nonlinear OMIT process are beyond any linear EP picture [48]. We note that the presence of nonlinearity can lead to a shift of the EP position [64] or even the emergence of high-order EPs [48]. We also note that η TP emerges at ∆ P = ±11 MHz, which is clearly different from the linear TP occurring at ∆ P = ±3 MHz. In fact, this frequency shift can also be identified by comparing the equations describing the linear process and its second-order sidebands, i.e., i∆ + γ 1 − 2i δa (2) with ∆ = ∆ L − gx s − Re(C 2 ), γ 1 = γ 1 + Im(C 2 ), and Here B, in terms of δx (1) + and δa (1) 1+ , can be taken as a constant. We see that in comparison with the linear process, the second-order sideband experiences a frequency shift |Re(C 2 ) − Re(C 1 )| ∼ 10 MHz, i.e., agreeing with our numerical results in the order of magnitudes.

Conclusions and Discussions
We theoretically investigate the impact of loss on OMIT in a passive compound COM system by coupling an external nanotip to the optical resonator. Loss-induced transparency is found at the EP in the OMIT transmission spectrum, which is reminiscent of that as reported in Ref. [56], but here corresponding to the off-resonance case (i.e., with ∆ P = ±3 MHz). For the resonance case, however, increasing the tip loss leads to very minor changes for the OMIT peak. We also find that a slow-to-fast light switch can happen in the vicinity of the loss-induced EP. A detailed comparison between the linear OMIT process and its second-order sidebands, in the presence of a tunable tip loss, is also given, indicating that more exotic EP-assisted effects may happen in a nonlinear COM system. In practice, our work provides a promising new way to manipulate or switch both light transmissions and optical group delays with various COM devices. Finally, in view of the sensitive change of the optical group delay at the EP, our work also indicates a new way to achieve EP-enhanced sensing [61][62][63].

Funding
This work is supported by NSF of China under Grants No. 11474087 and No. 11774086, and the HuNU Program for Talented Youth.