Optomechanically-induced transparency in parity-time-symmetric microresonators

Optomechanically-induced transparency (OMIT) and the associated slowing of light provide the basis for storing photons in nanoscale devices. Here we study OMIT in parity-time (PT)-symmetric microresonators with a tunable gain-to-loss ratio. This system features a sideband-reversed, non-amplifying transparency , i.e., an inverted-OMIT. When the gain-to-loss ratio is varied, the system exhibits a transition from a PT-symmetric phase to a broken-PT-symmetric phase. This PT-phase transition results in the reversal of the pump and gain dependence of the transmission rates. Moreover, we show that by tuning the pump power at a fixed gain-to-loss ratio, or the gain-to-loss ratio at a fixed pump power, one can switch from slow to fast light and vice versa. These findings provide new tools for controlling light propagation using nanofabricated phononic devices.

R ecent advances in steering a macroscopic mechanical object in the deep quantum regime [1][2][3] have motivated theoretical studies to understand the physics of photon-phonon interactions in cavity optomechanics (COM), and also led to exciting experimental studies on quantum nanodevices 4,5 . In particular, the experimental demonstration of OMIT allows the control of light propagation at room temperature using nano-and micro-mechanical structures [6][7][8] . The underlying physics of OMIT is formally similar to that of electromagnetically-induced transparency (EIT) in three-level L-type atoms 6-10 and its all-optical analogs demonstrated in various physical systems 10,11 . The resulting slow-light propagation provides the basis for a wide range of applications 9 . Mechanically-mediated delay (slow-light) and advancement (fast-light) of microwave pulses were also demonstrated in a superconducting nanocircuit [12][13][14][15] . These experimental realizations offer new prospects for onchip solid-state architectures capable of storing, filtering, or synchronizing optical light propagation.
As a natural extension of single-cavity structures, COM with an auxiliary cavity (compound COM: two cavities) has also attracted intense interest. The interplay between COM interactions and tunable optical tunnelling provides a route for implementing a series of important devices, such as phonon lasers 16 , phononic processors for controlled gate operations between flying (optical) or stationary (phononic) qubits 17,18 , and coherent optical wavelength converters [19][20][21] . Enhanced nonlinearities 22 and highly-efficient photon-phonon energy transfer [23][24][25] are other advantages of the compound COM. These studies were performed with passive (lossy, without optical gain) resonators.
Very recently, an optical system whose behavior is described by PT-symmetric Hamiltonians (i.e., the commutator [H, PT] 5 0) 26,27 was demonstrated in a system of two coupled microresonators, one of which has passive loss and the other has optical gain (active resonator) 28 . Observed features include: real eigenvalues in the PTsymmetric regime despite the non-Hermiticity of the Hamiltonian, spontaneous PT-symmetry breaking, as well as complex eigenvalues and field localization in the broken PT-symmetry regime. Moreover, nonreciprocal light transmission due to enhanced optical nonlinearity in the broken PT-symmetry regime was demonstrated 26 . Such a PT-symmetric structure provides unique and previously-unattainable control of light and even sound [26][27][28][29][30][31][32] . Manipulating the photon-phonon interactions in such systems opens new regimes for phonon lasing and quantum COM control 33 .
In this paper, we show that a compound COM with PT-symmetric microresonators leads to previously unobserved features and provides new capabilities for controlling light transmission in micro-and nano-mechanical systems. Particularly, we show: (i) a gain-induced reversed transparency (inverted-OMIT), i.e. an optical spectral dip between two strongly-amplifying sidebands, which is in contrast to the non-absorptive peak between strongly absorptive sidebands in the conventional passive OMIT; (ii) a reversed pump dependence of the optical transmission rate, which is most significant when the gain and loss are balanced (i.e., optical gain in one subsystem completely compensates the loss in the other); and (iii) a gain-controlled switching from slow (fast) light to fast (slow) light in the PT-symmetric (PT-breaking) regime, within the OMIT window. These features of the active OMIT enable new applications which are not possible in passive COM.
The inverted-OMIT observed here in an active COM, composed of a passive and an active optical microresonator, is reminiscent of the inverted-EIT observed in all-optical systems, composed of one active and one passive fiber loop 34 . In Ref. 34 a non-amplifying window accompanied with a negative group delay (fast light) was reported. However, our active OMIT, relies on hybrid photon-phonon interactions in a compound COM 6 . In the PT-symmetric regime, it provides the first OMIT analog of the optical inverted-EIT 34 . Distinct features of the inverted-OMIT that cannot be observed in the optical inverted-EIT are also revealed in the broken-PT-symmetric regime.

Results
The active COM system. We consider a system of two coupled whispering-gallery-mode microtoroid resonators 10,16,35,36 . One of the resonators is passive and contains a mechanical mode of frequency v m and an effective mass m 16 . We refer to this resonator as the optomechanical resonator. The second resonator is an active resonator which is coupled to the first one through an evanescent field. The coupling strength J between the resonators can be tuned by changing the distance between them. As in Ref. 28, the active resonator can be fabricated from Er 31 -doped silica and can emit photons in the 1550 nm band, when driven by a laser in the 980 nm or 1450 nm bands. The resonators can exchange energies only in the emission band of 1550 nm, so the gain photons can tunnel through the air gap between the resonators and provide a gain k to compensate the optical loss c in the passive resonator 28 .
Tuning the gain-to-loss ratio, while keeping J fixed, leads to two remarkably distinct regimes, i.e. broken-and unbroken-PT -symmetry regimes, that are characterized by distinct normal mode-splitting and linewidths 28,33 . Our aim here is to study OMIT in these two distinct regimes, focusing on the role of k/c. To this end, as in the conventional OMIT 6 , both a pump laser of frequency v L and a weak probe light of frequency v p are applied (see Fig. 1). The field amplitudes of the pump and probe are given by E L 5 (2P L c/ hv L ) 1/2 , e p 5 (2P in c/ hv p ) 1/2 , where P L and P in are the pump and probe powers.
The Hamiltonian of this three-mode COM system can be written as where a 1 and a 2 denote the annihilation operators of the bosonic fields in the microresonators with resonance frequency v c and radius R, g 5 v c /R is the COM coupling rate, is the mechanical position operator, x 0 5 [ h/(2mv m )] 1/2 , and b corresponds to the annihilation operator for the phonon mode. The pump-resonator, probe-resonator, and probe-pump frequency detunings are, respectively, denoted by The Heisenberg equations of motion (EOM) of this compound system are ( h 5 1) where C m is the mechanical damping rate. The optical or mechanical gain and damping terms are added phenomenologically into the EOM 24,33,35 ; they can also be incorporated into Eq. (1), resulting in a non-Hermitian Hamiltonian [26][27][28][29][30][31][32][33]36 . We note that k , 0 in Eq. (4) corresponds to a passive-passive COM; thus k/c , 0 and k/c . 0 define, respectively, a passive-passive COM and a passive-active COM.
The steady-state values of the dynamical variables are For D L 5 0, by choosing J 2 5 kc or k/c R 1 for J 5 c, one can identify a gain-induced transition from the linear to the nonlinear regime that significantly enhances COM interactions 33 . Here we focus on the effects of the gain-loss balance on the OMIT and the associated optical group delay, which, to our knowledge, has not been studied previously.
We proceed by expanding each operator as the sum of its steadystate value and a small fluctuation around that value, i.e. a 1 5 a 1,s 1 da 1 , a 2 5 a 2,s 1 da 2 , x 5 x s 1 dx. After eliminating the steady-state values, we obtain the linearized EOM, which can be solved using the ansatz (see the Method) The optical fluctuation in the optomechanical resonator A ; da 11 , the quantity of interest here, is www.nature.com/scientificreports where n 1 5 ja 1,s j 2 is the intracavity photon number of the passive resonator, m 6 5 2k2ij 6 iD L , and The expectation value of the output field can then be obtained by using where a in 1 and a out 1 t ð Þ are the input and output field operators. Then the optical transmission rate g(v p ) (i.e., the amplitude square of the ratio of the output field amplitude to the input probe field amplitude, We computed Eq. (11) with experimentally-accessible values of the system parameters 28  2 v c 5 j 2 v m . Now we discuss how the gain-loss ratio k/c, the coupling strength J, and the pump power P L affect the OMIT. Note that k/c and J are the tunable system parameters that allow one to operate the system in the broken-or unbroken-PT regimes.
Reversed-gain dependence. Figure 2 depicts the effect of k/c on the optical transmission rate. By introducing gain into the second microresonator, one can tune the system to transit from a conventional OMIT profile, quantified by a transparency window and two sideband dips, to the inverted-OMIT profile, quantified by a transmission dip and two sideband peaks (see Fig. 2a).
Increasing the loss ratio k/c , 0 in the passive-passive COM leads to shallower sidebands. When the amount of gain provided to the second resonator supersedes its loss and the resonator becomes an active one (amplifying resonator), increasing k/c . 0 helps to increase the heights of the sideband peaks until k/c 5 1, where g at sidebands is maximized. Increasing the gain further leads to the suppression of both the sideband peaks and the on-resonance (D p 5 0) transmission (Fig. 2b). This is in stark contrast with the observation of monotonically-increasing sideband peaks in the all-optical EIT system of Ref. 34.
This can be intuitively explained as follows. Under the condition of J/c 5 1, the system is in the PT-symmetric phase for k/c , 1, whereas it is in the broken-PT phase for k/c . 1. Thus, for k/c , 1, the provided gain compensates a portion of the losses, which effectively reduces the loss in the system and hence increases g 28 . Increasing the gain above the phase transition point k/c 5 1 puts the system in the broken-PT phase, with a localized net loss in the passive resonator 28 (i.e., the field intensity in the passive resonator is significantly decreased) which reduces the strength of the COM interactions and hence the value of g. The reduction of the transmission by increasing the gain provides a signature of the PT-breaking regime, and it is very similar to a recent experiment with two coupled-resonators where it was shown that increasing (decreasing) the loss of one of the resonators above (below) a critical level increases (decreases) the intracavity field intensity of the other, enhancing (suppressing) transmission 36 . Note that increasing (decreasing) loss is similar to decreasing (increasing) gain. We conclude here that only in the PTsymmetric regime (k/c , 1, with J/c 5 1), the active OMIT can be viewed as an analog of the optical inverted-EIT 34 . Figure 3 provides the fine features in the transmission rate, by numerically changed k/c . 0 by very small steps. One interesting observation in Fig. 3 is that when k 5 0, the second resonator has neither gain nor loss, and there still exists OMIT-like spectrum, i.e. small fluctuations around g , 1. In this case, the coupled-resonator system is still a passive-OMIT because one resonator is lossy while the other has neither loss nor gain. For k/c 5 0.01, we have a small resonance peak (see Fig. 3a). When the gain is increased, this peak tends to disappear (see e.g. Fig. 3b) and then evolves into a dip, e.g. g , 0 for k/c 5 0.2 (see Fig. 3c). This dip can also be manipulated by increasing the gain further [see Figs. 3(d-f)]. These results imply that one can tune the system from passive-OMIT to active-OMIT, or vice versa, by varying the gain-to-loss ratio k/c. Such transient behaviors have not been revealed previously.
Reversed-pump dependence. For the passive-passive COM, the transmission rate and the width of the OMIT window increase with increasing pump power P L 6,24 (see also Fig. 4a). For the passive-active COM, where we observe the inverted-OMIT, increasing the pump power P L leads to a significant decrease of the sideband amplifications (Fig. 4b). Here the pump power dependence of the OMIT profile is shown for k/c 5 1.5 (in the PT-breaking regime). We have also performed our calculations for k/c 5 1 and k/c 5 0.5, and similarly found that in these cases the sideband amplifications are also reduced as the pump is increased from P L 5 10 mW to 20 mW (not shown here). Nevertheless, the sideband amplification always reaches its maximum value at the gain-loss balance (see also Fig. 2b). We note that the counterintuitive effect of reversed pump dependence was also previously demonstrated in coupled optical systems (i.e., no phonon mode was involved) operating at the exceptional point 36,37 .
In addition, we have also studied the effect of the mechanical damping on the profiles of the conventional and inverted OMIT at different values of the gain-to-loss ratio. We confirmed that the profiles of both the conventional and the inverted OMIT are strongly  PT-breaking fast light. The light transmitted in an EIT window experiences a dramatic reduction in its group velocity due to the rapid variation of the refractive index within the EIT window, and this is true also for the light transmitted in the OMIT window in a conventional passive optomechanical resonator 6 . Specifically, the optical group delay of the transmitted light is given by We have confirmed that OMIT in the passive-passive COM leads only to the slowing (i.e., positive group delay: t g . 0) of the transmitted light, and that when the coupling J between the resonators is weak the reduction in the group velocity approaches to that experienced in a single passive resonator 6,24 . In contrast, in the active-passive COM, one can tune the system to switch from slow to fast light, or vice versa, by controlling P L or k/c, such that the COM experiences the PT-phase transition (Fig. 5).
In the regime k/c , 1, as P L is increased from zero, the system first enters into the slow-light regime (t g . 0), and t g increases until its peak value. Then it decreases, reaching t g 5 0, at a critical value of P L (Fig. 5a). The higher is the k/c, the sharper is the decrease. Increasing P L beyond this critical value completes the transition from slow to fast light and t g becomes negative (t g , 0). After this transition, the advancement of the pulse increases with increasing P L until it reaches its maximum value (more negative t g ). Beyond this point, a further increase in P L , again, pushes t g closer to zero.  In the regime k/c . 1, increasing P L from zero first pushes the system into the fast-light regime and increases the advancement of the pulse (t g , 0) until the maximum advance is reached (Fig. 5b). After this point, the advance decreases with increasing P L and finally t g becomes positive, implying a transition to slow light. If P L is further increased, t g first increases until its peak value, and then decreases approaching t g 5 0.
The P L value required to observe the transition from slow-to-fast light (when k/c , 1) or from fast-to-slow light (when k/c . 1) depends on the gain-to-loss ratio k/c if the coupling strength J is fixed (Fig. 5). This implies that, when P L is kept fixed, one can also drive the system from slow-to-fast or fast-to-slow light regimes by tuning k/c. A simple picture can be given for this numericallyrevealed feature: for D L , 0, j , 0, we simply have G 1~G Ã 2 < J 2 {kc À Á zikgx s , which is minimized for J 2 5 kc, or k/ c 5 1, J/c 5 1; therefore, in the vicinity of the gain-loss balance, the denominator of A is a real number, and Im(A)/Re(A) , (1-c/k) 21 , i.e. having reverse signs for k/c . 1 or k/c , 1. Correspondingly, arg (A) or arg[t(v p )] and hence its first-order derivative t g , (c/k 21) (for J/k 5 1). Clearly, the sign of t g can be reversed by tuning from the PT-symmetric regime (with k/c , 1) to the broken-PT regime (with k/c . 1). We note that the appearance of the fast light in the PT-breaking regime, where the gain becomes to exceed the loss, is reminiscent of that observed in a gain-assisted or inverted medium 38 .
In order to better visualize and understand how the switching from the slow-to-fast light and vice versa takes place, when the gain-to-loss ratio k/c is tuned at a fixed-pump power P L , or when P L is tuned at a fixed value of k/c, we present the phase of the transmission function t(v p ) in Figs. 6(a-c). For this purpose, we choose the values of P L and k/c from Fig. 5, where their effects on the optical group velocity t g were presented. These calculations clearly show that, near the resonance point (d p 5 0), the slope of the curves can be tuned from positive to negative or vice versa, by tuning P L or k/c, which agrees well with the slow-fast light transitions (see Fig. 5). In sharp contrast, Fig. 6d shows that for the passive-passive COM (e.g., k/c 5 21), no such type of sign reversal can be observed for the slope of the phase curves, corresponding to the fact that only the slow light can exist in that specific situation.

Discussion
In conclusion, we have studied the optomechanically-inducedtransparency (OMIT) in PT-symmetric coupled microresonators with a tunable gain-loss ratio. In contrast to the conventional OMIT in passive resonators (a transparency peak arising in the otherwise strong absorptive spectral region), the active OMIT in PT-symmetric resonators features an inverted spectrum, with a transparency dip between two sideband peaks, providing a COM analog of the all-optical inverted-EIT 34 . For this active-OMIT system, the counterintuitive effects of gain-or pump-induced suppression of the optical transmission rate are revealed. In particular, the transition from slow-to-fast regimes by tuning the gain-to-loss ratio or the pump power is also demonstrated. The possibility of observing the PT-symmetric fast light, by tuning the gain-to-loss ratio of the coupled microresonators 28 , has not studied previously. These exotic features of OMIT in PT-symmetric resonators greatly widens the range of applications of integrated COM devices for controlling and engineering optical photons. In addition, our work can be extended to study e.g. the OMIT in a quasi-PT system 36 , the OMIT cooling of mechanical motion 39,40 , the active-OMIT with two mechanical modes 19 , or the gain-assisted nonlinear OMIT [41][42][43] .

Methods
Derivation of the optical transmission rate. Taking the expectation of each operator given in Eqs. (2)-(4), we find the linearized Heisenberg equations as which can be transformed into the following form, by applying the ansatz given in Eq. (8), where A ; da 11 is given in Eq. (9). In order to receive some analytical estimations, we take v m /v c , 0, D L,p , 0, which leads to m 6 , 2k, G 1,2 *J 2 { c+igx s ð Þ k. For x s , 0, we have