High-order exceptional points in optomechanics

We study mechanical cooling in systems of coupled passive (lossy) and active (with gain) optical resonators. We find that for a driving laser which is red-detuned with respect to the cavity frequency, the supermode structure of the system is radically changed, featuring the emergence of genuine high-order exceptional points. This in turn leads to giant enhancement of both the mechanical damping and the spring stiffness, facilitating low-power mechanical cooling in the vicinity of gain-loss balance. This opens up new avenues of steering micromechanical devices with exceptional points beyond the lowest-order two.


Results
We consider a system of two coupled microresonators, one with an optical gain κ and the other with passive loss γ [see Fig. 1(a)], such that both the coupling strength J and the gain-to-loss ratio κ/γ of the resonators can be tuned, as experimentally demonstrated in ref. 36. The mechanical mode (frequency ω m and effective mass m), contained in the passive resonator, can be driven by an external field, with frequency ω L and input power P in 10 . The simplest Hamiltonian of this system can be written as (ħ = 1). where x and p are the mechanical position and momentum operators, respectively, J is the optical tunneling rate, a 1 and a 2 are the lowering operators for the optical modes in the resonators, ξ = ω c /R is the COM coupling coefficient, with R and ω c respectively denoting the radius and the resonance frequency of the resonator supporting the mechanical mode, and  η γ ω = P 2 /( ) L c in is the pump rate. For phonon cooling, we choose a laser whose frequency is red-detuned with respect to the resonator, Δ = ω L − ω c < 0.
The resulting equations of motion read  where γ and Γ m are the optical and mechanical damping rates, respectively, and ε th denotes the thermal force at finite environmental temperature T, with zero mean value and the Brownian noise correlation The steady-state solutions then become  The off-resonance case has an EP which is of order 2, due to the coalescing optical modes 36 . This features a pair of amplifying and decaying optical supermodes, = ± ± a a a ( ) / 2 1 2 , with frequencies ω ± (for details on ω 0,± , see the Method). (c) When the optical red detuning equals ω m , an EP of order 3 emerges, leading to low-power phonon cooling in the vicinity of gain-loss balance (see the text).
Scientific RepoRts | 7: 3386 | DOI:10.1038/s41598-017-03546-7 Eq. (1) can be further linearized as that describing non-cyclic coupled three oscillators, for Δ < 0 40 , resulting in . This linear three-mode model, including the optical gain and loss, can be exactly diagonalized (for more details, see the Method). The resulting supermode spectrum (ω ±,0 ) is shown in Fig. 2, with the experimentally achievable values, i.e. λ = c/ω c = 1550 nm, Q c ~ 10 6 , R ~ 20 μm, ω m ~ 2π × 500 MHz, m ~ 10 pg, Q m ~ 10 3 , and ξ = ω c /R ~ 10 GHz/nm, γ ~ 200 MHz, Γ m ~ 3.14 MHz 36,39 . This type of spectrum, by tuning both κ/γ and Δ/ω m , is unattainable in passive COM 41 . Figure 2(a,b,f) show that, for optical red-detunings larger or smaller than the mechanical frequency ω m , the lowest two-order EP is observed for the supermodes with frequencies ω ± , which is strongly reminiscent of what has been observed in all-optical devices 36 (hence we refer to ω ± as the optical-supermode frequencies), i.e., the imaginary parts of the eigenfrequencies bifurcate and lead to field localizations when surpassing the gain-loss balance. This feature, however, is radically changed near the COM resonance, where the optical red-detuning is equal to ω m . In this case an EP of order 3 emerges due to the interaction of the three modes of the system [16][17][18][19][20][21][22][23][24][25][26] . In particular, when surpassing the EP, energy transfer occurs among all the three supermodes [see  features are also seen for the near-resonance case, see Fig. 2(e) with Δ/ω m = −1.02. This indicates a new way for mechanical cooling: using the interplay between κ/γ and Δ/ω m , which is not possible at all in conventional COM.
To demonstrate COM cooling at h-EPs (i.e., 3-EP in our system), we study the linear response of the system to external noise, by expanding all the operators around their mean values, a i → a i,s + δa i (i = 1, 2). This yields the linearized equations in the matrix form is the vector representing the noise of the dynamical variables of the system, with the redefined fluctuations 2, . We note that the rotating-wave approximation is invoked only in Eq. (4), not in Eq. (5). In addition, we have chosen the parameter values as above to satisfy the Routh-Hurwitz criteria of stability (i.e., the eigenvalues of A have a non-positive real part), leading to stable solutions for Eq. (5) 28,30 . The fluctuations of the pump field are important only when the pump light is modulated near ω m , where a back-action force larger than ε th appears, e.g. in the process of optomechanically-induced transparency [42][43][44] . For thermal-force-driven system, the response of the system to the thermal force ε th can then be obtained by solving Eq. (5) in the frequency domain, leading to th with the Lorentzian-type mechanical susceptibility The effective mechanical frequency Ω eff and damping rate Γ eff are then given by where, 2 2 2 Figure 3 shows the exact results for Ω eff , Γ eff , by numerically solving Eqs (5)(6)(7)(8)(9)(10)(11). For comparison, the well-known results for a single lossy cavity are also shown in Fig. 3(a,b). We see that the conventional COM cooling is dominated by the optical damping, while the optical spring effect is negligible 45 . This situation is similar to that of the passive-passive resonators 39 , as shown in Fig. 3(c,d). In contrast, Fig. 3(e,f) shows that for the compound system, by approaching κ/γ = 1, the optical spring effect becomes dominant, e.g. for P in ~ 0.1 mW and κ/γ ~ 1, Ω eff of the compound system is at least 3 orders of magnitude larger than that of the conventional COM system. The sharp sign inversion for Γ eff , in the vicinity of κ/γ = 1 (Fig. 3f), means that when κ/γ < 1, the system operates in the amplified regime for the mechanical mode [see also Fig. 2(d)]; in contrast, when κ/γ > 1, it enters into the lossy regime (i.e., cooling of the mechanical mode). In other words, by surpassing the EP, the gain-to-loss compensation (for κ/γ < 1) evolves into field localizations (for κ/γ > 1), see also ref. 36. Clearly, approaching the exact balance of gain and loss (i.e., κ/γ = 1) from the right (i.e., κ/γ ≳ 1) results in efficient COM cooling due to the enhancement of both the optical spring and the effective mechanical damping.
In sharp contrast with conventional COM, the active-passive COM system features the low-power strong optical spring effect, in the vicinity of the gain-loss balance. This consequently results in a significant decrease in the phonon number For comparisons, we have calculated the phonon number n 0 in conventional COM with a lossy cavity (see the Method). When P in = 1 mW, Δ/ω m = −1 (red detuning), we find that, by starting from room temperature, n 0 can be decreased down to ~10 3 45 (see also the Method). A single active cavity, having the same COM parameters, is confirmed to exhibit similar cooling rates for blue detuning Δ/ω m = 1 (see also ref. 46). In contrast, for the compound system, the tunable parameter κ/γ provides a new way to enhance the cooling rate, without requiring blue detuning or strong COM coupling. Defining n as the phonon number in the active-passive compound system, we use β = n n / (11) 0 to evaluate the cooling efficiencies of COM systems with passive-passive cavities (i.e., κ/γ < 0) and with active-passive cavities (i.e., κ/γ > 0). When κ/γ < 1, cooling at the sidebands is only slightly enhanced, see Fig. 4(a). This situation, as aforementioned, is radically changed when surpassing the h-EP [see also Fig. 3(d)]. For example, the factor β is decreased by 3 orders of magnitude in the vicinity of the balance (κ/γ ~ 1) for P in = 0.1 mW [see Fig. 4(b)]. This cooling rate at 0.1 mW is two orders of magnitude higher than that of a single lossy cavity driven at P in = 1 mW, implying a significant benefit in terms of power budget. We have also calculated β for other values of P in , and found a reversed-dependence on P in (at ~ 0.12 mW), which is a general feature observed when a system is brought to its EPs (see also ref. 12). Nevertheless, for P in = 1 mW, more than 2-order enhancement of the cooling can still be achieved, see Fig. 4(c). The ground-state cooling, or n ~ 0.01, is accessible with P in as low as 0.1 mW, when the system is first cooled down to 650 mK [ Fig. 4(d)]. This is two orders of magnitude improvement in the cooling rate compared to ref. 8 which reported n ~ 1.7 with a power of 1.4 mW when the system temperature was 650 mK. Our approach utilizing an EP of order 3 also allows achieving n ≲ 1 with a power of 0.12 mW at temperatures as high as 20 K (Fig. 4(d), see also the Method). This opens up the prospect to engineer low-power micromechanical devices based on exotic high-order EPs.

Discussion
In conclusion, we have studied the emergence and applications of high-order EPs in a COM device composed of coupled active and passive cavities 36 . We find that for a pump laser that is red-detuned from the cavity resonance by the mechanical frequency (i.e., Δ/ω m ~ −1), the supermode structure of the system radically changes, featuring emergence of h-EP. We stress that in such a non-Hermitian COM system, the high-order EPs (i.e., the coalesce of three super-modes) appear only when the optical red detuning matches with the mechanical resonance frequency; otherwise, it is reduced to the lowest EP of order-2, i.e., the coalesce of two optical-like super-modes, which is similar to a purely optical system. This new possibility of observing the tunable high-order EPs in COM has not been reported previously. In addition, we find that in the vicinity of high-order EPs, significant enhancement of both the optical spring and the mechanical damping leads to a new route to achieve low-power mechanical cooling. Our findings open up novel prospects for applications of h-EP in realizing low-power COM or quantum acoustic devices. Future efforts along this direction include the study of e.g., h-EP in nonlinear COM [47][48][49][50] or phononic engineering by dynamically encircling the EPs 31, 32 .

Methods
More details about the supermode structure. By linearizing the COM coupling ξ † a a x 2 2 in the optical red-detuning regime, the non-Hermitian Hamiltonian of the system, including the optical gain and loss, can be written at the simplest level as (ħ = 1) where G = a 2,s ξx 0 , x 0 = (2mω m ) −1/2 , Δ = ω L −ω c , and b is the annihilation operator of the mechanical mode, with x = x 0 (b + b † ). The optical weak driving terms are not explicitly shown here. This linearized three-mode Hamiltonian, with noncyclic inter-mode interactions, can be exactly diagonalized in the super-mode picture [51][52][53] . The resulting super-mode frequency ω is determined by the determinant equation m or, more explicitly, the cubic equation Clearly, for ξ = 0, we have m and the familiar results in an all-optical two-mode system can be obtained 36 , where only an EP of the lowest-order 2 exists for the optical supermodes = ± ± a a a ( ) / 2 1 2 . The difference between the resonance frequencies and between the linewidths of the supermodes in the strong and weak coupling regimes are, respectively, given by As shown in Fig. 2(a,b,f), a similar supermode-splitting feature also appears for the far-off-resonance case, with ξ ≠ 0. For an analytical confirmation, we take as a specific example J/γ ~ 1 and Δ/ω m ~ −0.5. To solve the cubic equation Eq. (14), we introduce = − λ w x 3 1 , so that Eq. (14) can be written as By using Cardano's formula 54 , we have the solutions for Eq. (18) where ω κ γ ω  ω ω  This indicates that Except for an almost unchanged supermode (with frequency ω 0 ), the familiar two-order EP can be observed for the supermodes a ± , which is strongly reminiscent of what has been observed in all-optical devices 36 . Hence, we refer to a ± as the optical supermodes (i.e., the imaginary parts of the eigenfrequencies bifurcate and lead to field localization when surpassing the EP) [see also Fig. 2(a,b) of the main text]. We have confirmed that these analytical results agree well with the full numerical results as shown in Fig. 2 of the main text. The exact analytical solutions of the cubic equation, including the complicated results for the case with Δ/ω m ~ −1 and the cumbersome relations between (a ± , a 0 ) and (a 1,2 , b) [51][52][53] , are irrelevant to our work here. For the full numerical results of the supermode spectrum, see Fig. 2 of the main text.
Phonon number in a single passive resonator. For comparison purposes, we give here the minimum attainable phonon number n 0 in the conventional COM composed of a lossy cavity [ Fig. 5(a)] using the same system parameters given in the main text. We find that n 0 can be decreased down to ~10 3 for P in = 1 mW, when the system is initially at room temperature T = 300 K. This agrees well with the experiment reported in ref. 45. As shown in the main text, the cooling rate for the active-passive-coupled system driven by a lower power P in = 0.1 mW is two orders of magnitude higher than the cooling rate achieved for a lossy cavity (conventional COM) driven by a higher power P in = 1 mW. This implies a significant benefit in terms of power budget [see also