Defect-induced exceptional point in phonon lasing H. Lu¨1,5, S. K. O¨zdemir2,∗, Franco Nori3,6, L.-M. Kuang4, and H. Jing4,∗ 1Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China 2Electrical and Systems Engineering, Washington University, St. Louis, Missouri 63130, USA 3CEMS, RIKEN, Saitama 351-0198, Japan 4Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China 5University of Chinese Academy of Sciences, Beijing 100049, China and 6Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA 7 Exceptionalpoints(EPs)arenon-Hermitiandegeneracieswhichoccurwhentwoormoremodesof 1 aphysicalsystemcoalesceintheirresonantfrequencyandtheirrateofdecay. Insystemswherethe 0 2 loss of one mode is balanced by the gain in another mode, EPs are generally referred to as parity- time (PT) symmetry breaking point. EPs lead to many counter-intuitivephenomena such as loss- n induced transparency, invisible sensing, loss-induced optical revival, PT lasing or mode switching, a andtopological energytransfer. Ontheotherhand,theemerging field ofoptomechanics hasled to J many important applications, including the striking observation of a phonon laser, which provides 0 thecoretechnologytointegratecoherentphononsourcesanddevices. However,defectsinmaterials, 3 which have been traditionally considered to be detrimental for achieving phonon lasers, have been neglected in previous works. Here we show that contrary to this traditional view, material defects ] h can lead to the emergence of an EP, beyond which the mechanical gain and the phonon number p start to increase significantly with increasing loss. This indicates a novel and counterintuitive way - to improve the performance of a phonon laser i.e., by tuning lossy defects, instead of adding any t active dopants. In view of the controllability of defects via external electric fields, this also opens n a theway for electrically-tuned phonon lasing in optomechanical resonators. u q [ Rapid advances in cavity optomechanics (COM) [1, 2] to provide thresholdless -symmetric phonon lasing. PT inthepastdecadehaveledtonovelapplications,suchas More recently, we showed that [24] the same system can 2 quantumground-statecooling[3–5],single-photontrans- give rise to a high-order EP (i.e., with three coalesing v port [6], ultra-sensitive measurement [7–9], and phonon eigenfrequencies and eigenvectors) that significantly in- 0 0 lasing[10–16]. Asanacousticanalogofphotonlasers,co- creases phonon cooling rates. The required optical gain 0 herent stimulated amplification of sound (phonon laser insuch -symmetriccoupledCOMsystemsisprovided PT 8 or phaser) was demonstrated in various COM systems either by doping optically-active materials (e.g., rare- 0 [13, 17, 18]. Mode competition [17] and sub-Poissonian earth ions, dyes, etc) into the resonator or by using the . 1 distribution [18] have been observed for phonons, open- intrinsic gain mechanisms of the material the resonator 0 ing up the way to study nonlinear phononics and to is made from, such as Raman and parametric gain. 7 build functional phonon devices. The first COM exper- All experimental and theoretical analysis on phonon 1 iment on phonon lasing was based on two coupled opti- lasers have so far neglected the impact of two-level- : v cal microresonators,one of which supports a mechanical systems (TLSs) emerged due to defects, which naturally i mode[13]. Thiscompoundsystemhastwoopticalsuper- exist in bulk amorphous materials and mechanical res- X modes, with the frequency difference ∆ω determined by onators. Here we propose to use intrinsic TLS defect r a the inter-cavity coupling strength [19, 20]. When ∆ω is states for steering a coupled optomechanical system to equaltothefrequencyofthemechanicalmode,theCOM oneofitsEPstocontroloptomechanicalinteractionsand coupling enables energy exchange between the optical for obtaining -symmetric phonon lasing without in- supermodes through phonon-mediated transitions [13], troducingdopaPnTts. TLSdefectswerefirststudiedwithin leadingtolow-thresholdphononamplificationandlasing the context of thermal and acoustic properties of amor- [13, 21, 22]. This is analogous to a photon laser, where phous glasses at cryogenic temperatures [25–27]. They transitions between two electronic levels of an atom is are ubiquitous in amorphous materials which are used mediated by photon absorption and emission processes. in the fabricationof nanoelectromechanicaland optome- Itwasdemonstratedtheoretically[22,23]thatiftheopti- chanicalsystems,andquantumdevices. TLSscancouple callossoftheresonatorsupportingthe mechanicalmode to different modes of a system via different mechanisms, is balanced with gain provided by the other resonator, e.g., to superconducting qubits via electric dipole mo- optomechanical interactions and gain can be enhanced ment and to a mechanical mode via strain or deforma- tion force. For many years, they were considered as the source of loss and decoherence, and as such, techniques have been developed to decrease the number of TLS, if ∗Correspondingauthors: [email protected];[email protected] not to remove them completely, from the material sys- 2 a phonon lasing regime due to coupling between the lossy tapered fiber TLSmodeandtheactivemechanicalmode,whichcreates an effective parity-time ( )-symmetric system. This is PT a stronglyreminiscentofthe situationencounteredinloss- 1 induced suppression and revival of optical lasing in the vicinityofanEP[37]. TheunconventionalroleoftheEP J a2 m here is identified as the defect-induced enhancement of defect phonon lasing, including the enhanced mechanical gain and the lowered threshold power. Our results indicate R e that the performance of phonon lasers can be enhanced g and controlled by manipulating lossy defects and hence theTLSdefectsinthe resonatorsupportingthemechan- ical mode. This opens the possibility of tuning phonon b G - laserselectricallyduetothefactthatTLSdefectscanbe 0 g coherentlydrivenbymicrowavefields. Wenoteherethat a J a b d both of the resonators forming the COM device in this 2 1 - study are lossy optical resonators. This is different from theCOMdeviceusedforthe -symmetricphononlaser active Jaynes-Cummings model PT in Ref.[22], where one of the resonatorshad optical gain provided by dopants to compensate the loss of the other Figure 1: Defect-assisted phonon lasing. (a) The COM resonator. systemforphononlasingiscomposedoftwolossyopticalmi- croresonators coupled to each other with an inter-resonator tunnelingrate J. These microresonators havethesame opti- Results cal decay rate γ. The resonator coupled to a tapered optical fiberanddrivenbyapumpfieldwithfrequencyωl supportsa System. The system we consider in this study is de- mechanicalmodeatfrequencyωm [13]andcontainsadefect- induced TLS. The intracavity optical fields of the resonators picted in Fig.1. Two whispering-gallery-mode (WGM) are denoted by a1 and a2, the phonon annihilation operator microresonatorshavingthe sameresonancefrequency ωc isb,andRistheradiusoftheresonators. (b)Theinteraction andthe loss rate γ arecoupled to eachother with a cou- modelforthedefect-assistedphononlasing: theopticalmode pling strength of J which can be tuned by varying the a1, which is coupled to the mode a2 with strength J, inter- distance between the resonators. One of the resonators acts with the mechanical mode b, which, in turn, is coupled supports a mechanical mode with effective mass m, fre- to the defect TLS, represented by the Pauli operator σ−, by quencyω anddampingrateγ . Weconsiderresonators a coupling strength gd. The decay rates of thephonon mode made of smilica, silicon, or silicomn nitride, where intrinsic atenrdmtγhm′e d=efγemct−arGe0diesntohteedeffbeyctγivmeamnedchγaqn,irceaslpdeacmtivpeinlyg.rTathee, TLSs exist. Also, the resonatorsupporting the mechani- calmodehasadefectthatiscoupledtothelocalphonon where G0 corresponds to mechanical gain. mode via mechanical strain, with the coupling strength [34]: tem of interest [28]. However, recent studies have shown that TLS can play useful roles. For example, they have DT ∆0 ~ωm g , (1) been used as quantum memory in superconducting cir- d ≈ ~ ω r2YV q m cuits [29, 30] and as random-defect lasers [31]. In COM devices, the presence of TLS defects can affect COM in- where ωq is the tunable energy difference between the teractions [32], mechanical resonance and damping [9], excitedandgroundstatesofthedefect-inducedTLS[27], ground-state cooling [33–35], and nonlinear responses of DT denotes the potential of mechanical deformation, Y COM devices [36]. We note that all previous studies on is the Young’s modulus of the material, and Vm is the phononlasershavesofarneglectedtheroleofdefectnat- mechanical mode volume [36]. urally occurring in the materials used to fabricate COM In the rotating frame at the pump frequency ωl, the resonators. Hamiltonian of this system, which is composed of two Here,weexploretheimpactofintrinsicdefect-induced optical modes, one mechanical mode and one TLS, can TLSonphononlasingincompoundmicroresonators. We be written as demonstratetheoreticallythatwhenthedampingrateof the TLS defect is below a critical value, the mechanical H =H0+Hint+Hdr, gain of the system is suppressed with increasing damp- H = ∆(a†a +a†a )+ω b†b+ ωqσ , ing. Beyondthiscriticaldampingrate,however,increas- 0 − 1 1 2 2 m 2 z ing the damping rate enhances the mechanical gain. We H =J(a†a +a†a ) ξa†a x+g (b†σ +σ b), show that this counterintuitive phenomenon is the re- int 1 2 2 1 − 1 1 d − + H =i(ε a† ε∗a ), (2) sultoftheemergenceofanexceptionalpoint(EP)inthe dr l 1− l 1 3 wherea ,a orbdenotetheannihilationoperatorsofthe 1 2 a optical modes or the mechanical mode, x = x (b† +b) 0 is the mechanical displacement operator, ∆ ω ω denotes the frequency detuning between the p≡umpl −lasecr Hz) andthecavityresonance,ξ =ω /RistheCOMcoupling M c ( astnrdenσgzt,h,σR− aisndthσe+reasorenatthoerPraaduiluiso,pxe0ra=to(r1s/o2fmthωem)T1L/2S, gain (see Fig.1) given by σz = |eihe|−|gihg|, σ− = |gihe|, cial and σ+ = e g . The field amplitude of the pump light n is given by|εih=|(2Pγ/~ω )1/2, where P is the power of ha l l l l c the pump. e M Definingthesupermodeoperatorsfortheopticalfields asa± =(a1 a2)/√2andapplyingtherotating-waveap- lasing regime ± proximation,weobtaintheeffectiveinteractionHamilto- nian of the system: Δ/ω m ξx = 0(a†a b+b†a a† )+g (b†σ +σ b). (3) Hint − 2 + − + − d − + b The first term in this expression describes the phonon- ) z mediatedtransitionbetweenopticalsupermodes,andthe H M second term describes the coupling between the phonon ( and TLS defect. Thus, in the supermode picture, the n ai optomechanical coupling is transformed into an effective g coupling describing defect-assisted phonon lasing. The cial TLS can be excited by absorbing a phonon generated n a h fromthetransitionbetweentheupperopticalsupermode c e and the lower one, so that the behaviour of the phonon M lasing is strongly modified. Bysettingthetimederivativeoftheoperatorsa ,a , + − σ , andp=a† a tozero,wecanobtainthemechanical − − + γ / γ gain. As described in the Methods and the Supplement, q the mechanical gain of this system is given by Figure 2: Mechanical gain in the defect-assisted G=G +G , phonon laser. (a) The mechanical gains G0 and G as a 0 d function of the optical detuning ∆. (b) The mechanical gain ξ2x2γ ∆(2J ω )ε 2 G = 0 δn − m | l| , G as a function of theTLS decay rate γq. We choose γq =γ 0 2(2J −ωm)2+8γ2 (cid:20) − α2+4∆2γ2 (cid:21) and gd =1 MHz in (a). The inter-resonator coupling rate is g2γ set as J = 0.5ωm in (a,b) and the optical detuning is set as Gd =− γ2+(ω ωd q)2+2g2n , (4) ∆ = 0.5ωm in (b). Other tunable parameters are chosen as q q− m d b ωq =ωm and Pl =10µW. where γ is the defect TLSdecayrate,andδn=a†a q + +− a†−a− is the population inversion operator, and With the mechanical gain at hand, we can derive the threshold pump power P from the threshold condition ξ2x2 th α=J2+γ2 ∆2+ 0nb, G = γm (see the supplementary materials). By setting − 4 J =ω /2 and ω =ω , the threshold power P can be m q m th withthephononnumbern =b†b(seethesupplementary simplified to b materials). Ifthereisnodefect,i.e. g =0,wehaveG= d 8γ2(ω +J) g2γ G0,aswasalreadyobtainedinRef.[21]. Thesecondterm P c γ + d q . (5) ofG0,describingtheopticaldetuningeffect,canbemade th ≈ (ξx0)2 (cid:18) m γq2+2gd2nb(cid:19) positive or negative by tuning the optical tunneling rate J (i.e., coupling strength of the resonators) by changing Mechanical gain. Figure 2(a) shows the mechanical thedistancebetweentheresonators. Notethatthisterm gain G and G as a function of the optical detuning ∆ 0 ispositivewhen2J ω >0,negativewhen2J ω <0, with J = 0.5ω , using the experimentally feasible pa- m m m − − and zero when δ =0 or 2J =ω . In the latter case, G rameter values [13, 20], i.e. R = 34.5µm, m = 50ng, m 0 is described with only the first term, i.e. G = ξ2x2/8γ, ω = 193THz, ω = 2π 23.4MHz, γ = 6.43MHz and 0 0 c m × as given in Ref.[13]. The inevitable effect of material γ = 0.24MHz. Clearly, in the cooling regime (with m defects on the phonon lasing, as described by G , has ∆ < 0), G is negative and can be further enhanced by d not been studied before. the defects (as firstly revealed in Ref.[36]). The positive 4 gain in the lasing regime (with ∆ > 0) is also signifi- a cantly affected by the TLS-phononcoupling. In particu- lar,the phononlasing is mostpreferableat∆/ω 0.5, m ∼ where the defect-induced reduction in G is minimized. ) z If we further set J = 0.5ωm, optimized phonon lasing H G is achieved (see the supplementary materials). More in- ( terestingly, Fig.2(b) shows an unexpected evolution for E ) ± G with increasing defect loss rate: the mechanical gain ( e G first decreases with increasing TLS decay rate, until a R criticalvalue of γ . When this exceeds the criticalvalue, q Gincreaseswithincreasingdefectloss. Consequentlythe phononlaserthresholdpowerP firstincreasesandthen th decreases again with increasing TLS damping, as shown γ / γ in the supplementary materials. These counterintuitive q results are reminiscent of the loss-induced suppression and revival of lasing [37, 38] and the loss-induced trans- parency [39] demonstrated previously in coupled optical b resonators and coupled waveguides. Supermode spectrum. To understand this unconven- tional loss-induced effect intuitively, we use a simplified ) z modeltodescribethedefect-assistedlasingprocess,with H M only the active phonon mode and the lossy TLS. The ( effective Hamiltonian of this simplified model reads E ) ± ( m Heff =(ωm−iγm′ )b†b+(ωq−iγq)σ+σ−+gd(b†σ++σ−b), I (6) with the effective mechanical damping rate γ′ =γ G . (7) m m− 0 γq / γ We note that a similar route was adopted in a re- centexperimentfor an(anti-) -symmetric atomicsys- Figure3: Super-mode spectrum. Real(a)andimaginary tem [40] where, by starting frPoTm a Hermitian Hamilto- (b)partsoftheeigenvaluesofHeff. Theotherparametersare nian describing atom-lightinteractions,an effective non- chosen as J =0.5ωm, γq =γ,and Pl =7µW. Hermitian Hamiltonian was deduced for two spin-wave excitationchannels(seealsoRef.[41]). Choosingtwoba- γ increases beyond the critical value, the supermodes q sis states n ,g and n 1,e to diagonalize leads | b i | b− i Heff become increasingly localized such that one dominantly to the eigenvalues resides in the optomechanical resonator and the other in the TLS. This is the broken- phase. As a result, E = n 1 ω + ωq i [(2n 1)γ′ +γ ] the supermode in the optomechPanTical resonator experi- ± (cid:18) b− 2(cid:19) m 2 − 2 b− m q ences less and less loss with increasing γ , leading to an q 1 4n g2+[ω ω i(γ γ′ )]2 (8) increasing net mechanical gain. The supermode which ± 2q b d q− m− q− m is dominant in the TLS, on the other hand, experiences increasinglymorelossandhasnegligibleeffectontheme- The supermode spectrum (real and imaginary parts of chanical gain. The transition from the exact- to broken- theseeigenvalues)isshowninFig.3(a,b),whereanEPis phasetakesplaceatanEP[42–44],andwelabelthe clearly seen, at the positions close to the turning points PcrTitical value of γ at the EP as γEP. Thus in this com- in Fig.2(b). The emergence of an EP in this system can q q positesystemoftheoptomechanicalresonatoranddefect be explained as follows: The composite system corre- TLS, the region γ <γEP corresponds to the exact - spondstoa -symmetricsystemwherethemechanical q q PT PT phase, whereas the region where γ > γEP corresponds gain in the optomechanical resonator partially or com- q q to the broken- phase. Note that at ω =ω , the EP ipslestmelayllecromthpaennasactreistitchaelvlaolsuseo,fththeesydsetfeemctiTsLinSt.hWeuhnenbrγoq- of the system ePmTerges at γqEP =γm′ +2gdq√nb,mwhile the turning point of G is obtained, by setting ∂G/∂γ = 0, ken (exact) -symmetric phase where the supermodes q PT as arealmostequallydistributedbetweentheoptomechani- calresonatorandthe defect TLS, andthe active phonon γmin =√2n g . (9) q b d mode partially or completely compensates the loss in- ducedbythedefectTLS.Asaresult,thesystemhasless The slight shift of the turning point from the exact EP net mechanical gain as the defect loss γ is increased. If positionisduetothe factthatγmin depends on∆,while q q 5 the lossy defect mode, coupled via the mechanical strain [36],canbeenforcedintoanunconventional -breaking PT regime, where both the mechanical gain and the phonon EP number are enhanced despite increasing loss. Ourfindingssuggestanacousticanalogofloss-induced suppression and revival of an optical laser demonstrated EP inexperiments [37,38]. This indicates a novelandcoun- terintuitivewayto improvethe performanceofaphonon laser; that is by tuning the lossy defects, instead of adding any active dopants. This is important because the TLS energy splitting and damping rate are not only determined by the material strain [36, 46], but they can alsobe controlledby steering externalelectric fields [31], opening the way for electrically-tuned phonon lasing in COM systems. Our future works along this direction in- cludesthestudyofEP-assistedquantumnonlinearCOM [47, 48], the defect-induced transparency [2], and the defect-mediated sensing or transducing [49, 50]. Finally, Figure 4: Stimulated emitted phonon number. The we note that although we have developed the formalism phonon number Nb versus the pump power Pl (a) and the damping rate γq (b), with the fixed value ωq/ωm = 1. Also for COM systems, the same concept can be extended wetakePl =10µW in (a),andγq/γ =1,gd =1MHzin (b). tonanoelectromechanicaldevicesinwhichphononlasing has been achieved [15]. theEPdoesnot(seealsoRef.[37]andthesupplementary materials). Wenotethatthemechanismoutlinedherefor Methods ourcompositesystemformedbytheCOMresonatorand defect is similar to that which led to an EP observed in Optomechanics in the supermode picture. To de- anopenquantumsystemcomposedofa single atomand rivetheHamiltonianinthesupermodepicture,wedefine a high-Q optical cavity [45]. thesupermodeoperatora =(a a )/√2,whichtrans- ± 1 2 Phononnumberinthelasingregion. Figure4shows forms H and H into ± 0 dr the dependence of the phonon number ω =ω a†a +ω a†a +ω b†b+ qσ , N =exp[2(G γ )/γ ] H0 + + + − − − m 2 z b m m − i = ε (a† +a† ) h.c. , (10) on the defect loss and the pump power. Features sim- Hdr √2h l + − − i ilar to those observed for the mechanical gain also ap- pear for N , i.e. more loss leads to the suppression of with ω = ∆ J. Similarly, H becomes b ± int − ± N in the -symmetric regime but it is enhanced in b PT ξx ethxeacPtTco-rbrreesapkoinndgernecgeimweit.hTthheattuofnitnhgepmoeinchtaonficNalbgisaiinn, Hint =− 20 h(a†+a++a†−a−)−(a†+a−+H.c.)(b†+b)i as shown in Fig. 2(b) or the threshold power (see the +g (b†σ +σ b). (11) d − + supplementary material). Figure 4(b) shows that N is b strongly dependent on the optical detuning, and the op- In the rotating frame with respect to 0, we have H timizedcondition∆=0.5ω ,asinthesituationwithout m ξx any defect, still holds in this system. = 0 a†a bei(2J−ωm)t+H.c. Hint − 2 (cid:16) + − (cid:17) ξx 0 a†a b†ei(2J+ωm)t+H.c. Discussion − 2 (cid:16) + − (cid:17) ξx + 0 a†a +a† a b†eiωmt+H.c. In summary, we have studied the defect-assisted 2 (cid:16) + + − −(cid:17)(cid:0) (cid:1) phonon lasing in coupled passive resonators. We find +g b†σ ei(ωm−ωq)t+H.c. . (12) d − that,incontrasttointuitiveexpectations,themechanical h i gainis notalwayssuppressedwithmoredefectloss. The Considering the rotating-wave approximation exactevolutionsofthemechanicalgainandthethreshold power,startingfromtheHermitianfullsystem,depictsa 2J +ωm,ωm 2J ωm , ωq ωm , ≫| − | | − | turningpointasthelossisincreased. Thisisfoundtobe we have closely related to the emergence of an EP in an effective non-Hermitianphonon-defectsystem. Bysuitablychoos- ξx ing the system parameters,the active phonon mode and Hint =− 20(a†+a−b+b†a+a†−)+gd(b†σ−+σ+b). (13) 6 TheHeisenberg-Langevinequationsofmotionofthissys- Here a , b , Γ , and Γ denote environmental noises in in − z tem are given by corresponding to the operators a, b, σ and σ . We as- − z sume that the mean values of these noise operators are iξx ε 0 l zero, i.e. a˙ =( iω γ)a + a b+ +√γa , + + + − in − − 2 √2 a˙− =( iω− γ)a−+ iξx0a+b†+ εl +√γain, haini=hbini=hΓ−i=hΓzi=0. − − 2 √2 b˙ =( iω γ )b+ iξx0a†a ig σ + 2γ b , The fluctuations are small and we neglect the noise op- − m− m 2 + −− d − m in erators in our numerical calculations. Then the defect- p σ˙ =( iω γ )σ +ig bσ + 2γ Γ , assisted mechanical gain and the threshold power of the − q q − d z q − − − phonon lasing can be obtained, see the main text (for σ˙ = 2γ (σ +1) 2ig (σ b pb†σ )+ 2γ Γ . z q z d + − q z more details of the derivations and more related results, − − − p (14) see the Supplementary Materials). [1] Kippenberg,T.J. &Vahala, T.J. Cavityoptomechanics: Rev. Lett. 104, 083901 (2010). back-action at themesoscale. Science 321, 1172 (2008). [14] Khurgin,J.B.,Pruessner,M.W.,Stievater,T.H.&Ra- [2] Aspelmeyer, M., Kippenberg, T. 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H. 035417 (2011). J. is supported by the National Natural Science Foun- [35] You, J. Q., Liu, Y. X. & Nori, F. Simultaneous cooling dation of China under Grand numbers 11274098 and ofanartificialatomanditsneighboringquantumsystem. 11474087. S. K. O¨. is supported by ARO Grant No. Phys. Rev. Lett. 100, 047001 (2008). W911NF-16-1-0339. F. N. is supported by the RIKEN [36] Ramos, T., Sudhir, V., Stannigel, K., Zoller, P. & Kip- iTHESProject,theMURICenterforDynamicMagneto- penberg, T. J. Nonlinear quantum optomechanics via in- OpticsviatheAFOSRawardnumberFA9550-14-1-0040, dividual intrinsic two-level defects. Phys. Rev. Lett. 110, the IMPACT program of JST, CREST, and a Grant-in- 193602 (2013). Aid for Scientific Research (A). [37] Peng, B., O¨zdemir, S¸. K., Rotter, S., Yilmaz, H., Liertzer, M., Monifi, F., Bender, C. M., Nori, F. & Yang, L.Loss-induced suppression andrevivaloflasing. Science 346, 328 (2014). Author contributions statement [38] Brandstetter, M., Liertzer, M., Deutsch, C., Klang, P., Sch¨oberl, J., Tu¨reci, H. E., Strasser, G., Unterrainer, K. H.J.conceivedtheideaandperformedthecalculations &Rotter,S.Reversingthepumpdependenceofalaserat with the help from H. L.; H. J. and S. K. O¨. wrote the an exceptional point. Nat. Commun. 5, 4034 (2014). manuscript with input from F. N. and L. M. K.; all the [39] Guo, A., Salamo, G. J., Duchesne, D., Morandotti, authors discussed the content of the manuscript. R., Volatier-Ravat, M., Aimez, V., Siviloglou, G. A. & Christodoulides, D. N. Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. Additional information 103, 093902 (2009). [40] Peng, P., Cao, W., Shen, C., Qu, W., Wen, J., Jiang, L. & Xiao, Y. Anti-paritytime symmetry with flying atoms. Competing financial interests: The authors declare no Nat. Phys. 12, 1139 (2016). competing financial interests. [41] Kreibich, M., Main, J., Cartarius, H. & Wunner, G. Hermitian four-well potential as a realization of a PT- symmetric system. Phys. Rev. A 87, 051601 (2013). [42] Bender, C. M. & Boettcher, S. Real spectra in non- HermitianHamiltonianshavingPTsymmetry.Phys.Rev. Lett. 80, 5243 (1998). [43] Bender,C.M.,Boettcher,S.&Meisinger,P.N.,J.Math. Phys. (N.Y.) 40, 2201 (1999). 8 Supplementary Materials for “Defect-induced exceptional point in phonon lasing” I. DERIVATION OF THE MECHANICAL GAIN In the super-mode picture, a crucial term describing the phonon-lasing process can be resonantly chosen from the Hamiltonian, under the rotating-wave approximation [S1] (for J ω /2, ω ω , see the Methods). With the m q m ∼ ∼ supermode operators p=a† a , a =(a a )/√2, the reduced interaction Hamiltonian is given by − + ± 1± 2 ξx = 0(p†b+b†p)+g (b†σ +σ b). (S1) int d − + H − 2 The resulting Heisenberg equations of motion then read as [S1] iξx 1 p˙ =( 2iJ 2γ)p 0δnb+ (ε∗a +ε a† ), − − − 2 √2 l + l − iξx b˙ =( iω γ )b+ 0p ig σ , m m d − − − 2 − σ˙ =( iω γ )σ +ig bσ , − q q − d z − − σ˙ = 2γ (σ +1) 2ig (σ b b†σ ), (S2) z q z d + − − − − where δn = a†a a† a denotes the population inversion operator and γ is the decay rate of the TLS. The noise + +− − − q terms are negligible with a strong driven field. To obtain the mechanical gain, we set ∂p/∂t = 0,∂σ /∂t = 0, and − ∂a /∂t=0 with γ,γ γ , which leads to the steady-state values ± q m ≫ 1 iξx 1 p= 0δnb+ (ε∗a +ε a† ) , i(2J ωm)+2γ (cid:20)− 2 √2 l + l − (cid:21) − g (ω ω )+ig γ d q m d q σ = − b, − −γ2+(ω ω )2+2g2n q q− m d b ǫ (2iω +2γ+iξx b) l − 0 a = , + 2√2α i4√2γ∆ − ǫ (2iω +2γ+iξx b†) l + 0 a = , (S3) − 2√2α i4√2γ∆ − with n =b†b and b ξ2x2 ω = ∆ J, α=J2+γ2 ∆2+ 0n . ± b − ± − 4 Substituting Eq. (S3) into the dynamical equation of the mechanical mode results in b˙ =( iω +iω′+G γ )b+C, (S4) m m − − where g2(ω ω ) ξ2x2(2J ω ) ξ2x2∆ε 2 ω′ = d q− m 0 − m 0 | l| , γ2+(ω ω )2+2g2n − 16γ2+4(2J ω )2 − [2(2J ω )2+8γ2](α2+4∆2γ2) q q − m d b − m − m iε 2ξx (γ iJ)α+2∆2γ l 0 C = | | − , 2i(2J ω )+4γ · α2+4∆2γ2 m − and the mechanical gain G=G +G with 0 d ξ2x2γδn ∆γ(2J ω )ξ2x2 ε 2 G = 0 − m 0| l| , 0 2(2J ω )2+8γ2 − [2(2J ω )2+8γ2](α2+4∆2γ2) m m − − g2γ G = d q . (S5) d − γ2+(ω ω )2+2g2n q q− m d b Fig.S1(a)showsG asa function of∆ with differentinter-resonatortunneling rate J, indicating anoptimized phonon lasing can be achieved by choosing J/ω =0.5. m 9 II. THRESHOLD OF THE DEFECT-ASSISTED PHONON LASER With the mechanical gain at hand, we can derive the threshold pump power P from the threshold condition th G=γ [S1], i.e. m P =P +P th th,0 th,d 2 (2J ω )2+4γ2 (ω +J)γ ~∆(2J ω )(ω +J)ε 2 P = − m c m + − m c | l| , th,0 (cid:2) (ξx )2 (cid:3) α2+4∆2γ2 0 2g2γ (ω +J) (2J ω )2+4γ2 P = d q c − m . (S6) th,d ξ2x2 γ2+(ω(cid:2) ω )2+2g2n (cid:3) 0 q m− q d b (cid:2) (cid:3) The first term of P is exactly the same as given in Ref.[S1] for ∆ = 0. Clearly, even for this resonant case, both th,0 the mechanical gain G and the threshold power P can be obviously different due to the defects. th We plotthe thresholdvalue P as a function ofthe TLS decayrate inFig.S1(b). The thresholdfirstincreasesbut th then decreases again, which is reminiscent of the loss-induced suppression and revival of an optical Raman laser as demonstrated recently in coupled passive resonators [S2]. a b ) z H M ( n W) ai μ g ( al P th c ni a h c e M Δ/ω γ / γ m q Figure S1: (a) Mechanical gain in the defect-assisted phonon laser. (b) The threshold power of the defect-assisted phonon laser. The parameters are γq =γ, gd =1MHz in (a) and J =0.5ωm, ∆=0.5ωm in (b). Wechoose ωq =ωm and Pl =10µW in both thetwo figures . III. TURNING POINTS OF THE MECHANICAL GAIN Figure2(b) ofthe maintext showsthe mechanicalgainas a functionofthe TLSdecayrate,in whichthe minimum value ofmechanicalgaindoes notcorrespondto the EPofthe effective -symmetricsystemcomposedofthe active PT mechanical mode and the lossy TLS model. For ω =ω , we set ∂G/∂γ =0 and obtain the turning point, i.e. q m q γmin =√2n g , (S7) q b d while the EP emerges at γEP =γ′ +2√n g , (S8) q m b d whereγ′ =γ G ,andn is the stationaryvalue ofthe phononnumber. FigureS2 showsthe turningpoints ofthe m m− 0 b mechanicalgainand EPswith different detuning. We can see that the turning points and EPschanges with different detuning. This slight shift of the turning point from the exact position of the EP was also observed in Ref.[S2]. If γ < γmin, the system has less mechanical gain as the defect loss γ increases, which is due to the -symmetric q q q PT phase, where the active phonon mode partially or completely compensates the loss induced by the defect TLS. If 10 γ exeeds γmin until reaches γEP, the supermode distributions are strongly affected by γ , which results in a slight q q q q growthof the mechanical gain. If γ reaches γEP, the system enters into the broken- -symmetric phase, leading to q q PT the obvious growth of the mechanical gain. ) z H M ( n ai g al ci n a h c e M γ / γ q FigureS2: Mechanicalgaininthedefect-assisted phononlaserasafunctionoftheTLSdecayrateγq,fordifferentdetunings. The parameters are J =0.5ωm, ωq =ωm and Pl =10µW. [S1] Grudinin, I. S., Lee, H., Painter, O. & Vahala, K. J. Phonon laser action in a tunable two-level system. Phys. Rev. Lett. 104, 083901 (2010). [S2] Peng,B.,O¨zdemir,S¸.K.,Rotter,S.,Yilmaz,H.,Liertzer,M.,Monifi,F.,Bender,C.M.,Nori,F.&Yang,L.Loss-induced suppression and revival of lasing. Science 346, 328 (2014).