Observing exceptional point degeneracy of radiation with electrically pumped photonic crystal coupled-nanocavity lasers
Kenta Takata, Kengo Nozaki, Eiichi Kuramochi, Shinji Matsuo, Koji Takeda, Takuro Fujii, Shota Kita, Akihiko Shinya, Masaya Notomi
OObserving exceptional point degeneracy of radiation with electrically pumped photonic crystal coupled-nanocavity lasers
Kenta Takata , Kengo Nozaki , Eiichi Kuramochi , Shinji Matsuo , Koji Takeda , Takuro Fujii , Shota Kita , Akihiko Shinya , and Masaya Notomi Nanophotonics Center, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. NTT Device Technology Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan. Correspondence and requests for materials should be addressed to K. T. (email: [email protected]) or to M. N. ([email protected])
Non-Hermitian degeneracies with singularities, called exceptional points (EPs), incorporate unconventional functionalities into photonic devices. However, the radiative response under the EP degeneracy has yet to be observed due to both the limited controllability of pumping and the lifting of degeneracy by pump-induced cavity detuning. Here, we report the spontaneous emission at an EP degeneracy with two electrically pumped photonic-crystal coupled-nanocavity lasers. Systematically tuned and independent current injection to our active heterostructure nanocavities enables us to demonstrate the clear EP phase transition of their spontaneous emission, accompanied with the spectral coalescence of coupled modes and reversed pump dependence of the intensity. Furthermore, we find experimentally and confirm theoretically the peculiar squared Lorentzian emission spectrum very near the exact EP, which indicates the four-fold enhancement of the photonic local density of states induced purely by the degeneracy. Our results open a new pathway to engineer the light-matter interaction with optical non-Hermiticity. oupled optical cavities and waveguides with imaginary refractive index contrast, i.e. distributed gain and loss, can exhibit peculiar degeneracies called exceptional points (EPs), which divide their eigenmodes into two phases. One phase comprises extended supermodes with parity-time (PT) symmetry . Here, the real parts of their eigenvalues are split, and the imaginary parts are clamped at the average of the imaginary effective potential, canceling its local contribution over the unit cell (symmetric phase). In the other regime, PT symmetry is spontaneously broken, and thus the eigenstates localize at either the amplifying or de-amplifying elements (broken phase). Correspondingly, the imaginary spectrum bifurcates into two branches, after the real-part splitting is cancelled at an EP, i.e. a non-Hermitian degeneracy with a singularity. This EP transition leads to intriguing features, such as reversed pump dependence , single-mode oscillation , and enhanced sensitivity . There has also been rising interest in the photonic EP itself. Distinct from the accidental degeneracy of eigenvalues with orthogonal modes in Hermitian systems, the EP makes not only some eigenvalues but also corresponding eigenmodes identical. Thus, the effective non-Hermitian Hamiltonian becomes non-diagonalizable. The resultant nonorthogonal eigenstates surrounding the EP can enjoy optical isolation , coherent absorption , unidirectional reflectivity , and asymmetric mode conversion . Although many papers have studied phenomena around the EP, observing optical responses at the EP degeneracy has been a persistent technical challenge, even for basic two-cavity devices . The EP degeneracy is predicted to have significant influence on radiation processes . However, it is a single spot in the continuous parameter space for eigenfrequencies; therefore, fine and independent control of gain and loss is required for each cavity, which is emanding for passive scattering or optical pumping . To this end, preparing strongly coupled lasers with current injection is desirable. Meanwhile, carrier plasma and thermo-optic effects arising with asymmetric pumping induce detuning of their resonance frequencies. This active mismatch lifts directly the degeneracy of the eigenfrequencies . In addition, it results in the significant damping of one of the coupled modes , which hampers their coalescence and hence the EP response. Multiple cavity modes with comparative ππ factors are also subject to carrier-mediated mode competition that disrupts the pristine properties at the EP. In this article, we report the observation of spontaneous emission under the EP degeneracy with two current-injected photonic crystal lasers. Here, we establish the first nanocavity-based non-Hermitian platform with electrical pumping, which is based on our buried heterostructure technique . It is well known that electrically-pumped lasing in nanocavities is generally hard to achieve because of restricted gain and difficulty in thermal management. It should be noted that room-temperature continuous-wave lasing in nanocavities has been achieved only by the buried-heterostructure photonic crystals . In this study, we have achieved the first electrically-pumped coupled-nanocavity lasers, which successfully exhibit continuous-wave room-temperature oscillation, and employed them to a novel non-Hermitian platform. The buried-heterostructure design enables efficient carrier injection and high heat conductivity, which minimalize pump-induced resonance shifts and stable individual control of gain and loss for each cavity. Selective high ππ factors for their coupled ground modes are also achieved so that the mode competition is suppressed. By investigating the system with highly asymmetric pumping, we first clarify that whenever there is non-negligible cavity detuning, it is impossible for the lasing supermodes to reach any degree of non-Hermitian coalescence. In contrast, our elaborate easurement and analysis of the spontaneous emission demonstrate the distinct EP transition without severe detrimental effects and identify the fine EP location. Remarkably, we find the squared Lorentzian emission spectrum very near the exact EP, which signifies the unconventional enhancement of the photonic local density of states (LDOS) . Our results provide a new approach to handle the light-matter interaction and radiation.
Results
Theoretical backgrounds.
We consider two identically designed optical cavities with spatial proximity and imaginary potential contrast [Fig. 1(a)]. Their ground cavity modes exchange photons with evanescent waves, and thus the system eigenfrequencies are split by the mode coupling, π π . However, the gain and loss can counteract the coupling without shifting the local cavity resonances. The first-order temporal coupled mode equations (CMEs) for the complex cavity-mode amplitudes {ππ ππ (π‘π‘)} are derived as, ππππππ οΏ½ππ ππ οΏ½ = οΏ½i(ππ + πΏπΏ) β πΎπΎ iπ π iπ π i(ππ β πΏπΏ) β πΎπΎ οΏ½ οΏ½ππ ππ οΏ½, (1) where πΎπΎ ππ is the loss (positive) or gain (negative) for each cavity, and ππ is the average resonance frequency. Without loss of generality, small cavity detuning to ππ is introduced as Β±πΏπΏ β β . The model reduces to the eigenvalue problem with the ansatz (ππ ππ ) T = (π΄π΄ π΄π΄ ) T ππ ππππππ . The resultant eigen-detuning,
Ξππ ππ β‘ ππ β ππ = i(πΎπΎ + πΎπΎ )/2 Β± οΏ½π π β [(πΎπΎ β πΎπΎ )/2 β iπΏπΏ] , turns into the aforementioned EP when its second term vanishes [Fig. 1 (b)]. Here, the two eigenmodes become degenerate and chiral, (π΄π΄ π΄π΄ ) T = (1 βππ) T /β2 . Meanwhile, it is notable that πΏπΏ generally resolves the exact degeneracy and smooths the nearby singular spectrum. The EP exhibits peculiar radiation responses [Fig. 1(c)]. When the system is in the symmetric phase and the eigenfrequency splitting is large, the photonic LDOS of the two coupled odes are Lorentzian functions with ideally the same linewidth. At the non-Hermitian degeneracy, however, the two spectral peaks coalesce and interfere with each other. Thus, the resultant radiation power spectrum, which is directly relevant to the LDOS, takes on a squared Lorentzian shape. Without any net gain, the corresponding peak LDOS is increased purely by the effect of the coalescence. Such enhancement at this passive EP is four-fold, compared with each of the separate peaks in the large coupling limit. Namely, compared to the mere sum of the two Lorentzian modes (i.e. a Hermitian diabolic point of two orthogonal states), the EP with the common loss factor and the same integral intensity has a doubly high peak and οΏ½β2 β 1 β 0.644 times narrower linewidth [Fig. 1(d); see also Sec. VII of Supplementary Information]. Active cavities with spontaneous emission, i.e. flat spectral excitation via the pumped gain media, are well suited for its demonstration. In contrast, EPs with nonlinear processes can generate excess noise and result in their linewidth broadening . Instead of the theoretical set-up with a completely dark state (ππ =β) and a considerably leaky mode, we apply manageable alternatives for our experiment: pumped transparent (loss-compensated) and absorptive cavity modes. Experimental set-up and EP transition in lasing regime.
We prepared a sample comprising two coupled photonic crystal lasers based on buried heterostructure nanocavities [Fig. 2(a) and Methods]. Here, gain media with six quantum wells (colored red) were patterned by electron beam lithography and embedded in an air-suspended InP photonic crystal slab. Two line defects narrower than the lattice-matched width improve the cold ππ factors of the incorporated cavitiesβ ground modes. DC current is applied and controlled for each cavity via independent PIN junctions. Note that a single-laser device with a commensurate electric channel has a lasing threshold πΌπΌ th of about ΞΌA , at which it has a high ππ factor of 14,000 (Supplementary Sec. I). When symmetrically pumped below the threshold with 30 Β΅A for comparison, the two-laser sample gives spontaneous emission of the two coupled modes with ππ = 4,000 , slightly below that of the single diode (5,000). Their resonance peaks with a splitting of about 1.0 nm in reference to 1529.7 nm indicates π π =61 GHz (Supplementary Sec. III). It agrees well with the coupling of the simulated ground modes, π π sim =
65 GHz [Fig. 2(b) and Methods]. The near-field emission from both lasers is also observed [Fig. 2(b), inset]. We fix the injection current for one channel [ πΌπΌ for channel 2 here, left in Fig. 2(a)] and sweep that for the other ( πΌπΌ for channel 1, right), to vary the loss contrast πΎπΎ β πΎπΎ . As a result, the detected ground-mode power systematically recovers by the reduction of the local current πΌπΌ [Fig. 2(c)]. This reversed pump dependence indicates the EP transition (see also Methods and Supplementary Sec. II). Here, the effective retrieval ratio ππ R /ππ min , namely the power ππ R for zero bias along channel 1 ( ππ = 0 ) compared to the minimum ππ min , is maximized by heavy pumping πΌπΌ = 800 ΞΌA . Cavity 2 provides gain in this case; however, the system is critically affected by the cavity detuning πΏπΏ and hence misses the EP degeneracy. Fig. 2(d) depicts the device emission spectra in the lasing regime for constant πΌπΌ = 800 ΞΌA and different πΌπΌ , measured with an optical spectrum analyzer. Here, some leakage current from channel 2 induces a negative πΌπΌ β β6 ΞΌA for ππ = 0 . However, the data and hence loss πΎπΎ in cavity 1 consistently change under the reverse current. As πΌπΌ decreases from πΌπΌ = 100 ΞΌA and πΎπΎ hence increases, the blue-side peak |ππ β β© damps while the other red-side one |ππ + β© remains bright. This is a direct reflection of finite detuning πΏπΏ , with which the asymmetric pumping πΌπΌ β« πΌπΌ selectively excites the coupled mode closer to the solitary resonance of cavity 2, ππ β πΏπΏ . Eventually, the power f |ππ + β© also drops sharply around πΌπΌ = 5.4 ΞΌA , indicating the suppression of oscillation. However, it is |ππ β β© that undergoes the revival of lasing with a kinked rise in power and linewidth narrowing (Supplementary Sec. X). Such switching of the dominant mode has been observed in relevant studies and attributed to the pump-induced sign flip of πΏπΏ , namely the crossing of cavity-mode resonances. The restored peak moves toward the middle of the original supermodes with the growing contrast of pumping and hence evidences the EP transition in our device. The near-field patterns for selected πΌπΌ [Fig. 2 (e)] not only show the above-mentioned processes in the real space but also exhibit clear mode localization at cavity 2 in the intensity recovery, supporting the PT symmetry breaking. The steady oscillation condition Im ππ ππ = 0 enables us to estimate the eigenfrequencies ππ ππ for the lasing spectra , despite that the system here provides an adaptive (variable) gain πΎπΎ < 0 (Supplementary Sec. VIII). By considering an average effect of detuning and additional thermal and carrier shifts, our numerical analysis (black dots) shown in Fig. 2(d) successfully explains the major portion of the experimental data. Remarkably, one of the eigenmodes manifests itself as two different branches that correspond to different πΎπΎ . One is the observable coupled mode |ππ + β© in the symmetric phase. The other is the virtual middle branch, which is the same eigenstate in the broken phase, requires larger gain, and still satisfies ππ ππ β β . They are annihilated as a pair with a singularity, which does not represent an EP, and turn into a damping mode (Im ππ ππ β . This destabilization always occurs for finite πΏπΏ before the system obtains enough loss for the ideal PT-symmetric EP with πΏπΏ = 0 , i.e. πΎπΎ (πΌπΌ ) < πΎπΎ = π π . Our analysis hence means that it is infeasible for lasing coupled modes to be coalesced by gain and loss, as long as they have cavity detuning larger than their narrow linewidths. This is why the EP ransition with just a single mode is mostly observed in lasing systems , including our result here with revived |ππ β β© . Note that |ππ + β© actually splits into two subpeaks, and one remaining around 1530.15 nm is attributed to an unstable (non-steady) state . Additional data are shown in Supplementary Sec. IX. EP degeneracy of spontaneous emission.
The spontaneous emission (non-lasing) regime, in contrast, enables us to observe the clear EP transition with spectral coalescence of the two coupled modes, as shown in Fig. 3(a) for πΌπΌ = 100 ΞΌA and decreasing πΌπΌ . Here, the oscillation threshold for the case of pumping only one of them is about 200 Β΅A, because the other cavity behaves as an additional absorber (Supplementary Fig. S2). The radiation was measured by a spectrometer with a cryogenic InGaAs line detector (see Methods). In Fig. 3(a), the two distinct spectral peaks originally at 1529.3 and 1530.2 nm coalesce when πΌπΌ β 2 ΞΌA . In addition, the peak count of the merged ground resonance at 1529.9 nm increases back to the saturation level of about 55,000 for πΌπΌ = 0 , confirming the reversed pump dependence (Supplementary Sec. IV). Although weak higher-order modes are also found around 1523.4 nm [bottom of Fig. 3(a)] and 1521.5 nm (not shown), they are hardly affected by the change in πΌπΌ . This means that the mode competition is insignificant, because the ground coupled modes have ππ factors sufficiently higher than those of other modes. To analyze the system response theoretically, we performed the Fourier transform of the CMEs [Eq. (1)] for the spectral cavity amplitudes ππ ππ (ππ) = β±[ππ ππ (π‘π‘)] = β« ππ ππ (π‘π‘)ππ βππππππ πππ‘π‘ , together with net excitation terms {ππ ππ (ππ)} arising from the pumping. Here, because πΌπΌ is sufficiently larger than πΌπΌ over the entire measurement, we neglect the excitation ππ of cavity 1 for simplicity. By solving the esultant linear equation (shown in Methods), we reach, οΏ½ππ (ππ)ππ (ππ)οΏ½ = ππ (ππ)π π +[πΎπΎ +ππ(ΞππβπΏπΏ)][πΎπΎ +ππ(Ξππ+πΏπΏ)] οΏ½ πππ π πΎπΎ + ππ(Ξππ β πΏπΏ)οΏ½. (2) If the spontaneous emission has an ideal flat spectrum, the spectral intensity of the indirectly pumped cavity |ππ (ππ)| reflects directly the LDOS of the system, which was derived from a singular perturbation analysis . In contrast, |ππ (ππ)| is additionally affected by the relative resonance of cavity 1, i.e. Ξππ β πΏπΏ on the numerator in Eq. (2) (see also Methods and Supplementary Sec. VII). The theoretical fitting for the spectral data involves the detailed conditions of the optical collection system. Because ππ and ππ hold phase coherence with evanescent coupling, their radiation is expected to have a spatial (directional) intensity distribution due to interference . The detector signal hence depends on the position of the objective lens controlled by the three-axis nano-positioner. Here, it is aligned so that the out-coupled intensity at the coalescence is maximized. Considering that the degenerate eigenstate is (1 βππ) T /β2 , we take the analytic power spectrum for our measurement as ππ (ππ) = πππΎπΎ cav |ππ (ππ) + ππππ (ππ)| , under the premise that the identically designed cavity modes have the same radiation loss πΎπΎ cav and collection efficiency ππ . Note that our I-L data assure that the system detects the light from both cavity 1 and 2 [Fig. 2(c) and Supplementary Sec. II]. For other major possibilities like πππΎπΎ cav |ππ (ππ) Β± ππ (ππ)| , one of the coupled modes is cancelled out in the symmetric phase, and the other exhibits a Fano resonance with a peculiar spectral dip beside the main peak. We can exclude such cases since none of them were seen in our entire experiment. Figure 3(b) presents our least-square theoretical fitting for the emission spectra with ππ (ππ) . Here, because the pumped gain medium in cavity 2 is considered to be nearly transparent, we ssume a low πΎπΎ , setting it to to avoid any numerical problems like divergence. The data agree well with the experimental result, and the theoretical blue-side peak for πΌπΌ β³ 5 ΞΌA is slightly narrower mostly due to the neglected excitation of cavity 1 (Supplementary Sec. IV). The analysis enables us to estimate the physical fitting parameters in the model, such as π π , πΎπΎ and πΏπΏ , which include the effect of the mode confinement factor. The eigenfrequencies Re Ξππ ππ reconstructed with them, depicted by black points, ensure the correspondence between the sharp coalescence of the eigenmodes and the measured spectra. Figure 3(c) and (d) show the πΌπΌ dependence of estimated πΎπΎ and πΏπΏ . Here, the cavity coupling is found to be about π π = 58 GHz for the case of split resonances. Thus, π π is fixed as that value in fitting the coalesced peaks for πΌπΌ β€ 0.8 ΞΌA , which are of more complexity (Supplementary Sec. VI). The decline of the current for cavity 1 monotonically enhances its material absorption and hence πΎπΎ . On the other hand, the reduction of the local carrier plasma effect by decreasing πΌπΌ induces its red shift that continuously diminishes πΏπΏ . Ideally, the EP should be near πΎπΎ = 2π π β116 GHz . Our measurement points have an interval of ΞπΌπΌ = 0.2 ΞΌA when πΌπΌ is small, and πΌπΌ = 1.4 ΞΌA is considered the closest to the EP. By carrying proper current πΌπΌ = 100 ΞΌA for cavity 2, we can cancel the detuning πΏπΏ around the EP condition, which detrimentally lifts the degeneracy otherwise. Our buried heterostructure nanocavities enable the efficient and fine control of their imaginary potential, with the thermal and carrier effects suppressed enough. LDOS enhancement by the EP degeneracy.
Since we have identified the fine condition of the EP degeneracy in our system, we are now able to examine its effects on the light emission. Figure 4(a) shows the measured spectral peak count as a function of πΌπΌ . Its single-bottomed property may look imilar to the reversed pump dependence with the revival of lasing [Fig. 2(c)]. However, the peak intensity monotonically increases with πΌπΌ declining below ΞΌ A , not below the estimated EP ( πΌπΌ = 1.4 ΞΌ A ). This contradicts a naΓ―ve speculation for the reversed pump dependence via the phase transition of Im Ξππ at the EP. When we look at the spectra, two peaks approach each other as πΌπΌ decreases, and they are merged into a unimodal peak already at πΌπΌ = 2.4 ΞΌ A , as displayed in blue in Fig. 4(c). If two non-mixing Lorentzian peaks were to be simply summed here, the resultant contribution to the intensity must have saturated at the EP with their peak frequencies coalesced. In addition, because the eigenstates for πΌπΌ > 1.4 ΞΌ A are in the symmetric phase [ πΎπΎ < 2π π in Fig. 1 (b)], they become lossier, i.e. weaker when getting closer to the EP. Thus, the sharp growth of the peak count around the EP in Fig. 4(a) , rather than the formation of its local minimum there, suggests the LDOS enhancement by the EP degeneracy that was predicted theoretically . To confirm the experimental anomaly of the peak count, we plot the peak of the normalized CME spectral function
ππ (ππ) including the obtained fitting parameters for each πΌπΌ in Fig. 4(b). Importantly, our analysis involves the interference of the two nonorthogonal spectral peaks mediated by non-Hermiticity, as we can discuss the correspondence between the CME spectral response and LDOS around the EP (Supplementary Sec. VII). The theoretical peak intensity is consistent with the experimental data especially for πΌπΌ β€ 3 ΞΌ A , where the excitation for cavity 1 is sufficiently small. Although we ensure the unimodal merging of the two coupled-mode peaks at πΌπΌ = 2.4 ΞΌ A [blue curve in Fig. 4(c)], the intensity here is close to its minimal value because of the enhanced loss, πΎπΎ = 93.4 GHz ~ . In contrast, it increases by 30% until the near-EP condition with πΌπΌ = 1.4 ΞΌ A and a further larger πΎπΎ = 115.6 GHz ~ in both theory and experiment, as the corresponding spectral data (colored red) are shown in Fig. 4(c) for comparison. The onotonical increment in the peak intensity before the EP indicates the enhanced LDOS based on the EP degeneracy. Note that the enhancement ratio here is less than double seen in Fig. 1(d), because we control not π π but πΎπΎ and the eigenstates for πΌπΌ = 2.4 ΞΌ A are already nonorthogonal. Finally, the spontaneous emission spectra for πΌπΌ = 1.4 ΞΌA and ΞΌA are fit by some distinct trial functions and plotted in both linear and semi-logarithmic scales as Fig. 5(a) and (b), respectively. Again, our CME spectral function reproduces the experimental data well, and the apparent discrepancy between them is just seen in the region with 10% or less of the peak counts. The errors in their skirts can be mostly attributed to the background luminescence spectrumβs slightly ascending in frequency due to its peak located at around 1440 nm, and this non-ideal factor can be corrected within the first order, as shown in Supplementary Sec. VI. Remarkably, the entire section of the observed spectrum for πΌπΌ = 1.4 ΞΌA is in accordance with the squared Lorentzian function abovementioned in Fig. 1(c) and (d), β1 πΆπΆ{πΎπΎ /(Ξππ + πΎπΎ )} with coefficient πΆπΆ , rather than with the ordinary Lorentzian function (see Supplementary Sec. V for additional data). This evidences the resonance very near the exact EP and supports the enhancement of the photonic LDOS by the non-Hermitian degeneracy. We emphasize that the small difference between the CME analysis and squared Lorentzian response (LDOS) is rationalized by the fact that we measure not |ππ (ππ)| but |ππ (ππ) + ππππ (ππ)| (Methods and Supplementary Sec. VII). Here, we can exclude the Voigt fitting function , i.e. the convolution of the cavity Lorentzian factor and Gaussian noise, because it requires a too small average loss to have the EP (
26 GHz < πΎπΎ /2 = 58 GHz ), as well as persistent Gaussian noise (
27 GHz ) inconsistently larger than our lasersβ oscillation linewidths (< 4 GHz: our finest measurement resolution; see Supplementary Fig. S1). As πΌπΌ further decreases down to πΌπΌ = 0.2 ΞΌA ( πΎπΎ = 157.2 GHz ~ ), the experimental and best-fit CME spectra get ettled in more Lorentzian shapes [Fig. 4(b), Supplementary Sec. VII]. This indicates that the system loses the effect of the degeneracy on the LDOS for a large imaginary potential contrast, although the peak intensity further increases by the reduction of Im Ξππ and effective excitation of the dominant mode localizing at cavity 2.
Discussion
Enhancing the peak LDOS at the passive EP will drastically modulate the photonic responses of quantum emitters , coherent absorbers and nonlinear optical devices . It can also have assistance of local gain and get further enhanced at a higher order EP with a ratio of βππΞ(ππ + 1)/Ξ(ππ β1/2) ( = 4 for ππ = 2 ), where ππ is its order and Ξ(ππ) is the gamma function. Nonlinear optical effects will even be made hundreds of times more efficient by adopting the non-Hermitian degenerate states. In addition, the reversed power dependence in the EP transition also shows nonlinearity on the pumping. This property provides us with new possibilities for nanophotonic switches and regulators. Achieving the EP degeneracy is also essential in applications of its topological properties , such as the chirality of the degenerate supermode and the vortex charge around the EP frequency. Larger periodic devices in one and two dimensions exhibit more flexibility of parameters for EPs and rings of EPs in the Brillouin zone, respectively. Meanwhile, fabrication-induced defects make it challenging to handle the degeneracy in such systems. Further study of the radiative responses of corresponding cavity arrays will be of great significance. In conclusion, we showed the clear EP transition of spontaneous emission with our current-injected photonic crystal nanolasers. Our device has strong light confinement only for coupled ground modes and hence suppresses mode competition. The independent and efficient lectrical pumping enables the radiation near the exact EP, by limiting detrimental resonance shifts to the minimal level for active devices. In immediate proximity to the fine EP position elaborated by both our measurement and analysis, we found a squared Lorentzian emission spectrum, together with loss-induced growth of the peak power within the symmetric phase. They demonstrate the peak LDOS enhancement that is intrinsic to the EP degeneracy. Our results represent an important step toward controlling optoelectronic processes via non-Hermitian nanophotonics. Methods
Sample fabrication and design.
The sample [Fig. 2(a)] contains an air-suspended InP photonic crystal slab and two InGaAlAs-based buried heterostructure nanocavities (red) with six quantum wells embedded. Here, an InAlAs sacrificial layer, the active layer and an overcladding InP layer were grown by metal-organic chemical vapor deposition (MOCVD). The heterostructures and airholes were patterned by electron-beam lithography with SiO and SiN mask layers, respectively. The periodic air holes and narrow trenches (black lines) were opened by inductively coupled plasma reactive etching (ICP-RIE). Selective wet chemical etching was carried out to define the nanocavities. After the regrowth of the intrinsic InP layer over the heterostructures, Si ion implantation followed by activation annealing and Zn thermal diffusion was applied to diagonally pattern n-doped and p-doped layers. Each of the resultant lateral PIN junctions is in contact with an InGaAs contact layer and Au-alloy metal pads at its edges. The diagonal doped layers and current blocking trenches help suppress leakage current between the electric channels. Seven and five rows of airholes with a lattice constant of ππ = 437 nm are aligned on the outer sides of and between the line defects, respectively. A narrow line defect width of improves the modal confinement. The air hole radius and dimensions of the cavities are designed to be π π D =00 nm and . The slab thickness is 250 nm. Finite element simulation.
The sampleβs eigenmodes were simulated with a commercial electromagnetic solver based on the finite element method (COMSOL Multiphysics). The fine structure of the buried nanocavities including the quantum wells were taken into consideration by using their effective index of ππ BH = 3.41 , and the refractive index of InP is ππ InP = 3.16 . No finite imaginary part of the index was assumed over the entire device. The computational domain was halved by a two-dimensional perfect magnetic conductor placed along the middle of the slab ( π§π§ =0 ). An air layer with a height of 3.5 Β΅m was attached to the slab, and radiation loss was captured by the surrounding boundaries with the second-order scattering condition. We calculated the resonance frequencies and ππ factors of the coupled ground modes for different air-hole radii π π , which can change depending fabrication conditions. The simulated wavelengths close to our experimental result were found for π π = 104.4 nm (shown in the main text). The corresponding theoretical ππ factors are 60,000 and 62,000, while those of the first-order modes are less than 19,000. A similar device with a single nanocavity and the same parameters had the ground mode with a wavelength of 1530.8 nm and ππ = 2.9 Γ 10 . Measurement set-up.
We use a caged measurement system with a device stage, a probe station, and a nano-motion lens revolver. The sample was placed on a device holder with a vacuum contact. The temperature of the holder was maintained at
25 β with its Peltier unit and a feedback controller. Four electric microprobes were put on the metallic pads in contact with the ends of the doped layers. Independent DC currents for the right (channel 1) and left (channel 2) pairs of probes were applied and controlled with a precision source/measure unit. The device radiation was ollected at the top with a 20X objective lens with a numerical aperture (NA) of 0.26 and coupled to an optical fiber. The near-field patterns [Fig. 2(b) and (e)] were observed with another 50X lens with
NA = 0.42 and a near-infrared InGaAs camera. In the I-L measurement [Fig. 2(c)], the voltage (and hence current) for one channel was swept upward and backward, with the injection current to the other kept constant. The output light passed through a variable filter with a bandwidth of 3 nm around 1530 nm and was measured with a low-noise power meter. Owing to vibrational and mechanical stability over the measurement system, we have not seen notable hysteresis behavior in the signal. Thus, we show the data without the points in the backward sweep for a better view. The spontaneous emission (Fig. 3) was resolved with a spectrometer that has a grating with a groove number of 1000 g/mm and a spectral resolution of 0.12 nm. It was then detected by an InGaAs line detector array cooled down to β95β with liquid nitrogen. The detector integration time was 60 s. The measurement was performed in the wavelength range of 1517 to 1542 nm, and the data from 195.5 to 196.7 THz were extracted for analyzing the ground-mode spectra. The minimum count in each curve, detected far away from the ground-mode resonances, was subtracted as the background component. For the coherent radiation of the sample with the intense local pump (Fig. 4), we used a fiber-coupled optical spectrum analyzer. The wavelength sweep was performed under the lowest video bandwidth of 10 Hz for the maximum sensitivity, and every data point was averaged with 100 measurements. To compensate for negative detection currents (and power levels) caused by the spectral analyzerβs small calibration error, the data were offset with reference to their minimum value in each sweep.
Spectral coupled-mode analysis.
The Fourier transformation of Eq. (1) with the net external excitation terms ππππ οΏ½ππ (ππ)ππ (ππ)οΏ½ = οΏ½ππ(ππ + πΏπΏ) β πΎπΎ iπ π iπ π i(ππ β πΏπΏ) β πΎπΎ οΏ½ οΏ½ππ (ππ)ππ (ππ)οΏ½ + οΏ½ππ (ππ)ππ (ππ)οΏ½, is solved for ππ (ππ) and ππ (ππ) with ππ = 0 to obtain Eq. (2). Considering our experimental conditions, the fitting function for the measured spectra is expected to be ππ (ππ) = (πΎπΎ + π π ) + (Ξππ β πΏπΏ) (π π + πΎπΎ πΎπΎ β Ξππ + πΏπΏ ) + [(πΎπΎ + πΎπΎ )Ξππ + (πΎπΎ β πΎπΎ )πΏπΏ] πππΎπΎ cav |ππ (ππ)| , where Ξππ = ππ β ππ , and πππΎπΎ cav |ππ (ππ)| works as a proportional constant depending on the peak count. Here, ππ (Ξππ) is affected by the radiation of both the excited and absorptive cavities, including the term (Ξππ β πΏπΏ) in the numerator, which results in small deviation from the exact LDOS . However, its information is well reflected in the main part of the spectral peaks near the EP, namely (Ξππ β πΏπΏ) < (πΎπΎ + π π ) , for the trial function and hence the observed emission. In the fitting, we set πΎπΎ = 0.1 GHz because cavity 2 was considered to be transparent but not to provide gain. Note that the peak photon count started saturating, but the I-L data did not show any apparent reversed intensity (viz. oscillation) for (πΌπΌ , πΌπΌ ) = (0, 100 ΞΌA) . Although we had to reduce the number of parameters due to the complexity of the problem, we were able to find good values of πππΎπΎ cav |ππ (ππ)| ( β peak count) and ππ (center of the peak structure). Thus, we performed the nonlinear least-square fitting for the rest of the variables, i.e. π π , πΏπΏ , and πΎπΎ with OriginPro. Here, the initial value of π π was 60 GHz, considering the experimental and simulation result. Because π π , πΏπΏ , and πΎπΎ basically dominated the splitting, level difference and decaying tails of the spectral peaks, respectively, we were able to achieve a reliable estimation of them. Nonetheless, it became hard to have the consistent convergence of the fitting for the data after the P transition. We hence fixed π π as 58 GHz, as a value close to those obtained otherwise, and estimated πΎπΎ and πΏπΏ for πΌπΌ < 1.0 ΞΌA . References
1. Kato, T.
Perturbation Theory for Linear Operators (Springer Berlin Heidelberg, 1995). 2. Heiss, W. D. The physics of exceptional points.
J. Phys. A Math. Theor. , 444016 (2012). 3. Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parityβtime symmetry. Nat. Photonics , 752β762 (2017). 4. El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry.
Nat. Phys. , 11β19 (2018). 5. Γzdemir, K., Rotter, S., Nori, F. & Yang, L. Parityβtime symmetry and exceptional points in photonics. Nat. Mater. , 783β798 (2019). 6. Bender, C. M. & Boettcher, S. Real spectra in non-hermitian hamiltonians having PT symmetry. Phys. Rev. Lett. , 5243β5246 (1998). 7. Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Musslimani, Z. H. Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. , 103904 (2008). 8. Guo, A. et al.
Observation of PT-symmetry breaking in complex optical potentials.
Phys. Rev. Lett. , 93902 (2009). 9. RΓΌter, C. E. et al.
Observation of parityβtime symmetry in optics.
Nat. Phys. , 192β195 (2010). 10. Liertzer, M. et al. Pump-induced exceptional points in lasers.
Phys. Rev. Lett. , 173901 (2012). 11. Brandstetter, M. et al.
Reversing the pump dependence of a laser at an exceptional point.
Nat. Commun. , 4034 (2014). 12. Peng, B. et al. Loss-induced suppression and revival of lasing.
Science , 328β332 (2014). 13. Feng, L., Wong, Z. J., Ma, R.-M., Wang, Y. & Zhang, X. Single-mode laser by parity-time symmetry breaking.
Science , 972β975 (2014). 14. Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time-symmetric microring lasers.
Science , 975β978 (2014). 15. Hodaei, H. et al.
Enhanced sensitivity at higher-order exceptional points.
Nature , 187β191 (2017). 16.
Chen, W., Γzdemir, Ε. K., Zhao, G., Wiersig, J. & Yang, L. Exceptional points enhance sensing in an optical microcavity.
Nature , 192β195 (2017). 17. Peng, B. et al.
Parityβtime-symmetric whispering-gallery microcavities.
Nat. Phys. , 394β398 2014). 18. Chang, L. et al. Parity-time symmetry and variable optical isolation in active-passive- coupled microresonators.
Nat. Photonics , 524β529 (2014). 19. Longhi, S. PT-symmetric laser absorber. Phys. Rev. A , 031801 (2010). 20. Wong, Z. J. et al. Lasing and anti-lasing in a single cavity.
Nat. Photonics , 796β801 (2016). 21. Lin, Z. et al. Unidirectional invisibility induced by PT-symmetric periodic structures.
Phys. Rev. Lett. , 213901 (2011). 22. Regensburger, A. et al.
Parity-time synthetic photonic lattices.
Nature , 167β171 (2012). 23. Feng, L. et al.
Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies.
Nat. Mater. , 108β113 (2013). 24. Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching.
Nature , 76β79 (2016). 25. Yoon, J. W. et al.
Time-asymmetric loop around an exceptional point over the full optical communications band.
Nature , 86β90 (2018). 26. Zhu, Y. & Zhu, L. Accessing the Exceptional Points in Coupled FabryβPerot Resonators through Hybrid Integration.
ACS Photonics , 4920β4927 (2018). 27. Gao, Z., Thompson, B. J., Dave, H., Fryslie, S. T. M. & Choquette, K. D. Non-Hermiticity and exceptional points in coherently coupled vertical cavity laser diode arrays. Appl. Phys. Lett. , 061103 (2019). 28. Kim, K. H. et al.
Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains.
Nat. Commun. , 13893 (2016). 29. Gao, Z., Fryslie, S. T. M., Thompson, B. J., Carney, P. S. & Choquette, K. D. Parity-time symmetry in coherently coupled vertical cavity laser arrays. Optica , 323 (2017). 30. Liu, W. et al. An integrated parity-time symmetric wavelength-tunable single-mode microring laser.
Nat. Commun. , 15389 (2017). 31. Yao, R., Lee, C. Sen, Podolskiy, V. & Guo, W. Electrically Injected Parity TimeβSymmetric Single TransverseβMode Lasers. Laser Photonics Rev. , 1800154 (2018). 32. Hayenga, W. E. et al. Electrically Pumped Microring Parity-Time- Symmetric Lasers.
Proc. IEEE
Phys. Rev. A , 063833 (2011). 34. Lin, Z., Pick, A., LonΔar, M. & Rodriguez,
A. W. Enhanced Spontaneous Emission at Third-Order Dirac Exceptional Points in Inverse-Designed Photonic Crystals.
Phys. Rev. Lett. , 107402 (2016). 5. Pick, A. et al.
General theory of spontaneous emission near exceptional points.
Opt. Express , 12325 (2017). 36. Pick, A., Lin, Z., Jin, W. & Rodriguez, A. W. Enhanced nonlinear frequency conversion and Purcell enhancement at exceptional points. Phys. Rev. B , 224303 (2017). 37. Matsuo, S. et al. High-speed ultracompact buried heterostructure photonic-crystal laser with 13 fJ of energy consumed per bit transmitted.
Nat. Photonics , 648β654 (2010). 38. Takeda, K. et al. Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers.
Nat. Photonics , 569β575 (2013). 39. Takata, K. & Notomi, M. PT-Symmetric Coupled-Resonator Waveguide Based on Buried Heterostructure Nanocavities. Phys. Rev. Appl. , 054023 (2017). 40. Wijnands, F. et al. Greenβs functions for Maxwellβs equations: Application to spontaneous emission.
Opt. Quantum Electron. , 199 (1997). 41. Yariv, A., Xu, Y., Lee, R. K. & Scherer, A. Coupled-resonator optical waveguide:?a proposal and analysis. Opt. Lett. , 711 (1999). 42. Zhang, J. et al. A phonon laser operating at an exceptional point.
Nat. Photonics , 479β484 (2018). 43. Assawaworrarit, S., Yu, X. & Fan, S. Robust wireless power transfer using a nonlinear parity-time-symmetric circuit. Nature , 387β390 (2017). 44. Kominis, Y., Kovanis, V. & Bountis, T. Controllable asymmetric phase-locked states of the fundamental active photonic dimer.
Phys. Rev. A , 043836 (2017). 45. Limonov, M. F., Rybin, M. V., Poddubny, A. N. & Kivshar, Y. S. Fano resonances in photonics. Nat. Photonics , 543β554 (2017). 46. Bennett, B. R., Soref, R. A. & Del Alamo, J. A. Carrier-Induced Change in Refractive Index of InP, GaAs, and InGaAsP. IEEE J. Quantum Electron. , 113β122 (1990). 47. StΓ©phan, G. M., Tam, T. T., Blin, S., Besnard, P. & TΓͺtu, M. Laser line shape and spectral density of frequency noise. Phys. Rev. A , 043809 (2005). 48. Yang, J. et al. Diabolical points in coupled active cavities with quantum emitters.
Light Sci. Appl. , 6 (2020). 49. Matsuda, N. et al. Slow light enhanced optical nonlinearity in a silicon photonic crystal coupled-resonator optical waveguide.
Opt. Express , 19861 (2011). 50. Ota, Y. et al. Active topological photonics.
Nanophotonics , 547β567 (2020). 51. Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones.
Nature , 354β358 (2015). 52. Kremer, M. et al.
Demonstration of a two-dimensional PT -symmetric crystal.
Nat. Commun. , 435 (2019). Acknowledgements
We thank H. Sumikura and M. Takiguchi for their support with the measurement, H. Onji for CAD processing, and B. Zhen, A. Pick and J. W. Yoon for their kind discussions. This work was supported by Japan Science and Technology Agency (JST) through the CREST program under grant number JPMJCR15N4.
Author contributions
K. Takata designed the sample, built the measurement set-up, performed the measurement, analyzed the data, and wrote the manuscript. K.N. supported the sample design, measurement set-up and fabrication. E.K. supported the device design and discussion. S.M., K. Takeda, and T.F. fabricated the sample. S.K. supported the measurement and discussion. A.S. supported the sample design and organized the project. M.N. conceived the idea and led the project.
Figure 1 | Spontaneous emission properties of two coupled non-Hermitian nanolasers. a , Schematic of the system. Laser ππ has frequency detuning (β1) ππβ1 πΏπΏ to their average resonance ππ , local loss πΎπΎ ππ , and an evanescent coupling π π with the other. b, EP transition of the complex eigenfrequency detuning
Ξππ ππ = ππ ππ β ππ in reference to the coupling, π π = 1 , for πΎπΎ = 0 . The EP is at (πΎπΎ , πΏπΏ) = (2,0) , and finite πΏπΏ blurs the sharp coalescence of the two branches. c , Comparison of photonic LDOS for the system in the large coupling limit and that at the EP. The spectral LDOS of the EP resonance becomes a squared Lorentzian shape, and its peak is four times higher than that for one of the split Lorentzian supermodes far from the EP. d , Lorentzian and squared Lorentzian spectral functions with the same loss factor πΎπΎ and integrated intensity. By the non-Hermitian degeneracy, its peak power doubles compared to the accidental merging of two orthogonal states with the same linewidth (Hermitian diabolic point). a b (Lorentzian) Lorentzian-202 Ο Frequency detuning S pe c t r a l po w e r ( a . u . ) -1 0 1 24 Ο dc ~2 ΞΊ Lorentzian ΞΊ Ξ³ EP Γ4 Ο β L D O S Ο (Ξ³) (Lorentzian) I m Ξ Ο R e Ξ Ο Figure 2 | Electrically pumped photonic crystal lasers and EP transition of their lasing ground modes. a, False-color laser microscope image of the sample. Each of the two buried InGaAlAs heterostructure nanocavities (red squares) has six quantum wells. Diagonally patterned doping layers (purple: p-doped, green: n-doped) with contact pads (yellow) provide independent electric channels for the cavities (right and left: channel 1 and 2 with current πΌπΌ and πΌπΌ , respectively). b, One of the ground supermodes for
π π = 104.4 nm simulated by the p-InPn-InP Ch1Ch2a bc d W a v e l eng t h ( n m ) Channel 1, current (Β΅A) -96-92-88-84-80-76-72-68Power (dBm) e I inite-element method. It has a cavity coupling of π π sim β 65 GHz and wavelengths of 1529.25 and 1530.27 nm. Inset: near-field image of the device emission for πΌπΌ = πΌπΌ = 100 ΞΌA . c, Current-in and light-out (I-L) curves of the filtered ground modesβ emission for several fixed πΌπΌ values and swept πΌπΌ . The photodetector signal shows the systematic reversed pump dependence of the coherent device emission when πΌπΌ is small. d , Device emission spectra for fixed πΌπΌ = 800 ΞΌA and varied πΌπΌ , measured with an optical spectrum analyzer. As πΌπΌ drops, the spectrum features continuous decay of the lower branch |ππ β β© , followed by a discrete suppression of the upper branch |ππ + β© and the revival of |ππ β β© . Black dots: eigenvalue fitting with Ref. . e , Corresponding near-field patterns for different πΌπΌ , showing clear localization of emission at cavity 2 with decreasing πΌπΌ . The abruptly darkened signal from πΌπΌ = 5.5 to and glare spot with πΌπΌ = β5.0 ΞΌA support the suppression and revival of lasing by the transition. Figure 3 | Spectroscopy of the EP transition in the sampleβs spontaneous emission. a , Color plot of the observed spectra for constant πΌπΌ = 100 ΞΌA and varied πΌπΌ . Upper: coupled ground eigenmodes, clearly exhibiting the EP transition with the spectral peak coalescence and reversed pump dependence of the peak intensity. Lower: most clearly visible higher order mode, which is hardly affected by the ground-mode process. b , Result of theoretical fitting via the coupled-mode analysis. It reproduces the experimental spectra well and enables parameter estimation within the model. Here, π π β 58 GHz over the entire analysis. Black dots: eigen-wavelengths calculated with the obtained parameters, including their nearly exact coalescence. c, d, Estimated (c) loss rate πΎπΎ of cavity 1 and (d) cavity detuning πΏπΏ , dependent ΞΌ A) 01000020000300004000050000Count024681523.01523.41529.01529.41529.81530.21530.61523.8 W a v e l eng t h ( n m ) Channel 1, current ( ΞΌ A) a bc d ~ 2 ΞΊ Ξ³ Experiment CME fitting Ξ΄ = 0 n πΌπΌ . The EP condition for πΎπΎ is πΎπΎ = 2π π β 116 GHz , and the corresponding closest measurement point is πΌπΌ = 1.4 ΞΌA . Note that the detrimental detuning is almost cancelled out there, by the suppression of the carrier plasma effect with decreasing πΌπΌ . Figure 4 | Transition of the spectral peak photon count. a,
Measured peak count of the device emission spectrum depending on πΌπΌ . b, Peak of the theoretical spectral function
ππ (ππ) that is normalized with πππΎπΎ cav |ππ (ππ)| = 2π π /ππ and calculated with the estimated physical parameters for each πΌπΌ . c, Experimental photon count spectra (symbols) and their CME fitting (solid curves) at the apparent unimodal merging of the two peaks (πΌπΌ = 2.4 ΞΌA , blue) and the near-EP condition (πΌπΌ = 1.4 ΞΌA , red). When πΌπΌ is small, the excitation ππ (ππ) for cavity 1 is negligible. Thus, good consistency between ππ (ππ) with a constant coefficient and the experimental data is obtained. Despite that the net loss of the eigenstates is intensified until the systemβs reaching the EP condition ( πΌπΌ > 1.4 ΞΌA ), the peak count grows sharply by 30% from the two peaksβ uniting (πΌπΌ = 2.4 ΞΌA ) to the estimated EP (πΌπΌ = 1.4 ΞΌA ). This indicates the LDOS enhancement based on the EP degeneracy. P ea kc oun t Channel 1, current ( A) 8 6 4 2 00.60.81.01.2 P ea k s pe c t r a l f un c t i on ( a . u . ) Channel 1, current ( A)
Experiment CME analysis
Unimodal mergingEP
Unimodal mergingEP a b c
Figure 5 | Device emission spectra near and far from the EP degeneracy. a, b,
Observed photon count spectra (blue dots) and their theoretical fitting for (a) πΌπΌ = 1.4 ΞΌA and (b) πΌπΌ =0.2 ΞΌA . Left and right: their linear and semi-log plots, respectively. Our CME analysis (red line) explains both data well, and the plot in (a) agrees with a squared Lorentzian trial function (dotted orange curve) clearly better than a least-square Lorentzian trace (dashed purple curve), supporting the LDOS enhancement in the proximity of the EP. The emission with a smaller πΌπΌ (b) comes to have a more Lorentzian component, as it is a state localizing at the heavily pumped cavity in the broken phase. ab Linear Semi-logarithmic I = 1.4 ΞΌ A I = 0.2 ΞΌ A upplementary Information: Observing exceptional-point degeneracy of radiation with electrically pumped photonic crystal coupled-nanocavity lasers Kenta Takata , Kengo Nozaki , Eiichi Kuramochi , Shinji Matsuo , Koji Takeda , Takuro Fujii , Shota Kita , Akihiko Shinya , and Masaya Notomi Nanophotonics Center, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. NTT Device Technology Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan.
E-mail: [email protected] (Dated: September 4, 2020)
I. Single-laser sample
To estimate the optical properties of each diode in our two-cavity sample, we fabricated a device comprising a single nanocavity with the same structural parameter set, based on the previously reported technique . Its false-color image is shown in Fig. S1(a). To well confine the ground mode of our mode-gap nanocavity, seven rows of triangular-lattice air holes are placed on both the upper and lower sides of the line defects, whose width is ( ππ = 437 nm ). An active heterostructure designed to (red) is buried at the center of the line defect. Trapezoidal p-doped (purple) and n-doped (green) layers adjacent to the cavity form a PIN junction, and they are in contact with the edges of
50 Γ 50 ΞΌm metal pads at the top and bottom of the figure. An I-L curve of the device [Fig. S1(b)] indicates that its lasing threshold is πΌπΌ th β 37 ΞΌA . In its emission spectra around πΌπΌ th [Fig. S1(c), (d)], we see that, as the injection current πΌπΌ increases, an abrupt blue shift by the carrier plasma effect turns off and a linear red shift due to the thermal effect dominates. The consistent linewidth narrowing beyond the resolution ( ) of our optical spectral analyzer (OSA) together with the bright near-field patterns like ones seen in the two-laser experiment confirm that a high- ππ nanolaser is achieved. It is noteworthy that ππ =14,000 is obtained at
πΌπΌ = 37 ΞΌA with the corresponding raw spectral data. The optical coherence of the emission from a similar heterostructure nanolaser was assured in Ref. 2. The threshold current here is larger than that seen in Ref. 1, because the doped layers are located closer to the heterostructure. Remarkably, their material absorptions help broaden the range of the loss (i.e. the on-site effective imaginary potential) of the nanocavity.
Figure S1. (a) False-color image of the single-laser sample. (b) I-L curve of the device. The oscillation threshold is πΌπΌ th β 37 ΞΌA . (c) Spectral peak wavelength (black) and linewidth (red) of the emission, dependent on the injection current πΌπΌ . Filled markers: estimations with moving average of the spectra. (d) Device emission spectra around πΌπΌ th measured with the OSA. The data for πΌπΌ = 37 ΞΌA (green), without significant amplification by oscillation, indicate
ππ = 14,000 . (e) The linearly scaled spectrum for
πΌπΌ = 42 ΞΌA with offset Lorentzian and Gaussian fitting.
Figure S1(e) depicts the spectrum just above the threshold (
πΌπΌ = 42 ΞΌA ) and its fitting on a linear scale. The experimental data (orange points) contain non-negligible white spectral components and hence indicate that spontaneous emission is still effective, as well as low-power stimulated emission. Thus, the data match up well with an offset Lorentzian function, which has a FWHM of and a background level of (dashed purple curve). On the other hand, because its shape is asymmetric to the peak center, a Gaussian function β exp[β(ππ β ππ C ) /ππ ] , based on flicker noise, can also explain the plot (dot-dashed curve in green). Importantly, the noise in this case comes from the fluctuation of the active carrier population that is clamped around lasing . Thus, the effect of the Gaussian part on the line shape varies little and is well below the variance of the fitting in Fig. S1(e), ππ = 9.92 GHz . We can hence neglect this factor for fitting Fig. 3 and 4 in the main text, where the loss-biased resonances, including the EP, have linewidths as large as several tens of gigahertz. P ea k w a v e l eng t h ( n m ) L i ne w i d t h ( n m ) p-InPn-InP (a) P o w e r( p W ) Current ( ΞΌ A) (b) ΞΌAΞΌAΞΌAΞΌA P o w e r( d B m ) Wavelength (nm) I ΞΌA (d) (c)(e) ExperimentLorentzianGaussian P o w e r( p W ) Frequency (THz)
I. Current-in light-out curves
The I-L data for broader ranges of pumping under bidirectional sweeps are presented in Fig. S2. Because our measurement cage and equipment are connected with an optical fiber and a connector, mechanical and vibrational fluctuation in the set-up results in small discrepancies between the signal in the upward and downward sweeps. Nonetheless, we do not see any significant hysteresis behavior in the plots, showing that the mode competition and thermal nonlinearity are well suppressed in the sample. When πΌπΌ is fixed and πΌπΌ is swept [Fig. S2(a), also shown in the main text], we systematically see the steep suppression and revival of the coherent radiation power in diminishing πΌπΌ for πΌπΌ β₯ 300 ΞΌA . In comparison, the rise in power by pumping cavity 1 after oscillation is gentle. Fig. S2(a) also indicates that the system with πΌπΌ = 100 ΞΌA and πΌπΌ β² 10 ΞΌA is in the spontaneous emission regime. On the other hand, the reversed pump dependence for the opposite case [Fig. S2(b)] is less prominent. This is probably because the cavities are detuned more in this condition; for example, we have a solitary resonance of cavity 1 at 1529.47 nm for (πΌπΌ , πΌπΌ ) = (300 ΞΌA, 0) . This mode is away from the coalescence at 1529.9 nm under the constant πΌπΌ = 100 ΞΌA (Fig. 3). Figure S2. (a) I-L plots of the sample for swept πΌπΌ and fixed πΌπΌ and (b) those for varied πΌπΌ and constant πΌπΌ . The signal is filtered around 1530 nm with a bandwidth of 3 nm. The integration time of the power meter is 1 s. (a) (b) II. Symmetrically pumped two-laser sample
To estimate the cavity coupling π π , we also analyze the emission spectrum of the symmetrically pumped system with πΌπΌ = πΌπΌ = 30 ΞΌA, which is below the oscillation threshold (Fig. S3). The experimental data (blue dots) show the two coupled-mode peaks with ππ factors of about 4,000 based on their 3-dB linewidths. Because both channels have the same injection current, we need to take into consideration the net field excitation ππ (ππ) for cavity 1 that is neglected in the main text. The solution of the coupled mode theory (CMT) is hence modified as, ππ (ππ) = [πΎπΎ + ππ(Ξππ + πΏπΏ)]ππ (ππ) + πππ π ππ (ππ)π π + [πΎπΎ + ππ(Ξππ β πΏπΏ)][πΎπΎ + ππ(Ξππ + πΏπΏ)], (S1) ππ (ππ) = πππ π ππ (ππ) + [πΎπΎ + ππ(Ξππ β πΏπΏ)]ππ (ππ)π π + [πΎπΎ + ππ(Ξππ β πΏπΏ)][πΎπΎ + ππ(Ξππ + πΏπΏ)], (S2) and we again consider ππ (ππ) = πππΎπΎ cav |ππ (ππ) + ππππ (ππ)| as our trial function. If we assume that the net pumping is identical for the two cavities, we reasonably have πΎπΎ β πΎπΎ and |ππ (ππ)| β |ππ (ππ)| . In that case, however, finite πΏπΏ does not induce any level margins of the two peaks, despite that the experimental result exhibits a 20% discrepancy between the peak counts. This means that the two cavities have imbalanced pumping possibly due to inhomogeneous electric contact among the four DC probes and metal pads and/or due to a difference in current leak paths of the channels. A best-fit theoretical curve is obtained under a condition that the ratio of the field excitation rates is inversely proportional to that of the on-site field loss rates, namely ππ β‘ |ππ (ππ)||ππ (ππ)| = πΎπΎ πΎπΎ , (S3) which is a rational indication of the linear loss model. Another important factor is that, ππ (ππ) and ππ (ππ) are not expected to have any phase coherence, since they are spontaneous emissions from distinct quantum wells and cavities (in sharp contrast to the interference between the two cavity sources ππ and ππ under the single point excitation). Thus, we take the unweighted average of ππ (ππ) for their phase differences spanning over [0, ππ) at even intervals. We eventually find theoretical data that agree with the experiment in the peak counts, the ridge between the peaks, and their skirts outside (
Fig. S3) , which also suggests a good estimation of the parameters, π π = 61 GHz, πΏπΏ = 22 GHz, πΎπΎ = 18.3 GHz and ππ = 0.82. It is noteworthy that a small change in the coupling
Ξπ π up to a few gigahertz, together with a compensation of the detuning by
ΞπΏπΏ = β2Ξπ π , little affects the theoretical curve. We also see broader linewidths of the experimental peaks than those of the theory. This indicates small extra Gaussian noise, which becomes negligible in the case of asymmetric pumping and high local loss. Figure S3. The spectral photon counts for πΌπΌ = πΌπΌ = 30 ΞΌA in (a) linear and (b) semi-logarithmic plots. Blue dots: experimental data. Red curve: best CMT fitting with Eqs. (S1)-(S3) for π π =61 GHz, πΏπΏ = 22 GHz, πΎπΎ = 18.3 GHz and ππ = 0.82. The discrepancy between the experiment and analysis is attributed to small Gaussian random noise, which becomes minor in the spontaneous emission measurement with the asymmetric pumping and high local loss factors (Fig. 3 and 5).
IV. Measured spectra in spontaneous emission regime
In Fig. S4, we present experimental photon count spectra for some other πΌπΌ and πΌπΌ = 100 ΞΌA (blue dots) with their CME fitting (red curves), to show that our model consistently reproduces the measured data well. Here, there is small difference between them, in terms of the ridge lines and lower peaks, because we ignore ππ (ππ) . However, the fitting including it leads to significant complexity of the task, since the injection current to the channels does not necessarily reflect the ratio of the excitation amplitudes, as inferred in Sec. III. Nonetheless, the flat-top [Fig. S4(a)] and pointed [Fig. S4(b), (c)] peak structures in the experimental curves can simply be attributed to the fluctuation of the detection. The emission spectra around the EP condition, together with the fitting using different trial functions, are depicted in Fig. S5. Because the transition of the system spectral response is continuous, we can also observe its particular EP-based property nearby. For both πΌπΌ = 1.6 ΞΌA [Fig. S5(a)] and πΌπΌ = 1.2 ΞΌA [Fig. S5(b)], the Lorentzian curves obviously diverge from the experimental data in peak counts and decaying tails. In contrast, our CME and squared Lorentzian functions explain them significantly better and hence demonstrate the enhanced LDOS. In our measurement condition where both ππ (ππ) and ππ (ππ) are collected, the radiation at the EP has slightly broader spectral skirts than the squared Lorentzian function [see Eq. (2) and Methods]. Thus, Fig. S5(a) signifies the system ( πΎπΎ ) below the EP. The narrower peak shown in Fig. S5(b) is slightly more compliant with the Lorentzian fitting compared with that for πΌπΌ = 1.4 ΞΌA (Fig. 5), suggesting that the system with πΌπΌ = 1.2 ΞΌA (πΎπΎ = 119.7 GHz) is just above the EP. Figure S4. Selected spontaneous emission spectra of the sample (blue dots) and their CMT fitting (red lines) for (a) πΌπΌ = 6.0 ΞΌA, (b) πΌπΌ = 4.4 ΞΌA, (c) πΌπΌ = 3.0 ΞΌA and (d) πΌπΌ = 2.0 ΞΌA. πΌπΌ =100 ΞΌA. V. Correction of background counts in the spontaneous emission spectra
The background emission spectrum is not completely flat, because the electroluminescence of the sample has a global spectral peak located around 1440 nm. We can correct this non-essential factor within the first order in the following way, for clearer comparison between the theoretical and experimental results. We first pick a raw spectral curve from 195.0 to 197.0 THz and take its moving average for denoising. Next, we draw a smooth lower envelope of the resultant data, which avoids the effect of the resonant peaks. Linear fitting of the envelope gives the background gradient that should be compensated. The linear function with this slope and reference to the least point is subtracted from the entire raw data, before the additional offsetting with the minimum count. C oun t C oun t Frequency (THz) 195.8 196.0 196.2 196.40100002000030000 Frequency (THz) I = 6.0 ΞΌ A I = 4.4 ΞΌ A I = 3.0 ΞΌ A I = 2.0 ΞΌ A (a) (b)(c) (d) Figure S5. Measured spontaneous emission spectra near the EP condition and corresponding theoretical fitting. (a) πΌπΌ = 1.6 ΞΌA and (b) πΌπΌ = 1.2 ΞΌA . Left and Right: the data in linear and semi-logarithmic plots, respectively. Figure S6 shows the experimental spontaneous emission spectra with the linear background correction for (a) πΌπΌ = 1.4 ΞΌA and (b) πΌπΌ = 0.2 ΞΌA and their theoretical fitting. Compared with Fig. 5, the corrected data here agree better with our CMT fitting in both (a) and (b), especially on the right skirts of the peaks. The discrepancy between them is just within the order of hundreds of counts, and it is mostly attributed to the small oscillations superposed over the data, which have an amplitude of about 100 counts and correspond to the Fabry-Perot resonances by the line defects. We emphasize that the difference between the CMT and squared Lorentzian function around the EP condition [Fig. S6(a)] comes from the radiation from the excited cavity 2 that is subject to the reaction of cavity 1 and hence broadens the spectral tails via the quadratic term (Ξππ β πΏπΏ) in the response function. It is also noteworthy that the CMT also explains the data for πΌπΌ = 0.2 ΞΌA best I = 1.6 ΞΌ A I = 1.2 ΞΌ A (a)(b) f the considered trial functions [Fig. S6(b)], and it becomes closer to the Lorentzian function as the system gets away from the EP. VI. Estimated parameters
We discuss the rest of the parameters estimated in our theoretical analysis and shown in Fig. S7. The characteristics of the cavity-mode wavelengths ππ (πΌπΌ ) and ππ (πΌπΌ ) [Fig. S7(a)] include the information about the average frequency ππ and cavity detuning πΏπΏ . While the resonance of cavity 2 under the constant pumping ( πΌπΌ = 100 ΞΌA ) remains near ππ β 1529.9 nm in the symmetric phase, the other mode ( ππ ) undergoes a continuous red shift with decreasing πΌπΌ due to the Figure S6. Device spontaneous emission spectra with the linear background correction, to be compared with Fig. 5 in the main text. (a) πΌπΌ = 1.4 ΞΌA and (b) πΌπΌ = 0.2 ΞΌA . Linear functions with slopes of 240 and 200 counts/THz are subtracted from the raw data for (a) and (b), respectively. ab Linear Semi-logarithmic I = 1.4 ΞΌ A I = 0.2 ΞΌ A uppression of the carrier plasma effect. The detuning almost vanishes around πΌπΌ = 1.4 ΞΌA , i.e. the EP condition, and its sign is flipped beyond the degeneracy. The small but consistent red shift of ππ for πΌπΌ β² 1.6 ΞΌA is present possibly because of carrier depletion due to amplified spontaneous emission. Note that the absorption loss ( ππ factor) of the major localized eigenmode falls (scales) rapidly in the broken phase. As mentioned in the main text, the values of the coupling rate π π are mostly distributed between 58 and 59 GHz over our analysis [Fig. S7(b)]. As the fitting for the coalesced peaks with narrowing linewidths becomes difficult, we approximate the coupling as π π = 58 GHz considering the average and analyze the data by estimating πΎπΎ and πΏπΏ for πΌπΌ β€ 0.8 ΞΌA . VII. Some important properties of the systemβs spectral response
The photonic LDOS is directly relevant to how much a point-source dipole excitation couples with possible harmonic modes in the system . In the spectral coupled-mode model here, the field solutions {ππ (ππ) , ππ (ππ)} denote the response of the coupled eigenmodes to the net excitations {ππ (ππ), ππ (ππ)} for the tiny cavities. Because the cavity coupling π π is approximated as a constant for the considered frequency range, it is reasonable that the intensity of cavity 1 |ππ (ππ)| under the excitation only for transparent cavity 2 [ ππ (ππ) = 0, πΎπΎ = 0 ] exactly involves the peculiar squared Lorentzian LDOS at the EP condition ( πΏπΏ = 0, π π = πΎπΎ /2 ), Figure S7. Estimated dependence of the (a) resonance wavelengths of the cavity modes ( ππ , ππ ) and (b) cavity coupling rate ( π π ) on πΌπΌ in the spontaneous emission experiment. In the shaded area of (b), π π is approximated as 58 GHz to obtain a reliable fitting result. (a) (b) ππ (ππ)| EP2 = οΏ½πΎπΎ |ππ (ππ)| οΏ½οΏ½πΎπΎ + Ξππ οΏ½ , (S4) and vice versa for ππ (ππ) = 0, πΎπΎ = 0 , πΏπΏ = 0 and π π = πΎπΎ /2, |ππ (ππ)| EP,ππ =02 = οΏ½πΎπΎ |ππ (ππ)| οΏ½οΏ½πΎπΎ + Ξππ οΏ½ , (S5) as mentioned in Ref. . Remarkably, the full width at half maximum (FWHM) of Eq. (S4) [(S5)] is οΏ½β2 β 1πΎπΎ (πΎπΎ ) β 0.644πΎπΎ (πΎπΎ ) , where πΎπΎ (πΎπΎ ) is that for the Lorentzian function with the same loss factor. This shows the linewidth narrowing of the EP resonance. Without loss of generality, we focus on the former case as we have done in the experiment. Here, the excited cavity ( Ξππ , as the back action from the cavity 1βs resonance, |ππ (ππ)| EP2 = (πΎπΎ + Ξππ )|ππ (ππ)| οΏ½οΏ½πΎπΎ + Ξππ οΏ½ . (S6) Around the EP frequency ( ππ β ππ or Ξππ β 0 ), however, Eq. (S6) also features a squared Lorentzian shape, holding consistency with the result of the singular perturbation analysis . This is also the case for our fitting function, ππ (ππ) = πππΎπΎ cav |ππ (ππ) + ππππ (ππ)| . The device emission in the large loss limit ( πΎπΎ β« π π ) is another important property. Cavity 1βs intensity, with ππ (ππ) = 0, πΎπΎ = 0, and πΏπΏ = 0 for simplicity, can reduce to |ππ (ππ)| = π π πΎπΎ |ππ (ππ)| Ξππ + π π πΎπΎ οΏ½1 β Ξππ π π οΏ½ β π π πΎπΎ |ππ (ππ)| Ξππ + οΏ½π π πΎπΎ οΏ½ , (S7) when the main peak structure ( |Ξππ| β€ π π /πΎπΎ ) is sufficiently narrower than the width of the quadratic factor [ /π π ) ], i.e. πΎπΎ β« π π . Because the same derivation is applicable for ππ (ππ), a Lorentzian emission spectrum with a FWHM of /πΎπΎ is expected in the case of a high imaginary potential contrast. We also point out that the position of the emission peaks is not identical to the eigenfrequencies of the effective Hamiltonian. Approximate peak frequencies of ππ (ππ) are those giving extremal values of its denominator and found as real solutions for the following cubic equation, + (πΎπΎ β 2π π β 2πΏπΏ )Ξππ + πΎπΎ πΏπΏ = 0. (S8) hey successfully take into consideration the effect of the overlap between the two spectral peaks with their level difference and finite linewidths, and provide the exact result for |ππ (ππ)| under a flat (constant) excitation spectrum. We have numerically assured that Eq. (S8) and the estimated parameters reproduce the peak position of the measured spectra well. In contrast, the eigen-wavelengths of the lower peaks in Fig. 3(b) somewhat diverge from the peak locations of the experimental data, due to the skirts of the other peaks with larger peak counts. A simpler example is the case where πΏπΏ = 0. Here, we obtain the analytic solutions
Ξππ =Β±οΏ½π π β (πΎπΎ /2), which are certainly different from the corresponding eigenfrequencies, Ξππ ππ =Β±οΏ½π π β (πΎπΎ /4). VIII. Eigenvalue analysis for lasing states
To analyze the eigenfrequencies of the lasing states, the time-domain CMEs [Eq. (1) in the main text] are solved under the condition that cavity 1 has a loss πΎπΎ and cavity 2 provides a gain for oscillation, πΎπΎ = βππ . Here, while πΎπΎ depends on πΌπΌ , ππ is adaptively determined so that the net gain and loss of each eigenstate are cancelled, when πΌπΌ is large enough. If we assume that all the parameters and the eigenfrequencies ππ ππ are real (i.e. the eigenmodes are lasing), the characteristic equation for ππ ππ can be divided into its real and imaginary parts as, Re: (ππ ππ β ππ )(ππ ππ β ππ ) + πΎπΎ ππ β π π = 0, (S9) Im: (ππ ππ β ππ )ππ β (ππ ππ β ππ )πΎπΎ = 0, (S10) where ππ = ππ + πΏπΏ and ππ = ππ β πΏπΏ. Here, the latter equation gives the clamped oscillation-threshold gain that is supplied by cavity 2, ππ = ππ ππ β ππ ππ ππ β ππ πΎπΎ . (S11) This enables us to eliminate ππ in the former equation . The resultant cubic equation for ππ ππ is, (ππ ππ β ππ ) (ππ ππ β ππ ) + πΎπΎ (ππ ππ β ππ ) β π π (ππ ππ β ππ ) = 0. (S12) Our analysis for Fig. 2(d) in the main text takes into consideration the following parameter-dependent properties. First, we assume the saturable loss πΎπΎ (πΌπΌ ) in cavity 1 as a two-step linear model comprising straight lines connecting (πΌπΌ , πΎπΎ ) = (β5 ΞΌA, 160 GHz), ( ) and (50 ΞΌA, 0.2 GHz) , which is close to that estimated by our spontaneous emission measurement and fitting [Fig. 3(c)]. Second, we isolate the red shift ππ ππ (πΌπΌ ) caused by the sampleβs thermal expansion. This effect an be safely regarded as the proportional variation of all the eigenfrequencies on the pumping current , and the contribution of πΌπΌ is given by, ππ ππ (πΌπΌ ) = πΆπΆ ππ πΌπΌ = β8.547 Γ 10 β2 πΌπΌ (GHz). (S13) Here, πΌπΌ is in microamperes and the coefficient πΆπΆ T [GHz/ΞΌA] is calculated with the data points with the largest πΌπΌ , where πΎπΎ is well saturated. Finally, the reduction of the carrier plasma effect ππ πΆπΆ,πΏπΏ (πΌπΌ ) for the restored lasing state in cavity 2 is approximated as a proportional dependence on the peak power ππ ππ (πΌπΌ ) in pW [Fig. 5(b)], since πΌπΌ is constant and the steady active carriers in cavity 2 are consumed one by one by the stimulated photons localizing there. This is only applied for |ππ β β© with πΌπΌ β€ 5.4 ΞΌA and shown as ππ πΆπΆ,πΏπΏ (πΌπΌ : πΌπΌ β€ 5.4 ΞΌA) = πΆπΆ πΆπΆ ππ ππ (πΌπΌ ) = β2.726 Γ 10 β1 ππ ππ (πΌπΌ ) (GHz), (S14) where πΆπΆ πΆπΆ [GHz/pW] is estimated with the lasing frequencies with minimum πΌπΌ , where the eigenstates are deeply in the broken phase. With all of them, we eventually reach our trial function, ππ(πΌπΌ ) = ππ ππ [πΎπΎ (πΌπΌ )] + ππ ππ (πΌπΌ ) + ππ πΆπΆ,πΏπΏ (πΌπΌ : πΌπΌ β€ 5.4 ΞΌA), (S14) and we use π π = 71 GHz , ππ = 196.1143 THz (1529.72 nm) and ππ = 196.1284 THz (1529.61 nm) in our fitting. A somewhat large value of π π is adopted here, since we have to average the effect of the cavity detuning, which is hard to be precisely identified. Because the equation for ππ ππ β β is cubic, the solution can include the eigenstates in both the exact and broken phases. The analytic result is shown as the black dots in Fig. 2(d) and explains a major portion of the experimental data. In Fig. S8, we compare the transition of the theoretical eigen-wavelengths (measured in air) and threshold gain in the lasing regime for the case of no and finite cavity resonance detuning πΏπΏ . Here, we exclude the change of heat and carrier contributions to the resonances in the process and just measure the impact of gain, loss and constant (average) detuning, by solving the algebraic equation (S12) for the eigenfrequencies ππ e β β . We consider their dependence on πΎπΎ , without converting it into the injection current πΌπΌ . The cavity coupling here is π π = 71 GHz , as adopted for fitting Fig. 2(d). Note that the adaptive threshold gain ππ is given by Eq. (S11) for each ππ e . Complex solutions (Im ππ ππ β 0) are ruled out in the analysis, because they indicate unstable decaying states. When the local cavity-mode wavelengths are identical, ππ = ππ = 1529.665 nm , the system holds the singular EP degeneracy of the two dispersive solutions {|ππ + β©, |ππ β β©} corresponding to the coupled modes, even with the variable gain due to heavy pumping [Fig. S8(a)]. The red-side (|ππ + β©) and blue-side (|ππ β β©) branches in the symmetric phase have the same gain ππ = πΎπΎ , until hey reach the EP at πΎπΎ = π π = 71 GHz [Fig. S8(c)]. Thus, both states are ideally stable, and the device hence does not exhibit selective excitation for either |ππ + β© or |ππ β β© . On the other hand, the additional solution |ππ B β© , colored green in Fig. S8(a), has a fixed eigenvalue ππ B = ππ = ππ . When πΎπΎ < π π , |ππ B β© actually corresponds to the lossier eigenstate in the broken phase and requires the highest gain ππ = π π /πΎπΎ for oscillation. Thus, it never dominates over the coupled-mode solutions {|ππ + β©, |ππ β β©} in the symmetric phase. Beyond the EP ( πΎπΎ > π π ), however, |ππ B β© turns into the eigenstate that localizes at the cavity with gain in the broken phase and becomes the only stable branch with the lowest oscillation-threshold gain, ππ = π π /πΎπΎ < πΎπΎ . Figure S8. (a), (b) Steady eigen-wavelengths of the system in the lasing regime dependent on πΎπΎ , for (a) no cavity resonance detuning πΏπΏ = 0 and (b) finite detuning . (c), (d) The corresponding adaptive oscillation-threshold gain ππ for (c) πΏπΏ = 0 and (d) . Finite detuning always makes one of the lasing coupled-mode solutions {|ππ + β©, |ππ β β©} turn into a damping one (not shown), before the system obtains a local loss large enough to reach the EP, πΎπΎ = π π = 71 GHz . -B+ W a v e l eng t h ( n m ) -B+ (a) (b)(c) (d) Finite cavity detuning drastically alters the system response, as shown in Fig. S8(b) and (d) for ππ = 1529.72 nm and ππ = 1529.61 nm , which are also used in explaining Fig. 2 (d). In this case, the intense excitation to cavity 2 always couples better with |ππ β β© , because its wavelength ππ β is closer to ππ than that of |ππ + β© . Consequently, |ππ β β© always needs the smallest ππ [Fig. S8(d)] and undergoes the continuous but detuned EP transition without the degeneracy [Fig. S8(b)]. Here, ππ (πΎπΎ ) for |ππ β β© has a peaked structure. Thus, if πΌπΌ is not large enough to provide the peak threshold gain with |ππ β β© , the system undergoes the suppression and revival of lasing. We emphasize that the analysis is performed for tracing the revived lasing resonance by |ππ β β© in Fig. 2(d), and the effect of the detuning is averaged and constant, . The result is hence approximate and does not explain detailed experimental behavior such as the switching of the dominant mode, which is attributed to the pump-induced inversion of ππ and ππ . The red-side branch |ππ + β© exhibits peculiar properties in the process. It is stable but inferior compared to |ππ β β© and thus demands a slightly larger ππ , when πΎπΎ is small [Fig. S8(d)]. Remarkably, there is an apparent singular coalescence between |ππ + β© and |ππ B β© [Fig. S8(b)]. However, these two branches actually denote the single eigenstate with different variable gain ππ . |ππ + β© is one of the coupled modes in the symmetric phase. In contrast, |ππ B β© corresponds to the same eigenstate in the broken phase, which requires the largest ππ by localizing at the lossy cavity. Thus, their coalescence is not an EP. The combined solutions for larger πΎπΎ come to be a single damping state with Im ππ ππ > 0 , which is hence omitted from Fig. S8(b). This destabilization of |ππ + β© , together with the selective mode excitation by the singularly growing difference of ππ between |ππ + β© and |ππ β β© , always occurs before reaching πΎπΎ = π π for any finite πΏπΏ . Therefore, to achieve the EP of the lasing coupled modes, the detuning is required to be zero over the entire process with a fine accuracy. However, such elaborate control is not realistic in most cases, because there is no direct way to measure the local resonance wavelengths {ππ , ππ } for each {πΎπΎ , πΎπΎ } from coupled cavity systems in experiment. IX. Selected spectra in lasing regime
The destabilization of |ππ + β© shown in the last section would result in the unexpected experimental response with multiple peaks split from |ππ + β© just before the lasing of |ππ β β© . Here, we present and discuss the transition of the emission spectra in the lasing regime ( πΌπΌ = 800 ΞΌA ). When πΌπΌ is large [Fig. S9(a)], there are two spectral peaks corresponding to the coupled ground modes with a splitting of about 1.1 nm, which is reasonable for a system with a coupling of π π β 60 GHz and a cavity detuning πΏπΏ on an order of 10 GHz. The blue-side mode |ππ β β© decays faster [Fig. S9(b)] with ecreasing πΌπΌ and disappears for πΌπΌ < 30 ΞΌA. The significant peak power difference between the two modes here is attributed to a finite and non-negligible πΏπΏ.
The spectral property changes drastically at πΌπΌ = 5.4 ΞΌA [Fig. S9(c)]. The peak power of the red-side mode |ππ + β© falls by more than 10 dB from the preceding point ( πΌπΌ = 5.6 ΞΌA ), and the device comes to exhibit a weak and broad spectrum spanning from 1529.1 to 1530.2 nm, which is attributed to spontaneous emission. Remarkably, |ππ + β© is split into two peaks, and the minor one at 1529.9 nm shifts toward the middle of its higher counterpart and the revived |ππ β β© . Note that because cavity 2 is heavily pumped, the amplified spontaneous emission at resonance can have a linewidth below 0.1 nm (or the cold linewidth of each single cavity). Although we also find the first-order mode with an increased intensity (not shown), it never becomes dominant over the ground modes in the entire measurement. The peak power of the whole spectrum becomes minimum at πΌπΌ = 2.0 ΞΌA [Fig. S9(d)]. Here, |ππ β β© (left) exceeds the remaining peak around 1530.15 nm (right), which we denote as |ππβ² + β© . |ππβ² + β© has an unconventionally spiky structure and hence is not considered as a normal cavity resonance but as a non-steady state . The weaker itinerant bump coming off of |ππ + β© reaches the right verge of |ππ β β© and forms a visible plateau. |ππ β β© is further enhanced and |ππβ² + β© is damped by reducing πΌπΌ , as shown in Fig. S9(e). |ππ β β© is significantly red-shifted and eventually overlaps the shoulder on its recovery [Fig. S9(f)]. Interestingly, while the spectral resolution of the measurement is limited to 0.05 nm, the emission for πΌπΌ = β5.0 ΞΌA (near zero bias of channel 1) fits much better with the squared Lorentzian function than with the Lorentzian. As seen by the localized emission pattern [Fig. 2(e)], the state here is in the broken phase after the detuned EP transition. However, its peculiar spectral shape and position indicate the cancelling of the cavity detuning πΏπΏ by reducing πΌπΌ . It is noteworthy that the measurement equipment used here (OSA) is different from that for the spontaneous emission experiment (a spectrometer). X. Loss-induced lasing in detuned EP transition
The spectral variation with πΌπΌ for πΌπΌ = 800 ΞΌA is shown in terms of the peak powers and linewidths of the two major branches in Fig. S10(a) and (b), respectively. The abrupt suppression of |ππ + β© at πΌπΌ = 5.4 ΞΌA results in the broad spectrum plotted in Fig. S9(c), which includes the re-emergent blue-side peak |ππ β β© [Fig. S10 (a)]. The power of |ππ β β© then grows with a kink with decreasing πΌπΌ , and its linewidth is correspondingly narrowed from β 0.2 nm to below the minimum measurement resolution here, 0.05 nm [Fig. S10 (b)]. This indicates that the system undergoes the revival of lasing with |ππ β β© after the switching of the dominant eigenmode. Figure S9. Coherent emission spectra of the sample under high pumping for cavity 2 ( πΌπΌ = 800 ΞΌA ), for different πΌπΌ . (a) πΌπΌ = 100 ΞΌA , (b) πΌπΌ = 30 ΞΌA , (c) πΌπΌ = 5.4 ΞΌA , (d) πΌπΌ = 2.0 ΞΌA , (e) πΌπΌ =0.0 ΞΌA and (f) πΌπΌ = β5.0 ΞΌA . (a) (b) (c) (d)(e) (f) I = 100 ΞΌ A I = 30 ΞΌ A I = 5.4 ΞΌ A I = 2 ΞΌ A I = 0 ΞΌ A I = -5 ΞΌ A Figure S10. (a) Peak powers and (b) linewidths of the |ππ + β© , |ππβ² + β© (red) and |ππ β β© (blue) under small πΌπΌ . Filled blue markers: approximate linewidths estimated with the moving-average data. The plots for |ππβ² + β© after the drop of the power correspond to the immobile peak around 1530.15 nm in Fig. S9(c)-(e), which is attributed to the non-steady damping state. Another weaker and broader bump moving toward the middle is omitted. The peak power rising and linewidth narrowing of |ππ β β© indicates the revival of lasing. Here, the minimum linewidth resolution is 0.05 nm. It is notable that |ππβ² + β© for πΌπΌ β€ 5.4 ΞΌA denotes the immobile spiky peak remaining around 1530.15 nm, which is again considered unstable [Fig. S9(c)-(e)]. Although it seems to hold somewhat narrow linewidths, it soon decays as |ππ β β© oscillates. Its detailed properties are beyond the scope of this work. Finally, the spectra here do not have any hysteresis behavior. They are identical regardless of whether πΌπΌ is swept upward or downward, except for a small fluctuation coming from the long-term change in the electrical contact between the device and probes. References
1. Takeda, K. et al.
Few-fJ/bit data transmissions using directly modulated lambda-scale embedded active region photonic-crystal lasers.
Nat. Photonics , 569β575 (2013). 2. Takiguchi, M. et al. Systematic study of thresholdless oscillation in high- Ξ² buried multiple-quantum-well photonic crystal nanocavity lasers.
Opt. Express , 3441 (2016). 3. StΓ©phan, G. M., Tam, T. T., Blin, S., Besnard, P. & TΓͺtu, M. Laser line shape and spectral density of frequency noise. Phys. Rev. A - At. Mol. Opt. Phys. , 043809 (2005). (a) (b) -4-202468 -5-95-90-85-80-75-70 P ea k po w e r( d B m ) Channel 1, current ( ) -4-202468 -50.000.050.100.150.200.25 L i ne w i d t h ( n m ) |Ξ» β βͺ|Ξ» + βͺ , |Ξ»β² + βͺ Channel 1, current ( ) . Wijnands, F. et al.
Greenβs functions for Maxwellβs equations: Application to spontaneous emission.
Opt. Quantum Electron. , 199 (1997). 5. Pick, A. et al. General theory of spontaneous emission near exceptional points.
Opt. Express , 12325 (2017). 6. Pick, A., Lin, Z., Jin, W. & Rodriguez, A. W. Enhanced nonlinear frequency conversion and Purcell enhancement at exceptional points. Phys. Rev. B , 224303 (2017). 7. Assawaworrarit, S., Yu, X. & Fan, S. Robust wireless power transfer using a nonlinear parity-time-symmetric circuit. Nature , 387β390 (2017). 8. Kominis, Y., Kovanis, V. & Bountis, T. Controllable asymmetric phase-locked states of the fundamental active photonic dimer.
Phys. Rev. A96