1. INTRODUCTION

Coupled optical cavities and waveguides with imaginary refractive index contrast, i.e. distributed gain and loss, can exhibit peculiar degeneracies called exceptional points [1–5] (EPs). In such a system, eigenmodes undergo a transition between two phases that are divided by the EP. One phase comprises extended supermodes with parity–time (PT) symmetry [6–9]. Here, the real parts of their eigenfrequencies and propagation constants are split. In contrast, the imaginary parts of them 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; the eigenstates localize at either the amplifying or de-amplifying elements (broken phase). Correspondingly, the split real spectrum coalesces at the EP, and then the imaginary spectrum bifurcates into two or more branches, with a singular dependence on parameters involved. This EP transition leads to intriguing features, such as reversed pump dependence [10–12], single-mode oscillation [13,14], and enhanced sensitivity [15,16].

There has also been rising interest in photonic EP degeneracy itself. Distinct from the accidental degeneracy of characteristic eigenvalues in Hermitian systems with orthogonal modes, 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 [17,18], coherent absorption [19,20], unidirectional reflectivity [21–23], and asymmetric mode conversion [24,25].

 

Fig. 1. Spontaneous emission of two coupled non-Hermitian nanolasers. (a) Schematic of the system. Cavity $i$ has frequency detuning ${(- 1)^{i - 1}}\delta$ to their average resonance ${\omega _0}$, local loss ${\gamma _i}$, and an evanescent coupling $\kappa$ with the other. (b) EP transition of the complex eigen-detuning $\Delta {\omega _i} = {\omega _i} - {\omega _0}$ in reference to the coupling, $\kappa = 1$, for ${\gamma _2} = 0$. The EP is at $({\gamma _1},\delta) = (2,0)$, and finite $\delta$ blurs the sharp coalescence of the two branches. (c) Comparison of photonic LDOS for the system in the large coupling limit ($\kappa \gg {\gamma _1}$) and that at the EP ($2\kappa = {\gamma _1}$) for ${\gamma _2} = 0$. The spectral LDOS of the EP resonance has a squared Lorentzian shape, and its peak is four times higher than that of one of the split Lorentzian supermodes far from the EP. (d) Lorentzian and squared Lorentzian spectral functions based on the same loss factors $({\gamma _1} \gt 0,{\gamma _2} = 0)$ and integrated intensity. EP degeneracy doubles the peak power, compared to the sum of two orthogonal Lorentzian modes with a linewidth of ${\gamma _1}$ (Hermitian diabolic point).

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Although many papers have studied phenomena around the EP, observing optical responses at EP degeneracy has been a persistent technical challenge, even for basic two-cavity devices [11,12,14,17,18,26–32]. In fact, EP degeneracy is predicted to have significant influence on radiation processes [33–36]. 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 demanding for systems based on passive loss processes 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 [11,29]. In addition, it results in significant damping of one of the coupled modes [26–28], which hampers their coalescence and hence the EP response. Multiple cavity modes with comparative $Q$ factors [30–32] are also subject to carrier-mediated mode competition that can disrupt the pristine properties at the EP.

Here, we report the observation of spontaneous emission under EP degeneracy with two current-injected photonic crystal lasers. We establish the first nanocavity-based non-Hermitian platform with electrical pumping, by using our buried heterostructure technique [37–39]. It is generally hard to achieve lasing in electrically pumped nanocavities (i.e., cavities with wavelength- or subwavelength-scale mode volumes), because of restricted gain and difficulty in thermal management. Thus, the wavelength-scale active heterostructure with photonic crystals [38] operates as the only current-driven continuous-wave nanocavity laser at room temperature, in the present conditions. Now, we successfully integrate two of them with strong coupling, which also hold continuous-wave room-temperature oscillation, and explore the exact EP response of their emission. Efficient carrier injection and high heat conductivity in the tiny heterostructures enable minimal pump-induced resonance shift and stable control of gain and loss for each cavity. Selective high $Q$ factors for their coupled ground modes are also achieved so that the mode competition is suppressed. We first investigate the system with highly asymmetric pumping. Here, we clarify that whenever there is non-negligible cavity detuning, it is barely possible for the lasing PT-symmetric supermodes to reach any degree of non-Hermitian coalescence. In contrast, our elaborate measurement 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) [33–36,40]. Our results provide a new approach to handle the light–matter interaction and light emission.

 

Fig. 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 ${I_1}$ and ${I_2}$, respectively). (b) Magnetic field of a ground supermode for $R = 104.4\;{\rm nm}$ simulated by the finite-element method. The theoretical cavity coupling is ${\kappa _{{\rm sim}}} \approx 65\;{\rm GHz}$, and the eigenmodes’ wavelengths are 1529.25 and 1530.27 nm. Inset: near-field image of the device emission for ${I_1} = {I_2} = 100\;\unicode {x00B5}{\rm A}$. (c) Current-in and light-out (I-L) curves of the filtered ground modes’ emission for several fixed ${I_2}$ values and swept ${I_1}$. The photodetector signal shows the systematic reversed pump dependence of the coherent device emission when ${I_1}$ is small. (d) Device emission spectra for fixed ${I_2} = 800\;\unicode {x00B5}{\rm A}$ and varied ${I_1}$, measured with an optical spectrum analyzer. As ${I_1}$ drops, the spectrum features continuous decay of the lower branch $|{\lambda _ -}\rangle$, followed by a discrete suppression of the upper branch $|{\lambda _ +}\rangle$ and the revival of $|{\lambda _ -}\rangle$. Black dots: eigenvalue fitting with Ref. [43]. (e) Corresponding near-field patterns for different ${I_1}$, showing clear localization of emission at cavity 2 with decreasing ${I_1}$. The abruptly darkened signal from ${I_1} = 5.5$ to 2.0 µA and glare spot with ${I_1} = - 5.0\;\unicode {x00B5}{\rm A}$ support the suppression and revival of lasing by the transition.

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2. 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, $\kappa$. However, the gain and loss can counteract the frequency splitting by the EP transition. The first-order temporal coupled mode equations (CMEs) [41] for the complex cavity-mode amplitudes $\{{{a_i}(t)} \}$ are derived as

The EP exhibits peculiar radiation responses [33–36] [Fig. 1(c)]. When the system is in the symmetric phase and the eigenfrequency splitting is large, the spectral LDOSs [40] of the two coupled modes are Lorentzian functions with ideally the same linewidth. At non-Hermitian degeneracy, however, the two spectral peaks coalesce and constructively interfere with each other. Thus, the resultant radiation power spectrum, which is directly relevant to the LDOS, takes on a squared Lorentzian shape. When the system has a transparent cavity and a lossy cavity (e.g. ${\gamma _1} \gt 0$, ${\gamma _2} = 0$), the corresponding peak LDOS is increased purely by the effect of the degeneracy. Such enhancement at this passive EP is four-fold, compared to each of the separate peaks in the large coupling limit. Namely, compared to the mere sum of the two Lorentzian modes (i.e., Hermitian accidental degeneracy of two orthogonal states), the EP resonance with common loss and the same integral intensity has a doubly high peak and $\sqrt {\sqrt 2 - 1} \approx 0.644$ times narrower linewidth [Fig. 1(d) and Section 9 of Supplement 1]. 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 [42].

3. EXPERIMENTAL SETUP AND EP TRANSITION IN LASING REGIME

We prepared a sample comprising two coupled photonic crystal lasers based on buried heterostructure nanocavities [37–39] [Fig. 2(a) and Section 1 of Supplement 1]. Here, gain media with six quantum wells (red), which work as mode-gap cavities, are embedded in an air-suspended InP photonic crystal slab. Two line defects narrower than the lattice-matched width improve the cold $Q$ factors of the coupled ground cavity 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 low lasing threshold ${I_{{\rm th}}}$ of about 37 µA, at which it has a high $Q$ factor of 14,000 (Section 2 of Supplement 1). When symmetrically pumped below the threshold with 30 µA for comparison, the two-laser sample gives spontaneous emission of the two coupled modes with $Q = 4{,}000$, slightly below that of the single diode (5000). Their resonance peaks with a splitting of about 1.0 nm in reference to 1529.7 nm indicates $\kappa = 61\;{\rm GHz}$ (Section 4 of Supplement 1). It agrees well with the coupling of the simulated ground modes, ${\kappa _{{\rm sim}}} = 65\;{\rm GHz}$ [Fig. 2(b) and Section 1 of Supplement 1]. The near-field emission from both lasers is also observed [Fig. 2(b), inset].

We fix the injection current ${I_2}$ for channel 2 on the left and sweep that to the right, ${I_1}$ for channel 1, to vary the imaginary potential contrast ${\gamma _1} - {\gamma _2}$. As a result, the detected ground-mode power systematically recovers by the reduction of the local current ${I_1}$ [Fig. 2(c)]. This reversed pump dependence [10,11] indicates the EP transition (Sections 1 and 3 of Supplement 1). Heavy pumping ${I_2} = 800\;\unicode {x00B5}{\rm A}$ maximizes the ratio ${P_{R}}/{P_{{\min }}}$ between the power ${P_{R}}$ for zero bias along channel 1 (${V_1} = 0$) and the minimum value ${P_{{\min }}}$ in terms of ${I_1}$. Here, cavity 2 provides gain for achieving notable loss-induced revival of lasing [12]. However, the system is critically affected by cavity detuning $\delta$ and hence misses the EP degeneracy.

Figure 2(d) depicts the device emission spectra in the lasing regime for constant ${I_2} = 800\;\unicode {x00B5}{\rm A}$ and different ${I_1}$, measured with an optical spectrum analyzer. Here, some leakage current from channel 2 induces a negative ${I_1} \approx - 6\;\unicode {x00B5}{\rm A}$ for ${V_1} = 0$. However, the data and hence loss ${\gamma _1}$ in cavity 1 consistently change under the reverse current. As ${I_1}$ decreases from ${I_1} = 100\;\unicode {x00B5}{\rm A}$ and ${\gamma _1}$ hence increases, the blue-side peak $|{\lambda _ -}\rangle$ damps, while the other red-side one $|{\lambda _ +}\rangle$ remains bright. This is a direct reflection of finite detuning $\delta$, with which the asymmetric pumping ${I_2} \gg {I_1}$ selectively excites the coupled mode closer to the solitary resonance of cavity 2, ${\omega _0} - \delta$. Eventually, the power of $|{\lambda _ +}\rangle$ also drops sharply around ${I_1} = 5.4\;\unicode {x00B5}{\rm A}$, indicating the suppression of oscillation. However, it is $|{\lambda _ -}\rangle$ that undergoes the revival of lasing, accompanied with a kinked rise in power and linewidth narrowing (Section 12 of Supplement 1). Such switching of the dominant mode has been observed in relevant studies [28,43] and attributed to the pump-induced sign flip of $\delta$. The restored peak moves toward the middle of the original coupled-mode resonances by further reducing ${I_1}$ and hence evidences the EP transition in our device. The near-field patterns for selected ${I_1}$ [Fig. 2(e)] not only show the above-mentioned processes in real space but also exhibit clear mode localization at cavity 2 in the intensity recovery, supporting the PT symmetry breaking.

The steady oscillation condition ${\rm Im}\;{\omega _{e}} = 0$ enables us to estimate the eigenfrequencies ${\omega _{e}}$ for the lasing spectra [43], despite that the system here provides an adaptive (variable) gain ${\gamma _2} \lt 0$ (Section 10 of Supplement 1). By considering an average effect of detuning $2\delta = - 14.1\;{\rm GHz}$ 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 ${\gamma _2}$. One is the observable coupled mode $|{\lambda _ +}\rangle$ in the symmetric phase. The other is the virtual middle branch $|{\lambda _{B}}\rangle$, which is the same eigenstate in the broken phase, requires larger gain, and still satisfies ${\omega _{e}} \in {\mathbb R}$. They are annihilated as a pair with a singularity, which does not represent an EP, and turn into a damping mode (${\rm Im}\;{\omega _{e}} \ne 0$). This destabilization always occurs for finite cavity detuning $\delta$, before the system obtains the loss ${\gamma _{1,{\rm EP}}} = \kappa$ necessary to reach the only EP in oscillation with $\delta = 0$. Our analysis hence means that it is infeasible for lasing coupled modes to be coalesced by gain and loss, as long as $\delta$ is larger than their narrow linewidths. This is why the EP transition with just a single mode is observed mostly in lasing systems [11,27–29], including our result here with revived $|{\lambda _ -}\rangle$. Note that $|{\lambda _ +}\rangle$ in experiment actually splits into two subpeaks, and one remaining around 1530.15 nm is attributed to an unstable (non-steady) state [44]. Additional data are shown in Section 11 of Supplement 1.

4. EP DEGENERACY OF SPONTANEOUS EMISSION

The spontaneous emission (non-lasing) regime, in contrast, enables us to observe a clear EP transition with spectral coalescence of the two coupled ground modes, as shown in Fig. 3(a) for ${I_2} = 100\;\unicode {x00B5}{\rm A}$ and decreasing ${I_1}$. Here, the oscillation threshold for the case of pumping only one of them is about 200 µA, because the other cavity and its doped layers behave as additional absorbers (Fig. S2 of Supplement 1). The radiation was measured by a spectrometer with a cryogenic InGaAs line detector (Section 1 of Supplement 1). In Fig. 3(a), the two distinct spectral peaks originally at 1529.3 and 1530.2 nm coalesce when ${I_1} \approx 2\;\unicode {x00B5}{\rm A}$. In addition, the peak count of the merged resonance at 1529.9 nm increases back to the saturation level of about 55,000 for ${I_1} = 0$, confirming the reversed pump dependence (Section 5 of Supplement 1). 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 ${I_1}$. This means that the mode competition is insignificant, because the ground modes have $Q$ factors sufficiently higher than those of other modes. We emphasize that the eigenmodes observed here do not lase and are hence in the spontaneous emission regime, because their spectral linewidths (0.40 nm at least) are fairly broader than that for the single cavity on the lasing threshold (0.11 nm, Fig. S1). Since this is true of both the low-loss coupled modes for ${I_1} = 8\;\unicode {x00B5}{\rm A}$, ${I_2} = 100\;\unicode {x00B5}{\rm A}$ and the localized mode for ${I_1} = 0$, ${I_2} = 100\;\unicode {x00B5}{\rm A}$ in the broken phase, cavity 2 does not provide notable gain, i.e., ${\gamma _2} \approx 0$, despite ${I_2} \gt {I_{{\rm th}}}$ for the single laser.

 

Fig. 3. Spectroscopy of the EP transition in the sample’s spontaneous emission. (a) Color plot of the observed spectra for constant ${I_2} = 100\;\unicode {x00B5}{\rm A}$ and varied ${I_1}$. Upper: coupled ground eigenmodes, clearly exhibiting the EP transition with the spectral peak coalescence and reversed pump dependence of the peak intensity. Lower: most visible higher-order mode, which is hardly affected by the ground-mode process. (b) Result of theoretical fitting via coupled-mode analysis. It reproduces the experimental spectra well and enables parameter estimation within the model. Here, $\kappa \approx 58\;{\rm GHz}$ over the entire analysis. Black dots: eigen-wavelengths calculated with the obtained parameters, including their nearly exact coalescence. (c), (d) Estimated (c) loss rate ${\gamma _1}$ of cavity 1 and (d) cavity detuning $\delta$, dependent on ${I_1}$. The EP condition for ${\gamma _1}$ is ${\gamma _1} = 2\kappa \approx 116\;{\rm GHz}$, and the corresponding closest measurement point is ${I_1} = 1.4\;\unicode {x00B5}{\rm A}$. Note that the detrimental detuning is almost cancelled out there by the suppression of the carrier plasma effect with decreasing ${I_1}$.

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To analyze the system response theoretically, we performed the Fourier transform of the CMEs [Eq. (1)] for the spectral cavity amplitudes ${a_i}(\omega) = {\cal F}[{a_i}(t)] = {a_i}(t){e^{- i\omega t}}dt$, together with net cavity excitation fields $\{{{c_i}(\omega)} \}$ arising from the pumping. Because ${I_2}$ is sufficiently larger than ${I_1}$ over the entire measurement, we neglect the excitation of cavity 1 for simplicity, ${c_1} = 0$. By solving the resultant linear equation (Section 1 of Supplement 1), we reach

The theoretical fitting for the spectral data involves the detailed conditions of the optical collection system. Because ${a_1}$ and ${a_2}$ hold phase coherence with evanescent coupling, their radiation is expected to have a spatial (directional) intensity distribution due to interference [29]. 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 ${(1, - i)^{T}}/\sqrt 2$, we take the analytic power spectrum for our measurement as $P(\omega) = \eta {\gamma _{{\rm cav}}}{| {{a_1}(\omega) + i{a_2}(\omega)} |^2}$, under the premise that the identically designed cavity modes have the same radiation loss ${\gamma _{{\rm cav}}}$ and collection efficiency $\eta$. Note that our I-L data assure that the system detects the light from both cavities 1 and 2 [Fig. 2(c) and Section 3 of Supplement 1]. For other major possibilities such as $\eta {\gamma _{{\rm cav}}}{| {{a_1}(\omega) \pm {a_2}(\omega)} |^2}$, one of the coupled modes is cancelled out in the symmetric phase, and the other exhibits a Fano resonance [45] with a peculiar spectral dip beside the main peak. We can exclude such cases since none of them was seen in our entire experiment.

Figure 3(b) presents our least-square theoretical fitting for the emission spectra with $P(\omega)$. Because cavity 2 with ${I_2} = 100\;\unicode {x00B5}{\rm A}$ is considered nearly loss-compensated, we assume a low ${\gamma _2}$, setting it to 0.1 GHz to avoid any numerical problems such as divergence. The data agree well with the experimental result, and the theoretical blue-side peak for ${I_1} \mathbin{\lower.3ex\hbox{$\buildrel \gt \over{\smash{\scriptstyle\sim}\vphantom{_x}}$}} 5\;\unicode {x00B5}{\rm A}$ is slightly narrower mostly because of the neglected excitation of cavity 1 (Section 5 of Supplement 1). The analysis enables us to estimate the physical fitting parameters in the model, such as $\kappa$, ${\gamma _1}$, and $\delta$, which include the effect of the mode confinement factor. The eigenfrequencies ${\rm Re}\;\Delta {\omega _i}$ reconstructed with them, depicted by black points, ensure the correspondence between the sharp coalescence of the eigenmodes and the measured spectra.

Figures 3(c) and 3(d) show the ${I_1}$ dependence of estimated ${\gamma _1}$ and $\delta$. Here, the cavity coupling is found to be about $\kappa = 58\;{\rm GHz}$ for the case of split resonances. Thus, $\kappa$ is fixed as that value in fitting the coalesced peaks for ${I_1} \le 0.8\;\unicode {x00B5}{\rm A}$, which are of more complexity (Section 7 of Supplement 1). The decline of ${I_1}$ monotonically enhances the material absorption in cavity 1 and hence ${\gamma _1}$. On the other hand, the reduction of the local carrier plasma effect [46] by decreasing ${I_1}$ induces a red shift there, which continuously diminishes $\delta$. Ideally, the EP should be near ${\gamma _{1,{\rm EP}}} = 2\kappa \approx 116\;{\rm GHz}$. Our measurement points have an interval of $\Delta {I_1} = 0.2\;\unicode {x00B5}{\rm A}$ when ${I_1}$ is small, and ${I_1} = 1.4\;\unicode {x00B5}{\rm A}$ is considered the closest to the EP. By carrying proper current ${I_2} = 100\;\unicode {x00B5}{\rm A}$ for cavity 2, we can cancel the detuning $\delta$ around the EP condition, which detrimentally lifts the degeneracy otherwise. Our device enables efficient and fine control of its imaginary potential, with thermal and carrier effects suppressed enough.

5. LDOS ENHANCEMENT BY EP DEGENERACY

Since we have identified the fine condition of EP degeneracy in our system, we are now able to examine its effects on light emission. Figure 4(a) shows the measured spectral peak count as a function of ${I_1}$. Its single-bottomed property may look similar to the reversed pump dependence with the revival of lasing [Fig. 2(c)]. However, the peak intensity monotonically increases with ${I_1}$ declining below 2.4 µA, not below the estimated EP (${I_1} = 1.4\;\unicode {x00B5}{\rm A}$). This contradicts a naïve speculation for the reversed pump dependence via the phase transition of ${\rm Im}\;\Delta \omega$ at the EP. When we look closely at the spectra, two peaks approach each other as ${I_1}$ decreases, and they are merged into a unimodal peak already at ${I_1} = 2.4\;\unicode {x00B5}{\rm A}$, as displayed in blue in Fig. 4(c). If two non-mixing Lorentzian peaks were to be simply summed here as in a Hermitian system, the resultant contribution to the intensity must have saturated at degeneracy with their peak frequencies coincident (Fig. S8 in Supplement 1). In addition, because the eigenstates for ${I_1} \gt 1.4\;\unicode {x00B5}{\rm A}$ are supposed to be in the symmetric phase [${\gamma _1} \lt 2\kappa$ 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 LDOS enhancement by EP degeneracy that was predicted theoretically [34–36] (Section 8 of Supplement 1).

 

Fig. 4. Transition of the spectral photon count. (a) Measured peak count of the device emission spectrum depending on ${I_1}$. (b) Peak of the theoretical spectral function $P(\omega)$ normalized with $\eta {\gamma _{{\rm cav}}}{| {{c_2}(\omega)} |^2} = 2\kappa /\pi$ and calculated with the estimated physical parameters for each ${I_1}$. (c) Experimental photon count spectra (symbols) and their CME fitting (solid curves) at the apparent unimodal merging of the two peaks (${I_1} = 2.4\;\unicode {x00B5}{\rm A}$, blue) and the near-EP condition (${I_1} = 1.4\;\unicode {x00B5}{\rm A}$, red). When ${I_1}$ is small, the excitation ${c_1}(\omega)$ for cavity 1 is negligible. Thus, good consistency between $P(\omega)$ and the experimental data is obtained. Despite that the net loss of the eigenstates is intensified until the system reaches the EP condition (${I_1} \gt 1.4\;\unicode {x00B5}{\rm A}$), the peak count grows sharply by 30% from the two peaks’ uniting (${I_1} = 2.4\;\unicode {x00B5}{\rm A}$) to the estimated EP (${I_1} = 1.4\;\unicode {x00B5}{\rm A}$). This indicates the LDOS enhancement based on EP degeneracy.

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Fig. 5. Device emission spectra near and far from EP degeneracy. (a), (b) Observed photon count spectra (blue dots) and their theoretical fitting for (a) ${I_1} = 1.4\;\unicode {x00B5}{\rm A}$ and (b) ${I_1} = 0.2\;\unicode {x00B5}{\rm A}$. 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 ${I_1}$ (b) comes to have a more Lorentzian component, as it is a state localizing at the heavily pumped cavity in the broken phase.

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To confirm the experimental anomaly of the peak count, we plot the peak of the normalized CME spectral function $P(\omega)$ including the obtained physical parameters for each ${I_1}$, in Fig. 4(b). Importantly, our analysis involves the interference of the two nonorthogonal spectral peaks mediated by non-Hermiticity, as we can see the equivalence between the CME spectral response and LDOS around the EP [36] (Section 9 of Supplement 1). The theoretical peak intensity is consistent with the experimental data especially for ${I_1} \le 3\;\unicode {x00B5}{\rm A}$, where the excitation for cavity 1 is sufficiently small in experiment. Although we ensure the unimodal merging of the two coupled-mode peaks at ${I_1} = 2.4\;\unicode {x00B5}{\rm A}$ [blue curve in Fig. 4(c)], the intensity here is close to its minimal value because of the enhanced loss, ${\gamma _1} = 93.4\;{\rm GHz} \approx 1.6\kappa$. In contrast, it increases by 30% at the near-EP condition with ${I_1} = 1.4\;\unicode {x00B5}{\rm A}$ corresponding to a further larger ${\gamma _1} = 115.6\;{\rm GHz} \approx 2.0\kappa$, in both theory and experiment; the spectrum in this case is shown as the red plot in Fig. 4(c) for comparison. The monotonical increment in the peak intensity before the EP indicates the enhanced LDOS based on EP degeneracy. Note that the enhancement ratio here is less than double that in Fig. 1(d), because we control not $\kappa$ but ${\gamma _1}$, and the eigenstates for ${I_1} = 2.4\;\unicode {x00B5}{\rm A}$ are already nonorthogonal.

Finally, the spontaneous emission spectra for ${I_1} = 1.4\;\unicode {x00B5}{\rm A}$ and 0.2 µA are fit by some distinct trial functions and plotted in both linear and semi-logarithmic scales as Figs. 5(a) and 5(b), respectively. Again, our CME spectral function reproduces the experimental data well, and the apparent discrepancy between them is seen just in the region with 10% or less of the peak counts. The errors in their skirts can be attributed mostly to the slightly inclined background luminescence spectrum due to its peak located at around 1440 nm. This non-ideal factor can be corrected within the first order, as shown in Section 7 of Supplement 1. Remarkably, the entire section of the observed spectrum for ${I_1} = 1.4\;\unicode {x00B5}{\rm A}$ is in accordance with the squared Lorentzian function in Figs. 1(c) and 1(d), $4{\pi ^{- 1}}C{[{{\gamma ^2}/(\Delta {\omega ^2} + {\gamma ^2})}]^2}$ with coefficient C, rather than with the ordinary Lorentzian function (see Section 6 of Supplement 1 for additional data). This evidences the resonance very near the exact EP and supports the enhancement of the photonic LDOS by 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 ${| {{a_1}(\omega)} |^2}$ but ${| {{a_1}(\omega) + i{a_2}(\omega)} |^2}$ (Sections 1 and 8 of Supplement 1). Here, we can exclude the Voigt fitting function [47], i.e., the convolution of the cavity Lorentzian factor and Gaussian noise, because it requires a too small average loss to have the EP ($26\;{\rm GHz} \lt {\gamma _{1,{\rm EP}}}/2 = 58\;{\rm GHz}$), as well as persistent Gaussian noise (27 GHz) inconsistently larger than our lasers’ oscillation linewidths [38] ($ \lt $4 GHz: our finest measurement resolution; Fig. S1 of Supplement 1). As ${I_1}$ further decreases to ${I_1} = 0.2\;\unicode {x00B5}{\rm A}\;({\gamma _1} =157.2\;{\rm GHz} \approx 2.7\kappa$), the experimental and best-fit CME spectra get settled in more Lorentzian shapes [Fig. 4(b) and Section 9 of Supplement 1]. This indicates that the system loses the effect of degeneracy on the LDOS for a large imaginary potential contrast, although the peak intensity further increases by the reduction of ${\rm Im}\;\Delta \omega$ and effective excitation of the dominant mode localizing at cavity 2.

6. DISCUSSION AND CONCLUSION

The EP resonance also exhibits a peculiar transient response. In our ideal EP condition, namely, ${\gamma _1} = 2\kappa$, ${\gamma _2} = 0$, ${c_1}(\omega) = 0$, $|{c_2}(\omega {)|^2} = {\rm const}$., cavity 1’s radiation spectrum ${\gamma _{{\rm cav}}}|{a_{{1,{\rm EP}}}}(\omega {)|^2}$ is a squared Lorentzian function [Eq. (S7) of Supplement 1]. Its inverse Fourier transform directly reflects the autocorrelation function, ${C_{{1,{\rm EP}}}}(\tau) = {\langle a_{{1,{\rm EP}}}^*(t){a_{{1,{\rm EP}}}}(t + \tau)\rangle _t}$. This measures the temporal average of field decay in cavity 1 during an interval of $\tau$, in response to every incoherent photon excited at cavity 2. In fact, the analytic operation yields ${C_{{1,{\rm EP}}}}(\tau) \propto (1 + \kappa \tau)\exp (- {\gamma _1}\tau /2)$, while the coupled modes in the Hermitian limit undergo just exponential loss, ${C_{\kappa \gg \gamma}}(\tau) \propto \exp (- {\gamma _1}\tau /2)$.

Although the EP mode is distributed over both cavities, spontaneous emission occurs in cavity 2. As a result, it takes the time $1/\kappa$ for the fields to jump into cavity 1 and settle in the steady eigenstate. Here, the decay is prevented while the photons stay mostly in the loss-compensated cavity 2. Since $(1 + \kappa \tau) = (1 + {\gamma _1}\tau /2) \approx \exp ({\gamma _1}\tau /2)$ for ${\gamma _1}\tau /2 \ll 1$, the net damping term $\exp (- {\gamma _1}\tau /2)$ is indeed canceled within the first order of $\kappa \tau$. The EP hence enhances the peak spectral intensity, which corresponds to the integral of ${C_{{1,{\rm EP}}}}(\tau)$. Note that this mechanism is also valid for the fields ${a_{{2,{\rm EP}}}}$ of cavity 2. Thus, its radiation [Eq. (S9) of Supplement 1] and the entire device emission spectrum [Fig. 5(a), Fig. S6(a) of Supplement 1] hold squared Lorentzian shapes. Exploiting such EP dynamics is an intriguing future direction.

Enhancing the peak LDOS at the passive EP will drastically modulate the photonic responses of quantum emitters [48], coherent absorbers [20], and nonlinear optical devices [49]. It can also have the assistance of local gain [35] and get further enhanced at a higher-order EP [34] with a ratio of $\sqrt \pi \Gamma (n + 1)/\Gamma (n - 1/2)$ ($= 4$ for $n = 2$), where $n$ is its order, and $\Gamma (n)$ is the gamma function. Nonlinear optical effects will even be made hundreds of times more efficient [36] 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.

Coupled nanolasers with electrical pumping, despite leading to the integration of periodic and controllable non-Hermitian optical systems, have not been not reported. Our buried heterostructure technology can provide such a framework and will open the door for access to singularities of group velocity [39], reconfigurable photonic topological insulators [50], and vortex charges and chirality of EPs [51]. Large-scale passive devices [23,52,53] in one and two dimensions successfully relax the condition of parameters for achieving EPs and rings of EPs. Nonetheless, fabrication-induced defects make it somewhat challenging to handle the degeneracy in such systems. Further study of corresponding active cavity arrays will also be of great significance.

In conclusion, we showed the clear EP transition of spontaneous emission with our current-injected photonic crystal nanolasers. We first clarified that it was difficult for lasing PT-symmetric eigenmodes to reach EP degeneracy, because one of them was suppressed by the existence of cavity detuning. In contrast, the independent and efficient electrical pumping to our cavities enables the spontaneous emission 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. These features demonstrate the peak LDOS enhancement that is intrinsic to EP degeneracy. Our results represent an important step toward EP-based control of optoelectronic processes and large-scale non-Hermitian nanophotonic devices.