Majorana zero modes, a captivating concept originally proposed in the study of neutrinos, have intrigued physicists over the past eighty years. Being their own anti-particles and hosting non-Abelian braiding properties, their experimental demonstration is being actively pursued in high-energy physics and condensed matter physics [1–5]. The existence of these zero-energy excitations is warranted by particle-hole symmetry, in the form where the (Hermitian) Hamiltonian anti-commutes with an anti-linear operator [6].

Recently, there have been several proposals to realize particle-hole symmetry in non-Hermitian (NH) systems [7], especially on integrated photonic platforms, where the spatial arrangement of optical gain and loss [8], as well as asymmetric couplings between different elements [9], plays an important role. These findings are quite surprising because photons are bosons and cannot form particle-hole pairs in general. However, by realizing that these effective particles and holes have complex energies in an NH system, they do not need to adhere to Fermi–Dirac statistics as their condensed matter counterparts do. To highlight this difference, we will refer to such symmetries as NH particle-hole (NHPH) symmetry.

Although the resulting zero modes of NHPH symmetry differ from Majorana zero modes in several key aspects, they have two desirable properties in many photonic applications: their symmetry protection not only exists at the origin of the complex energy plane but also extends to the entire imaginary axis; they can also be conveniently excited in standard arrays of optical cavities or waveguides, without requiring the existence of Hermitian counterparts when non-Hermiticity is removed.

Thanks to the flexibility of designing optical elements, these properties of photonic zero modes can also be induced by pseudo-anti-Hermiticity [10]: ηH†η−1=−H, where H is the Hamiltonian and η a linear operator. If we consider a system consisting of two sublattices A and B, with K and L waveguides or cavities, respectively, where couplings only take place between two entities belonging to different sublattices, pseudo-anti-Hermiticity coincides with NHPH symmetry when H is symmetric, but it is distinct otherwise, such as in a topological insulator laser [11,12], where an effective gauge field is realized by staggered couplings in a two-dimensional array.

Despite these theoretical advances in the NH domain, the observations of photonic zero modes with the aforementioned properties have been restricted to arrays that resemble their Hermitian counterparts, such as the Su–Schrieffer–Heeger (SSH) lattice [13–16]. In contrast to topologically protected edge states, the NH zero modes protected by NHPH symmetry, including the ones we study here, are not restricted to interfaces and can be excited in different regions—both bulk and edges—of a coupled photonic array [8]. Let us stress also its simplicity from both the conceptual and technical points of view: an arbitrary number of zero modes can be generated in a large cavity array with arbitrary coupling coefficients by just pumping different sites of an array.

Active photonic crystal (PhC) cavity arrays are outstanding platforms to access the NH realm because they naturally enable in-situ realization of gain/loss configurations and coupling engineering [17,18]. A recent demonstration of laser mode switching in a coupled PhC cavity could be explained on the basis of Hermitian dark modes [19]. Remarkably, PhC platforms recently led to the observation of exceptional points (EPs) in two coupled nanocavities [20,21].

In order to show the potential of exciting and controlling a photonic zero mode in a broader range of systems, especially those without a topological origin [8,22], here we report on its observation in a minimal system consisting of three coupled PhC cavities with NHPH symmetry. We will show that the observability of zero modes strongly depends on the frequency detuning between the two extreme cavities (sublattice A) and the central one (sublattice B) in a linear array—which we refer to as sublattice detuning, Δω=ωB−ωA. In our coupled cavity system, the intercavity coupling g is modified by design, allowing us to feature both large (|g|≳|Δω|) and small (|g|<|Δω|) coupling regimes. Hereby, we will show that, when entering into the large detuning phase, the zero mode first looses its properties because it is no longer NHPH symmetry protected and eventually coalesces with another lattice mode through a parity-time (PT) phase transition.

This paper is organized as follows. In Section 2, we provide a simple theoretical framework based on coupled mode theory (CMT) to understand zero modes in gain/loss cavity arrays warranted by NHPH symmetry. In Section 3, we describe our PhC three-cavity array with controllable coupling by means of the so-called barrier engineering technique. We also provide an experimental characterization of the linear Hermitian modes through resonant scattering experiments. In Section 4, we move into NH mode characterization by incoherently pumping the system, and we report on the direct observation of the zero mode in the small sublattice detuning regime. Such observation is based on photoluminescence (PL) intensity maps under the spatial scanning of the pump spot combined with a Fourier imaging technique. Conclusions are given in Section 5.

A simple theoretical modal analysis of evanescently coupled cavity lattices can be carried out in the framework of the CMT formalism. In the case of N resonant optical cavity lattices, the CMT is valid under the hypothesis of negligible coupling between non-adjacent cavities and weak coupling overall. It assumes that the system can be accurately described with both the isolated (real) cavity frequencies ωn and the coupling strength to their neighboring cavities. The resulting hybrid mode frequencies and field distributions become the eigenvalues and eigenvectors of a Hamiltonian operator H˜, which does not need to be Hermitian. This is naturally the case of optical cavities in the presence of loss and/or gain [7]: a lattice can be described in CMT by a Hamiltonian H˜ whose matrix elements are H˜nn=ω˜n and H˜nm=gnm (n≠m), where ω˜n are the complex frequencies of the isolated cavities, and gnm are the nearest-neighbor intercavity coupling parameters. In a closed Hermitian system, ω˜n=ωn are real, while in an NH gain/loss optical system, ω˜n=ωn−i/τn, where τn is the nth cavity lifetime, which can be negative for net gain. Also, gnm can become complex in general in an NH framework [23]; here, we will consider real gnm for simplicity, i.e., we will neglect mode-loss splitting [17,24].

A zero mode that can be realized in photonics is often the result of the sublattice (or chiral) symmetry in Hermitian lattices, where eigenvalues are real. Indeed, it can be shown that, in Hermitian chiral arrays, the Hamiltonian anti-commutes with a diagonal matrix C consisting of K ones followed by L minus ones, i.e., {H˜,C}=0. This property ensures εk=−εj, and a zero mode with j=k verifies εj=0. These are known as dark modes, because the intensity in one of the sublattices vanishes [8]. On the other hand, eigenvalues are complex in general in NH arrays. While chiral symmetry can still be realized in this case [6], a more prevailing symmetry that leads to a zero mode in an NH lattice is NHPH symmetry. While sharing key features with their Hermitian counterparts such as the immunity to coupling disorder, they possess an important property: they can exist along the entire imaginary axis. This is warranted provided that {H˜,CT}=0, with T being the time-reversal operator, i.e., the system features NHPH symmetry. Zero modes in this framework (εk=−εj*) are not restricted to interfaces and can be excited in different regions—both bulk and edges—of a coupled photonic array. Table 1 summarizes similarities and differences between Hermitian and NH zero modes.Table 1.

Comparative Table with the Salient Properties of Hermitian and Non-Hermitian Photonic Zero Modes

Although our analysis can be extended to large cavity networks, in this work, we will focus on a small array of three coupled cavities as the minimal system containing a zero mode. Figure 1 illustrates a simple case where three cavities are aligned along the x direction, while the resonant intracavity field oscillates back and forth in the y direction [see schematics in Fig. 1(c)]. We assume they all have the same resonant frequency ω0 chosen as the reference frequency, nearest-neighbor coupling gnm=g, and intrinsic field decay time τ0 due to optical losses. Let us point out that the zero mode has the same frequency as a standalone cavity. One of the extreme cavities, say the top one, is incoherently pumped, therefore introducing a variable gain γ; see Fig. 1(a). The complex eigenvalues of H˜ and their evolution with increasing γ are depicted in Fig. 1(b). The black symbols correspond to the eigenvalues εj (j=1,2,3) evolving as the pump is increased from γ=0 (red square at the starting point) to γ=9.3 (black cross at the end point). We call M1 and M3 the lowest and highest frequency modes, respectively, both having a nearly symmetric field distribution, while M2 is the central mode featuring π/2 phase jumps between adjacent cavities [Fig. 1(b), inset].

A zero mode (j=k), which always exists in the case of an odd number of cavities, leads to Re(εj)=0. This is the case of M2, whose frequency does not depend on γ because of the NHPH symmetry protection, while M1 and M3 frequencies do. It is worthwhile noting that such a frequency change for M1 and M3 is a pure MH effect, not related to any nonlinear refractive index effect.

Unlike the Hermitian counterparts, the zero mode is no longer a dark one under single cavity pumping; see Table 1 and Fig. 1(b): no light extinction occurs in the central cavity. However, the exact π/2 phase shift between adjacent cavities distinguishes the wave function of a zero mode from all other modes and dramatically impacts the PL far-field imaging in experiments, as will be shown later on.

Here, we will restrict our analysis to the spontaneous emission regime—i.e., below laser threshold and neglecting amplification—in which a linear NH CMT is valid. We assume a (Gaussian) pump spot profile P(x;X) centered at a given X position; therefore, γn(X)=P(xn;X), where xn are the central positions of the cavities. As a result, εj depends not only on the pump power but also on the pump position, εj=εj(X). The spontaneous emission in the cavities is |f⟩ (fn∝γn), and the modal excitation amplitudes fj(X)=⟨Φj|f⟩, with |Φj⟩ being the left eigenvectors of H˜. In this spontaneous emission regime, the total emitted spectral intensity can be calculated as the incoherent superposition of the N mode intensities, each one contributing with a Lorentzian peak of amplitude fj(X), resonant frequency Re[εj(X)], and width Im[εj(X)]: I(ω;X)=|∑jfj(X){ω−Re[εj(X)]}i+Im[εj(X)]|2,(1)I(ω;X)≃∑j|fj(X)|2{ω−Re[εj(X)]}2+{Im[εj(X)]}2.(2)

Figure 1(c) shows the spectral intensity map I(ω;X) under spatial scanning of a Gaussian pump spot, computed from Eq. (2). The signatures of the zero mode are the two central lobes corresponding to M2 in Fig. 1(c). Such an intensity map, together with systematic far-field measurements, constitutes a tool to experimentally investigate zero-mode radiation in the active cavity array.

NH zero modes warranted by NHPH symmetry have the freedom to evolve along the Im(ε) axis. In particular, at γ=9.3, Im(ε2)=0 so that the gain compensates the losses. Thus, the linear CMT model predicts zero-mode lasing in three coupled cavities systems as one of the extreme cavities is pumped [8]. When dealing with active semiconductor media—such as InP membranes with quantum wells (QWs), as in our experiments—an important deviation from this simple CMT model is the absorption saturation and carrier-induced refractive index changes below the laser threshold, leading to frequency blue shifts of the cavity mode. We have incorporated those effects in the CMT model, which we call hereafter carrier-dependent CMT (CD-CMT) equations, derived in Appendix A. In essence, these are based on a logarithmic model for the absorption and the refractive index as a function of the charge carrier density in the QWs. The eigenvalues as a function of the carrier are superimposed to the linear CMT eigenvalues in Fig. 1(b) (red symbols and lines). It can be observed that it is now the highest energy eigenvalue M3 that reaches the threshold before the middle one M2. Qualitatively, lasing phenomena require a good spectral and spatial overlap between the gain and the optical mode. Since the pump-induced refractive index effect blue shifts the optical mode of the pumped cavity, it is likely that the closest hybrid mode to the blue-detuned pumped cavity will be more efficiently excited. This qualitative argument is translated in the evolution of the eigenvalues, as evidenced in Fig. 1(b).

PhC cavities with embedded QWs are a suitable platform to experimentally investigate zero-mode photonics. This is due to the multiple degrees of freedom provided by the design parameters, as well as the intrinsic and controllable gain/absoption features.

Three coupled PhC L3 cavities [three missing holes in the ΓK direction of a triangular air hole lattice, see Fig. 2(a)] are separated by three rows of holes in the ΓM direction, leading to evanescent coupling. The two extreme cavities of the linear array couple to the middle cavity through a coupling coefficient g. In order to control g, we implement the so-called barrier engineering technique, by virtue of which the coupling strength (and even its sign) can be changed, modifying the middle row separating two adjacent cavities [17,18]. We have designed the central hole row in the barriers with radius r3=r0(1+h), with r0 being the hole radius of the underlying PhC lattice. We call the parameter h the barrier perturbation. Importantly, h has a strong impact on the cavity frequencies due to its influence on the effective refractive index surrounding the cavities. Since the barrier induces a frequency detuning Δω(h) in a contiguous nanocavity, a good approximation is to consider the two extreme cavities as having the same frequency ω0+Δω(h) and ω0+2Δω(h) for the central cavity [see schematics in Fig. 2(a), bottom and Appendix C for further details]. Therefore, the sublattice detuning is Δω=ωB−ωA≈Δω(h).

In order to predict the influence of the barrier parameter in the coupled mode structure, we have carried out 3D finite difference time domain (FDTD) simulations. First, g(h) and Δω(h) have been obtained by polynomial fitting datasets of a two coupled cavity system separated by a barrier with perturbation h (see Fig. 5 in Appendix B). Importantly, two regions of h can be distinguished: the large detuning region, |Δω(h)|>|g(h)| for h≤−5% or h≥10%, and the small detuning region, |Δω(h)|<|g(h)| for −5%≲h≲10%. Subsequently, 3D-FDTD simulations with three cavities have been carried out. The mode frequencies as a function of h are depicted in Fig. 2(b), together with the Hermitian CMT predictions using the fitted parameters Δω(h) and g(h) (Appendix C), showing very good agreement.

We have fabricated the PhC cavity array of Fig. 2(a) in a suspended InP membrane of 280 nm thickness containing four InGa0.17As0.76P QWs, featuring a PL peak at λ≈1514nm [Fig. 2(c)]. The details on the fabrication can be found in Ref. [17]. The hole radius is r0=0.266a, and a, the period of the triangular lattice, lies in the range of 400–420 nm. The two holes limiting each cavity have a reduced radius of r1=r0−0.06a and are displaced away by s=0.16a, in order to increase the Q-factor [25]. Also, holes around the cavities are modified with a period 2a by r2=r0+0.05a in order to improve the beaming quality of the emitted light and hence the collection efficiency [26]; consequently, beaming holes inside the barrier have radius r4=r2(1+h). Only samples with h≤0 have been realized, which suffices to largely tune the intercavity coupling strength g. The resonance wavelengths of the samples range from 1500 to 1600 nm depending on the lattice period. The Q-factor of the cavity resonances is Q∼4000 at λ=1580nm, i.e., far from the QW absorption.

Two kinds of experimental characterizations have been carried out: reflectivity spectra and PL experiments, both with controlled spatial positions of the illumination spots. For the reflectivity spectra, a single mode tunable laser is used, where the reflected signal is coupled into a single mode optical fiber and sent to a femtowatt photodetector; the background reflectivity is highly suppressed using polarization optics (see Appendix D). In these experiments, the cavity wavelengths, lying in the range of 1560–1600 nm, are red-detuned from the QW absorption, and the illumination power is low enough to be considered as linear reflectivity experiments. Hence, these can be interpreted as optical characterizations of linear Hermitian modes [Fig. 2(d)]. Unlike standard resonant scattering experiments leading to Fano resonances, the reflectivity background suppression allows us to clearly identify modes as Lorentzian-like peaks.

The spectral position of the measured resonances is in very good agreement with the FDTD calculations of Fig. 2(b). Noticiably, three modes are clearly distinguished for h=0% and h=−5%, corresponding to the low sublattice detuning region, while the middle peak is not apparent for −25%≤h≤−10%. This interval is within a crossover region where g is small and changes signs [−25%≤h≤−5% corresponding to −2.17THz≤g≤1.26THz, dashed region in Fig. 2(b)], with a crossing point (g=0) at h≈−15%. As a consequence, M2 and M3 frequencies are slightly split in this region. Finally, M2 resonances re-emerge for h≤−30%. Note that although the system is no longer chiral for Δω≠0, M2 remains a zero mode due to geometric frustration, where the couplings from cavities 1 and 3 are canceled in cavity 2 (see Appendix E).

In addition to the reflectivity spectra of the previous section, which characterize Hermitian modes, we have also performed PL experiments, where the pump laser wavelength is now λ=800nm: the laser beam is mainly absorbed in the quantum barriers, and thus it can be considered as an incoherent pump. We use a pulsed laser (100 ps duration and 1 MHz repetition rate) in order to reduce thermal effects. As in the reflectivity experiments, the pump spot is focused down to nearly the diffraction limit so as to achieve a pump configuration with a localized gain profile across the cavity array, meaning that essentially one cavity is pumped when aligning the pump beam at its center. The radiated PL is collected in the free space and spectrally resolved with a spectrometer coupled to an InGaAs one-dimensional (1D) detector array. A piezo-electric-driven stage holding the sample allows us to externally control the sample position with respect to the pump spot with sub-micron resolution. The results are shown in Figs. 3 and 4.

The spectral intensity as a function of the sample position reveals two distinct typical patterns depending on the value of the barrier parameter h. For h<−5%, the detuning |Δω(h)| is larger than the coupling strength |g(h)|; we can then expect that the two extreme cavities become effectively decoupled from the central one, especially for h≤−20%. The spectral map of Fig. 3(a) (h=−20%) is consistent with this prediction: as long as the pump excites the QWs in one of the extreme cavities, a resonant mode is observed at λ0≈1558.2nm, and, when pumping the middle cavity, a mode red shifted by Δλ≈7.7nm∼6THz comes out, consistent with the |Δω(h=−20%)|≈7 THz detuning obtained from the numerical simulations. The near-field images [Figs. 3(a)-i to 3(a)-v] show that the emission essentially comes from the pumped cavity provided only one mode is excited. Note that two near-field lobes can be observed for intermediate positions, where two resonances are simultaneously present in the spectrum [Figs. 3(a)-ii and 3(a)-iv]. The far-field images [Figs. 3(a)-vi to 3(a)-x] confirm this observation of always revealing only one central lobe, consistent with a localized cavity mode. It is even the case for two near-field lobes, for which the radiation pattern becomes the incoherent superposition of two centered single cavity far-field lobes [Figs. 3(a)-vii and 3(a)-ix].

This analysis allows us to conclude that the two external cavities are decoupled from the central one for large sublattice detuning. The experimental results are in very good agreement with CD-CMT calculations that blue shift the cavity resonance as a function of the pump power.

Interestingly, although the zero mode would still exist in the case of large Δω in Hermitian systems, it is no longer observable in our NH system with detuning using a single localized pump spot. As pointed out before, this is because Hermitian zero modes are dark ones; therefore, they remain unaffected if the detuning takes place only in the cavities where the amplitude of the zero modes is zero. On the other hand, the NH zero modes are not dark ones in general. Here, in our NH system, a single localized pump spot results in an imaginary detuning that acts together with the real (frequency) detuning to eliminate the NHPH symmetry and its zero modes (see Appendix E). Therefore, within this large sublattice detuning regime, the NH zero mode is destroyed when pumping on one extreme cavity.

To experimentally address a zero mode, we change the barrier parameter to enter into the low detuning regime, h∼0%. Within this regime, the spectral intensity pattern totally differs from the large detuning case where localized modes prevailed [Fig. 3(a)]. For −5%≲h≲10%, |Δω(h)|<|g(h)| [Fig. 2(b)] and the three coupled cavities effectively behave as a whole. In the spectral intensity map [Fig. 4(a), h=0%], we can observe a pattern with the mode of highest energy M3 being excited independently of the sample position; it has a higher intensity compared to the two other modes. The central mode M2, on the other hand, attains two maxima in between the extreme cavities and the central one. The lowest energy mode M1 is the weakest one, and it is only observed when pumping near the central cavity, a feature that was already present in the simplified calculation of Fig. 1(c).

In these conditions, we have measured the near-field and far-field patterns, setting the pump spot positions at the local maxima of the modes and using pass-band filters to filter out all other spectral components. From the near and far-field images of the highest [M3, Figs. 4(a)-ii and 4(a)-vi] and lowest [M1, Figs. 4(a)-iii and 4(a)-vii] energy modes, we conclude that those are approximately symmetric modes, with M1 being the fundamental one, which is in agreement with the CMT and FDTD calculations. In particular, the near-field images [Figs. 4(a)-ii and 4(a)-iii] show that the emission of those two modes comes from all three cavities, with a higher intensity in the middle for M3 and in the extreme ones for M1. The central mode, on the other hand, features a far-field intensity node at the center [M2, Figs. 4(a)-v and 4(a)-viii]; its near field is more intense in the two extreme cavities [Figs. 4(a)-i and 4(a)-iv], while the intensity is below our detection limit in the central cavity region. These observations are compatible with the NH zero mode M2 of Section 2. In particular, the π/2 phase jump between adjacent cavities predicted for an NH zero mode is translated into a π phase difference between the two extreme cavities, giving rise to an anti-symmetric-like far-field profile, as it has already been observed for two coupled cavities [27].

Figure 4(b) shows the CD-CMT calculations. The carrier-induced refractive index effects have an important impact on the PL map, as compared to Fig. 1(c). Although the main qualitative features are already captured by a linear NH CMT, in the CD-CMT, the blue-detuned mode M3 strongly dominates over the other two modes. As discussed before, this can be explained as a consequence of the frequency blue shift of a cavity resonance under optical pumping: we can therefore expect that the blue most detuned hybrid mode will be more efficiently excited, since its spectral overlap with the pumped cavity resonance increases. This explains the enhancement of the high-energy mode M3 in this low sublattice detuning regime, even though the zero mode is clearly observable. Compared to the experimental result of Fig. 4(a), the two M2 lobes are less visible; this might be due to the simplification of our CD-CMT model that neglects dissipative coupling and amplification. Importantly, the zero mode might be brought to laser operation as long as a two spot pumping scheme is implemented (see Appendix E), as will be reported elsewhere. In addition, in our carrier-dependent model, we observe that the intercavity phase jumps in M2 become quite sensitive to frequency shifts; in the CD-CMT calculation of Fig. 1(b), the M2 cavity phases for N=0.3 are [0.85π,0,1.85π], thus deviating from the expected π/2. Such sensitivity is a further example of how active semiconductor media, featuring a carrier-induced refractive index, may have a significant impact in the NH realm. For instance, in PT-symmetry breaking experiments using semiconductor microcavities, PT symmetry gets explicitly broken in the sense that the real part of the permittivity is no longer symmetric under asymmetric pumping.

Let us now address the question of the extinction of the NH zero mode in this system for decreasing the h parameter. First, we notice that for very small |g| PT-symmetry breaking is predicted as one extreme cavity is pumped. Within this broken PT-symmetry phase (−18%<h<−11%), the central cavity (sublattice B)—already effectively decoupled from the two extreme ones—does not play an important role; at the same time, the two extreme ones (sublattice A) weakly interact with each other, such that a gain imbalance may undergo an EP. This is depicted in Figs. 7(l) and 7(m) of Appendix F: the real parts of the eigenvalues of modes M2 and M3 undergo EPs when pumping in the proximity of an extreme cavity. Within a pumped cavity of sublattice A, as it is apparent in the inset of Fig. 7(l), the real parts of the eigenvalues of M2 and M3 coalesce in a single branch; there, only one mode is observable—the one with higher gain—and it is localized in the pumped cavity. The otherwise EP bifurcation results in an imperfect symmetry breaking [Figs. 7(g) and 7(h)] [28], enhancing light localization in the pumped cavity. Therefore, within this broken PT-symmetry region, light localization in the real device is a combination of underlying PT-symmetry breaking and pump-induced frequency shift effects, while outside this region [Figs. 7(a), 7(f) and 7(k)] localization in one extreme cavity arises because of pure carrier-induced refractive index effects.

We have reported on the direct observation of NH zero modes warranted by NHPH symmetry in a minimal cavity array: three coupled PhC nanocavities in a gain/loss configuration under spatially localized optical pumping. Because the number of cavities is odd, Lieb’s theorem ensures the existence of a zero mode in the Hermitian limit, which evolves continuously into an NH zero mode. Therefore, there is no need to generate it through, for instance, NHPH spontaneous symmetry restoration. These two types of zero modes do not transform into each other in general (except at a higher-order EP [29]) and can hence be regarded as topologically distinct.

The nature and properties of these NH zero modes differ from their chiral Hermitian counterparts in two main aspects. First, unlike Hermitian zero modes, NH ones are more robust in the sense that they are not restricted to the origin of the complex plane, but they may exist along the imaginary axis, still benefiting from symmetry protection. More specifically, and analogously to chiral modes, they are immune to random coupling perturbations in the cavity array. Second, although NH zero modes are not dark ones in general, in the sense that there is no light extinction in one of the sublattices, these zero modes feature π/2 phase jumps between adjacent cavities. These constitute unambiguous physical signatures that distinguish their wavefunctions from any other ones in the array and enable experimental protocols to detect them. Here, we have shown that a PL intensity map under spatial scanning of the pump spot provides a clear fingerprint of zero modes in the form of PL maxima in between cavities. At those PL maxima, a Fourier imaging technique allows us to identify nodes at the far-field center (k=0) due to a π-phase difference between the extreme cavities, compatible with the predicted π/2 phase jumps.

We have identified different regimes that arise as the coupling barrier of the photonic molecule is systematically modified through the perturbation parameter h that varies the hole radius of a row within the photonic barrier. Such a barrier modification simultaneously changes the coupling g and the sublattice detuning Δω, giving rise to two important regimes: |Δω|>|g| leading to sublattice localization and |Δω|<|g| leading to sublattice delocalization and zero modes. Remarkably, we have identified the transition from zero-mode observation to its extinction as h is decreased as a combination of underlying PT-symmetry breaking and carrier-induced blue shift effects, which strongly localize the PL in a single cavity.

In spite of the tremendous theoretical advances in the NH domain, observations of photonic zero modes have been restricted so far to waveguide or cavity arrays that resemble their Hermitian counterparts, such as the SSH lattice. Although experimental demonstrations of NH phenomena such as PT-symmetry breaking and EPs are coming to maturity, with special focus in two resonator systems, the physical realization of NH symmetries beyond PT such as NHPH in large cavity arrays—a unique playground for NH photonics—is still in its infancy. We believe that further developments in this direction would enable promising applications of NH symmetry protected modes, ranging from laser array mode engineering, optical communications, and photonic computing [30]. For instance, one can imagine expanding the number of optical communication channels without requiring additional bandwidth, using instead the different spatial profiles of these modes as an additional degree of freedom. This approach provides an alternative or complement to encoding optical signals using beams with different orbital angular momentum [31], which has gained fast-growing interest in the past decade.

Acknowledgment. B. G. and A. M. Y. acknowledge the financial support from the European Union in the form of Marie Skłodowska-Curie Action. L.G. acknowledges support by NSF. This work has also received funding from the European Research Council (ERC) under the Horizon 2020 research and innovation programme (HYPNOTIC PROJECT Grant Agreement No. 726420).