Flore Hentinger

Melissa Hedir

Bruno Garbin

Mathias Marconi

Li Ge

Fabrice Raineri

Ariel Levenson

Alejandro M Yacomotti
email: alejandro.giacomotti@c2n.upsaclay.fr

Direct observation of zero modes in a non-Hermitian nanocavity array

Zero modes are symmetry protected ones whose energy eigenvalues have zero real parts. In Hermitian arrays, they arise as a consequence of the sublattice symmetry, implying that they are dark modes. In non-Hermitian systems, that naturally emerge in gain/loss optical cavities, particle-hole symmetry prevails instead; the resulting zero modes are no longer dark but feature π/2 phase jumps between adjacent cavities. Here we report on the direct observation of zero modes in a non-Hermitian three coupled photonic crystal nanocavity array containing quantum wells. Unlike the Hermitian counterparts, the non-Hermitian zero modes can only be observed for small sublattice detuning, and they can be identified through far-field imaging and spectral filtering of the photoluminescence at selected pump locations. We explain the zero mode coalescence as a parity-time phase transition for small coupling. These zero modes are robust against coupling disorder, and can be used for laser mode engineering and photonic computing.

Introduction

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 [START_REF] Hasan | Colloquium: Topological insulators[END_REF][START_REF] Qi | Topological insulators and superconductors[END_REF][START_REF] Alicea | New directions in the pursuit of Majorana fermions in solid state systems[END_REF][START_REF] Beenakker | Random-matrix theory of Majorana fermions and topological superconductors[END_REF][START_REF] Nayak | Non-Abelian anyons and topological quantum computation[END_REF] . The existence of these zero-energy excitations is warranted by particle-hole symmetry, in the form that the (Hermitian) Hamiltonian anticommutes with an anti-linear operator [START_REF] Rivero | Chiral symmetry in non-Hermitian systems: Product rule and Clifford algebra[END_REF] .

Recently, there have been several proposals to realize particle-hole symmetry in non-Hermitian systems [START_REF] Feng | Non-Hermitian photonics based on parity-time symmetry[END_REF] , especially on integrated photonic platforms where the spatial arrangement of optical gain and loss [START_REF] Ge | Symmetry-protected zero-mode laser with a tunable spatial profile[END_REF] , as well as asymmetric couplings between different elements [START_REF] Malzard | Topologically Protected Defect States in Open Photonic Systems with Non-Hermitian Charge-Conjugation and Parity-Time Symmetry[END_REF] , 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 a non-Hermitian 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 non-Hermitian 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 imagi-nary 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, where couplings only take place between two cavities 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 [START_REF] Bandres | Topological insulator laser: Experiments[END_REF][START_REF] Zhao | Non-Hermitian topological light steering[END_REF] , where an effective gauge field is realized by staggered couplings in a two-dimensional array.

Despite these theoretical advances in the non-Hermitian 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 [START_REF] Poli | Selective enhancement of topologically induced interface states in a dielectric resonator chain[END_REF][START_REF] Zhao | Topological hybrid silicon microlasers[END_REF][START_REF] Song | Breakup and Recovery of Topological Zero Modes in Finite Non-Hermitian Optical Lattices[END_REF][START_REF] Pan | Photonic zero mode in a non-hermitian photonic lattice[END_REF] . A recent demonstration of a laser mode switching in a coupled photonic crystal cavity could also be explained on the basis of Hermitian dark modes [START_REF] Pellegrino | Mode-field switching of nanolasers[END_REF] . Active photonic crystal cavity arrays are outstanding platforms to access the non Hermitian realm because they naturally enable in-situ realization of gain/loss configurations and coupling engineering [START_REF] Haddadi | Photonic molecules: tailoring the coupling strength and sign[END_REF] . Remarkably, they have recently led to the observation of exceptional points in two coupled nanocavities [START_REF] Kim | Direct observation of exceptional points in coupled photonic-crystal lasers with asymmetric optical gains[END_REF][START_REF] Takata | Observing exceptional point degeneracy of radiation with electrically pumped photonic crystal coupled-nanocavity lasers[END_REF] .

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 [START_REF] Ge | Symmetry-protected zero-mode laser with a tunable spatial profile[END_REF][START_REF] Qi | Defect States Emerging from a Non-Hermitian Flatband of Photonic Zero Modes[END_REF] , here we report on its observation in a minimal system consisting of three coupled photonic crystal cavities with NHPH symmetry. Interestingly, while Hermitian zero modes are robust against frequency detuning between the two extreme cavities (sublattice A) and the central one (sublattice B) in a linear array -that we refer to as sublattice detuning, ∆ω-, non Hermitian zero modes are not. In our coupled cavity system, the intercavity coupling g is modified by design, allowing to feature both large (|g| |∆ω|) and small (|g| < |∆ω|) 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 phase transition. This paper is organized as follows. In Section 1 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 2 we describe our photonic crystal 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 though resonant scattering experiments. In Section 3 we move into non Hermitian 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 intensity maps under the spatial scanning of the pump spot combined with a Fourier imaging technique. In Section 4 we provide a detailed analysis of the phase transitions in the system as both the coupling and the sublattice detuning are changed; a systematic comparison with linear and nonlinear CMT modeling allows us to explain the coalescence of the zero mode as a result of a parity time spontaneous symmetry breaking mechanism. Conclusions are given in Section 5.

Theoretical background: non-Hermitian zero modes

A simple theoretical modal analysis of evanescently coupled cavity lattices can be carried out in the framework of the Coupled Mode Theory (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 ω n∈ [1,N] 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 [START_REF] Feng | Non-Hermitian photonics based on parity-time symmetry[END_REF] : a lattice can be described in CMT by a Hamiltonian H whose matrix elements are H nn = ωn and H nm = g nm (n = m), where ωn are the complex frequencies of the isolated cavities and g nm are the nearest-neighbor intercavity coupling parameters. In a closed Hermitian system, ωn = ω n are real, while in a non-Hermitian gain/loss optical system ωn = ω ni/τ n , where τ n is the n-th cavity lifetime, which can be negative for net gain. Also g nm can become complex in general in a non-Hermitian framework [START_REF] Lalanne | Light interaction with photonic and plasmonic resonances[END_REF] ; here we will consider real g nm for simplicity, i.e. we will neglect loss splitting [START_REF] Atlasov | Wavelength and loss splitting in directly coupled photonic-crystal defect microcavities[END_REF][START_REF] Hamel | Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers[END_REF] .

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 ω 0 chosen as the reference frequency, nearest neighbor coupling g nm = g = 2, and intrinsic field decay time τ 0 = 1 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). Red symbols correspond to the eigenvalues ε j ( j = 1, 2, 3) evolving as the pump is increased from γ = 0 (red square at the starting point) to γ = 1.8 (red 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 that can be realized in photonics is often the result of the sublattice (or chiral) symmetry in Hermitian lattices, where eigenvalues are real. Hermitian chiral arrays ensure ε k = -ε j and a zero mode with j = k verifies ε j = 0. These are known as dark modes, because the intensity in one of the sublattices vanishes. On the other hand, eigenvalues are complex in general in non-Hermitian arrays. While chiral symmetry can still be realized in this case [START_REF] Rivero | Chiral symmetry in non-Hermitian systems: Product rule and Clifford algebra[END_REF] , a more prevailing symmetry that leads to a zero mode in a non-Hermitian lattice is NHPH symmetry, giving rise to ε k = -ε * j . A zero mode ( j = k), which always exists in the case of an odd number of cavities, leads to Re(ε j ) = 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 non Hermitian effect, not related to any nonlinear refractive index effect, as is usual in semiconductors; in Secs. 3 and 4 we will include such carrier-induced refractive index effects to model our experimental results more realistically.

Unlike the Hermitian counterparts, the zero mode is no longer a dark one under single cavity-pumping, see Fig. 1(b): no light extinction occurs in the central one. 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 photoluminescence far-field imaging in experiments, as will be shown later on.

Non-Hermitian zero modes warranted by NHPH symmetry 2/10 have the freedom to evolve along the Im(ε)-axis. In particular, at γ = 1.8, Im(ε 2 ) also becomes zero, so that the gain compensates the losses. Thus, the linear CMT model predicts zero-mode lasing in three coupled cavity systems as one of the extreme cavities is pumped [START_REF] Ge | Symmetry-protected zero-mode laser with a tunable spatial profile[END_REF] . Even though the zero mode can be clearly excited and identified by spatial localization of a single pump spot, semiconductor refractive index nonlinearities will generally prevent laser operation (see Section 3).

Here we will restrict our analysis to the spontaneous emission regime -i.e. below laser threshold and neglecting amplification-, in which a linear non-Hermitian CMT is valid. We assume a (gaussian) pump spot profile P(x; X) centered at a given X-position, therefore γ n (X) = P(x n ; X), where x n are the central positions of the cavities. As a result, ε j depends not only on the pump power but also on the pump position, ε j = ε j (X). The spontaneous emission in the cav-

ities is | f ( f n ∝ γ 1/2
n ), and the modal excitation amplitudes f j (X) = Φ j | f , Φ j | 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 f j (X), resonant frequency Re[ε j (X)] and width Im[ε j (X)]:

I(ω; X) = ∑ j f j (X) (ω -Re [ε j (X)]) i + Im [ε j (X)] 2 (1) ∑ j | f j (X)| 2 (ω -Re [ε j (X)]) 2 + (Im [ε j (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 intensity map, together with systematic far-field measurements, constitutes a tool to experimentally investigate zero-mode radiation in the active cavity array.

Three-coupled photonic crystal cavities: design and characterization of Hermitian modes

Photonic crystal (PhC) cavities with embedded quantum wells (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 only. In order to control the inter-cavity coupling 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 [START_REF] Haddadi | Photonic molecules: tailoring the coupling strength and sign[END_REF][START_REF] Hamel | Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers[END_REF] . We have designed the central holerow in the barriers with radius r 3 = r 0 (1+h), r 0 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 in-duces a frequency detuning ∆ω(h) in a contiguous nanocavity, a good approximation is to consider the two extreme cavities as having the same frequency ω 0 + ∆ω(h) and ω 0 + 2∆ω(h) for the central cavity [see schematics in Fig. 2(a), bottom, and Section III of the Supplementary Material for further details].

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 ∆ω(h) have been obtained by polynomial fitting datasets of a two coupled cavity system separated by a barrier with perturbation h (Fig. S1, Supplementary Material). Importantly, two regions of h can be distinguished: the large detuning region, |∆ω(h)| > |g(h)| for h < -5% or h > 10%, and the small detuning region, |∆ω(h)| |g(h)| for -5% h ≤ 10%. Subsequently, 3D-FDTD simulation with three cavities have been carried out. The mode frequencies as a function of h are 4/10 depicted in Fig. 2(b), together with the Hermitian CMT predictions using the fitted parameters ∆ω(h) and g(h), 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 InGa 0.17 As 0.76 P quantum wells (QWs), featuring a photoluminescence peak at λ ≈ 1514 nm [Fig. 2(c)]. The details on the fabrication can be found in Ref. [START_REF] Hamel | Spontaneous mirror-symmetry breaking in coupled photonic-crystal nanolasers[END_REF] . The hole radius is r 0 = 0.266a and a, the period of the triangular lattice, lies in the range 400 -420 nm. The two holes limiting each cavity have a reduced radius r 1 = r 0 -0.06a and are displaced away by s = 0.16a, in order to increase the Q-factor [START_REF] Akahane | Highq photonic nanocavity in a two-dimensional photonic crystal[END_REF] . Also holes around the cavities are modified with a period 2a by r 2 = r 0 + 0.05a in order to improve the beaming quality of the emitted light and hence the collection efficiency [START_REF] Tran | Directive emission from high-q photonic crystal cavities through band folding[END_REF] ; consequently, beaming holes inside the barrier have radius r 4 = r 2 (1 + h). Only samples with h ≤ 0 have been realized, which suffices to largely tune the inter-cavity coupling strength g. The resonance wavelengths of the samples range from 1500 nm to 1600 nm depending on the lattice period. The quality factor of the cavity resonances is Q ∼ 4000 at λ = 1580 nm, i.e. far form the QW absorption.

Two kind of experimental characterizations have been carried out: reflectivity spectra and photoluminescence experiments, both with controlled spatial positions of the illumination spots. For the reflectivity spectra a single mode tunable laser is used, 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 Methods). In these experiments the cavity wavelengths, lying in the range 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 sign [-25% ≤ h ≤ -5% corresponding to THz ≤ g ≤ 1.26 THz, dashed region in Fig. 2(b)], with a crossing point (g = 0) at h ≈ -15%. As a consequence, M1 and M2 frequencies are slightly split in this region. Finally, M2 resonances re-emerge for h ≤ -30%. While this M2 mode is expected to remain a dark mode and possess chiral symmetry in the Hermitian limit despite the large sublattice detuning, it will no longer be protected by NHPH symmetry, as will be discussed in the following.

Direct observation of the non-Hermitian zero-mode

In addition to the reflectivity spectra of the previous Section, which characterize Hermitian modes, we have also performed photoluminescence (PL) experiments where the pump laser wavelength is now λ = 980 nm: 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 1D detector array. A piezoelectric-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. 34. Interestingly, although the zero mode would still exist in the case of large ∆ω in Hermitian systems, it is no longer observable in our non-Hermitian system with detuning using a single localized pump spot. This is because Hermitian zero-modes warranted by sublattice symmetry 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 non-Hermitian zero modes are not dark ones in general. Here in our non-Hermitian 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 Supplementary material, Section I). Therefore, within this high sublattice detuning regime the system is generally not protected neither by sublattice nor by particle-hole symmetry and as a consequence there is no zero-mode that can be exploited.

To experimentally address a zero-mode, we change the barrier parameter so that 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 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, Fig. 4(a)-ii and (a)-vi] and lowest [M1, Fig. 4(a)-iii and (a)-vii] energy modes we conclude that those are approximately symmetric modes, with M1 being the fundamental one, in agreement with the CMT and FDTD calculations. In particular, the nearfield images [Figs. 4(a)-ii and (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, Fig. 4(a)-v,viii]; its near-field is more intense in the two extreme cavities, [Fig. 4(a)-i,iv], while the intensity is below our detection limit in the central cavity region. These observations are compatible with the non-Hermitian zero mode M2 of Section 1. In particular, the π/2 phase jump between adjacent cavities predicted for a non-Hermitian zero mode is translated into a π phase difference between the two extreme cavities, giving rise to an antisymmetric-like far-field profile, as it has already been observed for two coupled cavities [START_REF] Haddadi | Photonic crystal coupled cavities with increased beaming and free space coupling efficiency[END_REF] .

) c) d) e) f) g) h) i) j) k) l) m) n) o) CMT ∆𝜔 ≲ 𝑔 ∆𝜔 > 𝑔
Figure 4(b) shows the nonlinear 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 non Hermitian CMT, in the nonlinear CMT the blue-detuned mode M3 strongly dominates over the two other modes. 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 measurable. Importantly, the zero mode might be brought to laser operation as long as a two spot pumping scheme is implemented, as will be reported elsewhere.

Zero mode coalescence and phase transitions

In this section we unveil the underlying physical mechanisms that lead to the extinction of zero modes as the barrier parameter h is decreased from h = 0% to h = -25%, i.e. as the intercavity coupling |g| goes from above to below the sublattice detuning |∆ω|. Indeed, the non Hermitian zero mode is only observed in the |∆ω| |g| regime (Fig. 4), while it is missing in the |∆ω| > |g| regime (Fig. 3). First of all, let us recall that, in the large sublattice detuning regime, NHPH symmetry no longer warrants zero modes for single cavity pumping. But even if M2 looses its symmetry protection, the question arises of whether M2 still exists but remains undetectable, or it coalesces through a phase transition. Let us also recall that, in the Hermitian limit, the zero mode is observable for large sublattice detuning [Fig. 2(d), h = -30%, and

h = -35%].
The full picture of the different non Hermitian phases as a function of the barrier parameter is represented in Fig. 5; this completes the experimental PL maps for various h parameters [Fig. 5 5(e) (h = 0%). We first identify two phases, corresponding to effective coupling/decoupling of the central cavity with respect to the two extreme ones: the sublattice delocalized phase corresponds to |∆ω| |g| (-5% h ≤ 0%), in which M2 is observable and becomes the zero mode because of NHPH symmetry; conversely, the sublattice localized phase corresponds to |∆ω| > |g| (h ≤ -10%), in which M2 is no longer observable. Interestingly, within this sublattice localized phase, there is a sub-region corresponding to the crossover of g, from positive to negative, vanishing at h ≈ -15%. At such a crosssover M1 and M3 exchange symmetries, in the sense that M1 goes from a quasi symmetric mode with zero phase jumps between the cavities for g > 0, to a quasi symmetric mode with π phase flips between the cavities for g < 0.

More importantly, for very small g a parity time symmetry breaking is predicted as one extreme cavity is pumped. Within this broken PT symmetry phase (-20% h -10%), 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 unbalance may undergo an exceptional point (EP). This is depicted in Figs. 5(l) and (m): the real part 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 Fig. 5(m), 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.

Of course, in a semiconductor material under gain/loss operation, parameters can only be tuned to some extent near an EP. This is because, in addition to imperfect symmetries in real systems, semiconductors exhibit carrier-induced frequency shift that, in general, explicitly breaks the inversion symmetry of the real part of the dielectric constant. Therefore, the otherwise EP bifurcation results in an imperfect symmetry breaking [Figs. 5(g) and (h)] [START_REF] Garbin | Asymmetric balance in symmetry breaking[END_REF] . As already discussed, such a frequency shift enhances light localization in an extreme cavity when it is optically pumped. Therefore, within this broken PT symmetry region, light localization in the real device is a combination of both the underlying PT symmetry breaking and nonlinear effects, while outside this region [Figs.

Conclusion

We have reported on the direct observation of non Hermitian zero modes warranted by non Hermitian particle hole (NHPH) symmetry in a minimal cavity array: three coupled photonic crystal nanocavities in a gain/loss configuration under spatially localized optical pumping. Because the number of cavities is odd, the Lieb's theorem ensures the existence of a zero mode, and there is no need to generate them through, for instance, NHPH symmetry restoration. The nature and properties of these non Hermitian zero modes differ from their chiral Hermitian counterparts in two main aspects. Firstly, unlike Hermitian zero modes, non Hermitian 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. Secondly, although non Hermitian zero modes are not dark ones, 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 enables experimental protocols to detect them. Here we have shown that a photoluminescence (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 phase jumps as destructive interference in the far field. Specifically, the π/2 phase jumps between contiguous cavities result in a π phase difference between the extreme cavities, which can be detected as nodes at the far field center (k = 0).

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 ∆ω, giving rise to two important regimes: |∆ω| > |g| leading to sublaticce localization, and |∆ω| |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 both an underlying parity time symmetry breaking and carrier induced blue shift nonlinearities, that strongly localize the PL in a single cavity. Outside this PT-symmetry broken phase, and within the strong sublattice detuning region, the Hermitian zero modes do exist even though the system is no longer chiral, which we have shown from reflectivity resonances in a linear resonant scattering experiment (h ≤ -35%). On the other hand, the non Hermitian zero mode cannot be observed in this regime with a single pump spot because of symmetry mismatch under symmetric excitation. Moreover, carrirer induced blue shift effects enhance the PL emission within the highest energy mode, hindering the central resonance M2.

In spite of the tremendous theoretical advances in the non-

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Hermitian domain, observations of photonic zero modes have been restricted so far to waveguide or cavity arrays that ressemble their Hermitian counterparts, such as the Su-Schrieffer-Heeger (SSH) lattice. Although experimental demonstrations of non Hermitian phenomena such as PT symmetry breaking and exceptional points are coming to maturity, with special focus in two resonator systems, the physical realization of non Hermitian symmetries beyond PT such as NHPH in large cavity arrays -a unique playground for non Hermitian photonicsis still in its infancy. We believe that further developments in this direction would enable promising applications of non Hermitian symmetry protected modes, ranging from laser array mode engineering to photonic computing.

Figure 1 .

 1 Figure 1. Non Hermitian zero mode in a three coupled cavity array: CMT model. (a) Non-Hermitian Hamiltonian. (b) Eigenvalues and eigenvectors. Inset: intensity and phase spatial distribution at M2-laser threshold. (c) Logarithmic spectral intensity as a function of the center of a gaussian pump spot, computed from Eq. 2 in the text. The pump spot profile is P(x; X) = exp[-(x -X) 2 /σ 2 ]γ, with σ = 30 and γ = 1.5, i.e. below laser threshold. Solid lines correspond to Re(ε j ) for j = 1 (M1, blue), j = 2 (M2, burgundy) and j = 3 (M3, orange).

Figure 2 .

 2 Figure 2. Three coupled photonic crystal cavities. (a) Artist view of the system, featuring controllable coupling by means of the two barriers (highlighted with dashed boxes) in which holes are modified. Two sublattices A and B can be defined, where couplings only take place between cavities belonging to different sublattices. Bottom: schematic representation showing how the presence of the barriers modifies the cavity detuning. The sublattice detuning is then ∆ω. (b) 3D-FDTD simulation results of three coupled PhC cavities showing the evolution of the mode resonant wavelengths (symbols, left axis) as a function of the barrier parameter. Solid lines (right axis) are CMT predictions using polynomial approximations of g(h) and ∆(h) obtained from two-coupled cavity FDTD simulations (see Section III, Supplementary Material). Two regions can be distinguished: the low (light green, |g| |∆ω|) and high sublattice detuning regions (light blue, |g| < |∆ω|); the near zero coupling region is highlighted with striped background. (c) SEM image of a sample of three coupled L3 InP-based photonic crystal cavities. Inset: QW photoluminescence. (d) Normalized reflectivity spectra as a function of the barrier parameter, from resonant scattering experiments. These results can be interpreted as the optical response of the coupled cavity system in the Hermitian limit.

Figure 3 .

 3 Figure 3. Spatially-resolved photoluminescence measurements in the large detuning regime. (a) Experimental results showing spectral intensity maps upon spatial scanning of a pump spot for a = 416 nm and h = -20%, and (b) nonlinear CMT predictions. The horizontal position of the sample is changed by means of the piezo-electric voltage. Nearfield [(a)-i to (a)-v, (b)-i to (b)-iii] and farfield [(a)-vi to (a)-x, (b)-iv to (a)-vi] images are displayed at the selected pump spot positions marked with horizontal dashed lines.

Figure 4 .

 4 Figure 4. Observation of the zero mode in the low detuning regime. (a) Experimental results showing spectral intensity maps upon spatial scanning of a pump spot for a = 408 and h = 0%, and (b) nonlinear CMT predictions. The horizontal position of the sample is changed by means of the piezo-electric voltage. Nearfield [(a)-i to (a)-iv, (b)-i to (a)-iii] and farfield [(a)-v to (a)-viii, (b)-iv to (a)-vi] images are displayed at the selected pump spot positions marked with horizontal dashed lines. Spectral filters are used in order to remove contributions from other modes, the spectral bandwidth being represented by the horizontal extension of dashed boxes.

h = - 10 %Figure 5 .

 105 Figure 5. Phase diagram underlying the transition from sublattice delocalization and zero modes, to sublattice localization and mode coalescence. From (a) to (e): experimental PL intensity maps under pump spot position scanning across the coupled cavity system. (a)-(c) a = 416 nm, P pump = 0.8 µW; (d)-(e) a = 408 nm, P pump = 1.1 µW. From (f) to (j) nonlinear, and from (k) to (o) linear CMT calculations, with parameters Q = 4000, λ 0 = 1550 nm, α H = 3, σ = 30. (f) and (k) γ = 1.05, (g)-(h) and (l)-(m) γ = 1, (i) and (n) γ = 1.2, (j) γ = 1.45 and (o) γ = 1.55.

  (a)-(e)], and depicts nonlinear [Fig. 5(f)-(j)] as well as linear [Fig. 5(k)-(o)] CMT calculations. The experimental cases already analyzed in the previous section are reproduced in Fig. 5(b) (h = -20%) and in Fig.

  5(a), (f) and (k)], localization in one extreme cavity arises because of pure carrier induced refractive index effects. As a matter of fact, even if not detected in Figs. 5(a), the theory still predicts the M2 mode to be observable [see Figs. 5(f) and (k)].

Acknowledgments

This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (Labex NanoSaclay, reference: ANR-10-LABX-0035) and by the ANR UNIQ DS078. B. G. and A. M. Y. would like to acknowledge the financial support from the European Union in the form of Marie Skłodowska-Curie Action grant MSCA-841351. L.G. acknowledges support by National Science Foundation under grant No. PHY-1847240.

Methods

Linear reflectivity experiments

We characterize the behavior of the linear Hermitian modes in our system by performing linear reflectivity experiments as briefly described in the main text. The quasi-resonant injection beam is obtained from a Tunics T100s-HP, and its polarization is managed by means of a polarizing beam splitter combined to a subsequent half-wave plate that ensures linear horizontal polarization. The injection beam reaches the injection/detection beam splitter (80 % reflectivity), a halfwave plate and is finally injected through a 0.95 numerical aperture microscope objective (Olympus MPLAN x100 IR). The fast axis of the half-wave plate is rotated at 22.5 • from the injection beam allowing an injection at 45 • from the cavities polarization and an according polarization separation between the non-injected and the injected beams. An additional lens with 100 cm focal is located prior to injection so as to facilitate mode matching with the cavities. We then separate the non-injected and injected beams after the injection/detection beam splitter using a half-wave plate-in order to select the detected polarization-and a polarizing beam splitter. We finally inject a monomode optical fiber with the three cavity systems' emission and detect the signal using a Femtowatt Photoreceivers (New Focus 2153) connected to a 12-bits oscilloscope (Tektronix MSO64).

During a single realization of the experiment, the injection beam is first centered onto the considered nanocavity array, and subsequently sweep over either 20 or 40 nm with different starting wavelengths [as presented in Figure 2 (d)] depending on the barrier parameter.


Christelle Charpin-Nicolle

Nicolas Guillaume

Gauthier Lefèvre

Laurent Grenouillet

Tobias Vogel

Nico Kaiser

Eszter Piros

Stefan Petzold

C Trautmann

Sylvain David

Heavy ion irradiation hardening study on 4kb arrays HfO_2-based OxRAM

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

xide-based Resistive Random Access Memories (OxRAM), based on two resistive states, are promising candidates for replacing flash-memories, thanks to their fast switching speed, good endurance, high scalability potential and compatibility with CMOS Back-End-Of-Line (BEOL) technology. Moreover, they appear to be more resilient to radiation compared to flash memories and transistors [START_REF] Tan | Investigation on the Response of TaO$_{\rm x}$-based Resistive Random-Access Memories to Heavy-Ion Irradiation[END_REF][START_REF] Chen | Single-Event Effect Performance of a Commercial Embedded ReRAM[END_REF] because the information isn't stored thanks to electric charges, and so, it could be used for space applications. Previous studies have shown the impact of the irradiation on the pristine resistance of OxRAM single cells as well as on the formin g voltage, and the set voltage [START_REF] Alayan | Experimental and Simulation Study of the Effects of Heavy-ion Irradiation on HfO2-based RRAM Cells[END_REF][START_REF] Petzold | Heavy Ion Radiation Effects on Hafnium Oxide-Based Resistive Random Access Memory[END_REF][START_REF] Bi | The Impact of X-Ray and Proton Irradiation on ${\rm HfO}_2/{\rm Hf}$-Based Bipolar Resistive Memories[END_REF]. In this work, for the first time, the impact of heavy ion irradiation at fluences ranging from 10 9 ions/cm² to 10 12 ions/cm² were investigated on 4kbit arrays: with respect to distributions of the two states (HRS and LRS) and the endurance performances . Moreover, morphological characterizations (TEM, EDX and nano-diffraction) were carried out to investigate the OxRAM structure after irradiation .

II. PREPARATION OF T HE SAMPLES AND EXPERIMENTAL PROTOCOL

Fig. 1: FIB-SEM cross section revealing the OxRAM integration scheme OxRAM memories were integrated in 130nm CMOS BEOL between Metal 4 (M4) and Metal 5 (M5) [Fig. 1]. The transistors are NMOS types (with silicon oxide gate), and are 660 nm wide. Two types of resistive memories were fabricated: the first one is composed of a 10 nm thick HfO2 deposited by Atomic Layer Deposition (ALD) in-between a TiN Bottom Electrode (BE) and a Ti (10 nm)/TiN Top Electrode (TE), as shown in Figure 2. In the second one, the HfO2 active layer is Si-implant ed before the deposition of Ti/TiN top layers. Si implantation (3%) induces a reduction of the pristine resistance as well as a significant reduction of the forming voltage [START_REF] Barlas | Impact of Si/Al implantation on the forming voltage and pre-forming conduction modes in HfO2 based OxRAM cells[END_REF]. The top and bottom electrodes have been deposited by Physical Vapor Deposition (PVD). After depositing the top electrodes, the devices have been patterned by lithography and etched to produce cells of various diameters. In this study, we focus ed on 400nm diameter memory dots.

OxRAM devices were programmed in three different states before being irradiated: either LRS (Low Resistance State), HRS (High Resistance State) and one third left in pristine state, N. Guillaume is with CEA, LET I, Univ. Grenoble Alpes, 38000 Grenoble, France (e-mail: nicolas.guillaume@cea. For morphological investigation, specimens were prepared by focused ion beam etching and polishing using a FEI dualbeam Helios 450. Rough milling was applied using an operating voltage of 30kV, with subsequent reductions to 2-8kV, enabling to reduce the surface damage whereas keeping the sample with parallel sides. The TEM data were performed using a probecorrected FEI Titan Themis at an acceleration voltage of 200kV. Nano-beam electron diffraction (NBED) was performed with a convergence semi angle α = 0.5 mrad to get diffraction spots as small as possible.

III. RESULTS AND DISCUSSION

A. Electrical characterization

Transistors are used for the programming of OxRA M devices, providing a limitation of the current in the memories during the forming and set steps. It appears mandatory to check the functioning of the trans istors after irradiation. The evolution of the drain current (Id) as function of the gate voltage and of the drain voltage (Vg and Vd respectively) has been measured before and after irradiation for the three fluences (see Figure 3). Fig. 3 highlights the impact of irradiation on the behavior of transistors. At low fluence (10 9 ions/cm²), the transistor behavior is not impacted, regarding die to die variability . However, at higher fluences (5.10 10 ions/cm² and especially at 10 12 ions/cm²), a sharp decrease of the channel current is observed; moreover, the threshold voltage is clearly shifted when increasing the irradiation fluence. These results are consistent with previous results on silicon oxide gate transistor [START_REF] Schwank | Radiation Effects in MOS Oxides[END_REF] : the degradation of the channel conduction is attributed to the creation of defects at the interface between the channel and the gate oxide and the increase of the threshold voltage is linked to defects induced by the irradiation within the silicon oxide itself.

Considering the deficient behavior on the transistors at the highest fluence, we first focused on the evaluation of the OxRAM 4kbit array devices after irradiation at 10 9 ions/cm² and 5.10 10 ions/cm². Fig. 4 shows the evolution of the three states (HRS, LRS and pristine) for the two HfO2 stacks (classic HfO2 and Si-implanted HfO2) after irradiation at a fluence of 10 9 ions/cm². As shown in Fig. 4a), the impact of irradiation on LRS and HRS distributions is minor: no difference is seen on HRS distribution, and only four devices over 1536 switched from HRS to LRS (HRS values that crossed the line of the highest resistance value of LRS distribution). A small shift of the pristine resistance towards a lower resistance is observed after irradiation, overall remaining in a high resistance. It is pointed out that no major difference is seen between the classic HfO2 stack and the Si-implanted HfO2 stack, for all distributions (HRS, LRS and pristine). Then, endurance measurements have been performed on 4kbit irradiated arrays (see Fig. 5). The following programmin g conditions were applied: Vset=1.7V, Vgate_set=2.5V, Vreset= -2.5V and Vgate-reset=4.5V. A square time signal was used, trise=100ns, twidth=1µs, tfall=100ns. Endurance envelopes marked out with the quartile 1 and 3 for both HRS and LRS are shown in Fig. 5 for the two stacks (pure HfO2 and Si-implant ed HfO2). 10k cycles endurance is achieved on both stacks, meaning the memory arrays are not impacted by the irradiation . After 10k cycles, HRS and LRS normal distributions are compared for three different configurations, a reference array (no irradiation), an array irradiated at the fluence of 10 9 ions/cm² and an array irradiated at the fluence 5x10 10 ions/cm² [Fig. 6]. For arrays irradiated at a fluence of 5x10 10 ions/cm², LRS values are slightly shifted towards higher resistance; this can be attributed to a lower current flowing at a given gate voltage (see Fig. 3).

In order to get equivalent current to that of the non-irradiated array and lowest fluence, the gate voltage was set from 2.5V to 2.75V for the classic stack and the HfO2 Si-impacted stack [Fig. 6]. No impact on the resistance distribution is observed. In both cases, if we put aside the device failing due to intrinsic integration issues, it seems that going to a fluence of 5x10 10 ions/cm² has an effect on the HRS distribution. Since at the highest fluence, NMOS transistors are highly impacted [Fig. 3] and cannot deliver sufficient current for the programming of the OxRAM devices connected in series with transistors. Therefore, single devices (1R) have been tested after irradiation at a fluence of 10 12 ions/cm² and compared to 1R devices irradiated at the lowest fluence of 10 9 ions/cm² in their electroforming behavior in quasi-static mode [Fig. 7]. Fig. 7 shows pristine resistance values which are very similar in both cases: at the reading voltage of 0.1 V, the current is roughly equal to 10 -13 A. Anyway, an impact of the irradiation fluence can be enhanced: the shape of the current curves is somewhat different, indicating a conduction modificatio n inside the HfO2 active layer; this can be explained by leakage current induced by irradiation [START_REF] Li | Latent Reliability Degradation of Ultrathin Amorphous HfO 2 Dielectric After Heavy Ion Irradiation: The Impact of Nano-Crystallization[END_REF][START_REF] Hughes | Radiation effects and hardening of MOS technology: devices and circuits[END_REF]. Moreover, formin g voltages seem more spread in the case of the highest fluence compared to those in the case of the lowest fluence also increasing their mean value for the highest fluence.

Finally, a large gate width transistor (6700 nm) could be used in series with a 1R cell, so it could deliver sufficient current. Functionality of the 1R cell was demonstrated with good endurance performances reported up to 10 7 cycles (see Fig. 8).

In conclusion, irradiation fluence has an impact on the subthreshold current of the 1R cells during the forming step; however, even at fluence as high as 10 12 ions/cm², the cells are still functional, they can be formed, and even cycle when connected to a transistor which can deliver sufficient current.

B. Morphological characterization

Morphological measurements were performed on three pristine OxRAM devices: a reference device (no irradiation), and two devices irradiated at the lowest and highest fluences, i.e. 10 9 ions/cm² and 10 12 ions/cm²; Fig. 9 presents TEM images for the two irradiated samples: they are very similar, indicating same width, no distortion and no long-range order within the active layer. One can already appreciate the polycrystalline state of the upper and lower layers (titanium and titanium nitride, respectively). Elementary analysis by EDX was performed on the three samples [Fig. 10]: the composition as well as the distribution of the element through the active layer are identical in all cases, considering variations due to sample preparation. This indicates no noticeable interface mixing due to irradiation, with respect to a non-irradiated device.

Target damages have been simulated using the SRIM-2010 code [START_REF] Ziegler | SRIM -The stopping and range of ions in matter[END_REF] and following recommendations of Stoller et al. [START_REF] Stoller | On the use of SRIM for computing radiation damage exposure[END_REF]. According to the simulation, atoms mixing is extremely limit ed at the interfaces in terms of depth and concentration (at the highest fluence of 10 12 ions/cm², no more than 1 nm diffusion and 0,005% implantation respectively). This is in agreement with TEM-EDX analysis [Fig. 10]. Moreover, damage calculation indicates a vacancy density of 2x10 18 vacancies/cm 3 (corresponding to 0.008% target damage). Neither thermal effect, nor potential contribution of channeling, nor other orientation dependent phenomena are considered in these simulations, but it has been shown that trap density of this range has an impact on the current-voltage characteristics [START_REF] Gritsenko | Electronic properties of hafnium oxide: A contribution from defects and traps[END_REF] Such a low damage rate is not discernable for polycrystalline materials using TEM analysis, but is concordant with electrical characteristics changes as observed [Fig. 7]. The range and energy loss of ions in HfO2 were estimated using SRIM. The range and energy loss of ions in HfO2 were estimated using SRIM. The calculated ion range, electronic energy loss and nuclear energy loss for 1.635 GeV Au ions in HfO2 (after passing the different covering layers) were estimated to be 38.46 µm, 52.86 keV/nm and 70.28 keV/nm, respectively. Finally, nano-diffraction measurements have been performed on irradiated and non-irradiated devices [Fig. 11]. Results are similar for all samples, indicating no detectable change in the structure. Presence of numerous diffraction spots in the TiN and Ti layers surrounding HfO2 (areas 1 and 6 in Fig. 11) points out the polycrystalline structure of these conductive layers. On the contrary, absence of clear diffraction spot in the active layer (areas 3 to 5 in Fig. 11) confirms no long-range order within the atomic structure. Nonetheless, the bright halo corresponds to very small disordered crystallites. Further analysis with improved resolution could confirm whether a phase transition occurs, as already observed [START_REF] Benyagoub | Mechanism of the monoclinic-to-tetragonal phase transition induced in zirconia and hafnia by swift heavy ions[END_REF][START_REF] Vogel | Defect-induced phase transition in hafnium oxide thin films by heavy ion irradiation: The role of oxygen defects[END_REF].

IV. CONCLUSION In this study, heavy ion irradiation has been carried out on 4kbit OxRAM arrays utilizing three fluences, 10 9 ions/cm², 5x10 10 ions/cm² and 10 12 ions/cm². Despite a degradation of the transistors for fluences over 5x10 10 ions/cm², an appropriate monitoring of the programming conditions enables the functioning of 4kbits arrays: a proper memory window is maintained, even after 10k cycles. At the highest fluence, we showed 1R devices are impacted, but they were assessed to be still functionnal. TEM, EDX Nano-diffraction analyses showed no major difference between irradiated devices and nonirradiated devices, whatever fluence.

V.REFERENCES

-

  AbstractHfO2 based OxRAM devices integrated in Back En d Of Line (BEOL) of 130nm CMOS have been exposed to extreme irradiation conditions related to extensive journey in space, supernova or nuclear disaster exposure: 1.635 GeV Au ion energy and very high fluences, from 10 9 ions/cm² to 10 12 ions/cm². S ingle resistive devices as well as 4kbit 1T1R arrays have been studied, showing a very good resilience up to a fluence of 5.10 10 ions/cm². For higher fluences, the degradation of access transistors is identified as the main source for information loss. Without access transistor, single 1R devices were demonstrated to be functional even at the highest fluence of 10 12 ions/cm². TEM, EDX and nano diffraction analyses show no change in the HfO2 material, as well as in the repartition of the element in the layers, whatever the fluence. These results demonstrate the OxRAM structure is resilient to extreme irradiation conditions. Index Terms-Heavy ion irradiation, ReRAM, OxRAM, 1T1R, 4kbits arrays, Electrical characterization, TEM, nano-diffraction, EDX.

  fr) L. Grenouillet, C. Charpin-Nicolle, E. Nowak are with CEA, LETI, Univ. Grenoble Alpes, 38000 Grenoble, France. G. Lefèvre, S. David, C. Vallée are with Université Grenoble Alpes, CNRS, CEA-LETI, LTM, F-38054 Grenoble, France T . Vogel, N. Kaiser,E. Piros, S. Petzold, and L. Alff are with the Department Advanced T hin Film Technology, Institute of Materials Science, Technische Universität Darmstadt, Alarich-Weiss-Str.2. 64287 Darmstadt, Germany. C. T rautmann is with the Materials Research Department, GSI Helmholtzzentrum fuer Schwerionenforschung, 64291 Darmstadt, Germany, and also with the Institute of Materials Science, T echnische Universität Darmstadt, 64287 Darmstadt, Germany Heavy Ion Irradiation Hardening Study on 4kb arrays HfO 2 -based OxRAM N. Guillaume, G. Lefèvre, C. Charpin-Nicolle, L. Grenouillet, T. Vogel, N. Kaiser, E. Piros, S. Petzold, C. Trautmann, D. Sylvain, C. Vallée, L. Alff, E. Nowak O corresponding to the initial state of the memory stack before any forming step. More specifically, for each 4kbit 1T1R array, 1024 devices remained in pristine state, whereas 1536 devices were programmed in LRS and 1536 devices were programmed in HRS. Forming and set voltage were applied with a gate voltage on the word line (transistor gate) such that the compliance current is about 100 µA. Electrical programmin g and measurements are carriedout utilizing a cascade microtech bench with an Agilent B1500 parameter analyzer. Sample irradiation was performed at the UNILAC accelerator of the GSI Helmholtzzentrum (Darmstadt): Au ions were used with an energy of 1.635 GeV and three various fluences, 10 9 ions/cm², 5x10 10 ions/cm² and 10 12 ions/cm². No bias was applied during the irradiation. After irradiation, a thorough investigation has been performed: electrical measurements (4kbits resistance distribution and 10k cycles endurance), and also comparative morphological measurements: Trans mission Electron Microscope (TEM), Energy Dispersive X-ray (EDX) and nanodiffraction.

Fig. 2 :

 2 Fig. 2: Schematic Stack of the resistive memory studied (left) and SEM cross section of the stack (right).

Fig. 3 :

 3 Fig. 3: Measurements carried out on single NMOS transistors (Width=660nm and Length=500nm). a) Id (Vd) at Vg=1.3V after irradiation at three various fluences (10 9 ions/cm², 5x10 10 ions/cm², 10 12 ions/cm²) and with no irradiation (reference). b) Id(Vg) at Vd=1.3V after irradiation (three fluences).

Fig. 4 :

 4 Fig. 4: LRS and HRS (a) and Pristine resistance (b) normal distribution results on two memory stacks (pure HfO2 and Si-implanted HfO2) after irradiation at low fluence (10 9 ions/cm²) and with no-irradiation.

Fig. 5 :

 5 Fig. 5: 10k Cycles endurance succeeded on HfO2 stack (a) and Si implanted HfO2-s tack (b) on 3072 devices after irradiation at a fluence of 10 9 ions/cm².

Fig. 6 :

 6 Fig. 6: Resistance distribution at the 10kth cycle for Si-implanted HfO2 stack (a) and HfO2 classic stack (b) with the same parameters as in Fig. 5. In the case of the classic stack, at a fluence of 5x10 10 ions/cm², Vgate-set has been increased from 2.5V to 2.75V.

Fig. 7 :

 7 Fig. 7: Evolution of the current as function of the voltage when forming pristine OxRAM 1R devices irradiated at a fluence of 10 9 ions/cm² (a) and at a fluence of 10 12 ions/cm² (b).

Fig. 8 :

 8 Fig. 8: 10 7 cycles performed on 1T 1R HfO2 stack. T he device was irradiated at the fluence of 10 12 ions/cm².

Fig. 9 :

 9 Fig. 9: T EM imaging of devices irradiated with fluence of 10 9 ions/cm² (left) and 10 12 ions/cm² (right).

Fig. 10 :

 10 Fig. 10: Elementary profiles from EDX of a non-irradiated device and devices exposed to fluence of 10 9 ions/cm² and 10 12 ions/cm². Elementary profiles are the same.

Fig. 11 :

 11 Fig. 11: Nano-diffraction patterns (bottom) of 6 areas along the active layer (indicated in the top HAADF picture) of irradiated and non -irradiated devices.

  T his work was funded by European commission, French State and Auvergne-Rhône Alpes region through ECSEL project WAKeMeUP and French Nano 2022 program. T he work leading to this publication has received funding within the ECSEL Joint Undertaking project WAKeMeUP in collaboration with the European Union's H2020 Framework Program (H2020/2014-2020) and National Authorities, under grant agreement number 783176. Funding by the Federal Ministry of Education and Research (BMBF) under contract 16ESE0298 and by the DFG grant AL 560/21-1 is gratefully acknowledged. T he irradiation experiments for this publication were performed at the UNILAC X0 beam line at the GSI Helmholtz Center for Heavy Ion Research, Darmstadt (Germany), based on a UMAT experiment in the frame of FAIR Phase-0.


B Garbin

A Giraldo

K J H Peters

N G R Broderick

A Spakman

F Raineri

A Levenson

S R K Rodriguez

B Krauskopf

A M Yacomotti

Spontaneous symmetry breaking in a coherently driven nanophotonic Bose-Hubbard dimer

We report on the first experimental observation of spontaneous mirror symmetry breaking (SSB) in coherently driven-dissipative coupled optical cavities. SSB is observed as the breaking of the spatial or mirror Z2 symmetry between two symmetrically pumped and evanescently coupled photonic crystal nanocavities, and manifests itself as random intensity localization in one of the two cavities. We show that, in a system featuring repulsive boson interactions (U > 0), the observation of a pure pitchfork bifurcation requires negative photon hopping energies (J < 0), which we have realized in our photonic crystal molecule. SSB is observed over a wide range of the two-dimensional parameter space of driving intensity and detuning, where we also find a region that exhibits bistable symmetric behavior. Our results pave the way for the experimental study of limit cycles and deterministic chaos arising from SSB, as well as the study of nonclassical photon correlations close to SSB transitions.

The Bose-Hubbard dimer (BHD) is a paradigmatic system featuring quantum dynamics of bosons hopping across sites and interacting on-site [? ]. The competition between these processes in the closed BHD unveiled a wealth of quantum and classical nonlinear phenomena, including self-trapping [? ] and symmetry breaking [? ]. In most real BHDs, however, particle losses need to be compensated by external driving [? ? ? ? ]. These driven-dissipative BHDs have recently drawn much interest, as they display intriguing classical [? ? ] and quantum [? ] phenomena due to the balance between driving and dissipation. Driven-dissipative BHDs have been implemented on light-matter systems like semiconductor microcavities [? ? ? ] and superconducting circuits [? ? ].

SSB is a universal phenomenon occurring when a symmetric system ends up in an asymmetric state. SSB and its applications have attracted significant interest in optics [? ? ? ? ? ? ? ]. In single coherently driven optical cavities, SSB emerges from the nonlinear coupling between polarization or counter-propagating modes [? ? ]. SSB has also been observed in coupled active photonic crystal nano-resonators [? ], which share some properties with incoherently driven BHDs [? ]. However, optical gain saturation-a nonlinear mechanism reminiscent of inelastic two body collisions in atomic systems [? ]results in a fundamental difference with respect to BHDs. Compared to the incoherent pumping case, coherent driving offers a more versatile parameter control and suppressed spontaneous emission noise. Furthermore, coherently driven BHD have been predicted to undergo entan-glement close to the SSB transition [? ], thereby bridging classical nonlinear dynamics and quantum optics.

Despite the strong interest in coherently driven dissipative BHD physics, SSB remains unreported. This is likely due to two reasons. First, conventional driven-dissipative BHDs display symmetry broken phases restricted to narrow parameter regions, which coexist with other optical bistabilities [? ? ]. Thus, accessing SSB is challenging. Second, resonant optical excitation experiments require mode-matching, thereby leading to more stringent conditions than their incoherent pumping counterparts.

Here we demonstrate coherently driven SSB in a photonic crystal driven-dissipative BHD. Unlike standard BHDs, our system features a negative photon hopping energy that, in conjunction with a positive on-site interaction energy, results in a broken symmetry phase across large parameter regions [? ]. These are pure pitchforkbifurcated phases; they do not coexist with a bistable homogeneous state which might otherwise hinder the SSB observation.

We consider a BHD of two sites-"or cavities"-where photons interact on-site with energy U > 0, hop across sites with energy J < 0, and that is coherently driven symmetrically with amplitude F and frequency ω p [see Fig. 1(a)]. We apply the truncated Wigner approximation [? ], neglecting quantum correlations but accounting for quantum fluctuations via stochastic terms in semiclassical equations of motion. In a frame rotating at ω p , the expectation values α 1,2 of the bosonic operators â1 and â2 in the two sites satisfy:

i dα 1,2 dt = -∆ -i γ 2 + 2U |α 1,2 | 2 α 1,2 -Jα 2,1 + F 1,2 + γ 2 ζ 1,2 (t).
(1)

Here, ∆ = ω p -ω c is the detuning between ω p and the cavities resonance frequency ω c , and F 1 = F 2 = F are coherent driving amplitude applied to each cavity. γ is the loss rate. The last term in Eqs.

(1) models the fluctuations as complex Gaussian noise terms ζ j (t) = [ζ j (t) + iζ j (t)]/ √ 2 (j = 1, 2) which have zero mean and are delta-correlated in t.

In our experimental implementation, the sites are given by two evanescently-coupled photonic crystal (PhC) nanocavities with embedded quantum wells (QWs) illuminated by an optically resonant laser field. The bosons are intracavity photons that experience carrier-induced Kerr-like nonlinearities below the QW optical transparency, leading to a blue-shifting intensity-dependent refractive index (U > 0). Using Eqs. (1) to model the PhC dimer, we assume ultrafast carrier dynamics instantaneous Kerr nonlinearity.

For identical cavities with 2|J| > γ and in the linear regime, two hybrid modes arise [Fig. the symmetric mode with α 1 = α 2 , and the anti-symmetric mode with α 1 = -α 2 ; the mode splitting and their relative spectral position, respectively, depend on the magnitude and sign of J. Upon coherent in-phase symmetric illumination of the two cavities, only the symmetric mode is excited. The sign of J has a large impact on the observed behavior [Fig. 1(c)]: provided U > 0, pure SSB only exists for J < 0 (purple region and top inset). For J > 0, the bistability of the homogeneous symmetric states prevails (green region and bottom inset), see Fig. 1(c) for ∆ = 0.

Decreasing the value of ∆ from zero results in an overlap of these two regions near J = 0; this creates regions of e.g. periodic and chaotic behavior (see Supplementary Material). For J > 0, however, the SSB transition always lies inside the bistable symmetric zone. On the contrary, the pure super-critical pitchfork bifurcation is only warranted for J < 0, which we can implement experimentally.

Figure 2 shows our experimental setup. Two evanescently-coupled photonic-crystal nanoresonators stand on a free-standing InP membrane with four embedded InGaAs/InGaAsP QWs. The lattice period a is 435 nm (hole radii r 0 = 116 nm). Spatial mode matching of the illumination laser beam to the cavity mode is enhanced by means of far-field engineering of the PhC cavity [? ] (hole radii r 2 = 137 nm). Such mode matching is further improved using a lens with 100 cm focal length (L 1 in Fig. 2). Importantly, negative J with targeted splitting is achieved by controlling the radii of selected holes [? ] (hole radii r 0,2 -20%). The two cavities are coherently driven close to their resonant wavelength (≈ 1563.9 nm) with a narrow-linewidth widely-tunable laser, whose beam is split for injection power monitoring. Finally, a 100× magnification with 0.95 numerical aperture microscope objective (Olympus MPLAN x100 IR) focuses the laser beam to a 2.2 µm diameter spot on the sample from the top. The sample is mounted on a PZT driven 6-axis translation stage for controlling the relative position between the driving beam and the cavities and, hence, the relative power impinging on the two cavities. We use a half-wave plate to rotate the linearly polarized injection beam at input, by 45 • from the cavities polarization. The cavities outputs are subsequently separated from the reflected part of the injection beam with polarization optics, and split to allow spectral and temporal analysis. The spectral part is directed to a liquid-nitrogen-cooled spectrometer (Acton SP 2500).

|E1 | 2 = |E2 | 2 |E 1 | 2 (o r |E 2 | 2 ) |E2| 2 (or |E1|
To independently analyze three different regions of the sample, the temporal part is split and directed to: (i) a 750 Hz bandwidth Newfocus fW detector for linear spectroscopy (large dashed circle in Fig. 2), and (ii) two 550 MHz bandwidth low-noise APD detectors (Princeton Lightwave PLA-8XX, N.E.P.=250 fW/ √ Hz) for measuring the output intensity of each cavity (smaller dashed circles in Fig. 2).

We first perform linear reflectivity spectroscopy by sweeping the driving laser wavelength. Since J is negative, the symmetric mode is located at shorter wavelength (1563.2 nm) than the anti-symmetric one (1564.5 nm). Based on the measured mode splitting (1.3 nm) and mode linewidth (0.39 nm), we estimate the normalized coupling constant κ = -2J/γ ≈ 3.3; this is compatible with SSB observation for low excitation power. Increasing the splitting to |κ| 5, for instance without barrier hole modification, increases the injection amplitude of the SSB threshold and might hinder observation [see Fig. 1(c)].

After locating the illumination spot at the center of the dimer [line (c) in Fig. 2(e)], the excitation power (purple traces) is ramped up to about 1.5 mW within ∼ 300 ns to minimize thermal effects. Figures 3(a,b) show typical cavity intensity time traces (blue and orange curves) for two different values of the normalized detuning δ = -2∆/γ. At low driving powers, the two cavity output traces are coincident, indicating that a delocalized (symmetric) hybrid mode is being excited. At time of about 285 ns, however, the two output intensities part, which we identify as the SSB transition point. The spontaneous character of the SSB manifests itself as randomly selected asymmetric states (Fig. 3, top and middle traces). The bottom traces in Fig. 3, built by overlapping maxima of histograms of 500 successive realizations, evidence the mirror symmetry of the two asymmetric states (see Supplementary Material for comparisons with numerical simulations), and can be seen as the reconstruction of the underlying pitchfork bifurcation.

When increasing the magnitude of the detuning from Fig. 3(a) to (b), we need to realign the illumination spot to observe randomly selected asymmetric states. This may be understood from Eqs. (1) in the presence of asymmetries in the cavity frequencies (ω c,1 = ω c,2 ) and driving fields (F 1 = F 2 ). The first one is due to unavoidable fabrication imperfections (≈ 0.13 ± 0.07 nm), while the second one accounts for deviations of the illumination spot from the dimer center. Indeed, numerical simulations (not shown) indicate that the asymmetry parameter ξ ∼ F 1 -F 2 that induces random switching changes with ∆, consistent with the experimental observation; this explains the difference in cavity intensities when moving from panel (a) to (b) related to slight misalignment in the optical detection devices. The observation of SBB through compensation of two asymmetry parameters has recently been reported in [? ].

Figure 4 illustrates the three main regimes of operation observed in our experiment, as well as the associated SSB transition and bistability. Projections of the phase space onto the (I 1 , I 2 ) plane of the two measured intensities, for given values of injection power and detuning, show: (i) equal intensity in the two cavities [Fig. 4 |U |/γ) [? ] allows us to interpret γ/|U | ∼ 100 as the characteristic intracavity photon number; note that the thermodynamic limit corresponds to γ/|U | → ∞, for which the mean field solutions-i.e., these without stochastic fluctuationsare strictly justified. Analytical results of bifurcation points of the mean field BHD derived in [? ] give an important handle to analyse the stochastic simulation results. Firstly, the mean field pitchfork bifurcation associated with the SSB transition occurs at f ≈ 4.04 in panel (f); this is consistent with the stochastic numerical simulations, which also indicate the presence of random switching between asymmetric states from this value up to f ≈ 5.5. Secondly, the two predicted mean-field bifurcations of the symmetric state for 1.67 f 2.15 clearly set the limits of the observed hysteresis cycle in Figure 4(g).

We now identify in Fig. 5 the different regimes of operation in the two-dimensional parameter plane of driving strength and detuning. The experimental bifurcation diagram in panel (a) was created by color coding the number of maxima found for I + and I -. It reveals large regions of the parameter plane for which the system exhibits two asymmetric steady states; the SSB transition that bounds this region to the left moves towards lower injection power P in as the detuning δ is decreased; it is characterized by distinct maxima of I -. Notice also the region of symmetric bistability (identified by maxima of I + only) that exists for smaller δ; it may also overlap (dotted area) with asymmetric behavior: three states are observed (one lower symmetric and two higher asymmetric). We attribute the absence of asymmetric states for -7 δ -6 to large deviations of the sample's position with regard to the laser beam at larger detunings. furcation P [? ? ], yielding a region with two coexisting asymmetric steady states. As the detuning is decreased, the Hopf bifurcation curve H marks the onset of asymmetric oscillations, whereas PD indicates the first bifurcation of a period-doubling route to chaos, which eventu-ally creates a pair of asymmetric chaotic attractors that are each confined to one cavities [? ]; they eventually collide, forming a symmetric chaotic attractor that features irregular switching between the two cavities [? ].

In conclusion, we have reported the first experimental realization of SSB in a coherently driven photonic dimer. For repulsive nonlinearities (U > 0), we have shown that pure supercritical pitchfork bifurcations only exist for negative photon hopping energies (J < 0); we also predict this to hold for J > 0 provided that U < 0. The SSB transition consists of the splitting of the two intracavity intensities under symmetric driving conditions; this is observed as two different randomly-selected states where light is localized in either of the two cavities. We mapped out the SSB transition in a large parameter plane and identified the region of bistability between symmetric states. Our experiments are in good agreements with the mean field Bose-Hubbard dimer model and stochastic simulations validated the random nature of this process. The theory for the BHD also predicts the existence of limit cycle oscillations and deterministic chaos. Our results constitute an important step towards the study of symmetry breaking in the few photon regime [? ], at the crossroad between nonlinear dynamics and quantum optics.

B. G. and A. M. Y. would like to acknowledge the financial support from the European Union in the form of Marie Sk lodowska-Curie Action grant MSCA-841351. This work is part of the research programme of the Netherlands Organisation for Scientific Research (NWO). S.R.K.R. acknowledges an ERC Starting Grant with project number 85269.

  FIG. 1. Illustration of (a) two evanescently-coupled injected resonators and (b) the resulting split hybrid symmetric (ωS) and anti-symmetric (ωAS) modes. (c) Two-parameter bifurcation diagram for ∆ = 0 showing the regions of bistability of symmetric (green region) and asymmetric (purple region) steady states. The insets represent bifurcation slices showcasing the existence of pitchfork (top, -2J/γ = +3.3) and saddle-node (bottom, -2J/γ = -3.3) bifurcations.

  FIG. 2. Experimental setup and operational regime. (a) Schematic of the experimental setup. BS, beam splitter; PBS, polarizing beam splitter; L1, 100 cm focal lens; linear spectroscopy (dashed green circles) and individual cavity output (blue and orange dashed circles). (b)-(d) Selected spectra for given sample horizontal positions. (e) Evolution of the spectra with the sample horizontal position.

  FIG. 3. Temporal observation of SSB. (a), (b) Top and middle traces are subsequent realizations of the same experiment. Bottom traces are histogram maxima computed from 500 realizations; shown are the output powers of the two cavities (blue and orange, respectively) and the driving power (purple curve). (a) δ = -1.59. (b) δ = -2.86.

  FIG. 4. Identification of operational regimes. (a)-(c) Selected projections of the phase space showing monostable symmetric, bistable symmetric, and bistable asymmetric steady states, respectively. (d), (e) Experimental and (f), (g) numerical histograms computed from 500 realizations, showing I -[(d), (f)] and I + [(e), (g)] versus injection strength. (a) 0.22 mW. (b) 0.26 mW. (c) 1.03 mW. (a), (c) δ = -0.82. (d), (f) δ = -2.86. (b), (e), (g) δ = -6.19.

  Figures 4(f,g) show corresponding plots obtained by simulating Eqs. (1) while varying the normalized driving amplitude f = 4F |U |/γ 3/2 . We also set U = γ/100, which we estimated by comparing numerical and experimental standard deviations of the intensity around the SSB transition. Further rescaling the intracavity field as (A, B) = (-2iα * 1 |U |/γ, -2iα * 2

Figure 5

 5 Figure 5(b) shows the associated two-parameter bifurcation diagram of Eqs.(1) without noise (see Supplementary Material) computed using Auto07p [? ]. Overall, it is in good agreement with the experimental bifurcation diagram in terms of where symmetry-broken dynamics and bistability between symmetric states can be found. More specifically, the saddle-node bifurcation curve S with the cusp point CP at (f, δ) ≈ (1.24, -5.03) delimits the region with bistability between symmetric steady states. The SSB transition is identified as a pitchfork bi-