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Abstract
Recent years have witnessed a flurry of research activities in topological photonics, predominantly driven by the prospect for topological protection–a property that endows such systems with robustness against local defects, disorder, and perturbations. This field emerged in fermionic environments and primarily evolved within the framework of quantum mechanics which is by nature a Hermitian theory. However, in light of the ubiquitous presence of non-Hermiticity in a host of natural and artificial settings, one of the most pressing questions today is how non-Hermiticity may affect some of the predominant features of topological arrangements and whether or not novel topological phases may arise in non-conservative and out of equilibrium systems that are open to the environment. Here, we provide a brief overview of recent developments and ongoing efforts in this field and present our perspective on future directions and potential challenges. Special attention will be given to the interplay of topology and non-Hermiticity–an aspect that could open up new frontiers in physical sciences and could lead to promising opportunities in terms of applications in various disciplines of photonics.
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1. Introduction
Decades of progress carried out independently in non-Hermitian physics and topological insulators have met each other in the flourishing field of non-Hermitian topological photonics [1–22]. Studies in this area have been stimulated by the observations of counterintuitive topological phenomena that have no counterpart whatsoever in the Hermitian domain. Promising prospects in real-world open systems along with a flexibility for accommodating a variety of non-Hermitian sources on photonic platforms are among the most prominent driving forces in advancing this field. By deploying an arsenal of theoretical, numerical, and experimental tools and methodologies, research in non-Hermitian photonics is nowadays marching through a terrain that extends from mathematics and condensed matter physics to material science, and optics.
Symmetry is one of the essential cornerstones in topology given that it lays the foundations in understanding different topological phases. Classification schemes based on non-spatial internal symmetries are culminated in the celebrated Altland-Zirnbauer systems. These tenfold symmetry classes are categorized in terms of the presence or absence of time-reversal symmetry, particle-hole symmetry, and chiral symmetry. Even though the Altland-Zirnbauer scheme is well-defined in the Hermitian domain and it can be generalized to include spatial symmetries like for example that of reflection, understanding the implications of symmetry in non-Hermitian topological configurations is still a subject of intense investigation [23–29]. For instance, it has been found that in the presence of non-Hermiticity, distinct internal symmetries can be unified [30] and sublattice and chiral symmetries no longer coincide [24]. Non-Hermitian systems exhibiting exotic symmetries such as that of parity-time (PT) have no analogue in the Hermitian domain. A non-Hermitian Hamiltonian respects PT symmetry provided that its complex potential V satisfies the condition $V(x )= {V^\ast }({ - x} )$, which is necessary but not sufficient for supporting real eigenvalues. In this regard, one can switch between a PT-symmetric phase, exhibiting real eigenvalues, and a PT-broken phase associated with complex eigenspectra, a phase transition dictated by the magnitude of the complex potential V. To implement such PT ‘optical potentials’ in photonic platforms, an antisymmetric gain/loss profile with symmetric index guiding can be judiciously employed [1,3,31–35]. Enhanced modulation bandwidth in PT symmetric coupled lasers reveals how new paradigms in controlling laser characteristics can be introduced by such a non-Hermitian symmetry [36]. Exceptional semimetal–like crystals can be stabilized through novel types of spatial symmetries that are unique to non-Hermitian platforms [37]. These results suggest that there is a drastic departure from what one could expect in conservative systems to what may ensue in the presence of non-Hermiticity [24,25]. On another front, periodically driven non-Hermitian systems can display topological phases that cannot be attained under static conditions. The associated Floquet non-Hermitian topological bands in such structures have been systematically classified according to the internal symmetries based on the so-called K-theory [38].
Different classes of topological non-Hermitian systems have been also extended to include defects [39]. In this context, the existence of topologically protected edge states in PT symmetric arrangements, has been a matter of debate [40–43]. Such states were experimentally observed for the first time in passive Su–Schrieffer–Heeger (SSH) lattices inscribed in silica where the waveguide elements were intentionally wiggled as a function of propagation distance in order to fine-tune the loss [44]. Within the framework of time-binding formalism, the following Hamiltonian can be used to describe the dynamics of such SSH configurations:
Fig. 1. Topological lasers. (a) Robust lasing in an edge mode in a photonic implementation of the Haldane lattice [56]. Lasing in the protected midgap state of a Su–Schrieffer–Heeger (SSH) lattice implemented using (b) a zigzag chain of coupled polariton micropillars [59] and (c) a microring resonators InGaAsP/InP structure [60]. (d) A topological insulator laser using a two-dimensional array of site and intermediary link resonators that emulates the quantum spin Hall effect [64]. (e) Topological lasing in a quantum Hall platform realized by integrating a photonic crystal with a gyromagnetic substrate [68] (f) Topological insulator vertical-cavity surface-emitting laser (VCSEL) array [72]. (g) Multipolar lasing from corner topological modes in a generalized 2D SSH lattice [81]. (h) An electrically pumped topological insulator laser using a quantum spin Hall platform [87]. (i) topological temporal mode-locking between the temporal modes of a fiber lase cavity [88].
2. Topological lasers
Such topological lasers, indeed, constitute one of the most technologically relevant settings where topology synergistically interacts with non-Hermiticity and nonlinearity. Lasers, coherent light sources that rely on nonlinear gain saturation, provide a promising test bed for exploring the ramifications of non-Hermiticity and nonlinearity on topology. Along different lines, robust topological lasing was found to mitigate vulnerabilities because of disorder, defects, and operational malfunction – an aspect that makes topological coupled arrays of lasers even more appealing. As such, a vast body of theoretical and experimental studies has been devoted to this topic–reshaping our understanding of non-Hermitian topological systems. Research in this field was initiated by theoretical studies of the SSH model [58]. This was followed by three experiments demonstrating lasing in the associated topologically protected midgap defect state. St.-Jean et al. implemented an orbital version of the SSH Hamiltonian using a zigzag chain of coupled polariton micropillars (Fig. 1(b)) [59]. Parto et al. employed an array of coupled mirroring resonators fabricated on InGaAsP quantum wells that are subject to an optical pumping scheme that respects PT symmetry. Unlike the passive SSH system possessing chiral symmetry, the ensuing Hamiltonian in the non-Hermitian regime has acquired a chiral-time symmetry (Fig. 1(c)) [60]. In a parallel work, Zhao et al. reported lasing of a protected zero mode in coupled InGaAsP-silicon microring resonators on a silicon-on-insulator substrate [61]. Aside from these microscale demonstrations, the SSH model has been realized and lasing from its zero-dimensional topological states has been observed in photonic crystals with embedded nanocavities – a feature of particular interest in miniaturization and high-density integration schemes. In general, the lasing mode can survive onsite disorder as well as perturbations that affect the coupling strength, however, the spectral position of the zero-energy states is only immune to the latter [62,63].
Clearly, of importance would be to identify two–dimensional topological arrangements where robust lasing can be manifested through truly protected edge states. Unlike one-dimensional systems, in 2D configurations this can be achieved via protected unidirectional transport that is akin to that observed in actual fermionic systems. In this regard, Bandres et al. demonstrated robust edge mode lasing along the perimeter of a judiciously designed square lattice of coupled microcavities that employs a synthetic gauge magnetic field for photons. In this vein, it exhibits a quantum spin Hall effect where the unidirectional circulation of photons in the ring resonators (clockwise or counterclockwise) acts as a pseudo-spin. Even at pump powers significantly above threshold, the topological features of this system allow for single mode lasing with a slope efficiency considerably higher than that of the corresponding topologically trivial realization (Fig. 1(d)) [64–67]. Around the same time, Bahari et al. built a square photonic crystal on an yttrium iron garnet (YIG) substrate to break time reversal symmetry. The resulting Chern insulator was then interfaced with a topologically trivial triangular crystal to promote unidirectional lasing edge modes under optical pumping (Fig. 1(e)) [68,69]. Along these lines, Klembt et al. investigated lasing in topological edge states in a polariton honeycomb array in the presence of an external magnetic field through the Zeeman effect [70]. Moreover, Dikopoltsev et al. reported a topological insulator vertical-cavity surface-emitting laser (VCSEL) array. In this structure, VCSELs–each of them a circular micropillar–are arranged to form a topological interface between shrunken and expanded honeycomb lattices. The topological features of the array enable the injection locking of VCSELs along the interface–thus making them to lase coherently and operate like a single laser (Fig. 1(f)) [71,72]. In this spirit, proposals were made where a train of mode-locked pulses can be produced by forcing an ensemble of topological semiconductor laser resonators to mutually lock in a synthetic dimension [73]. Alternatively, topological insulator lasers have been realized by exploiting the valley degree of freedom in nanostructured photonic crystals [74]. Interestingly, it has also been suggested that the band-inversion-induced reflection in architected topological nanocavity arrays can be employed as a new type of feedback for developing topological bulk lasers [75]. Gain saturation has been shown to have a dramatic effect on topological insulator lasers by lifting the degeneracy of the clockwise/counterclockwise spin-momentum locked edge states–an aspect enabling single-mode topological vortex lasing in a cross shaped photonic crystal cavity [76].
Recently emerged higher-order topological phases are yet another theme that has generated a great deal of research in the quest for low threshold topological lasers at nanoscale [77,78]. Topological nanocavities based on corner states supported by the generalized 2D SSH model have so far been explored in this pursuit [79,80]. The radiative coupling between the topological corner states results in multipole corner lasing modes with distinct emission patterns [81]. Using an InGaAsP slab with embedded quantum wells as the gain medium [81] instead of quantum dots [79] enables the system to function at room temperatures (Fig. 1(g)) [81]. As an analog to Majorana bound states (MBS) in superconductors, Dirac-vortex states have opened up a new route to two-dimensional topological mid-gap defect cavities. Imposing generalized Kekulé modulations to honeycomb photonic crystals leads to a vortex gap – a photonic realization of Jackiw–Rossi zero modes in 2D Dirac and Hou–Chamon–Mudry (HCM) models used in condensed matter physics. The free spectral range of this Dirac-vortex cavity is found to be larger than any other optical resonator with scalable modal volume – an intriguing attribute for high-power single-mode surface-emitting lasers. The topological protection ensures that the lasing wavelengths are always near the Dirac frequency, regardless of cavity size [82–85].
In addition to these optically pumped lasing schemes, electrically pumped topological lasers have been investigated recently in an effort to eliminate the need for an external laser source. Examples includes demonstrations that have been made using topologically protected valley edge modes in the terahertz band [86] and a quantum spin Hall platform in the telecommunication window (Fig. 1(h)) [87]. While the former works at cryogenic temperatures, the latter operated at a room temperature. The robust transport of light along a triangular topological valley cavity without any morphological optimization of the corners and with sharp bends bypassed by the lasing edge mode, is a clear manifestation of topological protection in non-Hermitian systems. In an attempt to introduce topology into mode-locked lasers, Leefmans et al. reported how asymmetric dissipative coupling between temporal modes of a fiber laser cavity can conspire with gain saturation nonlinearity to attain a topological temporal mode-locking. Equidistant temporal modes in such a laser cavity are generated by sinusoidal modulation of the intracavity loss using intensity modulators, while two optical delay lines introduce asymmetric dissipative couplings between nearest-neighbor pulses. Given that the gain medium provided by erbium doped fiber amplifier (EDFA) saturates due to the average power of temporal cavity modes, gain saturation nonlinearity establishes nonlocal interactions between the laser pulses. This technique has been used to demonstrate a nonlinearity driven non-Hermitian skin effect that is fundamentally different from current conventional mechanisms that rely on linear transport processes (Fig. 1(i)) [88].
3. Non-Hermitian bulk-edge correspondence and non-Hermitian skin effect
Localization of all eigenstates at one of the open edges of finite-size (or semi-infinite) and non-Hermitian lattices, with asymmetric couplings as the underlying cause for non-Hermiticity, shows once again how the resulting non-Hermitian skin effect can challenge common wisdom about the topological bulk-edge correspondence [89–92]. The counterintuitive features of the non-Hermitian skin effect have sparked two main lines of research over the past few years. The first one focuses on developing non-Bloch frameworks that describe the topology of non-Hermitian bands [93,94]. Biorthogonal bulk-edge correspondence is one such example that replaces conventional eigensystems with a biorthogonal set of right and left eigenstates [95]. Meanwhile, the second one explains the nontrivial topological properties of non-Hermitian arrangements through their complex line and point gap structures–features that are associated with energy bands that can avoid the crossing of a reference line or point in the complex energy plane, respectively. In the presence of a line gap, the Hermitian topological phase has been shown to be resilient to non-Hermitian perturbations, as is the case for a group of topological lasers [64,68,87]. Note that the point gap is unique to non-Hermitian environments and is responsible for the topological origin of the non-Hermitian skin effect [24,25,96,97]. Along similar lines, auxiliary generalized Brillouin zone theory has been established as a self-consistent method to understand bulk-boundary correspondence in one dimensional non-Hermitian configurations [98]. Correspondingly, the topological invariants are either adapted from the Hermitian domain and then generalized to account for the non-Hermiticity or entirely new invariants are developed like for example that of vorticity. In the latter approach, topological invariants are defined not only based on the eigenfunctions but also on the associated complex eigenvalues. On the other hand, the non-Bloch Chern number was found to rely on the non-Bloch nature of eigenstates given that it accurately captures the topological phase transitions in the non-Hermitian regime [24,99]. In this context, the topological charge of the bound states in the continuum (BICs)– as polarization vortices in momentum space –is quantified with a winding number which is not conserved during band conversion. As such, a Skyrmion number has been introduced to describe the topology of symmetry-protected BICs experiencing band inversion in leaky photonic crystals [100].
The non-Hermitian skin effect was introduced in 1990s, when Hatano and Nelson proposed an archetypical disordered tight-binding lattice with asymmetric hoppings – as a non-Hermitian extension of the Anderson model. A mobility edge in this model appears due to the competition between non-Hermiticity and disorder and can be interpreted using the associated point gap topology. Indeed, as a canonical example where topology is induced by non-Hermiticity, the Hatano-Nelson model, possesses a non-trivial (trivial) winding number when the wave functions are extended (localized) [24,101,102]. The first experimental demonstration of the non-Hermitian skin effect was carried out in a time-synthetic photonic mesh lattice composed of two coupled fiber loops (Fig. 2(a)) [103]. In a parallel study, a single-photon interferometer was used that allows discrete-time non-unitary quantum-walk dynamics in order to explore the topological properties of the non-Hermitian skin effect [104]. Subsequently, a Hatano-Nelson model was implemented in a frequency synthetic dimension by simultaneously modulating the resonant modes in a fiber ring cavity in both amplitude and phase [105]. On the other hand, the first-ever realization of the photonic Hatano-Nelson model in real space has been achieved only recently using an array of unidirectional ring resonators and judiciously designed links, where the interplay of optical gain and non-Hermiticity provides a mechanism for asymmetric coupling [106]. The ensued non-Hermitian skin effect in this system has been successfully utilized for phase locking in laser arrays (Fig. 2(b)) [106].
Fig. 2. Non-Hermitian skin effect. (a). Efficient funnel for light based on non-Hermitian skin effect in time synthetic photonic mesh lattices [103]. (b). Unidirectional microring laser with adjustable asymmetric coupling that emulates a Hatano-Nelson model [106].
In terms of applications, the sensitivity of the energy spectrum of non-Hermitian Hamiltonians to boundary conditions has been suggested as a means for developing an entirely new class of topological sensors [107–109]. This sensitivity originates from the violation of the bulk-boundary correspondence in non-Hermitian environments that is known to grow exponentially with system size [107–109]. Parallel to these one-dimensional photonic demonstrations, there have been ongoing efforts in electric circuits [110], acoustics [111,112], mechanics [113], and cold atom systems [114] in producing non-Hermitian skin effects in higher dimensions. In electric circuits, it has been shown that the interaction between multiple and distinct directional hoppings can significantly change the localization of the eigenstates and even lead to complete disappearance of the non-Hermitian skin effect [115]. Of interest, the interplay between the non-Hermitian skin effect and dissipative gaps has unveiled an unexpected boundary-induced dynamical phenomenon. This so called non-Hermitian edge burst is robust against edge perturbations and is manifested as a pronounced peak [116].
The non-Hermitian skin effect has so far been explored as a means for squeezing bulk states towards the edges of a system. Recently however, Wang et al. reported an opposite process through which the localized topological modes can morph into more general diverse profiles. The resulting extended modes are still robust against local perturbations and can be reshaped by manipulating the non-Hermiticity distribution [117]. In a paradigm shift study, it has been recently shown that the gain/loss induced second order hybrid skin topological modes in two dimensional Chern insulators can be realized without imposing asymmetric hoppings. If the applied gain/loss does not close the line energy gap associated with these open topological systems, all the edge modes tend to localize at one corner, while the bulk states happen to remain extended. By increasing the gain/loss contrast above a certain threshold, a PT phase transition occurs, and an exceptional point (EP) emerges between pairs of skin topological modes. Interestingly, when the gain/loss contrast exceeds a certain threshold, a PT phase transition occurs, and an exceptional point emerges between pairs of skin topological modes [118,119]. Such platforms may open up new venues for exploring the synergistic operation of PT symmetry and topology in higher dimensional settings. Along similar lines, in the context of Floquet systems, it has been shown that periodic quenching can exhibit a non-Hermitian skin effect [120].
4. Non-Hermitian high-order topological insulators
In an effort to explain the observed corner localization, recent studies have indicated a link between higher- order skin effects and higher-order topological insulator phases (HOTI) [121,122]. Supporting N-2 or less protected topological modes, such that the bulk-edge correspondence is extended to a bulk-hinge or bulk-corner correspondence, is a characteristic feature of HOTIs. For instance, 2D (3D) structures manifesting second-order topology exhibit gapped surfaces and edges (gapped bulks and surfaces) and can support topologically protected gapless states at the intersection of two edges (surfaces) – the so called corner (hinge) states. It turns out that the dimensionality of these states can be, indeed, determined by the quantized electric multipole moments of the bulk [123–125]. HOTIs in Hermitian systems have been lately a subject of active research [126–129] while their non-Hermitian counterparts are only now being investigated. Introducing staggered on-site gain and loss, for example, has been shown to drive a trivial system into the topological domain and promote four degenerate corner states. Non-Hermitian topological invariants, built around nested Wilson loops for this framework, have been found to be protected by reflection or chiral symmetry [130]. Implementing this arrangement in an acoustic crystal suggests that the same behavior can be achieved simply by carefully designing the loss configuration [131]. Along these lines, non-Hermitian second-order topological insulators (SOTIs) are reported in spite of first order topologically trivial bulk bands. Given that the Hamiltonian of a two-dimensional case respects mirror-rotation and sublattice symmetry, the associated midgap robust states are only located at one corner. In three-dimensions, second-order topological modes are anomalously localized at a corner and not along the hinges–signifying the breakdown of the usual bulk-hinge correspondence in the presence of non-Hermiticity. If the underlying twofold mirror-rotation symmetry breaks, then, the modes are localized at more than one corner [132]. A concept unique to the non-Hermitian regime is the clock symmetry that was found to induce topological corner states when imposed on a breathing Kagome lattice [33]. Another example is the hosting of Wannier-type corner states by photonic graphene with mirror-symmetrical gain and loss patterns [133]. These tight binding predictions are still waiting for an experimental verification. On a different front, Yu et al. have explored the zero energy modes in a non-Hermitian quadrupole insulator using a Jordan decomposition. This study reveals the importance of replacing the conventional eigen decomposition with a Jordan decomposition when analyzing the response of non-Hermitian frameworks [134].
5. Topological features of non-Hermitian singularities
Exceptional points (EPs) are known to be among the most prominent and intriguing features of non-Hermitian systems. At an EP not only two or more eigenvalues tend to coalesce but also their corresponding eigenvectors simultaneously collapse on each other–thus leading to an abrupt reduction in dimensionality, an aspect that has no counterpart in a Hermitian environment [135–144]. These peculiar band singularities are characterized by a gap closing and as such they play a pivotal role in a rich array of anomalous topological phenomena [97,145,146]. For instance, by exposing a two-dimensional periodic photonic crystal slab to radiative losses, a pair of EPs emerges from a single Dirac point (DP) with a nontrivial Berry phase [147,148]. The associated complex band structure is endowed with a bulk Fermi arc while the polarization of the far field emission signifies the signature of a half integer topological index associated with EPs–both being byproducts of the non-Hermitian topological properties of exceptional points (Fig. 3(a)) [148]. A similar idea has been exploited in Weyl arrangements where a dissipative perturbation transforms isolated point degeneracies (i.e., Weyl points) into Weyl exceptional rings. Contrary to Weyl points and Weyl nodal rings, they possess a nontrivial Chern number and a quantized Berry phase [149–151]. Experimental demonstration of this effect provided the first evidence of a distributed Berry flux and demonstrated that the Berry charge and Fermi arc surface states survive in the presence of non-Hermiticity (Fig. 3(b)) [151]. This aspect is further enhanced by the appearance of a Weyl ring in thermal diffusive systems that could be potentially used in topological heat transfer [152].
Fig. 3. Topological features of non-Hermitian degeneracies or exceptional points. (a). A topologically protected bulk Fermi arc that connects two exceptional points emerges from a single Dirac point in a 2D-periodic photonic crystal when subjected to radiative losses [148]. (b). Weyl exceptional ring observed in the band structure of a bipartite helical waveguide array with tunable losses [151]. (c) Topological lasing in a coupled waveguide demonstrating dynamic encirclement of an EP [168]. (d) Monitoring the temporal evolution of light pulse during the parametric steering of a non-Hermitian Hamiltonian [169].
On a different front, the distinctive topology of self-intersecting Riemann manifolds around EPs leads to a remarkably different manifestation of the adiabatic theorem in non-Hermitian arrangements. Slow and cyclic transformations around Hermitian degeneracies return the system to itself apart from acquiring a geometric Berry phase. However, when non-Hermitian degeneracies are involved, two possible scenarios are now possible. If a non-Hermitian Hamiltonian steers around an exceptional point in a quasi-static fashion, the eigenvalues of a two-state system tend to swap with each other at the end of the loop [153,154]. In the course of such a process, one of the eigenstates picks up a topological phase shift of $\pi $. Evidently, this requires four encirclement rounds around an EP in order to retrieve the initial state. This peculiar behavior has been observed in two-state exciton-polariton systems [155] and also in a microwave billiard arrangement [156]. However, when an EP is dynamically encircled, non-hermiticity and the ensuing non-adiabatic transitions result in a counterintuitive chiral state transfer. In this regard, for any arbitrary input that happens to be a mixture of the two eigenstates, after winding around an EP in a clockwise (CW) fashion, the system generates in a robust and faithful manner the fundamental mode $|1\rangle$, while in the opposite direction, the output always happens to be the second mode $|2\rangle$, irrespective of the input state.
The dynamics of such systems are governed by the time dependent Schrödinger equation $i{\partial _t}|{\psi (t )\rangle + H(t )} |\psi (t )\rangle = 0$, where their associated Hamiltonian $H(t )$ is given by
6. Conclusion
Over the years, research at the interface of topology and non-Hermiticity has been enriched by a series of breakthroughs in these two fields. While topological physics had its birth within the framework of quantum mechanics, it is in optics that it found a reincarnation within the realm of non-Hermitian systems. Non-Hermiticity allows for the emergence of a new class of symmetries, ramifying the existent symmetry classifications substantially – an aspect enriching topological phases beyond their Hermitian counterparts. The interplay of non-Hermiticity with topology and nonlinearity in lasing schemes, has changed the common mindset about fundamental features of topological systems, thus generating a novel and robust class of lasers. The unique topological aspects of non-Hermitian degeneracies have opened another route to design photonic systems that are resilient against operational degradation, fabrication imperfection and environmental perturbations.
Yet, on most occasions, elements in optical systems tend to couple only to nearest neighbors. As such, new developments in realizing long range interactions and in employing synthetic dimensions of higher order are expected to dramatically accelerate the pace of research in this field [56,172–175]. Coupled waveguides [173], fiber loops [8,175], and integrated ring resonators [176] are amongst various promising platforms that have been successfully harnessed so far to realize synthetic dimensions. From a scientific standpoint, a universal and self-consistent theory of non-Hermitian bulk-edge-correspondence is still elusive. Another area of active research deals with the ramifications of non-Hermitian topology in open quantum systems. The prospect for topological phases in aperiodic media calls for the development of local and real-space definitions of topology [177,178]. It still remains to be seen whether new non-Hermitian symmetries can be classified as extended members within the broader topological family of insulators. Identifying non-Hermitian topological phases in the presence of many-body interaction is another emerging topic [179,180]. Non-Hermitian topological photonics also holds promise for manipulating the emission pattern of plasmonic nanoantennas, Purcell enhancement enabled by topological edge state, and developing topological beam emitters [181,182]. At a fundamental level, understanding various topological phases that are unique to Floquet and non-Hermitian arrangements is of immediate significance. In light of the recent progress, one can look forward to new research initiatives and paradigms in the field that could find their own niche applications for the next few years to come.
Funding
Air Force Office of Scientific Research (FA9550-20-1-0322, FA9550-21-1-0202); Office of Naval Research (N00014-19-1- 2052, N00014-20-1-2522, N00014-20-1-2789); Defense Advanced Research Projects Agency (D18AP00058); National Science Foundation (CBET 1805200); W. M. Keck Foundation; United States-Israel Binational Science Foundation (BSF: 2016381); Simons Foundation (Simons grant 733682); Air Force Research Laboratory (FA86511820019); Qatar National Research Fund (grant NPRP13S0121-200126).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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