Non-Hermitian chiral degeneracy of gated graphene metasurfaces

Non-Hermitian degeneracies, also known as exceptional points (EPs), have been the focus of much attention due to their singular eigenvalue surface structure. Nevertheless, as pertaining to a non-Hermitian metasurface platform, the reduction of an eigenspace dimensionality at the EP has been investigated mostly in a passive repetitive manner. Here, we propose an electrical and spectral way of resolving chiral EPs and clarifying the consequences of chiral mode collapsing of a non-Hermitian gated graphene metasurface. More specifically, the measured non-Hermitian Jones matrix in parameter space enables the quantification of nonorthogonality of polarisation eigenstates and half-integer topological charges associated with a chiral EP. Interestingly, the output polarisation state can be made orthogonal to the coalesced polarisation eigenstate of the metasurface, revealing the missing dimension at the chiral EP. In addition, the maximal nonorthogonality at the chiral EP leads to a blocking of one of the cross-polarised transmission pathways and, consequently, the observation of enhanced asymmetric polarisation conversion. We anticipate that electrically controllable non-Hermitian metasurface platforms can serve as an interesting framework for the investigation of rich non-Hermitian polarisation dynamics around chiral EPs.

A physical system describable with a non-Hermitian Hamiltonian may host exceptional points [1][2][3][4] (EPs), i.e., branching point singularities at which two or more eigenstates coalesce in parameter space.Unlike the degeneracies in Hermitian systems, for which an orthogonal set of eigenstates can be constructed, the eigenstates coalesce at the EP and become self-orthogonal, leading to a defective eigenspace of reduced dimensionality.These singular features have been observed and utilised in various quantum and classic systems, including electronic spins 5 , superconducting qubits 6 , condensed exciton-polaritons 7 , electronic circuits 8 , thermotic systems 9 and active matter 10 .Particularly in photonic systems [11][12][13][14] , the ease of precise loss and/or gain control has facilitated the discovery of a plethora of EP-associated exotic behaviours, with some representative examples including chiral mode transfer with or without encircling around EPs 15,16 , controlled electromagnetically induced transparency 17 , a ring (or a pair) of EPs in momentum space 18,19 , and coupling to the missing dimension at an EP 20 .In line with these advancements, we have also witnessed a series of promising EP-enabled functionalities, such as parity-time (PT) symmetry-broken lasing 21 , exceptional topological phase engineering 22 , electrical winding number switching 23 , exceptional sensing 24,25 and coherent perfect absorption 26 .
For the exploration of non-Hermitian physics and the application of EP-enabled functionalities, metasurfaces 22,23,[27][28][29] are now being considered one of the most versatile platforms because their constituent meta-atoms are inherently constructed from lossy coupled subwavelengthscale resonators.Generally, any change in the polarisation state of light transmitted through the non-Hermitian metasurface can be characterised by a non-Hermitian Jones matrix that plays the role of an effective Hamiltonian [28][29][30][31][32][33] .In contrast to the prevailing cases 34 , the non-Hermitian Jones matrix, of which the complex-valued elements can be engineered by geometrical and materials design of the meta-atoms, enables the utilisation of polarisation eigenstates for the examination of EP-related phenomena.Interestingly, at THz frequencies, the metasurface platform has been the only one that allows for the experimental observation of EPs, inheriting all the generic advantages of subwavelength-scale metaphotonics.However, until now, experimental probing of a branching point singularity in the parameter space has mostly been demonstrated in a passive way by repeatedly fabricating metasurfaces with varying meta-atom designs 28,29 .Furthermore, even with a series of repeated preparations, unavoidable errors from fabrications and/or measurements have made it difficult to observe relevant non-Hermitian dynamics around/at EPs.It is thus highly desirable to have precise real-time control of the parameters for access to an EP in a single metasurface platform 23,35 .
To circumvent the aforementioned problems, we hybridise gated graphene microribbons with non-Hermitian metasurfaces and demonstrate the electrically controlled probing of polarisation eigentransmission surfaces along with the corresponding eigenstates.Notably, this probing methodology utilises time domain spectroscopy that makes use of a broadband pulse, which in combination with a continuous gate tuning capability enables high-resolution access to chiral EPs in two-parameter space.Here, chiral EPs refer specifically to the non-Hermitian degeneracy at which a circularly polarised state becomes the only eigenstate as a result of coalescence.The measured non-Hermitian Jones matrix in the parameter space enables a systematic investigation of nonorthogonality between polarisation eigenstates and atypical linkage between input and output polarisation states at the chiral EP.Specifically, we show that, for a specific incident polarisation, augmenting dimensionality at the chiral EP can be solely revealed at the output.It is also found that the maximal nonorthogonality assured by the defective Jones matrix at the chiral EP leads to the observation of enhanced asymmetric polarisation conversion.Last but not least, the examination of polarisation eigenstates in parameter space reveals a vortex structure, from which half integer topological charges at the chiral EP are clarified.

Results
Design of non-Hermitian gated graphene metasurfaces.To map eigentransmission surfaces and investigate their structure near a chiral EP, we designed a non-Hermitian gated graphene metasurface consisting of an array of pairs of coupled split ring resonators (SRRs) with a graphene microribbon bridging the SRRs (Fig. 1a).The paired SRRs have their splits opened in orthogonal directions and are characterised by distinct external loss rates (Fig. 1b).Then, by employing temporal coupled-mode theory (TCMT) 31,33,36 , a parameter-dependent non-Hermitian Jones matrix of the designed metasurface can be derived (see Methods).The two coupled SRRs are modelled as two orthogonally oriented resonators with a resonance frequency of  0 , a coupling rate of , and intrinsic and external loss rates of   and   ( = , ), all of which can be adjusted to a certain degree by the geometry and materials constituting the unit cell (Fig. 1a, c).Under steady-state conditions, a 22 non-Hermitian Jones matrix   can be written in a linear polarisation basis.The matrix can be expressed as a sum of uncoupled and coupled parts (  =   +   ), only the latter of which is relevant for investigating the coalescing behaviour near/at the chiral EP.Specifically, the coupled part is found to be proportional to the following matrix: where the dimensionless parameters are introduced for the simplicity of expression (see Methods for details): and Κ = −/ √     .An inspection of the eigenvalues and eigenvectors of the above matrix shows the presence of a pair of chiral EPs when the following conditions are satisfied: The first equality specifies a one-dimensional subspace satisfying PT symmetry, while the second equalities further identify the two chiral EPs as singularities distinguishing a PT exact phase from a broken one on the subspace.The chirality of a coalesced eigenstate at each EP is determined by the sign in the second equality (i.e., + for RCP and − for LCP).In this work, the system is parameterised by two variables: the angular frequency  of an input wave and the gate voltage   (or the Fermi level   in simulations) that control the optical conductivity of graphene microribbons (Fig. 1c).The gating of graphene disproportionately adjusts the intrinsic loss rate of each SRR, resulting in a change in the Γ value and correspondingly the potential fulfilment of the second equality (here, in this experimental work, Γ = −Κ).From the simulations, it is found that the external loss rates are almost invariant within the gating range of interest.It is worthwhile to note that the pair of chiral EPs connected by a PT broken phase in the parameter space of our metasurface platform is analogous to the pair of EPs linked by a bulk Fermi arc in the momentum space of a two-dimensional non-Hermitian photonic crystal 19,37,38 .

Mapping of the eigentransmission surface and identification of chiral EP Non-Hermitian
graphene metasurfaces were prepared by standard microfabrication techniques and a CVDgrown graphene transfer method and characterised by terahertz time domain spectroscopy (THz-TDS, see Methods).Here, a charge-neutral point of the graphene is located at a gate voltage of ~ −1.1 V. From the measured co-and cross-polarised complex amplitude transmission through the fabricated metasurface, a set of non-Hermitian Jones matrices   , each of which is specified on a rectangular grid in the two-parameter space, can be obtained.
The eigentransmission of   clearly reveals self-intersecting Riemann surface structures (Fig. 2a, b); the cusp at the end of the line of intersection is identified as the chiral EP and found to be located at (  ,  , ) (0.55 THz, 0.1 V).To support the experimentally measured topological structure, we performed numerical simulations using a finite element method and extracted the eigentransmission (see Methods).As shown in Fig. 2c, d, the numerical simulations show qualitative agreement with the experimental results.Interestingly, sectioning of the reconstructed eigentransmission surfaces further captures the key features of coupling and phase transitions across the chiral EP.First, a transition from weak (Γ > −Κ,   >  , ) to strong (Γ < −Κ ,   <  , ) coupling between polarisation eigenstates can be seen by sampling eigentransmission surfaces at consecutively decreasing values of   across the chiral EP (corresponding to three cut lines on the Riemann surfaces shown in Fig. 2a).As more clearly seen in Fig. 2e, f, a crossing (anti-crossing) to anti-crossing (crossing) transition is clearly observable in the spectrally resolved eigentransmission amplitudes (phases).Second, an exceptional phase transition is observed in the gate-voltage-dependent eigentransmission amplitudes (phases) sampled along a one-dimensional subspace satisfying PT symmetry (Ω  = Ω  ); in these plots, exact and broken PT phases appear on either side of the chiral EP (Γ = −Κ, Fig. 2a-d).Here, it is also worth noting that the coupling crossover is observed to be concomitant with the exceptional phase transition across the chiral EP, as in other non-Hermitian systems 23,39 .

Nonorthogonality of eigenstates and half integer topological charge of chiral EP To
visualise the eigenstate coalescing behaviour, numerically calculated and experimentally extracted polarisation eigenstates are mapped on a Poincaré sphere (see Fig. 3a and Methods).
For clarity, the polarisation eigenstates corresponding to different values of gate voltage (or Fermi levels) are colour-coded.As seen in Fig. 3a, except at the chiral EP, the polarisation eigenstates exist in pairs and appear symmetrically with respect to the south pole.It is noteworthy that the paired polarisation eigenstates are not represented on the Poincaré sphere by antipodal points, which is indicative of the characteristic nonorthogonality of general non-Hermitian systems.As the chiral EP is approached in parameter space, the paired polarisation eigenstates move towards the south pole and eventually coalesce into the left circularly polarised state.To quantify the degree of nonorthogonality and coalescence, a Petermann factor (Fp) is calculated based on the left and right polarisation eigenstates extracted from the measurement (see Methods) 40,41 .Ideally, self-orthogonality and maximal nonorthogonality at the EP lead to the divergence of Fp, of which the experimental quantification can be done by plotting an inverse of Fp in parameter space (Fig. 3b).In the plot, smaller values of Fp -1 are seen along a one-dimensional subspace satisfying PT symmetry, and the value sharply drops down to ~310 −4 near the chiral EP.This sharp decrease illustrates a singular sensitivity near the chiral EP to a variation in parameters.It is interesting to note that in addition to the investigation of nonorthogonality, the polarisation eigenstate mapping in parameter space enables the characterisation of topological charges associated with the chiral EPs.For this purpose, we monitored the cyclic variation of the ellipse orientation of polarisation eigenstates along an encircling path on the Riemann surface around the chiral EP [42][43][44] (Fig. 3c).The cyclic variation of the ellipse orientation (Fig. 3d) reveals a polarisation vortex centre at the chiral EP along with a half integer topological charge (  = +1/2).While not observable in the measurements due to the maximum gate voltage limit, the existence of the other chiral EP in parameter space with a half integer topological charge of  = −1/2 can be confirmed in the simulations 37,38,45 .smaller than those of input states (Fig. 4a-c) due to the nonunitary transformation performed by the metasurface 46 .More interestingly, the perfect nulling of cross-polarised transmission   at the chiral EP leads us to observe the signature of a Pancharatnam-Berry phase during gate-controlled coupling crossover 47 (Fig. 5a, b).The phase of   at high frequencies sharply changes by 2π (see Fig. 5b), implying zero-to-one topological winding number switching 23 by gating around the chiral EP (Fig. 5c).This winding number switching and the associated phase jump across the chiral EP can also be employed for enhanced sensing and monitoring of chemical and biological events 48 .

Maximal asymmetric polarisation conversion
The asymmetric non-Hermitian Jones matrix of the fabricated graphene metasurface also leads to gate-controlled asymmetric polarisation conversion (Fig. 5d).When compared with a forward propagation case, a backward propagating wave is incident on the metasurface consisting of unit cells that are mirrorreflected with a line of symmetry connecting the centres of two constituting SRRs.This guarantees that the off-diagonal elements of the Jones matrix are exchanged, i.e.,    =    and    =    , where superscripts specify the direction of propagation.Specifically, at the chiral EP of the fabricated metasurface,    becomes zero, while    remains nonzero, which suggests that a large difference in off-diagonal elements can be observed.To quantitatively characterise the effect, a normalised difference in asymmetric polarisation conversion is defined here as ), the value of which generally ranges from -1 to 1. Figure 5e shows the values of parameter-dependent   extracted from the transmission measurement performed on the fabricated metasurface.It is clearly seen that the values are nonzero near the chiral EP and approach the maximum value of 1 at the chiral EP.
Therefore, the gate-controlled access to the chiral degeneracy enables us to reach the maximal asymmetric polarisation conversion that is hard to achieve in passive non-Hermitian metasurfaces due to the sensitivity of exceptional points to fabrication errors.

Conclusions
In this work, we experimentally demonstrated the potential of a non-Hermitian gated graphene metasurface platform for the clarification and characterisation of chiral EPs in parameter space.
The proposed platform stands among other recently implemented tunable non-Hermitian photonic systems 23 while distinguishing itself from others by utilising a non-Hermitian Jones matrix for the manipulation of polarisation states.Specifically, in addition to the well-known general features such as nonorthogonality and mode coalescence, the non-Hermitian Jones matrix, especially written in Jordan form at the chiral EP, leads to an unusual nonunitary relation between input and output polarisation states.One such manifestation is the preferential polarisation conversion into the state represented by a Jordan vector of the non-Hermitian Jones matrix.This implies that the output polarisation state can be made independent of the coalesced eigenstate of the metasurface being transmitted, which is contrary to our usual conception.We further experimentally clarified half-integer topological charges of a non-Hermitian chiral degeneracy and topological winding number switching by gating.We believe that the proposed tunable metasurface platform may become an essential tool in the investigation of dynamic phenomena related to non-Hermitian chiral degeneracies and serve as a testbed for realising artificial non-Hermitian effective matter.

Figure captions
Polarisation eigenstate coalescence and the corresponding reduction of an eigenspace dimensionality at the chiral EP also lead to singular behaviours in the transmission of waves through the non-Hermitian metasurface.More specifically, the left and right polarisation eigenstates become self-orthogonal at the chiral EP so that the output polarisation state |  ⟩ is described with the single eigenpolarisation state |⟩ and the associated Jordan vector, i.e., in our case, |⟩ = |⟩, |  ⟩ = ( , |⟩⟨| +  , |⟩⟨| +  , |⟩⟨|)|  ⟩ (3) where 's are elements of the 22 non-Hermitian Jones matrix at the chiral EP written in a circular polarisation basis and |  ⟩ is the input polarisation state.Note that the matrix is in Jordan form at the chiral EP with its elements indicating cross-and co-polarised transmission ( , =  , and  , = 0 ).In Fig. 4, three illustrative cases are schematically shown along with their corresponding Poincaré sphere representations: (i) For RCP incidence (|  ⟩ = |⟩, orthogonal to the polarisation eigenstate at the chiral EP, see Fig. 4a), the output polarisation state becomes a superposition of the polarisation eigenstate and the Jordan vector (|  ⟩ =  , |⟩ +  , |⟩ ).(ii) For LCP incidence (|  ⟩ = |⟩ , the polarisation eigenstate at the chiral EP, see Fig. 4b), the output polarisation state contains only the LCP component (|  ⟩ =  , |⟩).Note that the coalescence of polarisation eigenstates prohibits simultaneous nulling of both cross-polarised transmissions, which eventually leads to asymmetric polarisation conversion, as will be discussed below.(iii) Of particular interest is the case where the output polarisation state is completely devoid of the component parallel to the coalesced polarisation eigenstate (Fig. 4c); more specifically, preferential conversion to the Jordan vector (|  ⟩ =  , |⟩) can be achieved by setting the input polarisation states to |  ⟩ = − , / , |⟩ + |⟩ .This counterintuitive outcome is the accidental revelation of the missing dimension through the destructive interference of two LCP components: one from co-polarised transmission and the other from cross-polarised transmission of the prescribed input state.It is also worth mentioning that the solid angles subtended by output polarisation states are slightly

Fig. 2
Fig. 2 Eigentransmission amplitude and phase of the non-Hermitian gated

Fig. 3
Fig. 3 Electrical access to eigenpolarisation states at the EP. a Measured (square)

Fig. 4
Fig. 4 Peculiar linkage between input and output states at the chiral EP. a-c