Multiple Exceptional Points Research Articles

Tailoring exceptional points with one-dimensional graphene-embedded photonic crystals

We theoretically demonstrate that tunable exceptional points (EPs) can be realized by using graphene-embedded one-dimensional (1D) photonic crystals with optical pumping in the terahertz (THz) frequency range. By tuning the Fermi level of graphene sheet, the energy band are altered significantly and the EP appears. In particular, multiple EPs at different frequencies can be selectively produced via subtly adjusting the band structure. Furthermore, topological features of these EPs, such as crossing and anti-crossing of the real and imaginary parts of the eigenvalues, have been analyzed in detail. We expect that tunable EPs can provide an instructive method to design active optical devices based on photoexcited graphene sheets in the THz frequency range.

Frequency locking and controllable chaos through exceptional points in optomechanics

We engineer mechanical gain (loss) in system formed by two optomechanical cavities (OMCs), that are mechanically coupled. The gain (loss) is controlled by driving the resonator with laser that is blue (red) detuned. We predict analytically the existence of multiple exceptional points (EPs), a form of degeneracy where the eigenvalues of the system coalesce. At each EP, phase transition occurs, and the system switches from weak to strong coupling regimes and vice versa. In the weak coupling regime, the system locks on an intermediate frequency, resulting from coalescence at the EP. In strong coupling regime, however, two or several mechanical modes are excited depending on system parameters. The mechanical resonators exhibit Rabi-oscillations when two mechanical modes are involved, otherwise the interaction triggers chaos in strong coupling regime. This chaos is bounded by EPs, making it easily controllable by tuning these degeneracies. Moreover, this chaotic attractor shows up for low driving power, compared to what happens when the coupled OMCs are both drived in blue sidebands. This works opens up promising avenues to use EPs as a new tool to study collective phenomena (synchronization, locking effects) in nonlinear systems, and to control chaos.

Optical lattices with higher-order exceptional points by non-Hermitian coupling

Exceptional points (EPs) are degeneracies in open wave systems with coalescence of at least two energy levels and their corresponding eigenstates. In higher dimensions, more complex EP physics not found in two-state systems is observed. We consider the emergence and interaction of multiple EPs in a four coupled optical waveguides system by non-Hermitian coupling showing a unique EP formation pattern in a phase diagram. In addition, absolute phase rigidities are computed to show the mixing of the different states in definite parameter regimes. Our results could be potentially important for developing further understanding of EP physics in higher dimensions via generalized paradigm of non-Hermitian coupling for a generation of parity-time devices.

Non-Abelian nature of systems with multiple exceptional points

The defining characteristic of an exceptional point (EP) in the parameter space of a family of operators is that upon encircling the EP eigenstates are permuted. In case one encircles multiple EPs, the question arises how to properly compose the effects of the individual EPs. This was thought to be ambiguous. We show that one can solve this problem by considering based loops and their deformations. The theory of fundamental groups allows to generalize this technique to arbitrary degeneracy structures like exceptional lines in a three-dimensional parameter space. As permutations of three or more objects form a non-abelian group, the next question that arises is whether one can experimentally demonstrate this non-commutative behavior. This requires at least two EPs of a family of operators that have at least 3 eigenstates. A concrete implementation in a recently proposed $\mathcal{PT}$ symmetric waveguide system is suggested as an example of how to experimentally check the composition law and show the non-abelian nature of non-hermitian systems with multiple EPs.

Next-nearest-neighbor resonance coupling and exceptional singularities in degenerate optical microcavities

We report a specially configured non-Hermitian optical microcavity, imposing spatially imbalanced gain-loss profile, to host an exclusively proposed next nearest neighbor resonances coupling scheme. Adopting scattering matrix (S-matrix) formalism, the effect of interplay between such proposed resonance interactions and the incorporated non-Hermiticity in the microcavity is analyzed drawing a special attention to the existence of hidden singularities, namely exceptional points (EPs); where at least two coupled resonances coalesce. We establish adiabatic flip-of-states phenomena of the coupled resonances in the complex frequency plane (k-plane) which is essentially an outcome of the fact that the respective EP is being encircled in system parameter plane. Encountering such multiple EPs, the robustness of flip-of-states phenomena have been analyzed via continuous tuning of coupling parameters along a special hidden singular line which connects all the EPs in the cavity. Such a numerically devised cavity, incorporating the exclusive next neighbor coupling scheme, have been designed for the first time to study the unconventional optical phenomena in the vicinity of EPs.

Electrical tunability due to coalescence of exceptional points in parity-time symmetric waveguides.

We demonstrate theoretically the electric tunability due to the coalescence of exceptional points in PT-symmetric waveguides bounded by imperfect conductive layers. Owing to the competition effect of multimode interaction, multiple exceptional points and PT phase transitions could be attained in such a simple system, and their occurrences are strongly dependent on the boundary conductive layers. When the conductive layers become very thin, it is found that the oblique transmittance and reflectance of the same system can be tuned between zero and one by a small change in the carrier density. The results may provide an effective method for fast tuning and modulation of optical signals through electrical gating.

Cyclic permutation-time symmetric structure with coupled gain-loss microcavities

We study the coupled even number of microcavities with the balanced gain and loss between any pair of their neighboring components. The effective non-Hermitian Hamiltonian for such structure has the cyclic permutation-time symmetry with respect to the cavity modes, and this symmetry determines the patterns of the dynamical evolutions of the cavity modes. The systems also have multiple exceptional points for the degeneracy of the existing supermodes, exhibiting the "phase transition" of system dynamics across these exceptional points. We illustrate the quantum dynamical properties of the systems with the evolutions of cavity photon numbers and correlation functions. Moreover, we demonstrate the effects of the quantum noises accompanying the amplification and dissipation of the cavity modes. The reciprocal light transportation predicted with the effective non-Hermitian models for the similar couplers is violated by the unavoidable quantum noises.

Geometric phase around multiple exceptional points

We study the geometrical phase when multiple exceptional points (EPs) are involved. In an optical microcavity of a stadium shape, we find two EPs, connected through three interacting modes, in a two-dimensional parameter space. It is shown that the geometrical phase imprinted on wave functions becomes zero when the two EPs are encircled three times in the parameter plane. In addition, we examine geometrical phases when three EPs are involved in a $3\ifmmode\times\else\texttimes\fi{}3$ matrix model, and show that single- and double-loop return modes have the geometrical phase of $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ or zero, depending on the type of the three EPs encircled.

Analysis of multiple exceptional points related to three interacting eigenmodes in a non-Hermitian Hamiltonian

We have investigated the exceptional points (EPs) which are degeneracies of a non-Hermitian Hamiltonian, in the case that three modes are interacting with each other. Even though the parametric evolution of the modes cannot be uniquely determined when encircling more than two EPs once, we can recover the initial configuration of the modes by encircling two EPs three times or three EPs twice. We confirm our expectation by numerically calculating the modes of an open quantum system, two dielectric microdisks, and 3$\times$3 matrix model.