Exotic light dynamics around a fourth order exceptional point

Physical Review B

We present a novel approach and a theoretical framework for generating high order exceptional points of degeneracy (EPD) in photonic structures based on periodic coupled resonators optical waveguides (CROWs). Such EPDs involve the coalescence of Floquet-Bloch eigenwaves in CROWs, without the presence of gain and loss, which is in contrast to the requirement of Parity-Time (PT) symmetry to develop exceptional points based on gain and loss balance. The EPDs arise here by introducing symmetry breaking in a conventional chain of coupled resonators through periodic coupling to an adjacent uniform optical waveguide, which leads to unique modal characteristics that cannot be realized in conventional CROWs. Such remarkable characteristics include high quality factors (Q-factor) and strong field enhancement, even without any mirrors at the two ends of a cavity. We show for the first time the capability of CROWs to exhibit EPDs of various order; including the degenerate band edge (DBE) and the stationary inflection point (SIP). The proposed CROW of finite length shows enhanced quality factor when operating near the DBE, and the Q-factor exhibits an unconventional scaling with the CROW's length. We develop the theory of EPDs in such unconventional CROW using coupled-wave equations, and we derive an analytical expression for the dispersion relation. The proposed unconventional CROW concepts have various potential applications including Q-switching, nonlinear devices, lasers, and extremely sensitive sensors. I.

Physical Review A, 2014

The spectral, dynamical and topological properties of physical systems described by non-Hermitian (including PT -symmetric) Hamiltonians are deeply modified by the appearance of exceptional points and spectral singularities. Here we show that exceptional points in the continuum can arise in non-Hermitian (yet admitting and entirely real-valued energy spectrum) optical lattices with engineered defects. At an exceptional point, the lattice sustains a bound state with an energy embedded in the spectrum of scattered states, similar to the von Neumann-Wigner bound states in the continuum of Hermitian lattices. However, the dynamical and scattering properties of the bound state at an exceptional point are deeply different from those of ordinary von Neumann-Wigner bound states in an Hermitian system. In particular, the bound state in the continuum at an exceptional point is an unstable state that can secularly grow by an infinitesimal perturbation. Such properties are discussed in details for transport of discretized light in a PT -symmetric array of coupled optical waveguides, which could provide an experimentally accessible system to observe exceptional points in the continuum.

Physical review, 2022

Exceptional points (EPs) of both eigenvalue and eigenvector degeneracy offer remarkable properties of the non-Hermitian systems based on the Jordanian form of Hamiltonians at EPs. Here we propose the perturbation theory able to underpin the physics in the vicinity of the higher-order EPs. The perturbation theory unveils lifting of degeneracy and origin of the different phases merging at the EP. It allows us to analyze the photonic local density of states and resonance energy transfer, determining their spectral behaviors in a general form. Resonant energy transfer is investigated in analytical and numerical examples. We analytically find the resonance energy transfer rate near the third-order EP occurring in the system of three coupled cavities and reveal singularities caused by the interplay of the perturbation and frequency detuning from degenerate eigenfrequency. Numerical simulation of the coupled-resonator system reveals the vital role of a mirror for switching to the EP of the doubled order and corresponding enhancement of the resonance energy transfer rate. Our investigation sheds light on the behavior of nanophotonic systems in non-Hermitian environments.

Physical Review A, 2019

A two-mode optical parity-time (PT) symmetric system, with gain and damping, described by a quantum quadratic Hamiltonian with additional small Kerr-like nonlinear terms, is analyzed from the point of view of nonclassical-light generation. Two kinds of stationary states with different types of (in)stability are revealed. Properties of one of these are related to the presence of semiclassical exceptional points, i.e., exotic degeneracies of the non-Hermitian Hamiltonian describing the studied system without quantum jumps. The evolution of the logarithmic negativity, principal squeezing variances, and sub-shot-noise photon-number correlations, considered as entanglement and nonclassicality quantifiers, is analyzed in the approximation of linear-operator corrections to the classical solution. Suitable conditions for nonclassical-light generation are identified in the oscillatory regime, especially at and around exceptional points that considerably enhance the nonlinear interaction and, thus, the non-classicality of the generated light. The role of quantum fluctuations, inevitably accompanying attenuation and amplification in the evolution of quantum states, is elucidated. The evolution of the system is analyzed for different initial conditions.

Physical Review A, 2021

The implementation of exceptional points (EPs), a special type of topological singularities, has emerged as a new paradigm for engineering the quantum-inspired or wave-based photonic systems. Even though there exists a range of investigations on EPs of order two and three (say, EP2s and EP3s, respectively), the hosting of fourth-order EPs (EP4s) in any real system and the exploration of associated topological features are lacking. Here we have designed a simple Fabry-P\'erot type gain-loss-assisted open optical microcavity to host EPs up to order four. The scattering-matrix formalism has been used to analyze the microcavity numerically. With the appropriate modulation of the gain-loss profile in the same cavity geometry, we have encountered multiple different orders of EPs by investigating the simultaneous interactions among four coupled cavity states via level-repulsion phenomena. Besides affirming the second-order and third-order branch-point behaviors of the embedded EP2s and...

Physical Review Letters

Dynamically encircling an exceptional point in a non-Hermitian system can lead to chiral behaviors, but this process is difficult for on-chip PT-symmetric devices which require accurate control of gain and loss rates. Here, we experimentally demonstrated encircling an exceptional point with a fixed loss rate in a compact anti-PT-symmetric integrated photonic system, where chiral mode switching was achieved within a length that is an order of magnitude shorter than that of a PT-symmetric system. Based on the experimental demonstration, we proposed a topologically protected mode (de)multiplexer that is robust against fabrication errors with a wide operating wavelength range. With the advantages of simplified fabrication and characterization processes, the demonstrated system can be used for studying higher-order exceptional points and for exotic light manipulation.

Physical Review A

In the past few decades, many works have been devoted to the study of exceptional points (EPs), i.e., exotic degeneracies of non-Hermitian systems. The usual approach in those studies involves the introduction of a phenomenological effective non-Hermitian Hamiltonian (NHH), where the gain and losses are incorporated as the imaginary frequencies of fields and from which the Hamiltonian EPs (HEPs) are derived. Although this approach can provide valid equations of motion for the fields in the classical limit, its application in the derivation of EPs in the quantum regime is questionable. Recently, a framework [Minganti et al., Phys. Rev. A 100, 062131 (2019)], which allows one to determine quantum EPs from a Liouvillian EP (LEP), rather than from an NHH, has been proposed. Compared to the NHHs, a Liouvillian naturally includes quantum noise effects via quantum-jump terms, thus allowing one to consistently determine its EPs purely in the quantum regime. In this work we study a non-Hermitian system consisting of coupled cavities with unbalanced gain and losses, where the gain is far from saturation, i.e, the system is assumed to be linear. We apply both formalisms, based on an NHH and a Liouvillian within the Scully-Lamb laser theory, to determine and compare the corresponding HEPs and LEPs in the semiclassical and quantum regimes. Our results indicate that, although the overall spectral properties of the NHH and the corresponding Liouvillian for a given system can differ substantially, their LEPs and HEPs occur for the same combination of system parameters.

Journal of Physics: Conference Series, 2021

We review some recent work on the occurrence of coalescing eigenstates at exceptional points in non-Hermitian systems and their influence on physical quantities. We particularly focus on quantum dynamics near exceptional points in open quantum systems, which are described by an outwardly Hermitian Hamiltonian that gives rise to a non-Hermitian effective description after one projects out the environmental component of the system. We classify the exceptional points into two categories: those at which two or more resonance states coalesce and those at which at least one resonance and the partnering anti-resonance coalesce (possibly including virtual states as well), and we introduce several simple models to explore the dynamics for both of these types. In the latter case of coalescing resonance and anti-resonance states, we show that the presence of the continuum threshold plays a strong role in shaping the dynamics, in addition to the exceptional point itself. We also briefly discuss...

Physical Review A

One of the most intriguing topological features of open systems is that they exhibit exceptional point (EP) singularities. Apart from the widely explored second-order EPs (EP2s), the exploration of higher-order EPs in any system requires more complex topology, which is still a challenge. Here, we encounter a third-order EP (EP3) with the simultaneous presence of multiple second-order EPs in a simple fabrication feasible gain-loss assisted trilayer optical microcavity. Using the scattering-matrix formalism, we study the simultaneous interactions between three successive coupled states via avoided-resonance-crossing (ARC) phenomena, and we identify two EP2s near two ARC regimes. Such an occurrence of two EP2s inside a closed two-dimensional parametric space associated with an unbalanced gain-loss profile leads to the functionality of a cube-root branch point, i.e., an EP3. Following an adiabatic variation of two control parameters around the embedded EP3 in the presence of two identified EP2s, we present a robust successive-state-conversion mechanism among three coupled states. The proposed scheme indeed opens up a unique platform to manipulate light in integrated photonic devices.

Physical Review A, 2021

Recently, there has been intense research in proposing and developing various methods for constructing high-order exceptional points (EPs) in dissipative systems. These EPs can possess a number of intriguing properties related to, e.g., chiral transport and enhanced sensitivity. Previous proposals to realize non-Hermitian Hamiltonians (NHHs) with high-order EPs have been mainly based on either direct construction of spatial networks of coupled modes or utilization of synthetic dimensions, e.g., of mapping spatial lattices to time or photon-number space. Both methods rely on the construction of effective NHHs describing classical or postselected quantum fields, which neglect the effects of quantum jumps, and which, thus, suffer from a scalability problem in the quantum regime, when the probability of quantum jumps increases with the number of excitations and dissipation rate. Here, by considering the full quantum dynamics of a quadratic Liouvillian superoperator, we introduce a simple and effective method for engineering NHHs with high-order quantum EPs, derived from evolution matrices of system operators moments. That is, by quantizing higher-order moments of system operators, e.g., of a quadratic two-mode system, the resulting evolution matrices can be interpreted as alternative NHHs describing, e.g., a spatial lattice of coupled resonators, where spatial sites are represented by high-order field moments in the synthetic space of field moments. Notably, such a mapping allows correct reproduction of the results of the Liouvillian dynamics, including quantum jumps. As an example, we consider a U (1)-symmetric quadratic Liouvillian describing a bimodal cavity with incoherent mode coupling, which can also possess anti-PT-symmetry, whose field moment dynamics can be mapped to an NHH governing a spatial network of coupled resonators with high-order EPs.

Advances in Mathematical Physics, 2018

Non-Hermitian quantum physics is used successfully for the description of different puzzling experimental results, which are observed in open quantum systems. Mostly, the influence of exceptional points on the dynamical properties of the system is studied. At these points, two complex eigenvalues E of the non-Hermitian Hamiltonian H coalesce . We show that also the eigenfunctions of the two states play an important role, sometimes even the dominant one. Besides exceptional points, other critical points exist in non-Hermitian quantum physics. At these points in the parameter space, the biorthogonal eigenfunctions of H become orthogonal. For illustration, we show characteristic numerical results.

Physical Review A, 2019

Exceptional points (EPs) correspond to degeneracies of open systems. These are attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, nonunitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the Lindblad master-equation approach. Thus, we suggest to define EPs via degeneracies of a Liouvillian superoperator (including the full Lindbladian term, LEPs), and we clarify the relations between HEPs and LEPs. We prove two main theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs must have essentially different properties. Theorem 2 dictates a condition under which, in the "semiclassical" limit, LEPs and HEPs recover the same properties. In particular, we show the validity of Theorem 1 studying systems which have (1) an LEP but no HEPs, and (2) both LEPs and HEPs but for shifted parameters. As for Theorem 2, (3) we show that these two types of EPs become essentially equivalent in the semiclassical limit. We introduce a series of mathematical techniques to unveil analogies and differences between the HEPs and LEPs. We analytically compare LEPs and HEPs for some quantum and semiclassical prototype models with loss and gain.

Optics Communications

Dynamical parametric encirclement around an Exceptional Point (EP) and corresponding asymmetric state transfer phenomenon have attracted considerable attention recently. In this context, beyond the reported time-asymmetric state dynamics around a second-order EP (EP2) in a two-level system, the investigation of similar state dynamics around a third-order EP (EP3) in a multi-state system, having comparably complex topology with rich physics, is lacking. Here, we report a fabrication-feasible few-mode planar optical waveguide with a customized gain-loss profile and investigate the effect of dynamical parametric encirclement around an EP3 in the presence of multiple EP2s. The cube-root branch point behavior is established in terms of successive switching between the propagation constants of the coupled modes following an adiabatic encirclement process. Now, while considering the dynamical encirclement process, the breakdown in system adiabaticity around an EP3 leads to a unique light dynamics, where we have shown the breakdown of chirality of the device.

Physical Review Letters, 2017

We show that a two-level non-Hermitian Hamiltonian with constant off-diagonal exchange elements can be analyzed exactly when the underlying exceptional point is perfectly encircled in the complex plane. The state evolution of this system is explicitly obtained in terms of an ensuing transfer matrix, even for large encirclements, regardless of adiabatic conditions. Our results clearly explain the direction-dependent nature of this process and why in the adiabatic limit its outcome is dominated by a specific eigenstate irrespective of initial conditions. Moreover, numerical simulations suggest that this mechanism can still persist in the presence of nonlinear effects. We further show that this robust process can be harnessed to realize an optical omni-polarizer: a configuration that generates a desired polarization output regardless of the input polarization state, while from the opposite direction it always produces the counterpart eigenstate.

Physical Review Letters, 2020