Dynamically Encircling Exceptional Points: Exact Evolution and Polarization State Conversion

Abstract

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 eigenstateirrespective 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 omnipolarizer: 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. DOI: 10.1103/PhysRevLett.118.093002 Understanding the dynamics of time-dependent Hamiltonians is key in explaining a wide range of processes in many and diverse physical settings [1]. This ubiquitous class of problems is of significance since it allows one to tailor the evolution of a Hamiltonian towards certain outcomes. If a system is conservative or Hermitian, a cyclic adiabatic change in a multiparameter space can often lead to surprising results such as, for example, the emergence of gauge-invariant geometric phases, as first shown by Berry [2]. Of particular interest is the case where eigenvalue degeneracies are enclosed within the parameter loop. In this latter scenario, the geometric phase is robust against perturbations in the control path since it is related to the flux generated from the degeneracies that act as topological sources.While the Berry phase represents an intuitive and powerful unifying notion, it is by nature based on the adiabatic theorem [3]. Quite recently, a series of studies have critically reexamined these aspects in non-Hermitian environments where it was found that the system behavior can be significantly modified around degeneracies, better known as exceptional points (EPs) [4]. As opposed to conservative systems, in non-Hermitian arrangements both the eigenvalues and the corresponding eigenvectors tend to coalesce at an EP (while unfolding associated vectors of the Jordan form) [5]. In the last few years a number of intriguing possibilities have been realized in structures supporting EPs, including loss-induced transparency [6], single-mode lasing [7], band merging [8], asymmetric diffraction [9], and unidirectional invisibility [10], to mention a few. In other studies, the topological properties associated with the quasistatic encirclement of an EP were also investigated. Under such stationary conditions it was found that the instantaneous eigenstates swap with each other at the end of the parameter cycle with only one acquiring a geometric phase [11,12]. This behavior, attributed to the branch point character of the degeneracy that cause...

Floquet exceptional points and chirality in non-Hermitian Hamiltonians

“…For example, for a two-state system (N = 2) the flip of eigenvalues and corresponding eigenvectors is known to occur whenever the path in parameter space, described by a complex parameter R = R(t), encircles an EP, i.e. a point R = R 0 at which the stationary Hamiltonian H shows a static EP [16,17,18,27,40,41,42,44]. Here we avoid such a case and assume that each eigenvalue σ n (t) and corresponding eigenvector e (n) (t) are single-valued functions of time over each cycle, i.e.…”

“…They have been predicted and observed in a wide variety of physical systems, including atomic and molecular systems [13,14,15], microwave cavities and waveguides [16,17,18], electronic circuits [19], optical structures [20,21], Bose-Einstein condensates [22,23], acoustic cavities [24], non-Hermitian Bose-Hubbard models [25], exciton-polariton billiards [26], opto-mechanical systems [27] and many others. Besides of their theoretical interest, EPs can find important applications, for example in the design and realization of unidirectionally invisible media [28,29,30,31,32], for asymmetric mode switching [18,33] and topological energy transport [27], for the design of novel laser devices [34,35,36,37,38], for optical sensing [39] and polarization mode conversion [40]. The dynamical properties associated to the encircling of an EP and the chirality of EPs arising from breakdown of the adiabatic theorem have received a great attention in recent years [16,17,18,27,40,41,42,43,44,45].…”

“…Recent experiments [18,27] demonstrated chirality of EP cycling, originally predicted in Refs. [41,42], and raised a great interest owing to potential applications of EPs to topological energy transport [27], asymmetric mode switching [18,33] and polarization control of light [40]. Asymmetric non-adiabatic transitions observed when the EP is dynamically encircled are deeply rooted in the non-Hermitian nature of the underlying dynamics and have been explained in terms of the Stokes phenomenon of asymptotics [42], the stability loss delay phenomenon [44], and the asymmetric nature of transitions induced by non-Hermitian perturbations [47].…”

“…The dynamical properties associated to the encircling of an EP and the chirality of EPs arising from breakdown of the adiabatic theorem have received a great attention in recent years [16,17,18,27,40,41,42,43,44,45]. A remarkable property observed when encircling an EP is a topological-robust state-flip for quasi-static cycling [15,17,46], and the chiral behavior associated to breakdown of adiabaticity for dynamical circling [18,27,40,41,42,44]. In the former case state flip arises from to the branch point character of the degeneracy that causes a gradual transition between the intersecting complex Riemann sheets.…”

“…Besides of their theoretical interest, EPs can find important applications, for example in the design and realization of unidirectionally invisible media [28,29,30,31,32], for asymmetric mode switching [18,33] and topological energy transport [27], for the design of novel laser devices [34,35,36,37,38], for optical sensing [39] and polarization mode conversion [40]. The dynamical properties associated to the encircling of an EP and the chirality of EPs arising from breakdown of the adiabatic theorem have received a great attention in recent years [16,17,18,27,40,41,42,43,44,45]. A remarkable property observed when encircling an EP is a topological-robust state-flip for quasi-static cycling [15,17,46], and the chiral behavior associated to breakdown of adiabaticity for dynamical circling [18,27,40,41,42,44].…”