The role of exceptional points in quantum systems

Abstract

Exceptional points are known in the mathematical literature for many years. They are singular points at which (at least) two eigenvalues of an operator coalesce. In physics, they can be studied best when the Hamiltonian of the system is non-Hermitian. Although the points themselves can not be directly identified in physics, their strong influence onto the neighborhood can be traced. Here, the exceptional points are called mostly crossing points (of the eigenvalue trajectories) or branch points or double poles of the S matrix. In the present paper, first the mathematical basic properties of the exceptional points are discussed. Then, their role in the description of real physical quantum systems is considered (after solving the corresponding equations exactly). The Hamiltonian of these systems is non-Hermitian due to their embedding into an environment (continuum of scattering wavefunctions). Outside the energy window coupled directly to the continuum, the Hamiltonian is Hermitian but with corrections originating from the principal value integral of the coupling term via the continuum. Most interesting value of the non-Hermitian quantum physics is the phase rigidity of the eigenfunctions which varies (as function of a control parameter) between 1 (for distant non-overlapping states) and 0 (at the exceptional point where the resonance states completely overlap). This variation allows the system to incorporate environmentally induced effects. In the very neighborhood of a crossing (exceptional) point, the system can be described well by a conventional nonlinear Schrödinger equation. Here, the entanglement of the different states is large. In the regime of overlapping resonances, many eigenvalue trajectories cross or avoid crossing, and spectroscopic redistribution processes occur in the whole system. As a result, a dynamical phase transition takes place to which all states of the system contribute: a few short-lived resonance states are aligned to the scattering states of the environment by trapping the other states. The trapped resonance states are long-lived, show chaotic features, and are described well by means of statistical ensembles. Due to the alignment of a few states with the states of the environment, observable values (e.g. the transmission through the system) are enhanced. The dynamical phase transition breaks the spectroscopic relation of the short-lived and long-lived resonance states to the original individual states of the system. These results hold also for PT symmetric systems. The dynamical phase transition characteristic of non-Hermitian quantum physics, allows us to understand some experimental results which remained puzzling in the framework of conventional Hermitian quantum physics. The effects caused by the exceptional (crossing) points in physical systems allow us to manipulate them for many different applications.

Counting Majorana bound states using complex momenta

“…Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we also corroborate the fact that for systems with a chiral symmetry, these solutions can be identified with the so-called "exceptional points" (EP's) [23][24][25][26][27][28][29], where two or more eigenvalues of the complexified Hamiltonian coalesce. EP's are singular points at which the norm of at least one eigenvector vanishes, when certain real parameters appearing in the Hamiltonian are continued to complex values, and the complexified Hamiltonian becomes non-diagonalizable.…”

Localization effects induced by decoherence in superpositions of many-spin quantum states

“…However, once decoherence is considered, in the simplest two spin quantum channel, it was shown that when the effective systemenvironment interaction exceeds a given strength, it becomes impossible to perform even a simple SWAP operation [33,45]. Instead of having the expected oscillatory transfer of a state going forth and back between the two spins, an overdamped dynamics due to the appearance of a localized state appears at a critical value or exceptional point of the perturbation strength [45][46][47][48][49]. Similarly, this could be observed in a 3-spin chain.…”

Scattering states of a particle, with position-dependent mass, in a double heterojunction

“…more than a decade ago, the fact that a class of non-Hermitian Hamiltonians admit real and discrete spectrum under certain conditions, is well established [1,2]. Non-Hermitian hamiltonians having PT symmetry (P → parity, T → time reversal) form a special class in this category, as they admit real and discrete spectrum for exact PT symmetry and complex conjugate pairs of energy when this space-time symmetry is spontaneously broken, the transition occurring at the so-called exceptional point [3,4]. Naturally, numerous attempts have been made by various scientists to extend the framework of quantum mechanics into the complex domain [5,6,7].…”