Resonances in open quantum systems

Abstract

The Hamilton operator of an open quantum system is non-Hermitian. Its eigenvalues are, generally, complex and provide not only the energies but also the lifetimes of the states of the system. The states may couple via the common environment of scattering wavefunctions into which the system is embedded. This causes an external mixing (EM) of the states. Mathematically, EM is related to the existence of singular (the so-called exceptional) points (EPs). The eigenfunctions of a non-Hermitian operator are biorthogonal, in contrast to the orthogonal eigenfunctions of a Hermitian operator. A quantitative measure for the ratio between biorthogonality and orthogonality is the phase rigidity of the wavefunctions. At and near an EP, the phase rigidity takes its minimum value. The lifetimes of two nearby eigenstates of a quantum system bifurcate under the influence of an EP. At the parameter value of maximum width bifurcation, the phase rigidity approaches the value one, meaning that the two eigenfunctions become orthogonal. However, the eigenfunctions are externally mixed at this parameter value. The S-matrix and therewith the cross section do contain, in the one-channel case, almost no information on the EM of the states. The situation is completely different in the case with two (or more) channels where the resonance structure is strongly influenced by the EM of the states and interesting features of non-Hermitian quantum physics are revealed. We provide numerical results for two and three nearby eigenstates of a nonHermitian Hamilton operator which are embedded in one common continuum and are influenced by two adjoining EPs. The results are discussed. They are of interest for an experimental test of the non-Hermitian quantum physics as well as for applications.2

Spectra, current flow, and wave-function morphology in a modelPT-symmetric quantum dot with external interactions

“…Note that the phase rigidity increases between the two EPs and reaches its maximum at maximum width bifurcation. We note that we are not able to reproduce the results in [24,26,49], in which the phase rigidity almost reaches 1 between the EPs. When V I = 0 the system is Hermitian, so both the eigenvalues and eigenfunctions are real; these wave functions are shown in Fig.…”

Tunable Open‐Access Microcavities for Solid‐State Quantum Photonics and Polaritonics

“…Nevertheless, such interaction can be very important and lead to new scenarios of physics. Eleuch and Rotter systematically studied the quantum resonances in open systems, of which the Hamilton operator becomes non‐Hermitian due to the dissipative nature . The quantum states of the open system may couple via the common environment, exhibiting complex eigenvalues and biorthogonal eigenfunctions.…”

“…Eleuch and Rotter systematically studied the quantum resonances in open systems, of which the Hamilton operator becomes non-Hermitian due to the dissipative nature. [77] The quantum states of the open system may couple via the common environment, exhibiting complex eigenvalues and biorthogonal eigenfunctions. The most interesting property of the open system is the so-called exceptional point (EP) in the parameter space, a singularity at which the eigenvalues of two states coalesce, corresponding to the same eigenfunctions.…”

Bound states emerging from below the continuum in a solvablePT-symmetric discrete Schrödinger equation

“…For open systems this problem may be found clarified, e.g., in Ref. [46] where one works with the Hamiltonians describing, in general, unstable systems. For their genuine non-Hermitian Hamiltonians the spectrum of the energies is not real so that the experimentalists are allowed to speak about resonances.…”