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Do we need non-Hermitian Hamilton operators?

I. Rotter

Published Mar 31, 2018• 0 citations
OpenAlex

Abstract

The aim of the paper is to study the question whether or not equilibrium states exist in open quantum systems that are embedded in at least two environments and are described by a non-Hermitian Hamilton operator $\cal H$. The eigenfunctions of $\cal H$ contain the influence of exceptional points (EPs) as well as that of external mixing (EM) of the states via the environment. As a result, equilibrium states exist (far from EPs). They are different from those of the corresponding closed system. Their wavefunctions are orthogonal although the Hamiltonian is non-Hermitian.

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The aim of the paper is to study the question whether or not equilibrium states exist in open quantum systems that are embedded in at least two environments and are described by a non-Hermitian Hamilton operator H\cal HH. The eigenfunctions of H\cal HH contain the influence of exceptional points (EPs) as well as that of external mixing (EM) of the states via the environment. As a result, equilibrium states exist (far from EPs). They are different from those of the corresponding closed system. Their wavefunctions are orthogonal although the Hamiltonian is non-Hermitian.
The aim of this paper is to study the question of whether or not equilibrium states exist in open quantum systems that are embedded in at least two environments and are described by a non-Hermitian Hamilton operator H . The eigenfunctions of H contain the influence of exceptional points (EPs) and external mixing (EM) of the states via the environment. As a result, equilibrium states exist (far from EPs). They are different from those of the corresponding closed system. Their wavefunctions are orthogonal even though the Hamiltonian is non-Hermitian.
Calculations for open quantum systems are performed usually by taking into account their embedding into one common environment, which is mostly the common continuum of scattering wave functions. Realistic quantum systems are coupled however often to more than one continuum. For example, the conductance of an open cavity needs at least two environments, namely the input and the output channel. In the present paper, we study generic features of an open quantum system coupled to two channels. We compare the results with those characteristic of a one-channel system. Of special interest is the parameter range which is influenced by singular (exceptional) points. Here, the states of the system are mixed via the environment and the dynamics is determined, in any case, by nonlinear processes. These processes restore the resonance structure of the cross section in the one-channel case, such that there is no need to consider them in calculations for concrete systems. In the two-channel case, however, they cause new effects. Most interesting is the enhanced conductance of an open cavity in this parameter range. This enhancement follows directly from the properties of the eigenstates of a non-Hermitian Hamilton operator (and is not related to the existence of a superradiant state). The study of the conductance allows, among others, an experimental test of the non-Hermitian quantum physics.
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 Ei≡Ei+iΓi/2 of the non-Hermitian Hamiltonian H coalesce (where Ei is the energy and Γi is the inverse lifetime of the state i ). We show that also the eigenfunctions Φi 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 a=acr in the parameter space, the biorthogonal eigenfunctions of H become orthogonal. For illustration, we show characteristic numerical results.
Information on quantum systems can be obtained only when they are open (or opened) in relation to a certain environment. As a matter of fact, realistic open quantum systems appear in very different shape. We sketch the theoretical description of open quantum systems by means of a projection operator formalism elaborated many years ago, and applied by now to the description of different open quantum systems. The Hamiltonian describing the open quantum system is non-Hermitian. Most studied are the eigenvalues of the non-Hermitian Hamiltonian of many-particle systems embedded in one environment. We point to the unsolved problems of this method when applied to the description of realistic many-body systems. We then underline the role played by the eigenfunctions of the non-Hermitian Hamiltonian. Very interesting results originate from the fluctuations of the eigenfunctions in systems with gain and loss of excitons. They occur with an efficiency of nearly 100\%. An example is the photosynthesis.
Open photonic systems provide a versatile platform for non-Hermitian physics, enabling control over complex spectra, transport, and light-matter interactions. Exceptional points (EPs), at which eigenvalues and eigenvectors coalesce and the governing operator becomes defective, play a central role because they combine branch-point spectral topology, nonanalytic perturbative response, and controllable eigenstate conversion. This Review provides a unified framework for EP photonics by systematically distinguishing exceptional degeneracies according to the underlying operator, spectral variable, boundary conditions, and experimentally accessible observables. We discuss Hamiltonian EPs, absorbing EPs associated with scattering zeros, real-frequency scattering-matrix and Jones-matrix EPs, Bloch and Floquet EPs, and Liouvillian EPs in open quantum systems. We review their spectral topology, static and dynamical encircling, higher-order exceptional structures, and coexistence with bound states in the continuum, together with applications in sensing, lasing, coherent absorption, directional scattering, polarization and wavefront control, nonlinear optics, optical storage, nonreciprocal photonics, and quantum photonics. We also critically assess the current limitations, practical challenges, and future perspectives of EP-based photonic technologies, with particular attention to robustness, noise, scalability, and experimentally measurable performance.
The non-Hermitian formalism is used at present in many papers for the description of open quantum systems. A special language developed in this field of physics which makes it difficult for many physicists to follow and to understand the corresponding papers. We show that the characteristic features of the non-Hermitian formalism are nothing but nonlinearities that may appear in the equations when the Hamiltonian is non-Hermitian. They are related directly to singular points (called mostly exceptional points, EPs). At low level density, they may cause counterintuitive physical results which allow us to explain some puzzling experimental results. At high level density, they determine the dynamics of the system.