Avoided level crossings in open quantum systems

At high level density, two states avoid usually crossing at the critical value $a_{\rm cr}$ of the parameter $a$ by which the system is controlled. The wavefunctions of the two states are mixed in a finite parameter range around $a_{\rm cr}$. This holds true for discrete states as well as for narrow resonance states which are coupled via the environment of scattering wavefunctions. We study the influence of avoided level crossings onto four overlapping complex eigenvalues of a symmetric non-Hermitian operator. The mixing of the two wavefunctions around $a_{\rm cr}$ is simulated, in each case, by assuming a Gaussian distribution around $a_{\rm cr}$. At high level density, the Gaussian distributions related to avoided crossings of different levels may overlap. Here, new effects arise, especially from the imaginary part of the coupling term via the environment. The results show, moreover, the influence of symmetries onto the multi-level avoided crossing phenomenon.Comment: Contribution to the Special Issue "Quantum Physics with Non-Hermitian Operators: Theory and Experiment", Fortschritte der Physik - Progress of Physics 201

Clustering of exceptional points and dynamical phase transitions

The eigenvalues of a non-Hermitian Hamilton operator are complex and provide not only the energies but also the lifetimes of the states of the system. They show a non-analytical behavior at singular (exceptional) points (EPs). The eigenfunctions 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. When the parameters are tuned to the point of maximum width bifurcation, the phase rigidity suddenly increases up to its maximum value. This means that the eigenfunctions become almost orthogonal at this point. This unexpected result is very robust as shown by numerical results for different classes of systems. Physically, it causes an irreversible stabilization of the system by creating local structures that can be described well by a Hermitian Hamilton operator. Interesting non-trivial features of open quantum systems appear in the parameter range in which a clustering of EPs causes a dynamical phase transition.Comment: A few improvements; 2 references added; 28 pages; 7 figure

Resonances in open quantum systems

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 {\it 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 non-Hermitian Hamilton operator which are embedded in one common continuum and 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.Comment: Title of the paper is changed. The Introduction is broaden. The difference of the non-Hermitian formalism for the description of open quantum systems in our paper to the description of PT-symmetric systems is underlined. Paper published: Phys.Rev.A 95, 022117 (2017

Open quantum systems with loss and gain

We consider different properties of small open quantum systems coupled to an environment and described by a non-Hermitian Hamilton operator. Of special interest is the non-analytical behavior of the eigenvalues in the vicinity of singular points, the so-called exceptional points (EPs), at which the eigenvalues of two states coalesce and the corresponding eigenfunctions are linearly dependent from one another. The phases of the eigenfunctions are not rigid in approaching an EP and providing therewith the possibility to put information from the environment into the system. All characteristic properties of non-Hermitian quantum systems hold true not only for natural open quantum systems that suffer loss due to their embedding into the continuum of scattering wavefunctions. They appear also in systems coupled to different layers some of which provide gain to the system. Thereby gain and loss, respectively, may be fixed inside every layer, i.e. characteristic of it.Comment: Correction of a few misprints; addition of a few new references; paper will appear in a Special Issue of International Journal of Theoretical Physics related to several conferences on PHHQP in 2014; accepted 10 October 2014; available: DOI 10.1007/s10773-014-2375-

Gain and loss in open quantum systems

Photosynthesis is the basic process used by plants to convert light energy in reaction centers into chemical energy. The high efficiency of this process is not yet understood today. Using the formalism for the description of open quantum systems by means of a non-Hermitian Hamilton operator, we consider initially the interplay of gain (acceptor) and loss (donor). Near singular points it causes fluctuations of the cross section which appear without any excitation of internal degrees of freedom of the system. This process occurs therefore very quickly and with high efficiency. We then consider the excitation of resonance states of the system by means of these fluctuations. This second step of the whole process takes place much slower than the first one, because it involves the excitation of internal degrees of freedom of the system. The two-step process as a whole is highly efficient and the decay is bi-exponential. We provide, if possible, the results of analytical studies, otherwise characteristic numerical results. The similarities of the obtained results to light harvesting in photosynthetic organisms are discussed.Comment: Quality of figures is improved; a few improvements in the text. Paper is published in Phys. Rev.

Width bifurcation and dynamical phase transitions in open quantum systems

The states of an open quantum system are coupled via the environment of scattering wavefunctions. The complex coupling coefficients $\omega$ between system and environment arise from the principal value integral and the residuum. At high level density where the resonance states overlap, the dynamics of the system is determined by exceptional points. At these points, the eigenvalues of two states are equal and the corresponding eigenfunctions are linearly dependent. It is shown in the present paper that Im$(\omega)$ and Re$(\omega)$ influence the system properties differently in the surrounding of exceptional points. Controlling the system by a parameter, the eigenvalues avoid crossing in energy near an exceptional point under the influence of Re$(\omega)$ in a similar manner as it is well known from discrete states. Im$(\omega)$ however leads to width bifurcation and finally (when the system is coupled to one channel, i.e. to a common continuum of scattering wavefunctions), to a splitting of the system into two parts with different characteristic time scales. Physically, the system is stabilized by this splitting since the lifetimes of most ($N-1$) states are longer than before while that of only one state is shorter. In the cross section the short-lived state appears as a background term in high-resolution experiments. The wavefunctions of the long-lived states are mixed in those of the original ones in a comparably large parameter range. Numerical results for the eigenvalues and eigenfunctions are shown for $N=2, ~4$ and 10 states coupled mostly to 1 channel.Comment: 31 pages, 11 figure