Equilibrium states in open quantum systems

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.Comment: 12 page

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

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-

Nearby states in non-Hermitian quantum systems

In part I, the formalism for the description of open quantum systems (that are embedded into a common well-defined environment) by means of a non-Hermitian Hamilton operator $\ch$ is sketched. Eigenvalues and eigenfunctions are parametrically controlled. Using a 2$\times$2 model, we study the eigenfunctions of $\ch$ at and near to the singular exceptional points (EPs) at which two eigenvalues coalesce and the corresponding eigenfunctions differ from one another by only a phase. In part II, we provide the results of an analytical study for the eigenvalues of three crossing states. These crossing points are of measure zero. Then we show numerical results for the influence of a nearby ("third") state onto an EP. Since the wavefunctions of the two crossing states are mixed in a finite parameter range around an EP, three states of a physical system will never cross in one point. Instead, the wavefunctions of all three states are mixed in a finite parameter range in which the ranges of the influence of different EPs overlap. We may relate these results to dynamical phase transitions observed recently in different experimental studies. The states on both sides of the phase transition are non-analytically connected.Comment: Change of the title; Partition into two parts; Published in Eur. Phys. J. D 69, 229 (2015) and D 69, 230 (2015

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