arXiv subjects

Ingrid Rotter

Publications and source records attributed to Ingrid Rotter.

At least 19 recordsLinked to original sources

arXiv2018

Non-Hermitian formalism and nonlinear physics

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.

quant-ph↗

arXiv2018

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.

quant-ph↗

arXiv2017

Non-Hermitian Quantum Physics of Open Systems

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.

quant-ph↗

arXiv2017

Critical points in two-channel quantum systems

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 wavefunctions. 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 the transfer of particles through 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 points. Here, the states of the system are mixed via the environment. In the one-channel case, the resonance structure of the cross section is independent of the existence of singular points. In the two-channel case, however, new effects appear caused by coherence. An example is the enhanced conductance of an open cavity in a certain finite parameter range. It is anti-correlated with the phase rigidity of the eigenfunctions of the non-Hermitian Hamilton operator.

quant-ph↗

arXiv2017

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.

physics.bio-ph↗

arXiv2016

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.

quant-ph↗

arXiv2015

Quantum states talk via the environment

The states of an open quantum system interact ("talk") with one another via the extended environment into which the localized system is embedded. This interaction is mediated by the source term of the Schr\"odinger equation which describes the coupling between system and environment. The source term is nonlinear and causes width bifurcation and, respectively, level repulsion. It is strong only in the neighborhood of singular (exceptional) points. We provide typical results for the phase rigidity and the mixing of the biorthogonal eigenfunctions of the Hamiltonian. A completely unexpected result is that the phase rigidity approaches a value near to one (characteristic of orthogonal eigenfunctions) when width bifurcation (or level repulsion) becomes maximum. This behavior of the phase rigidity is caused exclusively by the nonlinearity of the source term of the Schr\"odinger equation. The eigenfunctions remain mixed in the set of original wavefunctions also under these critical conditions. Eventually, a dynamical phase transition occurs. This process is irreversible. It allows, among others, a physical interpretation of the well-known resonance trapping phenomenon. Our results for the eigenvalues and eigenfunctions of a non-Hermitian Hamiltonian are supported by experimental results obtained in different systems. The relation of our results for open quantum systems under critical conditions to those in optics and photonics with PT-symmetry breaking is considered. As a result, the balance between gain and loss is a very interesting general phenomenon that may occur in many different systems.

quant-ph↗

arXiv2014

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.

quant-ph↗

arXiv2014

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.

quant-ph↗

arXiv2013

Exceptional points in open and PT symmetric systems

Exceptional points (EPs) determine the dynamics of open quantum systems and cause also PT symmetry breaking in PT symmetric systems. From a mathematical point of view, this is caused by the fact that the phases of the wavefunctions (eigenfunctions of a non-Hermitian Hamiltonian) relative to one another are not rigid when an EP is approached. The system is therefore able to align with the environment to which it is coupled and, consequently, rigorous changes of the system properties may occur. We compare analytically as well as numerically the eigenvalues and eigenfunctions of a $2\times 2$ matrix that is characteristic of either open quantum systems at high level density or of PT symmetric optical lattices. In both cases, the results show clearly the influence of the environment onto the system in the neighborhood of EPs. Although the systems are very different from one another, the eigenvalues and eigenfunctions indicate the same characteristic features.

math.DS↗

arXiv2013

Open quantum systems and Dicke superradiance

We study generic features of open quantum systems embedded into a continuum of scattering wavefunctions and compare them with results discussed in optics. A dynamical phase transition may appear at high level density in a many-level system and also in a two-level system if the coupling $W$ to the environment is complex and sufficiently large. Here nonlinearities occur. When $W_{ij}$ is imaginary, two singular (exceptional) points may exist. In the parameter range between these two points, width bifurcation occurs as function of a certain external parameter. A unitary representation of the S-matrix allows to calculate the cross section for a two-level system, including at the exceptional point (double pole of the S-matrix). The results obtained for the transition of level repulsion at small (real) $W_{ij}$ to width bifurcation at large (imaginary) $W_{ij}$ show qualitatively the same features that are observed experimentally in the transition from Autler-Townes splitting to electromagnetically induced transparency in optics. Fermi's golden rule holds only below the dynamical phase transition while it passes into an anti-golden rule beyond this transition. The results are generic and can be applied to the response of a complex open quantum system to the action of an external field (environment). They may be considered as a guideline for engineering and manipulating quantum systems in such a way that they can be used for applications with special requirements.

quant-ph↗

arXiv2013

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.

quant-ph↗

arXiv2013

Enhancement of photon intensity in forced coupled quantum wells inside a semiconductor microcavity

We study numerically the photon emission from a semiconductor microcavity containing $N\ge 2$ quantum wells under the influence of a periodic external forcing. The emission is determined by the interplay between external forcing and internal interaction between the wells. While the external forcing synchronizes the periodic motion, the internal interaction destroys it. The nonlinear term of the Hamiltonian supports the synchronization. The numerical results show a jump of the photon intensity to very large values at a certain critical value of the external forcing when the number of quantum wells is not too large. We discuss the dynamics of the system across this transition.

nlin.CD↗

arXiv2012

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.

quant-ph↗

arXiv2012

Dynamical stabilization and time in open quantum systems

The meaning of time in an open quantum system is considered under the assumption that both, system and environment, are quantum mechanical objects. The Hamilton operator of the system is non-Hermitian. Its imaginary part is the time operator. As a rule, time and energy vary continuously when controlled by a parameter. At high level density, where many states avoid crossing, a dynamical phase transition takes place in the system under the influence of the environment. It causes a dynamical stabilization of the system what can be seen in many different experimental data. Due to this effect, time is bounded from below: the decay widths (inverse proportional to the lifetimes of the states) do not increase limitless. The dynamical stabilization is an irreversible process.

quant-ph↗

arXiv2011

Analysis technique for exceptional points in open quantum systems and QPT analogy for the appearance of irreversibility

We propose an analysis technique for the exceptional points (EPs) occurring in the discrete spectrum of open quantum systems (OQS), using a semi-infinite chain coupled to an endpoint impurity as a prototype. We outline our method to locate the EPs in OQS, further obtaining an eigenvalue expansion in the vicinity of the EPs that gives rise to characteristic exponents. We also report the precise number of EPs occurring in an OQS with a continuum described by a quadratic dispersion curve. In particular, the number of EPs occurring in a bare discrete Hamiltonian of dimension $n_\textrm{D}$ is given by $n_\textrm{D} (n_\textrm{D} - 1)$; if this discrete Hamiltonian is then coupled to continuum (or continua) to form an OQS, the interaction with the continuum generally produces an enlarged discrete solution space that includes a greater number of EPs, specifically $2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) [2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) - 1] $, in which $n_\textrm{C}$ is the number of (non-degenerate) continua to which the discrete sector is attached. Finally, we offer a heuristic quantum phase transition analogy for the emergence of the resonance (giving rise to irreversibility via exponential decay) in which the decay width plays the role of the order parameter; the associated critical exponent is then determined by the above eigenvalue expansion.

quant-ph↗

arXiv2010

The role of exceptional points in quantum systems

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. Most interesting value is the phase rigidity of the eigenfunctions which varies 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 an exceptional point, the system can be described well by a conventional nonlinear Schr\"odinger equation. In the regime of overlapping resonances, 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 show chaotic features. Due to the alignment of a few states with the states of the environment, observable values may be enhanced. The dynamical phase transition allows us to understand some experimental results which remained puzzling in the framework of conventional Hermitian quantum physics. The effects caused by the exceptional points in physical systems allow us to manipulate them for many different applications.

quant-ph↗

arXiv2009

Real eigenvalues in non-Hermitian Hamilton quantum physics

The dynamics of open quantum systems is determined by avoided and true crossings of eigenvalue trajectories of a non-Hermitian Hamiltonian. The phases of the eigenfunctions are not rigid so that environmentally induced spectroscopic redistribution processes may take place and a dynamical phase transition may occur. Due to the formal equivalence between the quantum mechanical Schrödinger equation and the optical wave equation in PT symmetric lattices, the dynamics of the system is determined also in this case by avoided and true crossings of eigenvalue trajectories of the non-Hermitian Hamiltonian. In contrast to the eigenvalues characterizing an open quantum system, the eigenvalues describing the PT symmetric optical lattice are real as long as the influence of the environment (lattice) onto the optical wave equation is small. In the regime of avoided level crossings, the symmetry is destroyed, the eigenvalues become complex and a dynamical phase transition occurs similar as in the open quantum system. The redistribution processes in the regime of avoided level crossings allow to design systems with desired properties in a broad parameter range.

quant-ph↗