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I. Rotter

Publications and source records attributed to I. Rotter.

At least 19 recordsLinked to original sources
arXiv2015

A review of progress in the physics of open quantum systems: theory and experiment

This Report explores recent advances in our understanding of the physics of open quantum systems (OQSs) which consist of some localized region that is coupled to an external environment. Examples of such systems may be found in numerous areas of physics including mesoscopic physics that provides the main focus of this review. We provide a detailed discussion of the behavior of OQSs in terms of the projection-operator formalism, according to which the system under study is considered to be comprised of a localized region ($Q$), embedded into a well-defined environment ($P$) of scattering wavefunctions (with $Q+P=1$). The $Q$ subspace must be treated using the concepts of non-Hermitian physics, and of particular interest here is: the capacity of the environment to mediate a coupling between the different states of $Q$; the role played by the presence of exceptional points (EPs) in the spectra of OQSs; the influence of EPs on the rigidity of the wavefunction phases, and; the ability of EPs to initiate a dynamical phase transition (DPT). DPTs occur when the quantum dynamics of the open system causes transitions between non-analytically connected states, as a function of some external control parameter. In addition to discussing experiments on mesoscopic quantum point contacts, we also review manifestations of DPTs in mesoscopic devices and other systems. Other possible manifestations of this phenomenon are presented. From these discussions a generic picture of OQSs emerges in which the environmentally-mediated coupling between different quantum states plays a critical role in governing the system behavior.

quant-ph↗
arXiv2015

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.

quant-ph↗
arXiv2007

Correlated behavior of conductance and phase rigidity in the transition from the weak-coupling to the strong-coupling regime

We study the transmission through different small systems as a function of the coupling strength $v$ to the two attached leads. The leads are identical with only one propagating mode $ξ^E_C$ in each of them. Besides the conductance $G$, we calculate the phase rigidity $ρ$ of the scattering wave function $Ψ^E_C$ in the interior of the system. Most interesting results are obtained in the regime of strongly overlapping resonance states where the crossover from staying to traveling modes takes place. The crossover is characterized by collective effects. Here, the conductance is plateau-like enhanced in some energy regions of finite length while corridors with zero transmission (total reflection) appear in other energy regions. This transmission picture depends only weakly on the spectrum of the closed system. It is caused by the alignment of some resonance states of the system with the propagating modes $ξ^E_C$ in the leads. The alignment of resonance states takes place stepwise by resonance trapping, i.e. it is accompanied by the decoupling of other resonance states from the continuum of propagating modes. This process is quantitatively described by the phase rigidity $ρ$ of the scattering wave function. Averaged over energy in the considered energy window, $ $ is correlated with $1-<ρ>$. In the regime of strong coupling, only two short-lived resonance states survive each aligned with one of the channel wave functions $ξ^E_C$. They may be identified with traveling modes through the system. The remaining $M-2$ trapped narrow resonance states are well separated from one another.

cond-mat.other↗
arXiv2006

Phase rigidity and avoided level crossings in the complex energy plane

We consider the effective Hamiltonian of an open quantum system, its biorthogonal eigenfunctions $ϕ_λ$ and define the value $r_λ= (ϕ_λ|ϕ_λ)/<ϕ_λ|ϕ_λ>$ that characterizes the phase rigidity of the eigenfunctions $ϕ_λ$. In the scenario with avoided level crossings, $r_λ$ varies between 1 and 0 due to the mutual influence of neighboring resonances. The variation of $r_λ$ may be considered as an internal property of an {\it open} quantum system. In the literature, the phase rigidity $ρ$ of the scattering wave function $Ψ^E_C$ is considered. Since $Ψ^E_C$ can be represented in the interior of the system by the $ϕ_λ$, the phase rigidity $ρ$ of the $Ψ^E_C$ is related to the $r_λ$ and therefore also to the mutual influence of neighboring resonances. As a consequence, the reduction of the phase rigidity $ρ$ to values smaller than 1 should be considered, at least partly, as an internal property of an open quantum system in the overlapping regime. The relation to measurable values such as the transmission through a quantum dot, follows from the fact that the transmission is, in any case, resonant with respect to the effective Hamiltonian. We illustrate the relation between phase rigidity $ρ$ and transmission numerically for small open cavities.

quant-ph↗
arXiv2006

Spectroscopic properties of large open quantum-chaotic cavities with and without separated time scales

The spectroscopic properties of an open large Bunimovich cavity are studied numerically in the framework of the effective Hamiltonian formalism. The cavity is opened by attaching leads to it in four different ways. In some cases, short-lived and long-lived resonance states coexist. The short-lived states cause traveling waves in the transmission while the long-lived ones generate superposed fluctuations. The traveling waves oscillate as a function of energy. They are not localized in the interior of the large chaotic cavity. In other cases, the transmission takes place via standing waves with an intensity that closely follows the profile of the resonances. In all considered cases, the phase rigidity fluctuates with energy. It is mostly near to its maximum value and agrees well with the theoretical value for the two-channel case. As shown in the foregoing paper \cite{1}, all cases are described well by the Poisson kernel when the calculation is restricted to an energy region in which the average $S$ matrix is (nearly) constant.

cond-mat.mes-hall↗
arXiv2004

Influence of branch points in the complex plane on the transmission through double quantum dots

We consider single-channel transmission through a double quantum dot system consisting of two single dots that are connected by a wire and coupled each to one lead. The system is described in the framework of the S-matrix theory by using the effective Hamiltonian of the open quantum system. It consists of the Hamiltonian of the closed system (without attached leads) and a term that accounts for the coupling of the states via the continuum of propagating modes in the leads. This model allows to study the physical meaning of branch points in the complex plane. They are points of coalesced eigenvalues and separate the two scenarios with avoided level crossings and without any crossings in the complex plane. They influence strongly the features of transmission through double quantum dots.

quant-ph↗
arXiv2004

S-matrix theory of single-channel ballistic transport through coupled quantum dots

We consider single-channel transmission through a double quantum dot system that consists of two single dots coupled by a wire of finite length L. In order to explain the numerically obtained results for a realistic double dot system we explore a simple model. It consists, as the realistic system, of two dots connected by a wire of length L. However, each of the two single dots is characterized by a few energy levels only, and the wire is assumed to have only one level whose energy depends on L. The transmission is described by using S-matrix theory. The model explains in particular the splitting of the resonant transmission peaks and the origin of the transmission zeros. The latter are independent of the length of the wire. When the transmission zeros of the single dots are of first order and both single dots are identical, those of the double dot are of second order. First-order transmission zeros cause phase jumps of the transmission amplitude by $π$, while there are no phase jumps related to second-order transmission zeros. In this latter case, a phase jump appears due to a resonance state whose decay width vanishes when crossing the energy of the transmission zero.

cond-mat.other↗
arXiv2003

Fano resonances in the overlapping regime

The line shape of resonances in the overlapping regime is studied by using the eigenvalues and eigenfunctions of the effective Hamiltonian of an open quantum system. A generalized expression $\tilde q_k(E)$ for the Fano parameter of the resonance state $k$ is derived that contains the interaction of the state $k$ with neighboured states $l\ne k$ via the continuum. It is energy dependent since the coupling coefficients between the state $k$ and the continuum show a resonance-like behaviour at the energies of the neighboured states $l\ne k$. Under certain conditions, the energy dependent $\tilde q_k(E)$ are equivalent to the generalized complex energy independent Fano parameters that are introduced by Kobayashi et al. in analyzing experimental data. Long-lived states appear mostly isolated from one another in the cross section, also when they are overlapped by short-lived resonance states. The $\tilde q_k(E)$ of narrow resonances allow therefore to study the complicated interplay between different time scales in the regime of overlapping resonance states by controlling them as a function of an external parameter.

quant-ph↗
arXiv2003

Effective Hamiltonian and unitarity of the S matrix

The properties of open quantum systems are described well by an effective Hamiltonian ${\cal H}$ that consists of two parts: the Hamiltonian $H$ of the closed system with discrete eigenstates and the coupling matrix $W$ between discrete states and continuum. The eigenvalues of ${\cal H}$ determine the poles of the $S$ matrix. The coupling matrix elements $\tilde W_k^{cc'}$ between the eigenstates $k$ of ${\cal H}$ and the continuum may be very different from the coupling matrix elements $W_k^{cc'}$ between the eigenstates of $H$ and the continuum. Due to the unitarity of the $S$ matrix, the $\TW_k^{cc'}$ depend on energy in a non-trivial manner, that conflicts with the assumptions of some approaches to reactions in the overlapping regime. Explicit expressions for the wave functions of the resonance states and for their phases in the neighbourhood of, respectively, avoided level crossings in the complex plane and double poles of the $S$ matrix are given.

quant-ph↗
arXiv2003

Dynamics of open quantum systems

The coupling between the states of a system and the continuum into which it is embedded, induces correlations that are especially large in the short time scale. These correlations cannot be calculated by using a statistical or perturbational approach. They are, however, involved in an approach describing structure and reaction aspects in a unified manner. Such a model is the SMEC (shell model embedded in the continuum). Some characteristic results obtained from SMEC as well as some aspects of the correlations induced by the coupling to the continuum are discussed.

nucl-th↗
arXiv2002

Exceptional points and double poles of the S matrix

Exceptional points and double poles of the S matrix are both characterized by the coalescence of a pair of eigenvalues. In the first case, the coalescence causes a defect of the Hilbert space. In the second case, this is not so as shown in prevoius papers. Mathematically, the reason for this difference is the bi-orthogonality of the eigenfunctions of a non-Hermitian operator that is ignored in the first case. The consequences for the topological structure of the Hilbert space are studied and compared with existing experimental data.

quant-ph↗
arXiv2002

Shot noise and transport in small quantum cavities with large openings

We present a dynamical analysis of the transport through small quantum cavities with large openings. The systematic suppression of shot noise is used to distinguish direct, deterministic from indirect, indeterministic transport processes. The analysis is based on quantum mechanical calculations of $S$ matrices and their poles for quantum billiards with convex boundaries of different shape and two open channels in each of the two attached leads. Direct processes are supported when special states couple strongly to the leads, and can result in deterministic transport as signified by a striking system-specific suppression of shot noise.

cond-mat.mes-hall↗
arXiv2001

Conductance of Open Quantum Billiards and Classical Trajectories

We analyse the transport phenomena of 2D quantum billiards with convex boundary of different shape. The quantum mechanical analysis is performed by means of the poles of the S-matrix while the classical analysis is based on the motion of a free particle inside the cavity along trajectories with a different number of bounces at the boundary. The value of the conductance depends on the manner the leads are attached to the cavity. The Fourier transform of the transmission amplitudes is compared with the length of the classical paths. There is good agreement between classical and quantum mechanical results when the conductance is achieved mainly by special short-lived states such as whispering gallery modes (WGM) and bouncing ball modes (BBM). In these cases, also the localization of the wave functions agrees with the picture of the classical paths. The S-matrix is calculated classically and compared with the transmission coefficients of the quantum mechanical calculations for five modes in each lead. The number of modes coupled to the special states is effectively reduced.

cond-mat.mes-hall↗
arXiv2001

Correlations in quantum systems and branch points in the complex plane

Branch points in the complex plane are responsible for avoided level crossings in closed and open quantum systems. They create not only an exchange of the wave functions but also a mixing of the states of a quantum system at high level density. The influence of branch points in the complex plane on the low-lying states of the system is small.

quant-ph↗
arXiv2001

Dynamics of quantum systems

A relation between the eigenvalues of an effective Hamilton operator and the poles of the $S$ matrix is derived which holds for isolated as well as for overlapping resonance states. The system may be a many-particle quantum system with two-body forces between the constituents or it may be a quantum billiard without any two-body forces. Avoided crossings of discrete states as well as of resonance states are traced back to the existence of branch points in the complex plane. Under certain conditions, these branch points appear as double poles of the $S$ matrix. They influence the dynamics of open as well as of closed quantum systems. The dynamics of the two-level system is studied in detail analytically as well as numerically.

quant-ph↗
arXiv2001

Whispering gallery modes in open quantum billiards

The poles of the S-matrix and the wave functions of open 2D quantum billiards with convex boundary of different shape are calculated by the method of complex scaling. Two leads are attached to the cavities. The conductance of the cavities is calculated at energies with one, two and three open channels in each lead. Bands of overlapping resonance states appear which are localized along the convex boundary of the cavities and contribute coherently to the conductance. These bands correspond to the whispering gallery modes appearing in the classical calculations.

cond-mat.mes-hall↗
arXiv2000

Branch points in the complex plane and information loss in quantum systems at high level density

The mechanism of avoided level crossings in quantum systems is studied. It is traced back to the existence of branch points in the complex plane which influence the properties of resonance states as well as of discrete states. An avoided level crossing of two states causes not only an exchange of the two wave functions but, above all, correlations between them. The correlations play an important role at high level density since they cause the loss of information on the individual properties of the states.

quant-ph↗