The decay of quantum systems with a small number of open channels

Dittes, Frank-Michael

doi:10.1016/s0370-1573(00)00065-x

Cited by 191 publications

(248 citation statements)

References 149 publications

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“…In these systems, a systematic way to achieve overcritical coupling for many states is to modulate the density of states, e. g. by considering systems with band spectra [3,24].…”

Section: Discussionmentioning

confidence: 99%

Effective coupling for open billiards

2001

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We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard which is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.

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“…In these systems, a systematic way to achieve overcritical coupling for many states is to modulate the density of states, e. g. by considering systems with band spectra [3,24].…”

Section: Discussionmentioning

confidence: 99%

Effective coupling for open billiards

2001

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“…Many properties are strongly dependent on the openness of the system and the way the system interacting with the environment around it. In order to take the influences of the environment into account, the effective non-Hermitian Hamiltonian approach has been used extensively in treating open systems [1][2][3][4][5][6]. By introducing imaginary parts to the Hamiltonian to represent the physical gain and loss of the system, one can study the open systems in an consistent way by analyzing the complex eigenvalues of the effective Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss

2017

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We investigate the Anderson localization in non-Hermitian Aubry-André-Harper (AAH) models with imaginary potentials added to lattice sites to represent the physical gain and loss during the interacting processes between the system and environment. By checking the mean inverse participation ratio (MIPR) of the system, we find that different configurations of physical gain and loss have very different impacts on the localization phase transition in the system. In the case with balanced physical gain and loss added in an alternate way to the lattice sites, the critical region (in the case with p-wave superconducting pairing) and the critical value (both in the situations with and without p-wave pairing) for the Anderson localization phase transition will be significantly reduced, which implies an enhancement of the localization process. However, if the system is divided into two parts with one of them coupled to physical gain and the other coupled to the corresponding physical loss, the transition process will be impacted only in a very mild way. Besides, we also discuss the situations with imbalanced physical gain and loss and find that the existence of random imaginary potentials in the system will also affect the localization process while constant imaginary potentials will not.

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“…Due to the presence of resonances the elements of the scattering matrix show irregular fluctuations as functions of the energy of incoming waves, see [24] and references therein. The main goal of the theory of quantum chaotic scattering is to provide an adequate statistical description of such a behaviour.The most natural framework for incorporating a random matrix description of the chaotic scattering (and hence for addressing statistics of resonances) is the so-called "Heidelberg approach" suggested in the pioneering paper [9] and described in much detail in [11,12]. The starting point of that approach is a particular formulation of the scattering theory based on the notion of an "effective Hamiltonian".…”

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confidence: 99%

Random matrices close to Hermitian or unitary: overview of methods and results

Fyodorov¹,

Sommers²

2003

J. Phys. A: Math. Gen.

168

237

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Abstract. The paper discusses recent progress in understanding statistical properties of eigenvalues of (weakly) non-Hermitian and non-unitary random matrices. The first type of ensembles is of the formĴ =Ĥ − iΓ, withĤ being a large random N × N Hermitian matrix with independent entries "deformed" by a certain anti-Hermitian N × N matrix iΓ satisfying in the limit of large dimension N the condition: TrĤ 2 ∝ N TrΓ 2 . HereΓ can be either a random or just a fixed given Hermitian matrix. Ensembles of such a type withΓ ≥ 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics. Related models are used to mimic complex spectra of the Dirac operator with chemical potential in the context of Quantum Chromodynamics.Ensembles of the second type, arising naturally in scattering theory of discrete-time systems, are formed by N × N matricesÂ with complex entries such thatÂ †Â =Î −T . ForT = 0 this coincides with the Circular Unitary Ensemble, and 0 ≤T ≤Î describes deviation from unitarity. Our result amounts to answering statistically the following old question: given the singular values of a matrixÂ describe the locus of its eigenvalues.We systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non-trivial crossover from WignerDyson statistics of real/unimodular eigenvalues typical for Hermitian/unitary matrices to Ginibre statistics in the complex plane typical for ensembles with strong non-Hermiticity: < TrĤ 2 >∝< TrΓ 2 > when N → ∞. Finally we discuss (scarce) results available on eigenvector statistics for weakly nonHermitian random matrices. IntroductionAs is well-known, the statistics of highly excited bound states of closed quantum chaotic systems of quite different microscopic nature is universal. Namely, it turns out to be independent of the microscopic details when sampled on energy intervals large in comparison with the mean level spacing ∆, but smaller than the so called Thouless energy scale. The latter is related by the Heisenberg uncertainty principle to the relaxation time necessary for the classically chaotic system to reach equilibrium in phase space [1]. Moreover, the spectral correlation functions turn out to be exactly those which are provided by the theory of large random Hermitian matrices with independent, identically distributed Gaussian entries. The correspondence holds in the limit of large matrix dimension on the so-called local scale. The Random matrices close to Hermitian or unitary2 local scale is determined by the typical separation ∆ = X i − X i−1 between neighbouring eigenvalues situated around a point X, with the brackets standing for the statistical averaging. Microscopic arguments supporting the use of random matrices for describing the universal properties of quantum chaotic systems have been provided recently by several groups, based both on traditional semiclassical periodic orbit expansions [2,3] and on advanced field-theoretical methods [4,5].In paralle...

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The decay of quantum systems with a small number of open channels

Cited by 191 publications

References 149 publications

Effective coupling for open billiards

Effective coupling for open billiards

Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss

Random matrices close to Hermitian or unitary: overview of methods and results

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