Dynamics and statistics of unstable quantum states

Соколов, В. В.; Zelevinsky, V. G.

doi:10.1016/0375-9474(89)90558-7

Cited by 280 publications

(360 citation statements)

References 30 publications

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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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Section: Introductionmentioning

confidence: 99%

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

2017

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“…n,λ is taken over the internal states n and all open scattering channels λ. For example, when the coupling constant (5) exceeds the critical value g = 1 and the number of scattering channels is small compared to the total number of states, a counterintuitive shrinking of the widths of most resonances with increasing coupling is observed [9,10,11]. For each attached scattering channel only one of the resonance widths grows further with the coupling g. The resulting redistribution of S-matrix poles was coined resonance trapping.…”

Section: Introductionmentioning

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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“…As to the parental discrete energy levels, they are converted into the resonances with finite lifetimes, since the original closed system becomes open (unstable). Such resonances manifest themselves in the energy-dependent S matrix as its poles in the complex energy plane, and can be analytically described as the complex eigenvalues of an effective non-Hermitian Hamiltonian [12][13][14]. Notably, the corresponding eigenfunctions are not orthogonal in the conventional sense but rather form a biorthogonal system.…”

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confidence: 99%

“…To this end let us stress that statis- [25] or system of randomly interacting fermions [26]. As to the theoretical framework, it mainly relies on studying the relevant non-Hermitian RMT [13,14], understanding of which has substantially improved over the last two decades; see, e.g., Ref.[27] and references therein. Note that the spatial properties related to the associated bi-orthogonal eigenvectors are known to a much lesser extent [18,[28][29][30].…”

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confidence: 99%

Statistics of Resonance Width Shifts as a Signature of Eigenfunction Nonorthogonality

2012

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We consider an open (scattering) quantum system under the action of a perturbation of its closed counterpart. It is demonstrated that the resulting shift of resonance widths is a sensitive indicator of the nonorthogonality of resonance wavefunctions, being zero only if those were orthogonal. Focusing further on chaotic systems, we employ random matrix theory to introduce a new type of parametric statistics in open systems, and derive the distribution of the resonance width shifts in the regime of weak coupling to the continuum. PACS numbers: 05.45.Mt, 03.65.Nk, 05.60.Gg The classical question of how energy levels of a quantum system get shifted under the action of a perturbation kept attracting renewed attention during the last two decades, mostly due to the established universality of such a parametric motion for systems with chaotic dynamics or intrinsic disorder [1,2]. In particular, the distributions and correlation functions of parametric derivatives of energy levels ("level velocities") [1][2][3] and their second derivatives ("level curvatures") [4] were found explicitly using the methods of random matrix theory (RMT) [5], and also verified, e.g., in microwave billiard experiments [6]. The other reason for such an interest is the recent development of the fidelity concept as the measure of the susceptibility of internal dynamics to perturbations [7].Experimentally, the energy levels are mostly accessible by means of a scattering setup [8]. From such a viewpoint, parametric dependencies of scattering characteristics, like phase shifts and time delays [9], conductances [10] and S matrix elements [11] were under intensive study. As to the parental discrete energy levels, they are converted into the resonances with finite lifetimes, since the original closed system becomes open (unstable). Such resonances manifest themselves in the energy-dependent S matrix as its poles in the complex energy plane, and can be analytically described as the complex eigenvalues of an effective non-Hermitian Hamiltonian [12][13][14]. Notably, the corresponding eigenfunctions are not orthogonal in the conventional sense but rather form a biorthogonal system. Their nonorthogonality plays an important role in many applications, e.g., describing interference in neutral kaon systems [15], influencing branching ratios of nuclear cross sections [16], and yielding excess noise in open laser resonators [17,18]. It also features in decay laws of quantum chaotic systems [19] and in dissipative quantum chaotic maps [20].In such a context the question of parametric motion of resonances and associated resonance states in open systems arises very naturally, but to the best of our knowledge has never been properly addressed. Our goal here is to begin filling in that gap by considering universal statistics of the shifts in the resonance widths under a generic perturbation in chaotic systems. In particular, we will demonstrate that such shifts are a clear manifestation of eigenstate nonorthogonality, thus providing a promising way to probe this ...

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Dynamics and statistics of unstable quantum states

Cited by 280 publications

References 30 publications

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

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

Effective coupling for open billiards

Statistics of Resonance Width Shifts as a Signature of Eigenfunction Nonorthogonality

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