Signatures of three coalescing eigenfunctions

Demange, Gilles; Graefe, Eva-Maria

doi:10.1088/1751-8113/45/2/025303

Cited by 199 publications

(215 citation statements)

References 58 publications

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“…Instead, the system can only be reduced to Jordan block form; as such Eq. (34) is complemented by [55,57] …”

Section: B Diagonalization Scheme For the Generalized Eigenvalue Promentioning

confidence: 99%

“…A given Hamiltonian is no longer diagonalizable at the exceptional point, but instead can only be transformed into a matrix containing at least one Jordan block [19,33,[55][56][57]. The dimension of the Jordan block is given by N , the number of coalescing levels.…”

Section: Introductionmentioning

confidence: 99%

“…The dimension of the Jordan block is given by N , the number of coalescing levels. Following the convention in the literature, we refer to an EP involving the coalescence of N levels as an EPN [56][57][58]; however, in the case of two coalescing levels (EP2) considered in this paper we find it useful to introduce two further subcategories, following the convention introduced in Ref. [52].…”

Section: Introductionmentioning

confidence: 99%

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Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Garmon

Ordóñez

2017

Journal of Mathematical Physics

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It has been reported in the literature that the survival probability P (t) near an exceptional point where two eigenstates coalesce should generally exhibit an evolution P (t) ∼ t 2 e −Γt , in which Γ is the decay rate of the coalesced eigenstate; this has been verified in a microwave billiard experiment [B. Dietz, et al., Phys. Rev. E 75, 027201 (2007)]. However, the heuristic effective Hamiltonian that is usually employed to obtain this result ignores the possible influence of the continuum threshold on the dynamics. By contrast, in this work we employ an analytical approach starting from the microscopic Hamiltonian representing two simple models in order to show that the continuum threshold has a strong influence on the dynamics near exceptional points in a variety of circumstances. To report our results, we divide the exceptional points in Hermitian open quantum systems into two cases: at an EP2A two virtual bound states coalesce before forming a resonance, anti-resonance pair with complex conjugate eigenvalues, while at an EP2B two resonances coalesce before forming two different resonances. For the EP2B, which is the case studied in the microwave billiard experiment, we verify the survival probability exhibits the previously reported modified exponential decay on intermediate timescales, but this is replaced with an inverse power law on very long timescales. Meanwhile, for the EP2A the influence from the continuum threshold is so strong that the evolution is non-exponential on all timescales and the heuristic approach fails completely. When the EP2A appears very near the threshold we obtain the novel evolution P (t) ∼ 1 − C1 √ t on intermediate timescales, while further away the parabolic decay (Zeno dynamics) on short timescales is enhanced.

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“…Instead, the system can only be reduced to Jordan block form; as such Eq. (34) is complemented by [55,57] …”

Section: B Diagonalization Scheme For the Generalized Eigenvalue Promentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Garmon

Ordóñez

2017

Journal of Mathematical Physics

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“…The resonances were metastable states associated with predissociation or autoionization phenomena and with leaking modes in waveguides [46,47]. A theoretical quest for multiple EP s [48], or for high order EP s in dissipative physical systems is pursued [49,50,51,52], specifically in the spectra of atoms in external fields [53].…”

Section: Discussionmentioning

confidence: 99%

Parameter estimation in atomic spectroscopy using exceptional points

2016

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Abstract. We suggest a method for accurate parameter estimation of atomic systems, employing the special properties of the exceptional points. The non-hermitian degeneracies at the exceptional points emerge from the description of the spontaneous emission of atomic system in the framework of an open quantum system, resulting in a non hermitian quantum master equation. The method is demonstrated for the atomic spectrum of S → P transitions of 85 Rb and 40 Ca + .

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“…The occurrences of EPs have been experimentally and numerically demonstrated in a variety of systems [35][36][37][38][39][40][41][42]. If the system with two coalescing eigenvalues at an EP is subjected to a perturbation of strength , the magnitude of the resulting eigenvalue splitting is typically proportional to 1/2 [27,32,[43][44][45][46]. For weak perturbation of strength 1, the 1/2 -order splitting is much greater than the -order splitting of normal degenerate eigenmodes.…”

Section: Introductionmentioning

confidence: 99%

Large Sagnac frequency splitting in a ring resonator operating at an exceptional point

Sunada

2017

Phys. Rev. A

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In rotating ring resonators, resonant frequencies are split because of the Sagnac effect. The rotation sensitivity of the frequency splitting characterizes the sensitivity of resonator-based optical gyroscopes. In this paper, it is shown that the sensitivity of frequency splitting can be significantly enhanced in a ring resonator operating at an exceptional point (EP), which is a non-Hermitian degeneracy where two eigenvalues and the corresponding eigenmodes coalesce. As an example, a ring resonator with a periodic structure is proposed and theoretically and numerically studied. It is numerically demonstrated that in the resonator operating near an EP, the rotation-induced frequency splitting can be more than two orders greater than that in conventional ring resonators. In addition, this paper discusses the influence of the resonator loss on the measurement sensitivity of the frequency splitting and a method of rotation detection based on rotation-induced changes of eigenmodes near an EP.

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Signatures of three coalescing eigenfunctions

Cited by 199 publications

References 58 publications

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Parameter estimation in atomic spectroscopy using exceptional points

Large Sagnac frequency splitting in a ring resonator operating at an exceptional point

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