Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems

Brink, Alec Maassen van den; Young, K.

doi:10.1088/0305-4470/34/12/308

Cited by 15 publications

(4 citation statements)

References 48 publications

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“…Since the J µ ⟩⟩ and M µ ⟩⟩ are proper eigenvectors and there are no generalized eigenvectors (Lemma 17 of [40]), the Jordan block with eigenvalue zero of Λ will be simply a D-by-D matrix of zeros. The respective transformed left and right eigenvectors, M µ ⟩⟩ = S M µ ⟩⟩ and ⟨⟨J µ = ⟨⟨J µ S −1 , will be linearly independent and orthogonal to all other basis vectors of L. Thus they are dual bases and can be made to be biorthogonal [50], i.e., such that ⟨⟨J µ M ν ⟩⟩ = δ µν . It is clear that once the transformed vectors are biorthogonal, the original ones are also: ⟨⟨J µ M ν ⟩⟩ = δ µν .…”

Section: Discussionmentioning

confidence: 99%

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Symmetries and conserved quantities in Lindblad master equations

2014

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This work is concerned with determination of the steady-state structure of time-independent Lindblad master equations, especially those possessing more than one steady state. The approach here is to treat Lindblad systems as generalizations of unitary quantum mechanics, extending the intuition of symmetries and conserved quantities to the dissipative case. We combine and apply various results to obtain an exhaustive characterization of the infinite-time behavior of Lindblad evolution, including both the structure of the infinite-time density matrix and its dependence on initial conditions. The effect of the environment in the infinite time limit can therefore be tracked exactly for arbitrary state initialization and without knowledge of dynamics at intermediate time.As a consequence, sufficient criteria for determining the steady state of a Lindblad master equation are obtained. These criteria are knowledge of the initial state, a basis for the steady-state subspace, and all conserved quantities. We give examples of two-qubit dissipation and single-mode d-photon absorption where all quantities are determined analytically. Applications of these techniques to quantum information, computation, and feedback control are discussed. Environmental/reservoir interaction features rather prominently in the design, engineering, and realization of quantum systems. Many models exist for simulating the environment, with the most prominent being the framework of GKS-L (Gorini-Kossakowsi-Sudarshan-Lindblad, or just Lindblad ) master equations [1]. Such master equations are valid only for specific Markovian environments (e.g. [2,3]). Nevertheless, Lindblad master equations continue to be implemented in a multitude of systems, lying in a nexus between quantum optics, quantum information, mesoscopic physics, and dynamical systems theory. We briefly list some notable works.While system-environment interaction in the case of cavity (circuit) quantum mechanics usually consists of optical (microwave) photon loss, recent efforts have been to design the cavity such that other forms of dissipation can be realized. This can be done in order to control the state of either the qubit [4] [38].The intractable literature on Lindblad systems begs the question of why this work is useful. While the properties of abstract Lindblad systems are garnering interest from physicists due to increasing ability to engineer previously un-physical Lindblad models, a gap in accessibility and nomenclature nevertheless remains (resonating with note 1.4 in [39]). In the spirit of bridging this gap using physical intuition, this work points out the utility of symmetries and conserved quantities from ordinary quantum mechanics in the Lindblad formalism. We answer the following questions: (1) How are symmetries and conserved quantities different in Lindblad systems when compared to unitary systems? and (2) Despite Lindblad evolution being irreversible, what information from an initial state is preserved in the infinite-time limit? In answering these questions, we ap...

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Section: Discussionmentioning

confidence: 99%

“…This result can be derived from the generalized eigenvector decomposition of exponentials of linear operators (Eq. (10.23) in [47]; see also [49,50]). It is proven here using properties of Lindblad operators and linear algebra.…”

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confidence: 93%

Symmetries and conserved quantities in Lindblad master equations

2014

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“…In contrast to level degeneracy points in Hermitian systems, the EP is associated with level coalescence, in which not only the eigenenergies but also the eigenstates become identical [13,14]. The similar mode coalescence phenomenon was discovered in classical damped oscillator systems and coined as the 'critical point' [15,16]. Many distinctive effects without Hermitian counterparts arise around the EP, such as the square root frequency dependence [8] and the nontrivial topological property resulting from the Riemann sheet structures of the EP-ended branch-cut in the complex parameter plane [17][18][19][20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 90%

Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system

2019

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The exceptional points (EPs) of non-Hermitian systems, where n different energy eigenstates merge into an identical one, have many intriguing properties that have no counterparts in Hermitian systems. In particular, the n 1  dependence of the energy level splitting on a perturbative parameter ò near an nth order EP stimulates the idea of metrology with arbitrarily high sensitivity, since the susceptibility dò 1/ n /dò diverges at the EP. Here we theoretically study the sensitivity of parameter estimation near the EPs, using the exact formalism of quantum Fisher information (QFI). The QFI formalism allows the highest sensitivity to be determined without specifying a specific measurement approach. We find that the EP bears no dramatic enhancement of the sensitivity. Instead, the coalescence of the eigenstates exactly counteracts the eigenvalue susceptibility divergence and makes the sensitivity a smooth function of the perturbative parameter. between these eigenenergies can be excited to measure the eigenvalue susceptibility. However, non-Hermitian systems are fundamentally different. Because different eigenstates of non-Hermitian systems are in general nonorthogonal and even become identical at the EP, exciting the transitions between different eigenstates near the EP to measure the eigenvalue susceptibility is infeasible.In this paper, we study the sensitivity around the EP of a coupled cavity system for its immediate relevance to recent experimental studies [34,35]. Nonetheless, the theoretical formalism and the main conclusion-no dramatic sensitivity enhancement at the EP-are applicable to a broad range of systems, such as magnon-cavity systems [37,38] and opto-mechanical systems [39,40]. We use the exact formalism of quantum Fisher information (QFI) [41] to characterize the sensitivity of parameter estimation. The QFI formalism enables us to evaluate the sensitivity without referring to a specific measurement scheme-be it phase, intensity, or any other complicated measurements of the output from the system. We find that no sensitivity boost exists at the EP. The reason boils down to the coalescence of the eigenstates around the EP. Due to the indistinguishability of different eigenstates around the EP, not one but all eigenstates are equally excited by an arbitrary detection field. The average of all eigenstates exactly cancels out the singularity in the susceptibility divergence of the eigenenergies and makes the sensitivity normal around the EP. The cancellation may also be understood by the divergent Pertermann excess-noise factor at the EP (or critical point) [42,43].The article is organized as follows: in section 2, the basic concept of EPs is introduced using a simple coupled-cavity model; in section 3, we introduce the definition of sensitivity. Sensitivity analysis for general linear systems is presented in this section; In sections 4 and 5, two specific experimentally relevant schemes, namely, coupled passive-passive cavities and coupled active-passive cavities, are considered and the numerical simul...

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“…The interest of the research community concentrates on both the practical computational issues [11][12][13] and on the better understanding of the general properties and the mathematics underlying the description of physical systems using quasi-normal modes [14][15][16][17][18][19] in various geometries. [20][21][22] However, despite the long history of leaky modes and non-Hermitian quantum mechanics 23 and electromagnetics, the proper theory and even the physical concepts are still somewhat obscure (see, e.g., Ref.…”

Section: Introductionmentioning

confidence: 99%

Leaky-mode expansion of the electromagnetic field inside dispersive spherical cavity

Jakobsen

Mansuripur

Kolesík

2018

Journal of Mathematical Physics

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Rigorous justification is presented for a recently introduced method to construct leakymode expansions of electromagnetic fields excited inside a spherical cavity filled with a dispersive, lossy medium. In a departure from the traditional approaches, our construction does not rely on Green's functions, rather it starts from a judiciously chosen auxiliary meromorphic function. Convergence of both the series expansions and of the over-completeness relations for the leaky modes is proven for a realistic model of chromatic dispersion.

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Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems

Cited by 15 publications

References 48 publications

Symmetries and conserved quantities in Lindblad master equations

Symmetries and conserved quantities in Lindblad master equations

Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system

Leaky-mode expansion of the electromagnetic field inside dispersive spherical cavity

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