Eigenvalues and eigenvectors of a class of irreducible tridiagonal matrices

Hu, Zhiguang

doi:10.1016/j.laa.2021.03.014

Cited by 10 publications

(10 citation statements)

References 6 publications

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“…We elaborate in detail on such a procedure in Section ??. Interestingly, when dealing with dimer systems, that reduction always leads to the formation of the effective evolution matrices with a Sylvester matrix shape [63,64,68]. The Sylvester matrix is a tridiagonal matrix, whose elements obey certain relations (see Ref.…”

Section: Discussionmentioning

confidence: 99%

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Tensor Product States in the Synthetic Space of Quadratic Bosonic Systems: Emergent Interplay Between Diabolic and Exceptional Degeneracy

Arkhipov¹,

Miranowicz²,

Nori³

et al. 2022

Preprint

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Tensor product states (TPS), which enable decomposing the states of large-dimensional Hilbert space in terms of lower-dimensional elementary tensors, are fundamental tools to capture the quantum nature of condensed matter phenomena. Here, we show how TPS can naturally emerge in the synthetic space of operator moments of bosonic systems described by quadratic (non-)Hermitian Hamiltonians. Beyond their theoretical interest, allowing to simulate the many-body physics of nontrivial large-dimensional Hamiltonians, we show how such construction explicitly reveals the interplay between the diabolic (DPs) and exceptional points (EPs) in the quantum regime. That is, the emergent diabolic degeneracy due to the noncommutative nature of the bosonic creation and annihilation operators profoundly impacts the structure of the evolution matrices governing higherorder field moments. This results in the appearance of highly degenerate DPs and EPs, whose interplay enables the existence of hybrid DPs-EPs, which cannot be captured by a "classical" (i.e., commutative) treatment of the bosonic fields. These findings can also be utilized for constructing effective Hamiltonians with similar spectral characteristics, possessing hybrid DPs-EPs. Our results can be exploited in various quantum protocols based on EPs, and can pave a new direction of research in this field.

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Section: Discussionmentioning

confidence: 99%

“…(1) j are the eigenvalues of the matrix M 1 . Intriguingly, but this unfolding TPS structure in the FMs eigenspace of quadratic systems can be used for solving the eigendecomposition problem for other tridiagonal matrices, so-called Sylvester-shaped matrices [68] (see also Appendix B).…”

Section: Tensor Product States and Their Degeneracy In The Higher-ord...mentioning

confidence: 99%

Tensor Product States in the Synthetic Space of Quadratic Bosonic Systems: Emergent Interplay Between Diabolic and Exceptional Degeneracy

Arkhipov¹,

Miranowicz²,

Nori³

et al. 2022

Preprint

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“…The formula plays an important role in showing (1.1) in [2]. It is proved by showing the eigenvalues and eigenvectors directly in [2], or by computing the eigenvalues of tridiagonal matrices in [7]. Here we present a proof by avoiding these computations.…”

Section: Conjugation Automorphisms On Sl(2 C)mentioning

confidence: 98%

“…is well studied in [3] and [4], where they prove that the characteristic polynomial of π is irreducible when π is an irreducible representation of G. It is quite natural to consider similar topics for Lie algebras of finite dimension. In [2], [9] and [7], the characteristic polynomial f π (z 0 , z 1 , z 2 , z 3 ) = det(z 0 I + z 1 φ(h) + z 2 φ(e 1 ) + z 3 φ(e 2 )) in sl(2, C) on its irreducible representation π of dimension m + 1 is obtained, which is…”

Section: Introductionmentioning

confidence: 99%

Characteristic polynomials and finitely dimensional representations of $\mathfrak{sl}(2, \mathbb{C})$

Jiang¹,

Liu²

2021

Preprint

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“…On the other hand, the eigenvectors in both approaches naturally differ. Their mutual relations [45] were discussed in relation to the quantum versus classical descriptions in [38].…”

Section: Spectral Eigenfrequencies Of a Two-mode System With Damping ...mentioning

confidence: 99%

Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach

Perina,

Miranowicz,

Chimczak

et al. 2022

Preprint

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Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine and induced) and hidden exceptional points not recognized directly in amplitude spectra are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in open quantum infinitely dimensional systems.

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Eigenvalues and eigenvectors of a class of irreducible tridiagonal matrices

Cited by 10 publications

References 6 publications

Tensor Product States in the Synthetic Space of Quadratic Bosonic Systems: Emergent Interplay Between Diabolic and Exceptional Degeneracy

Tensor Product States in the Synthetic Space of Quadratic Bosonic Systems: Emergent Interplay Between Diabolic and Exceptional Degeneracy

Characteristic polynomials and finitely dimensional representations of $\mathfrak{sl}(2, \mathbb{C})$

Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach

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