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@ARTICLE{Heib:1048888,
      author       = {Heib, Tim and Goia, Andreea Silvia and Baghiyan, Sona and
                      Zeier, Robert and Bruschi, David Edward},
      title        = {{F}inite-dimensional {L}ie algebras in bosonic quantum
                      dynamics: {T}he single-mode case},
      publisher    = {arXiv},
      reportid     = {FZJ-2025-04993, 2511.06940},
      year         = {2025},
      abstract     = {We study, classify, and explore the mathematical properties
                      of finite-dimensional Lie algebras occurring in the quantum
                      dynamics of single-mode and self-interacting bosonic
                      systems. These Lie algebras are contained in the real
                      skew-hermitian Weyl algebra $\hat{A}_1$, defined as the real
                      subalgebra of the Weyl algebra $A_1$ consisting of all
                      skew-hermitian polynomials. A central aspect of our analysis
                      is the choice of basis for $\hat{A}_1$, which is composed of
                      skew-symmetric combinations of two elements of the Weyl
                      algebra called monomials, namely strings of creation and
                      annihilation operators combined with their hermitian
                      conjugate. Motivated by the quest for analytical solutions
                      in quantum optimal control and dynamics, we aim at answering
                      the following three fundamental questions: (i) What are the
                      finite-dimensional Lie subalgebras in $\hat{A}_1$ generated
                      by monomials alone? (ii)~What are the finite-dimensional Lie
                      subalgebras in $\hat{A}_1$ that contain the free
                      Hamiltonian? (iii) What are the non-abelian and
                      finite-dimensional Lie subalgebras that can be faithfully
                      realized in $\hat{A}_1$? We answer the first question by
                      providing all possible realizations of all
                      finite-dimensional non-abelian Lie algebras that are
                      generated by monomials alone. We answer the second question
                      by proving that any non-abelian and finite-dimensional
                      subalgebra of $\hat{A}_1$ that contains a free Hamiltonian
                      term must be a subalgebra of the Schrödinger algebra. We
                      partially answer the third question by classifying all
                      nilpotent and non-solvable Lie algebras that can be realized
                      in $\hat{A}_1$, and comment on the remaining cases. Finally,
                      we also discuss the implications of our results for quantum
                      control theory. Our work constitutes an important stepping
                      stone to understanding quantum dynamics of bosonic systems
                      in full generality.},
      keywords     = {Quantum Physics (quant-ph) (Other) / Mathematical Physics
                      (math-ph) (Other) / FOS: Physical sciences (Other)},
      cin          = {PGI-12 / PGI-8},
      cid          = {I:(DE-Juel1)PGI-12-20200716 / I:(DE-Juel1)PGI-8-20190808},
      pnm          = {5221 - Advanced Solid-State Qubits and Qubit Systems
                      (POF4-522) / EXC 2004:  Matter and Light for Quantum
                      Computing (ML4Q) (390534769) / Verbundprojekt: German
                      Quantum Computer based on Superconducting Qubits (GEQCOS) -
                      Teilvorhaben: Charakterisierung, Kontrolle und Auslese
                      (13N15685) / BMBF 13N16210 - SPINNING –
                      Spin-Photon-basierter Quantencomputer auf Diamantbasis
                      (BMBF-13N16210) / PASQuanS2.1 - Programmable Atomic
                      Large-scale Quantum Simulation 2 - SGA1 (101113690)},
      pid          = {G:(DE-HGF)POF4-5221 / G:(BMBF)390534769 / G:(BMBF)13N15685
                      / G:(DE-Juel1)BMBF-13N16210 / G:(EU-Grant)101113690},
      typ          = {PUB:(DE-HGF)25},
      doi          = {10.48550/arXiv.2511.06940},
      url          = {https://juser.fz-juelich.de/record/1048888},
}