Lie algebraic techniques are powerful and widely used tools for studying dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie algebra with finite dimension, the unitary evolution can be expressed as a finite product of exponentials using the Wei-Norman expansion. The system is then exactly described by a finite set of scalar differential equations, even if the Hilbert space is infinite. However, the differential equations provided by the Wei-Norman expansion are nonlinear and often have singularities that prevent both analytic and numerical evaluation. We derive a new Lie algebra expansion for the quantum Fisher information, which results in linear differential equations. Together with existing Lie algebra techniques this allows many metrology problems to be analysed entirely in the Heisenberg picture. This substantially reduces the calculations involved in many metrology problems, and provides analytical solutions for problems that cannot even be solved numerically using the Wei-Norman expansion. It also allows us to study general features of metrology problems, valid for all quantum states. We provide detailed examples of these methods applied to problems in quantum optics and nonlinear optomechanics. Published by the American Physical Society 2024Lie algebraic techniques are powerful and widely used tools for studying dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie algebra with finite dimension, the unitary evolution can be expressed as a finite product of exponentials using the Wei-Norman expansion. The system is then exactly described by a finite set of scalar differential equations, even if the Hilbert space is infinite. However, the differential equations provided by the Wei-Norman expansion are nonlinear and often have singularities that prevent both analytic and numerical evaluation. We derive a new Lie algebra expansion for the quantum Fisher information, which results in linear differential equations. Together with existing Lie algebra techniques this allows many metrology problems to be analysed entirely in the Heisenberg picture. This substantially reduces the calculations involved in many metrology problems, and provides analytical solutions for problems that cannot even be solved numerically using the Wei-Norman expansion. It also allows us to study general features of metrology problems, valid for all quantum states. We provide detailed examples of these methods applied to problems in quantum optics and nonlinear optomechanics. Published by the American Physical Society 2024

arXiv:2311.12446Physical Review ResearchNov 14, 20244 citations

An implementation of the polynomial Lie algebra methods for solving a class of nonlinear models in quantum optics

V. P. Karassiov, Alexander A. Gusev, Sergue Il'ich Vinitsky

We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras $su_{pd}(2)$ . These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the $su_{pd}(2)$ defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras $su_{pd}(2)$. These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the $su_{pd}(2)$ defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.

arXiv:quant-ph/0105152May 31, 20010 citations

Polynomial deformations of the Lie algebrasl(2) in problems of quantum optics

V. P. Karasev

Theoretical and Mathematical PhysicsApr 1, 199318 citations

Atom-photon cluster in nonlinear and quantum optics

Ashat M. Basharov

We described the constructed models of radiative quantum systems analogous of multiphoton and Raman resonances of classical field quanta on an atomic system with the participation of resonator mode quanta. A distinctive feature of the models is the possibility to describe an atomic ensemble and quanta using either the generators of polynomial algebra or the two-mode Jordan-Schwinger representation of the su(2) algebra, that could point to atomic-photon and/or photon clusters. The algebras are arising that are mathematically insoluble, in contrast to the Heisenberg — Weyl algebra, which makes it possible to implement and observe fundamental effects of coherent nonlinear optics, such as photon echo, optical nutation, and superradiance in ensembles of the described clusters.We described the constructed models of radiative quantum systems analogous of multiphoton and Raman resonances of classical field quanta on an atomic system with the participation of resonator mode quanta. A distinctive feature of the models is the possibility to describe an atomic ensemble and quanta using either the generators of polynomial algebra or the two-mode Jordan-Schwinger representation of the su(2) algebra, that could point to atomic-photon and/or photon clusters. The algebras are arising that are mathematically insoluble, in contrast to the Heisenberg — Weyl algebra, which makes it possible to implement and observe fundamental effects of coherent nonlinear optics, such as photon echo, optical nutation, and superradiance in ensembles of the described clusters.

Известия Российской академии наук Серия физическаяJun 15, 20241 citation

Polynomial Lie Algebras

\hat E_{R_1 }^\mathcal{P} (u(n);(m))

in Nonlinear Models of Quantum Optics: Basic Ideas and Cluster Dynamics in the Heisenberg Picture

We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras

su_{pd}(2)

. These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the

su_{pd}(2)

defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.