New Lie-algebraic structures in nonlinear problems of quantum optics and laser physics

Jules Beckers

Igor Leonidovich Kirilyuk, Leonid E. Kon’kov, Sergey V. Prants

Ruvi Lecamwasam, Tatiana Iakovleva, Jason Twamley

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 2024

Mark S. Hillery, Leonard D. Mlodinow

Shun-Jin Wang

Contrary to the usual view that quantum mechanics in configuration space has, in general, only one algebaic representation( the Heisenberg algebraic representation), we have proved that quantum mechanics in configuration space has, in general, alternative algebraic representations: (i) In a finite domain of configuration space, it can be expressed in terms of su(2) algebra; (ii) in an infinite domain of configuration space, it can be expressed in terms of su(l,l) algebra.The above results open a new possibility to reformulate quantum mechanics and provide more mathematical tools to solve diverse physical problems.Nonlinear relations of different Lie algebas may imply a unification of Lie algebras and their physical implications ( the quantum motion modes ) in a deeper nonlinear domain.This observation raises challenging mathematical and physical questions.

Masashi Ban