A new class of Lie-algebraic structures gd is revealed in some multi-particle processes of quantum physics having internal symmetry groups Ginv. They are extensions of some Lie algebras h (via coset construction) by G-invariant h-tensors v which are polynomials in boson operators. Applications of these algebras are briefly discussed for solving spectral and evolution tasks with gd as their dynamic symmetry algebras.

https://iopscience.iop.org/article/10.1088/0305-4470/27/1/009/pdf

G-invariant polynomial extensions of Lie algebras in quantum many-body physics

A new consequent description of the polarization structure of light within the framework of quantum optics is given by using the polarization gauge SU(2) invariance of free electromagnetic fields and a related concept of the polarization (P) spin. Within this approach the quantum light may be thought as a mixture of the usual polarized photons and unpolarized biphotons. New classes of unpolarized light states generated by P-scalar biphotons are examined. Possible applications are briefly discussed.

http://iopscience.iop.org/article/10.1088/0305-4470/26/17/040/pdf

Polarization structure of quantum light fields: a new insight. I. General outlook

New Lie-algebraic structures (polynomial deformations of Lie algebras) are revealed in some problems of quantum optics and laser physics. Specifically, deformations of oscillator algebras due to extensions of unitary algebras by their symmetric and skew-symmetric tensors are shown to be algebras of dynamic symmetry (ADS) in models of n-photon processes with internal symmetries. Similarly, deformed algebras sud(2) are found as ADS in the context of generalized Dicke models and frequency conversion models. We also briefly discuss some possible schemes of employing the results to solving physical problems.

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

Polarization properties of biphoton light, emitted by an optical parametric oscillator, are studied both theoretically and experimentally. The theoretical treatment is performed using the Stokes operator formalism; polarization density matrix of the light fields; and the quasi-probability function, which represents an analog of the field Wigner function. The hidden polarization of biphoton light is demonstrated experimentally with the help of elements of polarization tomography.

Biphoton light with hidden polarization and its polarization tomography

We proposed the procedure of measuring the unknown state of the three-level system - the qutrit, which was realized as the arbitrary polarization state of the single-mode biphoton field. This procedure is accomplished for the set of the pure states of qutrits; this set is defined by the properties of SU(2) transformations, that are done by the polarization transformers.

https://arxiv.org/pdf/quant-ph/0207096.pdf

Measurement of qutrits

A general approach is formulated for analyzing algebraic models of quantum composite systems with an internal symmetry described by group G. The case G = SU(2) is examined in detail as applied to the analysis of polarization invariance in quantum optics. A new class of fully depolarizable quantum states of light (scalar biphotons) is defined and investigated. Certain interactions of scalar biphoton light with material media are considered in the context of Dicke and Jaynes-Cummings models.

V. P. Karasev

Papers

Invariant-algebraic approach to problems of quantum optics