Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

Abstract

h i g h l i g h t s• Ladder and shift operators for the Hermite-Gaussian modes are constructed.• These operators are used to obtain the generators of the dynamical algebras.• The Barut-Girardello and Perelomov coherent states are determined.• The uncertainty relations are considered in connection to the beam quality factor. a r t i c l e i n f o A group-theoretical approach to the paraxial propagation of Hermite-Gaussian modes based on the factorization method is presented. It is shown that the su(1, 1) and the su(2) algebras generate the spectrum of propagation constants at any fixed transversal plane. The complete set of HG modes is decomposed into hierarchies that are used to establish the representation spaces of SU(1, 1) and SU(2). The corresponding families of generalized coherent states are constructed and the variances of the quadratures and canonical variables are determined.

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

“…13 Its features in the complex domain 14 are the source of new challenges in controlling the time evolution of quantum wave packets [15][16][17] as well as the paraxial propagation of structured light in optical media with quadratic index profile. 18 The complex version of the Riccati equation is also useful to strengthen the systematic search for non-Hermitian quantum systems with real spectrum. 19 On the other hand, the stationary form of Schrödinger equation for 1-dimensional systems is also connected with a nonlinear second-order differential equation introduced by Ermakov 20 and revisited by Milne 50 years after.…”

“…Besides the monograph, 13 some exceptions can be found in previous studies. 18,19,[30][31][32][33][34][35][36][37][38] Quite remarkably, only recently, such relationship has been exploited to obtain integrable quantum models with real energy spectrum but described by non-Hermitian Hamiltonians. 19 A generalization of the oscillation theorem applies to study the zeros of the real and imaginary parts of the corresponding eigenfunctions, 31 and the introduction of bi-orthogonal bases permits to work with these systems in much the same way as for the Hermitian ones.…”

Position dependent mass Scarf Hamiltonians generated via the Riccati equation

“…For each fixed value of ℓ m i n , the complete space of wave functions splits into the direct sum of the set of subspaces spanned by the corresponding hierarchies. This fact is relevant, eg, in the construction of different families of coherent states and in the generation of the dynamical algebra underlying the corresponding system …”

“…The operators intertwine the wave functions corresponding to the same k ‐hierarchy. Thus, for each fixed value of ℓ m i n , the whole set of wave functions can be decomposed as the direct sum of the subspaces each one spanned by the corresponding k ‐hierarchy …”

“…Thus, for each fixed value of min , the whole set of wave functions n can be decomposed as the direct sum of the subspaces {  k , k = 1, 3 2 , 2· · · } each one spanned by the corresponding k-hierarchy. 41,42…”