An algebraic view to the operatorial ordering and its applications to optics

Dattoli, G.; Gallardo, J.; Torre, A.

doi:10.1007/bf02724503

Cited by 91 publications

(71 citation statements)

References 59 publications

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“…In the review paper [60] one can find references to some early applications of ME to Optics. For example to Hamiltonians involving the generators of SU (2), SU(1, 1) and Heisenberg-Weyl groups with applications to laser-plasma scattering and pulse propagation in free-electron lasers.…”

Section: Opticsmentioning

confidence: 99%

The Magnus expansion and some of its applications

Blanes¹,

Casas²,

Oteo³

et al. 2009

Physics Reports

1,085

1,300

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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution.The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.

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Section: Opticsmentioning

confidence: 99%

The Magnus expansion and some of its applications

Blanes¹,

Casas²,

Oteo³

et al. 2009

Physics Reports

1,085

1,300

View full text Add to dashboard Cite

show abstract

“…Метод, который мы применяем, можно в общих чертах описать следующим образом: при заданной группе Ли симметрий константы движения можно выбрать подходящий базис для алгебры Ли этой группы, при этом группа Ли является также подобной группой симметрий для дифференциального уравнения в частных производных инвариант-ной системы [33], [34]. Поскольку группа Ли изоморфна матричной группе Ли G ′ ,…”

Section: квантовый подход и подход на основе алгебры лиunclassified

Ли-Алгебраический Подход К Квадратичным Инвариантам Квантовых Систем

Абдалла¹,

Abdalla²,

Лич³

2009

ТМФ

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“…Dattoli et al have also studied the Lie-algebraic structures and time evolutions of the most of the above illustrative examples [37]. It should be noted that most of above systems and models in the literature were only considered in the stationary cases where the coefficients of the Hamiltonians were totally time-independent ( or partly time-dependent ).…”

Section: The Algebraic Structures Of Various Three-generator Quanmentioning

confidence: 99%

Exact solutions and geometric phase factor of time-dependent three-generator quantum systems

Shen

Zhu

Chen

2003

The European Physical Journal D - Atomic, Molecular and Optical

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By using the Lewis-Riesenfeld theory and the invariant-related unitary transformation formulation, the exact solutions of the time-dependent Schrödinger equations which govern the various Lie-algebraic quantum systems in atomic physics, quantum optics, nuclear physics and laser physics are obtained. It is shown that the explicit solutions may also be obtained by working in a subHilbert-space corresponding to a particular eigenvalue of the conserved generator ( i. e., the timeindependent invariant ) for some quantum systems without quasi-algebraic structures. The global and topological properties of geometric phases and their adiabatic limit in time-dependent quantum systems/models are briefly discussed. The more systematic approach to obtaining the formally exact solutions for the spin-j system was proposed by Gao et al [9] who made use of the Lewis-Riesenfeld quantum theory [10]. In this spin-j system, the three Lie-algebraic generators of the Hamiltonian satisfy the commutation relations of SU (2) Lie algebra. In addition to the spin model, there exist many quantum systems whose Hamiltonian is also constructed in terms of three generators of various Lie algebras, which we will illustrate in the following.The invariant theory that can be applied to solutions of the time-dependent Schrödinger equation was first proposed by Lewis and Riesenfeld in 1969 [10]. This theory is appropriate for treating the geometric phase factor. In 1991, Gao et al generalized this theory and put forward the invariant-related unitary transformation formulation [11]. Exact solutions for time-dependent systems obtained by using the generalized invariant theory contain both the geometric phase and the dynamical phase [12][13][14]. This formulation was developed from the Lewis-Riesenfeld's formal theory and proven useful to the treatment of the exact solutions of the time-dependent Schrödinger equation and geometric phase factor. In the present paper, based on these invariant theories we obtain exact solutions of various time-dependent quantum systems with the three-generator Lie-algebraic structures. *

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An algebraic view to the operatorial ordering and its applications to optics

Cited by 91 publications

References 59 publications

The Magnus expansion and some of its applications

The Magnus expansion and some of its applications

Ли-Алгебраический Подход К Квадратичным Инвариантам Квантовых Систем

Exact solutions and geometric phase factor of time-dependent three-generator quantum systems

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