Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

Abstract

We develop a Lie algebraic approach to systematically calculate the evolution operator of the generalized two-dimensional quadratic Hamiltonian with time-dependent coefficients. Although the development of the Lie algebraic approach presented here is mainly motivated by the two-dimensional quadratic Hamiltonian, it may be applied to investigate the evolution operators of any Hamiltonian having a dynamical algebra with a large number of elements. We illustrate the method by finding the propagator and the Heisenberg picture position and momentum operators for a two-dimensional charge subject to uniform and constant electro-magnetic fields.2

Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

“…where C(a, q , ωt/2) is the even Mathieu function with a = 4ω 2 1 /ω 2 and q = −2ω 2 0 /ω 2 . Notice that this set of coefficients had been found before by us [34,35] and others. [40,41] Nevertheless, these coefficients themselves are insufficient because they only permit to write the evolution operator in the form U A (see Equation (4)) whereas the calculation of H e requires the evolution operator to be written in the form of U B (see Equation (5)).…”

“…[5,33] This approach is based on our earlier results on the use of Lie algebras to find the evolution operator of time-dependent Hamitlonians. [34,35] However, the method presented here has major improvements and differences with respect to our previous work. First, it does not require the prior knowledge of the transformation rules arising from the commutators between the algebra's elements.…”

“…In order for U † A to be the evolution operator, the condition u = 0 must be fulfilled. [35] This condition translates into a system of ordinary differential equations (ODE) for the α(t) parameters that one could in principle attempt to solve. However, specially for algebras with large dimensions, these equations might be very complex.…”

Dinámica de la polarización del espín electrónico y nuclear en trampas paramagnéticas de Ga en GaAsN

“…Estos generadores cuánticos poseen un álgebra de Lie de dimensión finita ante la conmutación [52,53]. Es decir,…”

“…16) donde C i jk son las constantes de estructura del álgebra de Lie[52,53,59] y tienen un valor C i jk = 0 si i, j = 0 y C i jk = i jk para i, j = 1, 2, 3, con i jk el tensor de Levi-Civita.Usando estos operadores (3.15), podemos encontrar el bloque de generadores cuánticos para las TP's de la siguiente manerâ…”