Lie algebraic solution of Bloch equations with time-dependent coefficients

Maciej Kuna, Jan Naudts

Christopher Feuillade

D. Martínez-Tibaduiza, Carlos González-Arciniegas, Carlos Farina +2 more

Using the Wei-Norman theory, we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the $\mathrm{su}(1,1)$, $\mathrm{su}(2)$, and $\mathrm{so}(2,1)$ Lie algebras. Using a recently developed solution for the time evolution of these quantum systems, we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra $\mathrm{su}(2)$. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.

Jing‐Bo Xu, Lan Wang, Xiao-Chun Gao

In this letter, the matrix invariant of the Bloch equation with time-dependent coefficients is found, the time evolution of the system is discussed and the exact solution of the system is obtained. We then proceed to study the cyclic evolution of the system, obtaining the Aharonov-Anandan phase. Finally, it is shown that the study of noncyclic evolution of the system can be reduced to the cyclic evolution.

J. Aliaga, J. L. Gruver, Araceli N. Proto

Latévi M. Lawson, Gabriel Y. H. Avossevou, Laure Gouba

In this paper, we study a two-dimensional rotationally symmetric harmonic oscillator with time-dependent frictional force. At the classical level, we solve the equations of motion for a particular case of the time-dependent coefficient of friction. At the quantum level, we use the Lewis-Riesenfeld procedure of invariants to construct exact solutions for the corresponding time-dependent Schrödinger equations. The eigenfunctions obtained are in terms of the generalized Laguerre polynomials. By means of the solutions, we verify a generalization version of Heisenberg’s uncertainty relation and derive the generators of the su(1, 1) Lie algebra. Based on these generators, we construct the coherent states à la Barut-Girardello and à la Perelomov and respectively study their properties.