Abstract
Using the Hubbard representation for S U ( 2 ) , we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of nonlinear coupled equations. In order to find exact solutions, we use an inverse approach and find families of time-dependent Hamiltonians whose off-diagonal elements are connected with the Ermakov equation. A physical model with the so-obtained Hamiltonians is discussed in the context of the nuclear magnetic resonance phenomenon.
Highlights
The dynamical manipulation of two-level systems has been a long-standing issue in quantum mechanics [1,2,3]
Requiring that the corresponding time-evolution operator is written as a product of exponential operators, we find analytical solutions to the dynamical law using an inverse approach
The control fields are determined by means of the solutions of an Ermakov equation, which is obtained once the time-dependent frequency of the related parametric-oscillator equation is specified
Summary
The dynamical manipulation of two-level systems has been a long-standing issue in quantum mechanics [1,2,3]. It is well known that the solution of the Schrödinger equation with time-dependent Hamiltonians represents, in general, a difficult task and the number of exactly solvable cases is very limited These include rotating fields [6,7], nonlinear modulated Rabi fields [8], periodic driving fields [9,10], and pulse shaped fields [11]. Of particular interest in the solution of the evolution equation and in the generation of exactly solvable models is the Lie algebraic technique, which takes advantage of the fact that the Hamiltonian can be written as a linear combination of the generators of a Lie algebra [15,16,17,18,19,20,21,22] In this scheme, the Wei Norman theorem establishes that the corresponding evolution operator can be factorized (or disentangled) into independent exponential factors involving only one generator of the algebra [23].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
