Recursive evaluation of interaction pictures

Abstract

‘The interaction picture is a useful tool in quantum mechanics, allowing one to investigate slow and possibly complicated processes in the presence of fast motions whose behavior is well understood. Important examples in NMR are the rotatingFrame transformation (I) and the “togglingfi-ame” picture used in multiple-pulse NMR (2, 3) and composite pulses (4). The interaction picture is related to the laboratory frame by a timedependent unitary transformation operator. If this operator commutes with itself at different times, as in the rotating-frame transformation, the evaluation of the transformation is usually quite simple. In multiple-pulse NMR, however, the pu.lses are usually applied with different phases, leading to noncommuting transformation operators. In this case the transformation is conventionally performed in a straightforward fashion which requires a number of transformations proportional to the square of the number of pulses in the sequence. In this paper we show that this method is not inherent to the transformation and demonstrate a simplification which is recursive, thereby making the number of necessary transformations linear in the number of pulses in the sequence. The Hamiltonian describing a multiple-pulse sequence can be written in the laboratory frame as A?(t) = x0 + 2FRF(t), [II wlhere XRF(t) describes the effect of the radiofrequency pulses, while A?” contains all the other interactions. The general solution of the Liouville-von-Neumann equation p(t)= Texp[ -i~A?(t’)dt’]p(O)relp[ i~Z(f’)dfl,