I recently found out that "the interaction picture" comes in two variants--"the Schrodinger interaction picture" and "the Heisenberg interaction picture".
I am looking for a good reference on the complete details on how operators and states behave in the "Heisenberg interaction picture".
One famous example of such operator evolution can be found in eq. (2.22) of https://arxiv.org/abs/1208.5174
a) I am trying to find the details on eq. (2.2.3) which is given the name "the Heisenberg interaction picture"! (page 18 of the book preview) https://www.google.com/books/edition/Methods_in_Theoretical_Quantum_Optics/Gw4sxyr6UhMC?hl=en&gbpv=1&printsec=frontcover
b) The relationship between operators in the Heisenberg picture and the Dirac(interaction) picture can be found in eqs. (5.2) or (5.3) here.https://projecteuclid.org/download/pdf_1/euclid.cmp/1103839389 I am trying to work out how a matrix element would be the same in the two pictures.