Baker-Campbell-Hausdorff expansion

The effective coupling tensor between two coupled spins in the toggling frame is only a good approximation of the effective coupling tensor in the (doubly) rotating frame if the higher order contributions in the Baker-Campbell-Hausdorff expansion [see Eq. (119)] can be neglected. This is the case if the term... 

A Baker-Campbell-Hausdorff expansion of the exponential time-evolution operator gives for the density (and similarly for other operators)... 

Reinserting this form of Uj t, — ) in the expectation value (63) and with the aid of the Baker-Campbell-Hausdorff expansion we arrive at the final expression... 

Note that when a symplectic integration method is used, we have, from the discussion in Chap. 3, a perturbed energy function and, moreover, from Theorem 3.1, the error in energy H is OQf), nonetheless the perturbations are large for a large stepsize h. In the Shadow Hybrid Monte-Carlo (SHMC) method [2, 3, 188], the accept-reject test is based on the modified Hamiltonian Hh (see Chap. 3), derived from the Baker-Campbell-Hausdorff expansion. SHMC can improve efficiency by decreasing the rejection rate. 

In cases where the Hamiltonians (typically due to phase or amplitude switching in the rf fields) are discontinuously time-dependent, the average Hamiltonian may conveniently be set up using the semi-continuous Baker-Campbell-Hausdorff (scBCH) expansion [56] as... 

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

The terms of the perturbation expansion for D can be computed using the Baker-Campbell-Hausdorff (BCH) expansion already introduced in Chap. 3. Recall that for linear operators X and Y, we can write the composition of their exponentials as... 

The Baker-CampbeU-Hausdorff formula is a fundamental expansion in elementary Linear Algebra and Lie group theory (J. E. Campbell, Proc. London Math. Soc. 29, 14 (1898) H. F. Baker, Proc. London Math. Soc. 34, 347 (1902) F. Hausdorff, Ber. Verhandl. Saechs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 58, 19 (1906)). 

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