Effective adiabatic guiding of quantum systems: Continuous tracking vs.\ stroboscopic hopping

Time-dependent Hamiltonians generally induce transitions between their corresponding instantaneous eigenstates. In cases where parameters in the Hamiltonian change slowly enough (compared with intrinsic dynamical timescales), the adiabatic theorem tells us that transitions are strongly suppressed. When parameters change more quickly, Berry has shown that the addition of a specific term $H_1$ to the Hamiltonian suppresses transitions completely, thus recovering transition- less driving. We discuss a convenient reformulation of Berry's approach in which $H_1$ is given in terms of an integral formula involving only operator quantities, including the nonabelian extension required for degenerate systems. We show that the integral formula is well suited to many-body problems and approximation schemes. Finally, we address a complementary issue: how to hop to an instantaneous eigenstate without necessarily tracking it. To do this, we construct a variational approach to seeking paths in parameter space that optimize the overlap between the time-evolved state and a given instantaneous eigenstate. This approach has the advantage that one can limit the types of operators appearing in the Hamiltonian which is useful when limiting the search to local or readily applicable operators.