Dynamics of quantum systems driven by time-varying Hamiltonians: Solution for the Bloch-Siegert Hamiltonian and applications to NMR

Abstract

Comprehending the dynamical behavior of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood as the usual methods including diagonalization of the Hamiltonian do not work in this setting. In fact, since the inception of Magnus' expansion [Commun. Pure Appl. Math. 7, 649 (1954)10.1002/cpa.3160070404], no fundamentally novel mathematical approach capable of solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here on an entirely different nonperturbative approach, termed path sum, which is always guaranteed to converge, yields the exact analytical solution in a finite number of steps for finite systems, and is invariant under scale transformations of the quantum state space. Path sum can be combined with any state-space reduction technique and can exactly reconstruct the dynamics of a many-body quantum system from the separate, isolated, evolutions of any chosen collection of its subsystems. As examples of application, we solve analytically for the dynamics of all two-level systems as well as of a many-body Hamiltonian with a particular emphasis on nuclear magnetic resonance applications: Bloch-Siegert effect, coherent destruction of tunneling, and N-spin systems involving the dipolar Hamiltonian and spin diffusion.