Quantum integrability in the multistate Landau–Zener problem

We analyze Hamiltonians linear in the time variable for which the multistate Landau–Zener (LZ) problem is known to have an exact solution. We show that they either belong to families of mutually commuting Hamiltonians polynomial in time or reduce to the $2\times 2$ LZ problem, which is considered trivially integrable. The former category includes the equal slope, bow-tie, and generalized bow-tie models. For each of these models we explicitly construct the corresponding families of commuting matrices. The equal slope model is a member of an integrable family that consists of the maximum possible number (for a given matrix size) of commuting matrices linear in time. The bow-tie model belongs to a previously unknown, similarly maximal family of quadratic commuting matrices. We thus conjecture that quantum integrability understood as the existence of nontrivial parameter-dependent commuting partners is a necessary condition for the LZ solvability. Descendants of the $2\times 2$ LZ Hamiltonian are e.g. general SU(2) and $SU(1,1)$ Hamiltonians, time-dependent linear chain, linear, nonlinear, and double oscillators. We explicitly obtain solutions to all these LZ problems from the $2\times 2$ case. (paper)