Quantum integrability in the multistate LandauZener problem
Abstract
We analyze Hamiltonians linear in the time variable for which the multistate LandauZener (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× 2 LZ problem, which is considered trivially integrable. The former category includes the equal slope, bowtie, and generalized bowtie 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 bowtie 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 parameterdependent commuting partners is a necessary condition for the LZ solvability. Descendants of the 2× 2 LZ Hamiltonian are e.g. general SU(2) and SU(1,1) Hamiltonians, timedependent linear chain, linear, nonlinear, and double oscillators. We explicitly obtain solutions to all these LZ problems from the 2× 2 case.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 June 2015
 DOI:
 10.1088/17518113/48/24/245303
 arXiv:
 arXiv:1412.4926
 Bibcode:
 2015JPhA...48x5303P
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Quantum Physics
 EPrint:
 17+ pages, no figures