Quantum integrability in the multistate Landau-Zener problem

Abstract

We analyze Hamiltonians linear in the time variable for which the multistate Landau-Zener 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 x 2 Landau-Zener 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 Landau-Zener solvability. Descendants of the 2 x 2 Landau-Zener 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 Landau-Zener problems from the 2 x 2 case.

arXiv comment17+ pages, no figures

References in corpus (18)

• Single artificial-atom lasing
• Anderson localization casts clouds over adiabatic quantum optimization
• Non-adiabacity and large flucutations in a many particle Landau Zener problem
• Representations of the Weyl group and Wigner functions for SU(3)
• Multiparticle Landau-Zener problem
• Cavity-Mediated Entanglement Generation Via Landau-Zener Interferometry
• Large-amplitude driving of a superconducting artificial atom: Interferometry, cooling, and amplitude spectroscopy
• Landau-Zener transitions in a linear chain
• The Origin of Degeneracies and Crossings in the 1d Hubbard Model
• Quantum integrability in systems with finite number of levels
• Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach
• Landau-Zener Transitions in Chains
• The Link between Integrability, Level Crossings, and Exact Solution in Quantum Models
• Classification of Parameter-Dependent Quantum Integrable Models, Their Parameterization, Exact Solution, and Other Properties
• SU(1,1) symmetry of multimode squeezed states
• Unitary irreducible representations of SL(2,C) in discrete and continuous SU(1,1) bases
• A Class of Parameter Dependent Commuting Matrices
• Parameter Dependent Commuting Matrices, Plücker relations and Related Quantum Glass Models

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