SU(3) Landau - Zener Interferometry

We consider a general theory of Landau-Zener transitions in a three-level system. Based on a classification of three level crossings we express the Landau - Zener Hamiltonians in terms of two bases: i) spin S=1 SU(2) operators and ii) SU(3) Gell - Mann matrices. We show that the generic Hamiltonians being non-linear in terms of the SU(2) group generators become linear in the SU(3) basis. If the diabatic states of the SU(3) Landau - Zener Hamiltonian form a triangle, the interference between two paths results in formation of "beats" and "steps" pattern in the time-dependent transition probability. The characteristic time scales describing the "beats" and "steps" depend on a dwell time through the triangle. These scales are related to the geometric size of the interferometer. We formulate the SU(3) Landau - Zener problem in terms of Bloch dynamics of a unit vector and find a solution of eight-dimensional Bloch equations in the limit of a non-adiabatic transition. Possible experiments in triangular and linearly arranged triple quantum dots where Landau - Zener interferometry can be used for finding manifestations of SU(3) symmetry are discussed.Comment: 6 pages, 3 figure

Semi-fermionic representation of SU(N) Hamiltonians

We represent the generators of the SU(N) algebra as bilinear combinations of Fermi operators with imaginary chemical potential. The distribution function, consisting of a minimal set of discrete imaginary chemical potentials, is found for arbitrary N. This representation leads to the conventional temperature diagram technique with standard Feynman codex, except that the Matsubara frequencies are determined by neither integer nor half-integer numbers. The real-time Schwinger-Keldysh formalism is formulated in the framework of complex distribution functions. We discuss the continuous large N and SU(2) large spin limits. We illustrate the application of this technique for magnetic and spin-liquid states of the Heisenberg model.Comment: 11 pages, 7 EPS figures included, extended versio