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Mathematical Notes

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

Владимир Владимирович Белов, M. F. Kondrat’eva

Published Dec 1, 1995• 5 citations
Publisher
DOI
10.1007/bf02304883

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We develop a formally exact master equation to describe the evolution of the average density matrix of a closed quantum system driven by a stochastic Hamiltonian. The average over stochastic processes generally results in decoherence effects in closed system dynamics, in addition to the unitary evolution. We then show that, for an important class of problems in which the Hamiltonian is proportional to a Gaussian random process, the 2nd-order master equation yields exact dynamics. The general formalism is applied to study the examples of a two-level system, two atoms in a stochastic magnetic field and the heating of a trapped ion, where we find phenomena such as decoherence-induced disentanglement.
Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.