Special solutions of nonlinear von Neumann equations

TL;DR

This paper explores special solutions to nonlinear von Neumann equations involving elliptic functions, identifying unique cases where these solutions can be explicitly constructed and linked to known physical models.

Contribution

It introduces a method to construct state-dependent Hamiltonians for specific nonlinear von Neumann equations and proves these are the only such solutions under certain conditions.

Findings

Two unique cases for solutions involving elliptic functions

Connection to solutions of Maxwell-Bloch equations

Reproduction of known 3-dimensional solutions

Abstract

We consider solutions of the non-linear von Neumann equation involving Jacobi's elliptic functions sn, cn, and dn, and 3 linearly independent operators. In two cases one can construct a state-dependent Hamiltonian which is such that the corresponding non-linear von Neumann equation is solved by the given density operator. We prove that in a certain context these two cases are the only possibilities to obtain special solutions of this kind. Well-known solutions of the reduced Maxwell-Bloch equations produce examples of each of the two cases. Also known solutions of the non-linear von Neumann equation in dimension 3 are reproduced by the present approach.