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arXiv:quant-ph/9904110

Nonlinear von Neumann-type equations

Marek Czachor, Maciej Kuna, Sergey Leble, Jan Naudts

Published Apr 30, 1999• 14 citations
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Abstract

We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate the technique of binary Darboux integration of nonlinear density matrix equations and special attention is payed to the problem of how to find physically nontrivial `self-scattering' solutions.

DOI
10.48550/arxiv.quant-ph/9904110

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We review some recent developments in the theory of nonlinear von Neumann equations. We distinguish between the von Neumann equation (which can be nonlinear) and the Liouville equation (which should be linear). Explicit examples illustrate the technique of binary Darboux integration of nonlinear density matrix equations and special attention is payed to the problem of how to find physically nontrivial `self-scattering' solutions.
A Darboux-type method of solving the nonlinear von Neumann equation idotrho=[H,f(rho)]i\\dot \\rho=[H,f(\\rho)]idotrho=[H,f(rho)], with functions f(rho)f(\\rho)f(rho) commuting with rho\\rhorho, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities f(rho)f(\\rho)f(rho) are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian HHH of a one-dimensional harmonic oscillator and f(rho)=rhoq−2rhoq−1f(\\rho)=\\rho^q-2\\rho^{q-1}f(rho)=rhoq−2rhoq−1 with arbitary real qqq. It is shown that self-scattering solutions possess the same asymptotics for all qqq and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.
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A nonlinear Lie-Nambu generalization of the Liouville-von Neumann equation is considered [M.~Czachor, Phys.~Lett.~A 225, 1 (1997)]. It is shown that spectrum of rho(t)\\rho(t)rho(t) is conserved by the Lie-Nambu dynamics if rho(t)\\rho(t)rho(t) is a self-adjoint and Hilbert-Schmidt solution of a nonlinear triple-bracket equation. This generalizes to arbitrary separable Hilbert spaces the previous result which was valid for finite-dimensional Hilbert spaces. The result shows that conservation of eigenvalues of density matrices may be a general property which holds independently of linearity of dynamics. The corresponding spectral decomposition plays a role of a nonlinearly generalized convexity principle.
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Starting with a time-independent Hamiltonian h and an appropriately chosen solution of the von Neumann equation $i\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)=[h,\ensuremath{\rho}(t)]$ we construct its binary-Darboux partner h1(t){h}_{1}(t)h1​(t) and an exact scattering solution of $i{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}}_{1}(t)=[{h}_{1}(t),{\ensuremath{\rho}}_{1}(t)],$ where h1(t){h}_{1}(t)h1​(t) is time dependent and not isospectral to h. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where h corresponds to a one-dimensional harmonic oscillator. The resulting h1(t){h}_{1}(t)h1​(t) represents a scattering of a solitonlike pulse on a three-level system.