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arXiv:math-ph/0506020

Special solutions of nonlinear von Neumann equations

Jan Naudts, Maciej Kuna

Published Jun 9, 2005• 5 citations
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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.

DOI
10.48550/arxiv.math-ph/0506020

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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.
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.
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.
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.
The nonlinear generalization of the von Neumann equation preserving convexity of the state space is studied in the nontrivial case of the qutrit. This equation can be cast into the integrable classical Riccati system of nonlinear ordinary differential equations. The solutions of such system are investigated in both the linear case corresponding to the standard von Neumann equation and the nonlinear one referring to the generalization of this equation. The analyzed dynamics of the qutrit is rich and includes quasiperiodic motion, multiple equilibria and limit cycles.