Von Neumann equations with time-dependent Hamiltonians and supersymmetric quantum mechanics

Czachor, Marek; Doebner, Heinz-Dietrich; Syty, Monika; Wasylka, Krzysztof

Von Neumann equations with time-dependent Hamiltonians and supersymmetric quantum mechanics

Authors

Czachor, Marek

Doebner, Heinz-Dietrich

Syty, Monika

Wasylka, Krzysztof

Description

Starting with a time-independent Hamiltonian $h$ and an appropriately chosen solution of the von Neumann equation $i\dotρ(t)=[ h,ρ(t)]$ we construct its binary-Darboux partner $h_1(t)$ and an exact scattering solution of $i\dotρ_1(t)=[h_1(t),ρ_1(t)]$ where $h_1(t)$ 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 1-D harmonic oscillator. The resulting $h_1(t)$ represents a scattering of a soliton-like pulse on a three-level system.
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