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arXiv · 1908.03699

Nonlinear Dynamics from Linear Quantum Evolutions

Florio M. Ciaglia·Fabio Di Cosmo·Armando Figueroa·Vladimir I. Man'ko·Giuseppe Marmo·Luca Schiavone·Franco Ventriglia·Patrizia Vitale
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Abstract

Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group $ϕ_t$ of transformations on a Hilbert space, $\mathcal{H}$, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schrödinger equation to a submanifold invariant under the flow $ϕ_t$. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow $ϕ_t$. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.

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Florio M. Ciaglia, Fabio Di Cosmo, Armando Figueroa, Vladimir I. Man'ko, Giuseppe Marmo, Luca Schiavone, Franco Ventriglia, Patrizia Vitale. 2019-08-10. Nonlinear Dynamics from Linear Quantum Evolutions. https://doi.org/10.1016/j.aop.2019.167957

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