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We use analytic (current) density-potential maps of time-dependent (current)density functional theory (TD(C)DFT) to inverse engineer analytically solvabletime-dependent quantum problems. In this approach the driving potential (thecontrol signal) and the corresponding solution of the Schr\"odinger equationare parametrized analytically in terms of the basic TD(C)DFT observables. Wedescribe the general reconstruction strategy and illustrate it with a number ofexplicit examples. First we consider the real space one-particle dynamicsdriven by a time-dependent electromagnetic field and recover, from the generalTDDFT reconstruction formulas, the known exact solution for a driven oscillatorwith a time-dependent frequency. Then we use analytic maps of the latticeTD(C)DFT to control quantum dynamics in a discrete space. As a first example weconstruct a time-dependent potential which generates prescribed dynamics on atight-binding chain. Then our method is applied to the dynamics of spin-1/2driven by a time dependent magnetic field. We design an analytic control pulsethat transfers the system from the ground to excited state and vice versa. Thispulse generates the spin flip thus operating as a quantum NOT gate.

Dynamics of observables and exactly solvable quantum problems: Using time-dependent density functional theory to control quantum systems