  The unitary generation of coherence from an incoherent thermal state is
investigated. We consider a completely controllable Hamiltonian allowing to
generate all possible unitary transformations. Optimizing the unitary control
to achieve maximum coherence leads to a micro-canonical energy distribution on
the diagonal energy representation. We demonstrate such a control scenario
starting from a Hamiltonian utilizing optimal control theory for unitary
targets. Generating coherence from an incoherent initial state always costs
external work. By constraining the amount of work invested by the control,
maximum coherence leads to a canonical energy population distribution. When the
optimization procedure constrains the final energy too tightly local suboptimal
traps are found. The global optimum is obtained when a small Lagrange
multiplier is employed to constrain the final energy. Finally, we explore
constraining the generated coherence to be close to the diagonal in the energy
representation.
