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In the framework of nuclear magnetic resonance, we consider the generalproblem of the coherent control of a spin coupled to a bath by means ofcomposite or continuous pulses of duration $\tau_\mathrm{p}$. We show explicitythat it is possible to design the pulse in order to achieve a decoupling of thespin from the bath up to the third order in $\tau_\mathrm{p}$. The evolution ofthe system is separated in the evolution of the spin under the action of thepulse and of the bath times correction terms. We derive the correction termsfor a general time dependent axis of rotation and for a general couplingbetween the spin and the environment. The resulting corrections can be madevanish by an appropriate design of the pulse. For $\pi$ and $\pi/2$ pulses, wedemonstrate explicitly that pulses exist which annihilate the first and thesecond order corrections even if the bath is fully quantum mechanical, i.e., itdisplays internal dynamics. Such pulses will also be useful for quantuminformation processing.

Optimized pulses for the perturbative decoupling of spin and decoherence bath