  A three-level quantum system having two energy gaps presents a nontrivial
working medium for a quantum heat engine. Our focus lies in understanding the
constraints on the ability to modulate these gaps relative to the changes in
probability distributions at the two given heat reservoirs. It is seen that an
Otto engine in the quasistatic limit is feasible if at least one energy gap
shrinks during the first quantum adiabatic stage. We analyze operating
conditions under different variations of the gaps, revealing that a definite
majorization relation between the hot and cold distributions serves as a
sufficient criterion for the engine when both gaps are shrinking. Further, Otto
efficiency is enhanced in case majorization holds. On the other hand,
majorization becomes a necessary condition when only one of the gaps is
shrinking. In the special case where one gap remains fixed, majorization is
both necessary and sufficient for engine operation. For an $n$-level system, we
note that a well defined change in energy gaps aligns with the majorization
relation, thus characterizing the operation of the engine.
