Vigilantly is the adverb of vigilant: doing something while watching carefully and continuously for danger or wrongdoing. It tells you how an action is carried out — 'she watched the baby vigilantly', 'the team monitored the system vigilantly through the night'. Because the meaning is about sustained watching, it sits most naturally with verbs of guarding, watching and checking: watch, guard, monitor, patrol, observe, look out. It sounds odd with verbs that have nothing to watch for. It is a formal word and much rarer than 'vigilant' itself; in speech people are far more likely to say 'kept a close eye on' or 'watched carefully'. Position is flexible — usually after the verb or after the object ('guarded the gate vigilantly'), sometimes before the verb ('they vigilantly checked every bag').

警覺地、戒備地