  We give a perturbative analysis of Crooks relation and Jarzynski equality in
an arbitrary driven quantum system weakly coupled to a heat bath. Invoking no
efficient Hamiltonian nor any restriction on the form of the coupling, we
derive the first-order correction to Crooks relation and Jarzynski equality in
terms of the interacting Hamiltonian. A Crooks type of relation about energy
increment in the system and nonvanishing heat expenditure to the heat bath up
to the second order of interaction would be given by the way. Our results tell
us that deviation of the quantum fluctuation relations is mainly caused by
energy trapped interaction rather than the heat flow to (or from) heat bath. We
also show a way of defining work in weak coupling regime which is consistent to
the work definition in closed systems and thus deepen our understanding of work
in quantum realm.
