  Quantum work fluctuation theorem (FT) commonly requires the system initially
prepared in an equilibrium state. Whether there exists universal exact quantum
work FT for initial state beyond equilibrium needs further discussions. Here, I
initialize the system in a near-equilibrium state, and derive the corresponding
modified Jarzynski equality by using the perturbation theory. The correction is
nontrivial because it directly leads to the principle of maximum work or the
second law of thermodynamics for near-equilibrium system and also gives a much
tighter bound of work for a given process. I also verify my theoretical results
by considering a concrete many-body system, and reveal a fundamental connection
between quantum critical phenomenon and near-equilibrium state at really high
temperature.
