  We consider a single Josephson junction in the presence of time varying gate
charge, and examine the nonequilibrium work done by the charge control in the
framework of fluctuation theorems. We obtain the probability distribution
functions of the works performed by forward protocol and by its time reversed
protocol, which from the Crooks relation gives the estimation of the free
energy changes \Delta F =0. The reliability of \Delta F estimated from the
Jarzynksi equality is crucially dependent on protocol parameters, while
Bennett's acceptance ratio method confirms consistently \Delta F=0. The error
of the Jarzynski estimator either grows or becomes saturated as the duration of
the work protocol increases, which depends on the protocol rapidity determining
the existence of the oscillatory motion of the phase difference across the
junction. The average of the work also shows similar behaviors and its
saturation value is given by the relative weight of the oscillatory trajectory
with respect to running trajectories with constant acceleration. We discuss
non-negativity of the work average and its relation to heat and entropy
production associated with the circuit control.
