  The well-known Jarzynski equality, often written in the form $e^{-\beta\Delta
F}=\langle e^{-\beta W}\rangle$, provides a non-equilibrium means to measure
the free energy difference $\Delta F$ of a system at the same inverse
temperature $\beta$ based on an ensemble average of non-equilibrium work $W$.
The accuracy of Jarzynski's measurement scheme was known to be determined by
the variance of exponential work, denoted as ${\rm var}\left(e^{-\beta
W}\right)$. However, it was recently found that ${\rm var}\left(e^{-\beta
W}\right)$ can systematically diverge in both classical and quantum cases. Such
divergence will necessarily pose a challenge in the applications of Jarzynski
equality because it may dramatically reduce the efficiency in determining
$\Delta F$. In this work, we present a deformed Jarzynski equality for both
classical and quantum non-equilibrium statistics, in efforts to reuse
experimental data that already suffers from a diverging ${\rm
var}\left(e^{-\beta W}\right)$. The main feature of our deformed Jarzynski
equality is that it connects free energies at different temperatures and it may
still work efficiently subject to a diverging ${\rm var}\left(e^{-\beta
W}\right)$. The conditions for applying our deformed Jarzynski equality may be
met in experimental and computational situations. If so, then there is no need
to redesign experimental or simulation methods. Furthermore, using the deformed
Jarzynski equality, we exemplify the distinct behaviors of classical and
quantum work fluctuations for the case of a time-dependent driven harmonic
oscillator dynamics and provide insights into the essential performance
differences between classical and quantum Jarzynski equalities.
