  A result of great theoretical and experimental interest, Jarzynski equality
predicts a free energy change $\Delta F$ of a system at inverse temperature
$\beta$ from an ensemble average of non-equilibrium exponential work, i.e.,
$\langle e^{-\beta W}\rangle =e^{-\beta\Delta F}$. The number of experimental
work values needed to reach a given accuracy of $\Delta F$ is determined by the
variance of $e^{-\beta W}$, denoted ${\rm var}(e^{-\beta W})$. We discover in
this work that ${\rm var}(e^{-\beta W})$ in both harmonic and an-harmonic
Hamiltonian systems can systematically diverge in non-adiabatic work protocols,
even when the adiabatic protocols do not suffer from such divergence. This
divergence may be regarded as a type of dynamically induced phase transition in
work fluctuations. For a quantum harmonic oscillator with time-dependent
trapping frequency as a working example, any non-adiabatic work protocol is
found to yield a diverging ${\rm var}(e^{-\beta W})$ at sufficiently low
temperatures, markedly different from the classical behavior. The divergence of
${\rm var}(e^{-\beta W})$ indicates the too-far-from-equilibrium nature of a
non-adiabatic work protocol and makes it compulsory to apply designed control
fields to suppress the quantum work fluctuations in order to test Jarzynski
equality.
