  In this work, we study the time-dependent behaviour of quantum correlations
of a system of an inverted oscillator governed by out-of-equilibrium dynamics
using the well-known Schwinger-Keldysh formalism in presence of quantum
mechanical quench. Considering a generalized structure of a time-dependent
Hamiltonian for an inverted oscillator system, we use the invariant operator
method to obtain its eigenstates and continuous energy eigenvalues. Using the
expression for the eigenstates, we further derive the most general expression
for the generating function as well as the out-of-time-ordered correlators
(OTOC) for the given system using this formalism. Further, considering the
time-dependent coupling and frequency of the quantum inverted oscillator
characterized by quench parameters, we comment on the dynamical behaviour,
specifically the early, intermediate and late time-dependent features of the
OTOC for the quenched quantum inverted oscillator. Next, we study a specific
case, where the system of inverted oscillator exhibits chaotic behaviour by
computing the quantum Lyapunov exponent from the time-dependent behaviour of
OTOC in presence of the given quench profile.
