  We investigate the classical-quantum correspondence in the inverted harmonic
oscillator (IHO) system. It is shown that the out-of-time-order correlators
(OTOCs) which the initial states are located at any position in the IHO system
possess the same exponential growth rates (EGRs) as that at the saddle point,
and their EGRs are twice the classical lyapunov exponent (CLE) of the saddle
point. Through the time evolution of mean photon number and the OTOCs, we
exhibit that the classical-quantum correspondence in the IHO system not only
depends on the initial system photon number, but also on the central positions
of the initial states in the phase space. Moreover, we use the Husimi Q
function to visualize the quantum wave packets during the OTOCs grow
exponentially.
