  A quenched second order phase transition is modeled by an effective
$\Phi^4$-theory with a time-dependent Hamiltonian $\hat{H} (t)$, whose symmetry
is broken spontaneously in time. The quantum field evolves out of equilibrium
(nonequilibrium) during the phase transition as the density operator
significantly deviates from $\hat{\rho} (t) = e^{- \beta \hat{H} (t)}/Z_H$. The
recently developed Liouville-von Neumann (LvN) method provides various quantum
states for the phase transition in terms of a complex solution to the
mean-field equation, which is equivalent to the Gaussian effective potential in
the static case and to the time-dependent Hartree-Fock equation in the
nonequilibrium case. Using the multiple scale perturbation theory (MSPT) we
solve analytically the mean-field equation to the first order of coupling
constant and find the quantum states during the quenched second order phase
transition. We propose a renormalization scheme during the process of phase
transition to regularize the divergences, which originate from the mode
coupling between hard and hard modes or between the soft and hard modes. The
effect of mode coupling is discussed.
