Dynamical phase transition in the transverse field Ising chain characterized by the transverse magnetization spectral function

We study the response of a quantum Ising chain to transverse field oscillations in the asymptotic state attained after a quantum quench. We show that for quenches across a quantum phase transition, the dissipative part of the response at low frequencies is negative, corresponding to energy emission up to a critical frequency omega*. The latter is found to be connected to the time period t* of the singularities in the Loschmidt echo (t* = 2 pi/omega*) signaling the presence of a dynamical quantum phase transition. This result suggests that a linear-response experiment can be used to detect this kind of phenomenon.