  We study the emission spectra of dissipative Rabi and Jaynes-Cummings models
in the non-Markovian and ultrastrong coupling regimes. We have derived a
polaron-transformed Nakajima-Zwanzig master equation (PTNZE) to calculate the
emission spectra, which eliminates the well known limitations of the Markovian
approximation and the standard second-order perturbation. Using the
time-dependent variational approach as benchmark, the PTNZE is found to yield
accurate emission spectra in certain ultrastrong coupling regimes where the
standard second-order Nakajima-Zwanzig master equation breaks down. It is shown
that the emission spectra of the dissipative Rabi and Jaynes-Cummings models
are in general asymmetric under various initial conditions. Direct comparisons
of spectra for the two models illustrate the essential role of the qubit-cavity
counter-rotating term and the spectra features under different qubit-cavity
coupling strengths and system initial conditions.
