  We study the vacuum Rabi splitting of a qubit ultrastrongly coupled to a
high-$Q$ cavity mode and a radiation reservoir. Three methods are employed: a
numerically exact variational approach with a multiple Davydov ansatz, the
rotating-wave approximation (RWA), and the transformed RWA. Agreement between
the variational results and the transformed RWA results is found in the regime
of validity of the latter, where the RWA breaks down completely. We illustrate
that the Lamb shift plays an essential role in modifying the vacuum Rabi
splitting in the ultrastrong coupling regime, leading to off-resonant
qubit-cavity coupling even though the cavity frequency equals to the bare
transition frequency of the qubit. Specifically, the emission spectrum exhibits
one broad low-frequency peak and one narrow high-frequency peak in the presence
of relatively weak cavity-qubit coupling. As the cavity-qubit coupling
increases, the low-frequency peak narrows while the high-frequency peak
broadens until they have similar widths.
