We consider a system consisting of a qubit and a microwave transmission line that are coupled by a capacitor which, in turn, is modulated sinusoidally. The Unruh effect is the simultaneous production from vacuum of a pair of photons, one in the qubit and the other in the cavity. The dynamical Casimir effect is the production from vacuum of a pair of photons in the cavity. We analyze this qubit–cavity system and show that the system can be viewed as a pair of coupled quantum-mechanical oscillators and that both the Unruh effect and the dynamical Casimir effect are resonances of this coupled oscillator system. For the case where the cavity supports two propagating modes, in addition to the Unruh and dynamical Casimir effect at each of the supported modes, we predict a “paired Casimir effect,” where one photon is emitted in the cavity in each of two allowed modes, at the appropriate driving frequency. We also calculate analytical approximations to the driving frequencies for all three effects.