Extracavity quantum vacuum radiation from a single qubit

We present a theory of the quantum vacuum radiation that is generated by a fast modulation of the vacuum Rabi frequency of a single two-level system strongly coupled to a single cavity mode. The dissipative dynamics of the Jaynes-Cummings model in the presence of antirotating-wave terms is described by a generalized master equation including non-Markovian terms. Peculiar spectral properties and significant extracavity quantum vacuum radiation output are predicted for state-of-the-art circuit cavity quantum electrodynamics systems with superconducting qubits.

Figure 1

(Color online) Left panel: sketch of the system under consideration. A single two-level system (qubit) is strongly coupled to a single cavity mode. A possible realization of such a device consists of a Cooper pair quantum box embedded in a microwave resonator. Right panel: schematic representation of the JC ladder of eigenstates of the isolated system in the absence of modulation, dissipation and anti-rotating-wave terms. In this limit, the eigenstates $|n,\pm ⟩=\left(|n,\text{g}⟩\pm |n-1,\text{e}⟩\right)/\sqrt{2}$ have energies $E_{n,\pm }=n{\omega }_0\pm \sqrt{n}{g}_0$.

Figure 2

(Color online) Extracavity photon emission rate $R_{\text{em}}$ (units of ${\omega }_0$) as a function of the modulation frequency, ${\omega }_{\text{mod}}$, for a modulation amplitude of the vacuum Rabi frequency $\Delta g/\gamma =0.1$. Parameters: ${\omega }_0={\omega }_{\text{ge}}$; $\gamma ={\gamma }_{\text{cav}}={\gamma }_{\text{ge}}=0.002{\omega }_0$; $g_0=0.02{\omega }_0$. For comparison, the dashed line shows the extracavity emission rate ${\gamma }_{\text{cav}}{N}_{\text{in}}$ (where $N_{\text{in}}$ is the steady-state intracavity photon number) that would be predicted by the Markovian approximation: note the unphysical prediction of a finite value of the emission even far from resonance. The inset shows the dependence of the photon emission rate on the modulation amplitude, calculated both numerically and analytically (see later in the text), for ${\omega }_{\text{mod}}={\omega }_{2,-}$.

Figure 3

(Color online) Spectral density $\mathcal{S}\left(\omega \right)$ of extracavity photons emitted per unit time for given values of the modulation frequency: ${\omega }_{\text{mod}}={\omega }_{2,-}$ (top panel), ${\omega }_{\text{mod}}={\omega }_{2,+}$ (bottom panel). The modulation amplitude is $\Delta g/\gamma =0.1$. The insets illustrate the optical transitions responsible for the different emission lines that are visible in the main panels. For the sake of clarity, the level spacings in the insets are not in scale.