Ancillary Qubit Spectroscopy of Vacua in Cavity and Circuit Quantum Electrodynamics

We investigate theoretically how the spectroscopy of an ancillary qubit can probe cavity (circuit) QED ground states containing photons. We consider three classes of systems (Dicke, Tavis-Cummings, and Hopfield-like models), where nontrivial vacua are the result of ultrastrong coupling between $N$ two-level systems and a single-mode bosonic field. An ancillary qubit detuned with respect to the boson frequency is shown to reveal distinct spectral signatures depending on the type of vacua. In particular, the Lamb shift of the ancilla is sensitive to both ground state photon population and correlations. Backaction of the ancilla on the cavity ground state is investigated, taking into account the dissipation via a consistent master equation for the ultrastrong coupling regime. The conditions for high-fidelity measurements are determined.

Figure 1

A sketch of a cavity (circuit) QED system $S$ consisting of a single-mode resonator coupled to $N$ two-level (artificial) atoms. An ancillary qubit $M$ is coupled to the resonator boson mode. The spectroscopy of the ancilla is used to probe quantum ground state properties of $S$.

Figure 2

Left panels: excitation energies for the three considered systems $S$ versus the coupling $\lambda$ between the boson field and the $N$ atoms. Right panels: the Lamb shift of the ancillary qubit transition. The red dashed lines are calculated via Eq. (3). Top panels: Dicke system with $N=3$, ${\omega }_c={\omega }_0$, ${\omega }_M=2.75{\omega }_c$, $g_M=0.1{\omega }_c$. Middle panels: Tavis-Cummings system with $N=3$, ${\omega }_c={\omega }_0$, ${\omega }_M=2.75{\omega }_c$, $g_M=0.1{\omega }_c$. Bottom panels: Hopfield system with $N=3$, ${\omega }_c={\omega }_0$, ${\omega }_M=6.75{\omega }_c$, $g_M=0.1{\omega }_c$, $D={\lambda }^2/{\omega }_0$.

Figure 3

Top panels: Lamb shift of the ancillary qubit (black dots) versus $\lambda$ for $\mathrm{N}=1$, 3, 10, and 30. Red dashed lines are obtained via Eq. (3). Bottom panel: Ground state fidelity ${\mathcal{F}}_G$. Left panels: Dicke model with ${\omega }_c={\omega }_0$, ${\omega }_M=2.75{\omega }_c$, $g_M=0.1{\omega }_c$. Right panels: Hopfield model with ${\omega }_c={\omega }_0$, ${\omega }_M=6.75{\omega }_c$, $g_M=0.1{\omega }_c$, $D={\lambda }^2/{\omega }_0$.

Figure 4

Excited state population of the ancilla qubit $M$ versus the coherent drive frequency ${\omega }_p$ for different values of collective coupling $\lambda$. Left panel: Dicke model with ${\mathrm{\Omega }}_p=0.5{\gamma }_M$. Right panel: Tavis-Cummings model with ${\mathrm{\Omega }}_p=0.2{\gamma }_M$. Dissipation parameters: ${\gamma }_M={\gamma }_c={\gamma }_0=0.01{\omega }_c$ for the left panel and $0.005{\omega }_c$ for the right panel. The other parameters are as in Fig. 2. The white line corresponds to the analytical curve in Eq. (3).

Figure 5

Measurement fidelity $\mathcal{F}$ (see definition in the text) for different values of $\lambda$ for the Dicke model and for different dissipation rates ${\gamma }_c={\gamma }_0=\eta {\gamma }_M$, and ${\gamma }_M=0.01{\omega }_c$. Solid line: $\eta =0$. Dashed line: $\eta =1$. Dot-dashed line: $\eta =10$. Other parameters as in Fig. 4.