Sideband Transitions and Two-Tone Spectroscopy of a Superconducting Qubit Strongly Coupled to an On-Chip Cavity

Sideband transitions are spectroscopically probed in a system consisting of a Cooper pair box strongly but nonresonantly coupled to a superconducting transmission line resonator. When the Cooper pair box is operated at the optimal charge bias point, the symmetry of the Hamiltonian requires a two-photon process to access sidebands. The observed large dispersive ac-Stark shifts in the sideband transitions induced by the strong nonresonant drives agree well with our theoretical predictions. Sideband transitions are important in realizing qubit-photon and qubit-qubit entanglement in the circuit quantum electrodynamics architecture for quantum information processing.

Figure 1

(a) Dispersive dressed states energy level diagram for qubit states $|g⟩$ and $|e⟩$ with separation ${\omega }_a$ and single mode cavity photon states $|n=0,1,2,\dots ⟩$ with separation ${\omega }_r$. Red (${\omega }_{\mathrm{red}}$) and blue (${\omega }_{\mathrm{blue}}$) sideband transitions are indicated. (b) Two-tone scheme for sideband transitions. Fixed ac-tone ${\omega }_{\mathrm{ac}}$ detuned by $\delta$ from cavity combined with spectroscopy tone ${\omega }_s={\omega }_a\pm \delta$ induces sideband transitions. (c) Measurement setup for two-tone spectroscopy, compare with setups in Refs. 14, 18. An extra phase coherent microwave source at the ac tone is applied to the cavity input.

Figure 2

(a) Calculated Lorentzian cavity transmission spectrum over large frequency range. The cavity drives at frequencies ${\nu }_s\approx {\nu }_a$, ${\nu }_{\mathrm{rf}}\approx {\nu }_r$, and ${\nu }_{\mathrm{ac}}$ and the cavity transmission at these frequencies are indicated. (b) Measured cavity spectrum (dots) and fit (line) around cavity frequency.

Figure 3

(a) Density plot of measured cavity phase shift $\varphi$ (white: 70°, black: 0°) vs ${\omega }_s$ and ${\omega }_{\mathrm{ac}}$. (b) Measured $\varphi$ vs ${\omega }_s$ for $\delta /2\pi =50\mathrm{MHz}$ (dots) as indicated by arrows in (a) and fit to linear combination of three independent Lorentzian line shapes (solid line).

Figure 4

(a) Two-tone red and blue sidebands (red/blue dots) and fundamental (black dots) qubit transition frequencies for fixed $P_s$ and $P_{\mathrm{ac}}$ varying ${\omega }_{\mathrm{ac}}$. Lines are fits to Eqs. (3, 4). (b) For fixed ${\omega }_{\mathrm{ac}}$ detuned by 50 MHz from ${\omega }_r$ and varying $P_{\mathrm{ac}}$. Measurements are done at fixed measurement frequency ${\omega }_{\mathrm{rf}}={\omega }_r$ and power $P_{\mathrm{rf}}$ weakly populating the resonator with $n_{\mathrm{rf}}\approx 2$ photons.

Figure 5

(a) As Fig. 4b but for fixed ${\omega }_{\mathrm{ac}}$, $P_{\mathrm{ac}}$, and varying $P_s$. (b) Master equation simulation of the qubit inversion $⟨{\sigma }_z⟩$ as a function of spectroscopy frequency ${\omega }_s/2\pi$ and the spectroscopy power $\log ({ϵ}_s/2\pi {)}^2$ on a logarithmic scale. Blue: $⟨{\sigma }_z⟩=-1$; red: $⟨{\sigma }_z⟩=0$. The white lines are obtained from Eqs. (3, 4). The parameter values used in the simulation are those quoted in the text.