Observation of the Bloch-Siegert Shift in a Qubit-Oscillator System in the Ultrastrong Coupling Regime

We measure the dispersive energy-level shift of an $LC$ resonator magnetically coupled to a superconducting qubit, which clearly shows that our system operates in the ultrastrong coupling regime. The large mutual kinetic inductance provides a coupling energy of $\approx 0.82\mathrm{GHz}$, requiring the addition of counter-rotating-wave terms in the description of the Jaynes-Cummings model. We find a 50 MHz Bloch-Siegert shift when the qubit is in its symmetry point, fully consistent with our analytical model.

Figure 1

Circuit layout and images of the device. (a) Schematic of the measurement setup. The interdigitated capacitor of the $LC$ resonator can be seen in the center of the optical image, with the circuitry of the two SQUIDs next to it (top left and bottom right); $C_r/2\simeq 0.25\mathrm{pF}$ and $L_r\simeq 1.5\mathrm{nH}$. (b) Scanning electron micrograph (SEM) picture of the SQUID circuit. The readout line is made to overlap with a big volume of AuPd and Au to thermalize the quasiparticles when the SQUID switches. (c) SEM picture of the qubit with the SQUID on top. On the right of the picture the coupling wire to the resonator of length $l$ can be seen.

Figure 2

Measurement scheme and energy-level diagram. (a) Schematic of the measurement protocol to perform qubit spectroscopy. (b) JC ladder depicting the energy-level structure of the system of a flux qubit coupled to an $LC$ resonator. The levels are drawn for the case $\delta ={\omega }_q-{\omega }_r<0$. The arrows represent the level-transitions that are visible in the spectrum. The dashed lines represent the uncoupled qubit and resonator states. ${\delta }_q$ and ${\delta }_r$ are the dispersive shifts that the qubit and the resonator induce to each other.

Figure 3

Spectrum of the flux qubit coupled to the $LC$ resonator. An avoided-level crossing is observed at a frequency of 8.13 GHz. The weak transition near 8 GHz is associated with excited photons due to thermal population of the qubit excited state ($T_{\mathrm{eff}}\sim 100\mathrm{mK}$ at $\sim 4–5\mathrm{GHz}$ energy splitting). (Inset) Zoom in around the resonance between qubit and oscillator. The splitting on resonance is $2g\sin (\theta )/2\pi \simeq 0.9\mathrm{GHz}$.

Figure 4

Bloch-Siegert shift. (a) Spectrum in proximity to the resonator frequency obtained using lower driving power than in Fig. 3 and flux pulses [16]. The solid black line is the fit of Eq. (1) and the dashed green line is a plot of the JC model (Eq. (1) without counterrotating terms). The dotted line indicates the bare resonator frequency ${\omega }_r$. A clear deviation between the dashed line and the data can be observed around the symmetry point of the qubit. A transition associated with thermal population of the qubit excited state can be observed around 8 GHz. (b) Difference between measurement (blue dots) and the prediction of the JC model (dashed green line). The solid black curve is the same as the solid black curve in (a) and the dashed red curve represents ${\lambda }_{1,g}-{\lambda }_{0,g}$. All the curves are subtracted from the JC model. The blue dots are peak values extracted from Lorentzian fits to frequency scans at fixed flux, with the error bars representing the full width at half maximum of each Lorentzian.