Observation of the Bloch-Siegert shift in a driven quantum-to-classical transition

We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for a strong drive (classical phase) it presents an oscillatory behavior as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum.

Figure 1

Schematic of the experimental setup. (a) Energy levels of a dispersive JC system. The vacuum ac Stark shift is given by ${\chi }_0=g^2/({\omega }_0-{\omega }_{\mathrm{c}})$. (b) Optical micrograph of the main elements of the sample (false colors). (c) We drive the cavity with a detuned drive frequency ${\omega }_{\mathrm{d}}<{\omega }_{\mathrm{c}}$. The spectrum of the system is monitored by a weak probe with frequency ${\omega }_{\mathrm{p}}$, which is swept within the window $[4.36,4.39]$ GHz. (d) In the $T_1$ measurements, the cavity is driven continuously at ${\omega }_{\mathrm{d}}$. In the steady state, a long $2\mu \mathrm{s}$ microwave pulse at the ac Stark shifted transmon frequency ${\omega }_{\mathrm{q}}\gg {\omega }_{\mathrm{c}}$ is applied to the gate line of the transmon. When the pulse ends, the decay traces of the system relaxing back to the steady state are monitored in the time domain using a homodyne detection scheme.

Figure 2

Numerical simulation of the spectrum for a two-level system, corresponding to experimental values ${\omega }_{\mathrm{c}}/\left(2\pi \right)=4.376$ GHz, ${\omega }_0/\left(2\pi \right)=5.16$ GHz, $g/\left(2\pi \right)=80$ MHz, ${\omega }_{\mathrm{d}}/\left(2\pi \right)=4.350$ GHz, and $\kappa /\left(2\pi \right)=4.0$ MHz. (a) Probe reflection coefficient $\left|\mathrm{\Gamma }\right|$ as a function of probe frequency ${\omega }_{\mathrm{p}}$ and cavity occupation $N_{\mathrm{c}}$. The locations of resonance in the RWA (blue) and in the CHRW (green) approximations are shown as well. (b) Reflection coefficients for the three different probe frequencies indicated in (a) with dashed vertical lines.

Figure 3

Frequency renormalization and Bloch-Siegert effect. (a) Measured reflection coefficient $\left|\mathrm{\Gamma }\right|$ as a function of the probe frequency ${\omega }_{\mathrm{p}}$ and the input power $P_{\mathrm{in}}$ at the sample (or equivalent average number of cavity quanta $N_{\mathrm{c}}$). Numerically calculated resonance frequencies are shown with colored dots for $N=2,5,7$ transmon states. (b) Measured relaxation time $T_1$ (left axis) as a function of $N_{\mathrm{c}}$. On the right axis, we show the ac Stark shifted qubit transition frequency ${\omega }_{\mathrm{q}}$ in the same interval. For $N_{\mathrm{c}}\le 1$, a standard perturbative quantum treatment [47, 48, 49] can be used to explain the data (continuous lines), while above these values the increase of $T_1$ can be modeled using a classical approach [33, 50, 51]. (c) Quantum-to-classical transition in the driven transmon-cavity system. We show the numerically calculated average transmon state occupation $\langle N\rangle$, occupation number fluctuations $\langle \delta N^2\rangle$, and the order parameter $\mathrm{\Xi }$ as a function of the cavity occupation.

Figure 4

The Bloch-Siegert effect in the quantum and classical regimes. (a) Measured reflection coefficient $\left|\mathrm{\Gamma }\right|$ as a function of ${\omega }_0$ and ${\omega }_{\mathrm{p}}$, taken at high power (upper panel) and low power (lower panel). The strongest resonance minima extracted from the data are marked with dots. The lower panel demonstrates the vacuum BS shift of ${\chi }_{\mathrm{BS}}=0.7$ MHz, displayed as the difference between the calculated cavity frequency with only the analytic vacuum Stark shift (${\omega }_{\mathrm{c}}-{\chi }_0$, dashed white line) and with the analytic vacuum BS shift included (${\omega }_{\mathrm{c}}-{\chi }_0-{\chi }_{\mathrm{BS}}$, solid black line). (b) Bloch-Siegert shift as a function of driving power (corresponding to $N_{\mathrm{c}}=10–1650$) in the classical regime for ${\omega }_{\mathrm{d}}/\left(2\pi \right)=4.350$, 4.355, and 4.360 GHz. We show the experimental contour for the reflection coefficient and compare the resonance locations with the classical-approximation formula for ${\omega }_{+}$ in Eq. (3) and with the numerical results with $N=5,7$ ($N=7$ is calculated also in the RWA).