Stark Effect and Generalized Bloch-Siegert Shift in a Strongly Driven Two-Level System

A superconducting qubit was driven in an ultrastrong fashion by an oscillatory microwave field, which was created by coupling via the nonlinear Josephson energy. The observed Stark shifts of the “atomic” levels are so pronounced that corrections even beyond the lowest-order Bloch-Siegert shift are needed to properly explain the measurements. The quasienergies of the dressed two-level system were probed by resonant absorption via a cavity, and the results are in agreement with a calculation based on the Floquet approach.

Figure 1

Schematic of the experimental setup. (a) The single-Cooper-pair transistor qubit consists of a superconducting loop interrupted by two Josephson junctions that separate a small island. The two relevant charge states correspond to zero or one extra Cooper pairs on the island; (b) Except near the anticrossings, the energy eigenvalues depend nearly sinusoidally on the applied flux $\Phi$. The flux is driven likewise sinusoidally in time; (c) Detailed view of the circuit shows the microwave reflectometry readout via an $LC$ resonator formed by on-chip lumped circuit elements [27, 32].

Figure 2

Calculated quasienergies as a function of the bias flux ${\Phi }_b$ at ${\Phi }_L=0.26{\Phi }_0$, and with the experimental parameters obtained by microwave spectroscopy [27]; $E_{J0}/h=27.0\mathrm{GHz}$, ${\omega }_L/2\pi =6.11\mathrm{GHz}$, and $d=0.19$. The fully dressed states (red solid lines) are the eigenvalues of the Floquet matrix. The quasienergies repeat periodically in energy with period $\hslash {\omega }_L$. The level splittings are denoted by $\Delta$ and (${\omega }_L-\Delta$). The vertical green [medium gray] and blue [dark gray] arrows denote transitions induced by 3.5 GHz probe field. We also show the quasienergies of the bare states (${\Phi }_L=0$, dotted lines) and longitudinally dressed states (${\Omega }_k=0$, dashed lines).

Figure 3

(a) The measured resonance absorption of the probe signal plotted in the ${\Phi }_b-{\Phi }_L$ plane. Dark corresponds to higher absorption. The ${\Phi }_L$ scale is obtained by comparison to theory; (b) The landscape of the quasienergy splitting $\Delta$ calculated from the Floquet matrix of Hamiltonian (2) with the experimental parameters. The $k$-photon Rabi resonances appear as light (odd $k$) or dark (even $k$) tracks that are marked by index $k$ at small ${\Phi }_L$ and curve to the right with increasing ${\Phi }_L$. Figure 2 shows a cut along the horizontal line denoted by the black arrow. The blue [dark gray] and green [medium gray] lines denote the resonances in Eq. (3). The solid dots are the experimental resonances picked up from panel (a), or from the measured phase shift (two rightmost lines).

Figure 4

The shifts of the spectral lines due to the qubit driven by a strong field. The data points for each curve A–C are picked from the resonance lines in Fig. 3 marked correspondingly. The theoretical curves were produced with the rotating-wave approximation (red [light gray] lines), or with the full numerical result (green [medium gray]). For curve A, we show also the counterrotating Bloch-Siegert correction to the RWA result (blue [dark gray]). For curve C, the black line is the analytical solution for the Stark shift, $(3{\omega }_L+{\omega }_P)\{1-[J_0(\pi {\Phi }_L/{\Phi }_0){]}^{-1}\}$.