Pulse imaging and nonadiabatic control of solid-state artificial atoms

Transitions in an artificial atom, driven nonadiabatically through an energy-level avoided crossing, can be controlled by carefully engineering the driving protocol. We have driven a superconducting persistent-current qubit with a large-amplitude radio-frequency field. By applying a biharmonic wave form generated by a digital source, we demonstrate a mapping between the amplitude and phase of the harmonics produced at the source and those received by the device. This allows us to image the actual wave form at the device. This information is used to engineer a desired time dependence, as confirmed by the detailed comparison with a simulation.

Figure 1

(Color online) (a) Schematic experimental setup. The device was placed at 20 mK (see Ref. 13 for details). (b) Energy diagram determined by amplitude spectroscopy (Ref. 16). The energy gaps are small on the scale of the figure (${\Delta }_{0,q}/h=0.013$, 0.090, 0.40, and 2.2 GHz for $q=0,\dots ,3$, respectively). Below: flux excursion $\delta f\left(t\right)$ during the short biharmonic pulse that probes the time-dependent dynamics in Fig. 3. (c),(d) Biharmonic wave-form examples (1). $\theta$ values are indicated by dashed vertical lines; solid blue (dotted red) lines denote maxima (minima). The lower left wave form corresponds to the one in Fig. 3b.

Figure 2

(Color online) Calibration of rf amplitude and phase using long pulses. (a) Spectroscopic diamonds from single-harmonic driving. The points $A_{1,2}^{\ast }$ of optimal cooling (arrows) are used for amplitude calibration at each frequency. (b) Diamonds when driving with the biharmonic wave form $\left\{1/4,\pi \right\}$ at $\omega /2\pi =45\text{MHz}$ [see Eq. (1)]. The arrows point out the diamond edges corresponding to the quartic extremum reaching the ${\Delta }_{0,q}$ crossings and the quadratic reaching ${\Delta }_{p,0}$. (c) Measurement along the dashed line in (b) for varying ${\theta }_g$. The quartic extremum $\theta =\pi$ is obtained at a cusp (solid white line) [cf. Fig. 1d].

Figure 3

(Color online) Time-dependent response. (a),(b) Population in state $|L⟩$ at the final time $\Delta t$ of a variable-length flattop pulse $\left\{1/4,\pi \right\}$, reaching through ${\Delta }_{0,2}$ as in Fig. 1b. The amplitude $A$ is indicated by the rightmost dotted line in Fig. 2b. The edges of the curved fringes provide a direct image of the pulse (black dotted line), obtained experimentally by comparing (a) and (b). (c) Population in state $|L⟩$ after a return trip $\left(\Delta t=22\text{ns}\right)$ through ${\Delta }_{0,2}$. (d),(e) Like (b) and (c) but with a wave form calibrated using the long pulses in Fig. 2, thus failing to give the correct wave form. (f) The lower (red) and upper (blue) sets of data show the spacings of the Stückelberg-interference fringes in (c) and (e), respectively, with fits to the $N^{4/5}$ power law (fringes $N=4–35$ and III–XXI fitted). (g) The residuals indicate that the fringes in (e) (blue ×) do not fit well, while those in (c) (red $+$) fit better; this confirms that the calibrated pulse better approximates the $\left\{1/4,\pi \right\}$ wave form.