Fig. 3: Multi-mode optimal control (MMOC) by single-port driving.

a MMOC pulses. A multi-section pulse with target equilibrium time t_f = 60 ns is applied to shortcut thermal equilibrium of both resonator and filter modes for different qubit states simultaneously. A different pulse is used to reset to the vacuum state in the end. b, c Measured output amplitudes (in mV) for the MMOC pulses with and without reset. The steady output signal is achieved for qubit state \(\left|0\right. \rangle\) (b) and \(\left|1\right. \rangle\) (c) about 30 ns later than the target time, likely due to the high driving amplitude and low-Q energy-storing components in the feedline, such as the impedance-matched Josephson parametric amplifier. This explanation is reinforced by the observation of a similar 30 ns tail with a far-detuned and high-amplitude driving which, in principle, will not excite the multi-mode cQED system (Supplementary Note 9). In c, the steady output decays over time due to the T₁ decay of the qubit. The error bar is the standard deviation of the points in the equilibrium states at 930 ns. The good final reset performance for the mixed qubit states demonstrates that MMOC works for both qubit states simultaneously. The corresponding IQ trajectories are plotted inset, from which we can infer the system undergoes highly non-equilibrium dynamics during the MMOC pulse. Trajectories begin at points indicated by black crosses. All data are processed in the same way as in Fig. 2.