Extended Data Fig. 2: PReSPA spectroscopy.
From: Protecting a bosonic qubit with autonomous quantum error correction
a, b, Control pulse sequence for two-dimensional spectroscopy to find the resonance conditions for the PReSPA mixing comb and transmon comb. We prepare an even-parity Fock state (\(|0\rangle \), \(|2\rangle \), \(|4\rangle \), or \(|6\rangle \)), apply PReSPA for a fixed time (12 μs) with varying detunings of the transmon comb (Δq) and the mixing comb (Δm) in an attempt to activate dissipative photon addition. After a 1 μs wait time for the reservoir to relax, we either selectively π-pulse the transmon conditioned on cavity A being in the targeted final state (\(|1\rangle \), \(|3\rangle \), \(|5\rangle \), or \(|7\rangle \)) (a) or skip this pulse (for a background measurement, b), and proceed to read out the transmon state. The difference between the two measurements informs the likelihood of successful photon addition. c, Two-dimensional PReSPA spectroscopy data: probability of photon addition (colour scale) as a function of the comb detunings (Δq and Δm) for the \(|0\rangle \) to \(|1\rangle \) transition. Note that the linewidth of the four-wave-mixing transition is an order of magnitude greater than that of the transmon excitation owing to the short reservoir T1R. We can repeat this procedure to find all four sets of transition frequencies. d, Cartoon spectrum of PReSPA drive frequencies. Four transmon drives, left, and four mixing drives, right, compose PReSPA. The coloured ticks indicate the actual transition frequencies whereas the vertical black bars show the microwave drive frequencies in PReSPA. The transmon drive for the \(|0\rangle \) to \(|1\rangle \) conversion process is approximately at the Stark-shifted transmon frequency, \({\omega }_{{\rm{q}}}-{\varDelta }_{{\rm{Stark}}}\), and the \(|0\rangle \) to \(|1\rangle \) mixing drive is near \({\omega }_{{\rm{A}}}+{\omega }_{{\rm{R}}}-{\omega }_{{\rm{q}}}+{\varDelta }_{{\rm{Stark}}}\). Because of the equal frequency spacing η in each comb and the unequal frequency spacing between the transitions with different photon numbers (owing to the 6th-order nonlinearity, \({\chi }_{{\rm{q}}}^{{\prime} }\)), not all drives can be placed exactly on resonance. Experimentally, we settle for η slightly greater than 2χq, and Δq = Δm slightly smaller than ΔStark to compensate for the effect of \({\chi }_{{\rm{q}}}^{{\prime} }\).
