Fig. 2: Phonon-mediated quantum state transfer and process tomography.

a Measured Q₁ excited state population \({P}_{e}^{{Q}_{1}}\) as a function of time and Q₁ bare frequency, with coupler G₁ at an intermediate coupling κ₁/2π = 2.4 MHz (measured at 3.976 GHz) and G₂ set to zero coupling. In this configuration, Q₁’s energy relaxation is dominated by phonon emission via UDT₁, followed by traveling phonon dynamics. The white and red dashed lines indicate the unidirectional and bidirectional working frequencies, respectively (see text); inset shows the qubit excitation and measurement pulse sequence. b Quantum state transfer via a traveling phonon at the unidirectional (left) and bidirectional (right) operating frequencies. Q₂’s final population is 4.5 times smaller for the bidirectional transfer compared to the unidirectional transfer, in line with simulations. Green solid lines are from a master equation simulation. Inset: pulse sequence. For either process, Q₁’s emission rate is set to \({\kappa }_{c}^{{{{{{\rm{uni| bi}}}}}}}/2\pi =10| \,{{\mbox{6}}}\,\ \,{{\mbox{MHz}}}\,\), corresponding to a 81∣138 ns full-width-at-half-maximum (FWHM) phonon wave packet. c Quantum process tomography for the unidirectional and bidirectional regimes, with process fidelities of \({{{{{{\mathcal{F}}}}}}}_{{{{{{\rm{uni}}}}}}}={{{{{\rm{Tr}}}}}}\,({\chi }_{\exp } \cdot {\chi }_{{{{{{\rm{ideal}}}}}}})=82\pm \,{{\rm{0.3}}}{\%}\) and \({{{{{{\mathcal{F}}}}}}}_{{{{{{\rm{bi}}}}}}}=39\pm \,{{\rm{0.3}}}{{{\%}}}\,\), respectively. Red solid lines show values expected for an ideal transfer; black dashed lines show master equation simulations, taking into account finite qubit coherence and phonon channel losses. Uncertainties are standard deviations from the mean.