Fig. 4.

Theoretical predictions for the observation of coherent exchange oscillations. a The value of exchange energy, J, as a function of tunnel coupling, t_c and detuning, $ϵ$. The boundary separating the two-electron product states with singlet-triplet states occurs where the difference in magnetic field between the two qubits, ΔB_z is equal to the exchange energy. For donor-based systems ΔB_z is dominated by the donor hyperfine strength, and is equal to A for a 1P−1P system (solid green line), and can take the two values A/2 or 3A/2 for a 2P−1P system (dashed and dotted green lines respectively) dependent on the nuclear spin orientation (examples shown in inset). We assume the bulk 1P value for the hyperfine, A = 117.53 MHz. The dashed blue line indicates the values of J accessible for the current device with t_c ~ 200 MHz. b Theoretical prediction of coherent exchange oscillations for a 2P−1P device in natural silicon with tunnel coupling t_c = 2.5 GHz. The two-electron state is initialised as ${∣\uparrow \downarrow ⟩}_{}^{}$ at a point where the exchange energy is negligible, and subsequently a non-adiabatic detuning pulse is applied to $ϵ$ = −25 GHz (circle marker in a). We have assumed voltage noise equivalent to 850 MHz along the detuning axis, $ϵ$ (obtained from measurements) as well as a single electron $T_2^{*}=55\mathrm{ns}$ measured in previous works³². From this result an oscillation frequency ν and dephasing time τ_d are extracted. c The product of oscillation frequency, ν and dephasing time, τ_d as a function of tunnel coupling and detuning. The green dashed line represents the boundary between product and singlet-triplet eigenstates of the two-electron system. The Bloch sphere cross sections indicate the relative magnitudes of ΔB_z (purple) and J (blue) in different regions. d Theoretical prediction of ντ_d along the line ΔB_z = J as a function of tunnel coupling for a 2P−1P double quantum dot. Solid (dashed) line shows results including (excluding) the ²⁹Si Overhauser field