Low-temperature behavior of transmission phase shift across a Kondo correlated quantum dot

We study the transmission phase shift across a Kondo correlated quantum dot in a GaAs heterostructure at temperatures below the Kondo temperature ($T < T_{\rm K}$), where the phase shift is expected to show a plateau at $π/2$ for an ideal Kondo singlet ground state. Our device is tuned such that the ratio $Γ/U$ of level width $Γ$ to charging energy $U$ is quite large ($\lesssim 0.5$ rather than $\ll 1$). This situation is commonly used in GaAs quantum dots to ensure Kondo temperatures large enough ($\simeq 100$ mK here) to be experimentally accessible; however it also implies that charge fluctuations are more pronounced than typically assumed in theoretical studies focusing on the regime $Γ/U \ll 1$ needed to ensure a well-defined local moment. Our measured phase evolves monotonically by $π$ across the two Coulomb peaks, but without being locked at $π/2$ in the Kondo valley for $T \ll T_{\rm K}$, due to a significant influence of large $Γ/U$. Only when $Γ/U$ is reduced sufficiently does the phase start to be locked around $π/2$ and develops into a plateau at $π/2$. Our observations are consistent with numerical renormalization group calculations, and can be understood as a direct consequence of the Friedel sum rule that relates the transmission phase shift to the local occupancy of the dot, and thermal average of a transmission coefficient through a resonance level near the Fermi energy.