Do single-shot projective readouts necessarily estimate the $T_1$ lifetime ?

Abstract

When single-shot qubit readout protocols are adapted for multilevel systems, theoretical $T_1$ lifetime calculations often fall short of capturing the experimental lifetime trends. We identify {\it extrinsic} population dynamics as the fundamental origin of this disparity, establishing that the lifetime estimates can, in certain operating regions, be distinct from the intrinsic $T_1$ time. We clarify these aspects with an integrated theory to address recent measurements [Nat. Nano, 20, 494, (2025)] on spin-valley states in bilayer graphene. While confirming that phonon and Johnson noise are the dominant intrinsic sources, we show that the inclusion of extrinsic factors provide the critical match to the experimental estimates. The extrinsic factors also effectuate violations of generalized Mathiessen's rules also. With an improved handle on the design space, a revised readout protocol to estimate the $T_1$ lifetime of the valley qubit is proposed.

Figures (7)

• Figure 1: Preliminaries. (a) ERT based on energy-selective tunneling between the dot and the reservoir. (b) Representative energy spectrum of the four spin--valley states as a function of $B_{\perp}$. The first anticrossing occurs when $E_Z$ equals $\Delta_{\mathrm{SO}}$, with $t_v$ inducing hybridized eigenstates. A second, weaker anticrossing appears at higher fields due to intravalley mixing, leading to a modified state composition. The intrinsic relaxation channels, spin ($\gamma_s$), valley ($\gamma_v$), and Kramers ($\gamma_k$) are indicated. (c) Intrinsic pathway away from, and near the anticrossing, which involves state--mixing. (d) Extrinsic factors: population dynamics which modify the experimentally inferred lifetime.
• Figure 2: Experimental match, intrinsic and extrinsic factors. The dependence of intrinsic spin, valley and Kramers lifetimes on $B_\perp$ with contributions from phonon and Johnson noise in the presence of $t_v$. The experimentally measured lifetimes ($T_{1m}$) and the corresponding readout protocols (inset). Shaded regions mark the regions where $t_v$ and thermal broadening become relevant. The effective lifetimes ($\gamma_{\mathrm{effective}}^{-1}$) obtained from the integrated framework capture the observed behavior. Parameters used are $g_1=50~\mathrm{eV}$ and $g_2=3~\mathrm{eV}$ (s, k), $g_1=50~\mathrm{eV}$ and $g_2=2.8~\mathrm{eV}$ (v), $J_s=1.8~\mathrm{Hz}$, $J_v=2.5~\mathrm{Hz}$, $J_k=0.02~\mathrm{Hz}$, $R=30~\mathrm{nm}$, $U_0=40~\mathrm{meV}$, $\Delta_{\mathrm{SO}}=64~\mu\mathrm{eV}$, $t_v=1~\mu\mathrm{eV}$, and $g_v=14.5$denisov2025spin.
• Figure 3: Extrinsic effects. Bi-exponential distribution of the ES probability versus $T_{\mathrm{load}}$ at $B_{\perp}=55~\mathrm{mT}$. Charge jumps occasionally load the electron into the excited state $\lvert K^{-}\!\downarrow\rangle$ (top), producing fast valley-mediated relaxation. In the absence of such jumps, the system relaxes through the Kramers channel $\lvert K^{+}\!\downarrow\rangle$ (bottom).
• Figure 4: Effect of $T_e$ and $t_v$ on the $1/T_1$ spikes. (a) Simulated $1/T_1$ versus $T_e$ at the spin/valley operating point where a spike is observed, which is not a typical signature. The $T_e$ sweeps confirm a smooth increase of $1/T_1$ without any spike if the potential fluctuation is not included. (b) The $1/T_1$ versus $B_{\perp}$ around the spin--valley anticrossing. In this region, the sharp rise in $1/T_1$ is governed by $t_v$. Spectral constraints result in $t_v \lesssim 2~\mu\mathrm{eV}$.
• Figure 5: Designing readouts. Predicted valley $1/T_{1}$ versus $B_{\perp}$ upto $1~\mathrm{T}$. The $B_{\perp}$ dependence follows the $\gamma_v$ with a weak non-monotonic shoulder near the second anticrossing due to $t_v$. The inset depicts the readout protocol.