Thermal and Residual Excited-State Population in a 3D Transmon Qubit

Remarkable advancements in coherence and control fidelity have been achieved in recent years with cryogenic solid-state qubits. Nonetheless, thermalizing such devices to their milliKelvin environments has remained a long-standing fundamental and technical challenge. In this context, we present a systematic study of the first-excited-state population in a 3D transmon superconducting qubit mounted in a dilution refrigerator with a variable temperature. Using a modified version of the protocol developed by Geerlings et al., we observe the excited-state population to be consistent with a Maxwell-Boltzmann distribution, i.e., a qubit in thermal equilibrium with the refrigerator, over the temperature range 35–150 mK. Below 35 mK, the excited-state population saturates at approximately 0.1%. We verified this result using a flux qubit with ten times stronger coupling to its readout resonator. We conclude that these qubits have effective temperature $T_{\text{eff}}=35\mathrm{mK}$. Assuming $T_{\text{eff}}$ is due solely to hot quasiparticles, the inferred qubit lifetime is $108\mu \mathrm{s}$ and in plausible agreement with the measured $80\mu \mathrm{s}$.

Figure 1

(a) The readout signal versus readout frequency. Three well-separated peaks are visible, corresponding to different qubit states. We read out state $|e⟩$. (b) Modified experiment protocol using $e\text{−}f$ Rabi driving. The excited state is populated by a ${\pi }^{g\text{−}e}$ pulse (left panel) or environmental excitation (right panel). (c) Observed $e\text{−}f$ Rabi oscillations at 150 mK. The blue trace determines $A_{\text{ref}}$, while the red trace determines $A_{\text{sig}}$. When $A_{\text{sig}}$ is small, only two points on the blue trace ($R1$, $R2$) and the red trace ($S1$, $S2$) are measured. See text for details.

Figure 2

Calibration measurement of excited state population (percent), pumped using a small fraction of a ${\pi }^{g\text{−}e}$ pulse. The data can be fit to a linear $y=x+b$ function, where $b=0.067\%$, with 95% confidence bounds of (0.025%,0.011%).

Figure 3

(a) $P_{|e⟩}^{\exp }$ ratio [Eq. (1)] versus temperature, 15–150 mK. Experimental data are obtained through fitting a $1\text{−}\mu \mathrm{s}$ Rabi trace (blue points) or the two-point method (red points). Solid lines are calculated $P_{|e⟩}^{\exp }$ (blue line) and $P_{|e⟩}$ (red line) based on the Maxwell-Boltzmann distribution for the lowest four energy levels (see text). (b) Zoom: $P_{|e⟩}^{\exp }$ ratio versus temperature, 15–60 mK. In this limit, $P_{|e⟩}^{\exp }$ is a good estimator for $P_{|e⟩}$ The data saturate to 0.1% at lower temperatures (purple dashed line) with an inferred effective temperature of 35 mK.