Readout-induced degradation of transmon lifetimes: interplay of TLSs and qubit spectral reshaping

Abstract

Measurement backaction degrades dispersive readout of superconducting qubits even at modest drive strengths, often via the reduction of qubit lifetimes during readout. In this work, we theoretically and experimentally study this degradation and show how it can result from the interplay between detuned two-level systems (TLSs) and a drive-renormalized qubit spectrum. For modest to strong readout, the qubit emission spectrum becomes non-Lorentzian and depends sensitively on the readout drive frequency (even when measurement rate is fixed). We combine the readout-modified qubit emission spectrum with time-dependent perturbation theory to predict qubit lifetimes in the presence of a TLS bath. Master equation simulations and experimental measurements on a frequency-tunable transmon confirm these predictions quantitatively. In particular, we find that driving at the resonator frequency associated with the qubit ground state yields the narrowest qubit emission spectrum and the least lifetime degradation for a fixed measurement rate, providing a practical guideline for optimizing readout protocols in future quantum processors.

Figures (5)

• Figure 1: Qubit linewidth broadening in different qubit-resonator coupling regimes. (a) A schematic of the system comprising a superconducting qubit coupled to a readout resonator, which is driven by a readout (RO) pulse and connected to an environmental bath containing TLSs. (b) In the weak-pull limit ($|\chi|\ll \kappa$), the qubit spectrum under readout broadens with width proportional to the photon number and the dispersive shift. This broadening enables interaction with previously detuned TLSs in the bath (gray shading). (c) For a conventional $|\chi|/\kappa$ ratio, the qubit spectrum already deviates from the Lorentzian prediction (dashed line for Lorentzian prediction for center readout frequency). Different colors correspond to three choices of readout frequencies. (d) For a large $|\chi|/\kappa$ ratio, the qubit spectrum exhibits distinctly different lineshapes for each readout frequency choice. For all cases in (c) and (d), the measurement rate is kept constant across the three frequency choices.
• Figure 2: Numerical verification of the analytical model. We consider a transmon qubit coupled to two TLSs and dispersively coupled to a driven, lossy resonator. (a,b) Qubit emission spectra $S_q(\omega)$ under readout drive for (a) $\chi=\kappa$ and (b) $\chi=4\kappa$. Different colors correspond to different readout frequencies: $\omega_d = \omega_r(g)$ (blue), $\omega_d = \omega_r(e)$ (red), and $\omega_d = [\omega_r(g) + \omega_r(e)]/2$ (orange). Gray shading indicates the TLS noise spectrum, with TLS-transmon coupling strengths set as $g_{\mathrm{tls}}/2\pi = 0.5$ MHz (left) and $0.4$ MHz (right) and loss rates $\gamma_{2,\mathrm{tls}}/2\pi = 0.5$ MHz. (c,d) Qubit decay rate $\Gamma_{e \to g}$ as a function of pointer-state separation $|\delta\alpha|$. Solid lines: predictions from Eqs. \ref{['eq:rate']} and \ref{['eq:spectrum']}; open circles: master equation simulation results. Dashed lines: predictions using the simple Lorentzian broadening model for (c) $\omega_d = \omega_r(g)$ and (d) $\omega_d = \omega_r(e)$, which deviate significantly from simulations. The excellent agreement between solid lines and markers validates our analytical model across both coupling regimes.
• Figure 3: Quantitative comparison between experimental data and theoretical calculation. (a-c) Data for a TLS far-detuned from the qubit frequency ($\Delta_{\mathrm{tls}}\approx 2\chi$). (d-f) Data for a TLS near-resonant with the qubit frequency ($\Delta_{\mathrm{tls}}\lesssim \chi$). (a, d) Experimental qubit relaxation time $T_1$ (colorscale, in $\mu$s) as a function of readout drive frequency (horizontal axis) and SNR rate (vertical axis, in MHz). The drive frequency is swept across the $g$ state resonant frequency $\omega_r(g)$ and $e$ state frequency $\omega_r{(g)} + \chi$, where $\omega_r{(g)}$ is the bare resonator frequency and $\chi$ is the dispersive shift. (b, e) Corresponding theoretical calculations using the filter function formalism, reproducing the key features observed in the experimental data including the $T_1$ suppression near specific readout drive frequencies. (c, f) Horizontal linecuts comparing experimental data (open circles) and theoretical predictions (solid lines) at SNR rates indicated by the colored arrows in (b) and (e), demonstrating quantitative agreement across the full frequency range. Insets in (c) and (f) display the TLS spectral density $S_{B}(\omega)$ (gray) and filter functions $S_q(\omega)$ (other coloring) for three representative operating points marked by crosses with same coloring in (a, d) and (c, f). The degree of spectral overlap between the filter function and TLS spectrum directly determines the qubit relaxation rate: larger overlap leads to enhanced $T_1$ degradation.
• Figure 4: Readout system characterization. (a) Qubit-state-dependent resonator transmission spectrum $|S_{21}|$ with the transmon prepared in $|g\rangle$ (blue) and $|e\rangle$ (red). Solid lines are fits to a single-mode Purcell-filtered readout model Swiadek_2024. (b) Steady-state intracavity photon number $|\alpha_g|^2$ as a function of drive amplitude at several drive frequencies (colored curves), extracted from the ac-Stark shift of the transmon $g$-$e$ transition measured via qubit absorption spectroscopy. (c) Signal-to-noise ratio (SNR) as a function of steady-state integration time for several readout drive amplitudes, demonstrating the linear dependence used to extract the SNR rate, $\frac{d}{dt}\text{SNR}$. Inset: extracted $\frac{d}{dt}\text{SNR}$ as a function of drive amplitude, used to level the measurement rate across different drive detunings. The quantum efficiency of the measurement chain is characterized to be $12.94\,\%$Bultink_2018.
• Figure 5: Inversion recovery measurements of the two TLS configurations observed during the experiment. Blue circles: measured excited-state population $P_e(t)$ as a function of delay time; red lines: fits to a double-exponential decay model [Eq. \ref{['eq:double_exp']}]. (a) First TLS configuration: $\Delta_\mathrm{tls}/2\pi = -16.3\,\mathrm{MHz}$, $g_\mathrm{tls}/2\pi = 0.20\,\mathrm{MHz}$, $\gamma_{2,\mathrm{tls}}= 0.85\,\mathrm{MHz}$, $\gamma_1 = 0.15\,$MHz. (b) Second TLS configuration: $\Delta_\mathrm{tls}/2\pi = -6.0\,\mathrm{MHz}$, $g_\mathrm{tls}/2\pi = 0.19\,\mathrm{MHz}$, $\gamma_{2,\mathrm{tls}}= 1.35\,\mathrm{MHz}$, $\gamma_1 = 0.11\,$MHz. The extracted parameters are used to construct the bath spectral density $S_B(\omega)$ entering the theoretical predictions shown in Fig. \ref{['fig:data']}.