Evidence for unexpectedly low quasiparticle generation rates across Josephson junctions of driven superconducting qubits

Abstract

Recent studies find that even drives far below the superconducting gap frequency may cause drive-induced quasiparticle generation (QPG) across Josephson junctions (JJs) of superconducting qubits (SCQs), posing a serious concern for fault-tolerant superconducting quantum computing (FTSQC). Nonetheless, quantitative experimental estimation on QPG rates has remained vague. Here, we investigate QPG using strongly driven SCQs, reaching qubit drive amplitudes up to $2π\times$300 GHz by applying intense drive fields through the readout resonators. The resonator nonlinear responses enable quantification of the energy loss at SCQs, including the contribution from QPG. Surprisingly, the estimated total energy loss rates are far lower than those expected by the Floquet-Markov formalism with QPG as the sole loss mechanism. Meanwhile, calculations that incorporate high-frequency cutoffs (HFCs) in the QPG conductance at approximately 17-20 GHz effectively explain the experimental observations. These results suggest limitations in either the QPG conductance model or the Markovian treatment of the QPG processes. Both possibilities possess crucial implications for handling QPG problems toward FTSQC and for a more deeper understanding of Josephson junctions.

Figures (6)

• Figure 1: Nonlinear dissipation in the bare resonator response.(a) A transmon is coupled to a resonator mode of the bare frequency $\omega_r$. The arrow indicates a resonator probe of frequency $\omega_d$ and amplitude $\Omega_r$. (b) Equivalent schematic in the semi-classical approximation. The arrow indicates an effective drive on the transmon. Here, $g$ and $N_r$ are transmon–resonator coupling strength and resonator mean photon number, respectively. (c) Observed resonator transmission ($S_{21}$) with respect to resonator probe power $P_r$. The collapse of dressed resonator frequencies ($\widetilde{\omega}_r$) to the bare one ($\omega_r$) is identified (white arrow). The dashed line indicates where the bare resonator response appears. (d) Cross-sections corresponding to red, blue, and green arrows in (c) near $\omega_r$. Corresponding resonator photon numbers for resonant drives are presented beside. (e) Extracted resonator dissipation rates $\kappa$ (dots) for resonant drives neglecting resonator inhomogeneous broadening. $\widetilde{V}_{\textup{JJ}}$ refers to an estimated mean amplitude of voltage across the JJ in the transmon. The gray area represents scaling errors in the $x$-axes considering $\pm$20% calibration error in $P_r$. (f) Power spectrum density of the resonator output when $N_r\approx2000$ (blue), background (gray), and fits (dashed lines). The sharp spike at the center is the transmitted probe. The broad peak corresponds to the spectrum of $\delta\hat{a}$. (g) Estimated $\left<\delta\hat{a}^\dagger\delta\hat{a} \right>$ versus $N_r$. Error bars represent scaling errors assuming $\pm$20% calibration errors in $\left<\delta\hat{a}^\dagger\delta\hat{a} \right>$ and $N_r$. Statistical errors are negligible in all the cases.
• Figure 1: Experimental data from Q2–Q4 and theoretical calculations based on QPG mechanism. The circles indicate the corrected $\kappa$ subtracted by $\kappa_{ex}$ with respect to $\Omega_q/2\pi$ from the readout resonators of the other transmons (Q2–Q4). Here, the correction means subtracting $\mathcal{R}/\hbar\omega_d N_r$ from $\kappa$. The lines indicate the theoretically calculated $\mathcal{T}/\hbar\omega_d N_r + \kappa_o$ subtracted by $\kappa_{ex}$ when only taking QPG mechanism into consideration. As with Q1 in the main text, we assume that $\mathcal{T}$ mainly contribute to nonlinear dissipation in the regime $\Omega_q/2\pi \gtrsim 40$ GHz. We manually set $\omega_{\textbf{QPG}}^c/2\pi$ and $\kappa_{o}$ such that the theoretical curves consistently fit the experimental data over this regime. In the calculations, $\kappa_{o}$ is 16.79, 27.59, and 6.43 MHz, and $\omega_{\textbf{QPG}}^c/2\pi$ is 17, 19, and 20 GHz for Q2, Q3, and Q4, respectively. These values can be interpreted as approximate upper bounds of $\omega_{\textbf{QPG}}^c/2\pi$. We obtain clear and stable $S_{21}$ only above $\Omega_q/2\pi\approx50$ GHz for Q4, and thus, the range of $\Omega_q/2\pi\lesssim50$ GHz remains blank. See Supplementary Table 2 for the specifications of Q2–Q4 and their readout resonators. The comparably weak nonlinearity in $\kappa$ of Q4 arises from the significantly smaller transmon–resonator coupling compared to the others (see Supplementary Note 8 for further explanation). Systematic and statistical errors are negligible in all the cases.
• Figure 2: Floquet modes of the transmon in the semi-classical approximation.(a–b) Energy diagrams of the undriven and driven transmons, respectively. Each horizontal line indicates $\overline{H}_q$, the averaged energies of the eigenstates ${\ket{i}}$ or Floquet modes $\ket{\phi_{i}(t)}$. $\varphi$ and $E_J$ indicate superconducting phase and Josephson energy, respectively. In the strong drive limit, chaotic Floquet modes (gray lines) grow down to the ground state. The regular states (blue) lie above them. (c) Numerically calculated $\overline{H}_q$ with respect to $\Omega_q$. The dashed line indicate the threshold $\Omega_q$ and corresponding approximate $N_r$ above which the ground state falls into the chaotic layer. (d) Calculated $|\braket{j|\phi_i (0)}|^2$ for $\Omega_q/2\pi$=2.2, 20, and 60 GHz (from left to right). $N_{\textup{ch}}$ is the roughly estimated size of chaotic subspace, which linearly increases with $\Omega_q$. We use experimental parameters while setting $n_g=0.25$ and $\omega_d=\omega_r$ in the calculations.
• Figure 3: Theoretical description for drive-induced energy loss at the transmon in the semiclassical approximation ($\mathcal{T}$).(a) Transitions between two Floquet modes ($\ket{\phi_i(t)}$ and $\ket{\phi_j(t)}$) of the driven transmon in the semi-classical approximation. $\Gamma_{ij}$ and $\Gamma_{ji}$ indicates each transition rate. In the steady state, the relation $\Sigma_{i}p_{i}\Gamma_{ij} = \Sigma_{i}p_{i}\Gamma_{ji}$ is satisfied, where $p_i$ is the probability to find the system in $\ket{\phi_i(t)}$. (b) The resonator photons assist the transition between $\ket{\phi_i(t)}$ and $\ket{\phi_j(t)}$. The upward dashed arrows indicates a process assisted by $k$-photon absorption from the resonator photons (multiple upward solid arrows) at a rate of $\Gamma_{ij,k}$. The downward dashed arrows indicates a process assisted by $l$-photon emission into the resonator (multiple downward solid arrows) at a rate of $\Gamma_{ji,-l}$. The multiple solid arrows represent the resonator photons participating in the processes. $\Gamma_{ij}$ is then given by $\sum_{k}\Gamma_{ij,k}$. For each scenarios, the transmon emits a photon of energy $\Delta_{ij,k} = \epsilon_i - \epsilon_j + k\hbar\omega_d$ ($\Delta_{ji,-l} = \epsilon_j - \epsilon_i - l\hbar\omega_d$) into its environments (wavy arrows). Here, $\epsilon_i$ is the quasienergies in the first Brillouin zone. Eventually, $\mathcal{T}$ is given by $\hbar{\sum_{ijk}\left( p_{i}{\Gamma_{ij,k} \Delta_{ij,k}}\right)}$.
• Figure 4: Numerical calculations of drive-induced energy loss at the transmon in the semiclassical approximation ($\mathcal{T}$). The Markovinity of the baths is assumed. Radiative (rad), dielectric (diel) and drive-induced quasiparticle generation (QPG) mechanisms are taken into consideration. We use $n_g=0.25$ and experimental values in $\hat{H}_q$ for the calculations. $J^{(\textbf{rad})}(\omega)$ is the Ohmic upper bound. In $J^{(\textbf{diel/QPG})}(\omega)$, we use $\omega_{\textbf{diel}}^c/2\pi=1$ THz and $\omega_{\textbf{QPG}}^c/2\pi=17$ GHz. (a–c) The drive amplitude $\Omega_q$ in the calculations is $2\pi \times$ 40 GHz. (a) Calculated $\Gamma_{ij,k}$ for each dissipation mechanisms with respect to $\Delta_{ij,k}$. Only the elements with $i,j\leq N_{\textup{ch}}$ that dominantly affect the transmon dynamics are presented. (b) Histogram of calculated transition rates between Floquet modes $\ket{\phi_i(t)}$ and $\ket{\phi_j(t)}$ for each dissipation mechanisms. (c) Calculated steady state population $p_i$ when $\Omega_q/2\pi=40$ GHz. (d–e) Calculated $\mathcal{T}\approx\hbar\omega_d N_r(\kappa - \kappa_o)$ when $\Gamma_{ij,k}$ is $\Gamma_{ij,k}^{(\textbf{rad})}$,$\Gamma_{ij,k}^{(\textbf{diel})}$, and $\Gamma_{ij,k}^{(\textbf{QPG})}$, respectively. The radiative calculations stop at $\Omega_q/2\pi=120$ GHz for a technical reason (See Supplementary Note 4).