Nonequilibrium Quasiparticles in Superconducting Circuits: Energy Relaxation, Charge and Flux Noise

TL;DR

This work presents a unified impedance framework to quantify energy relaxation and decoherence in superconducting circuits caused by resident quasiparticles in quasiequilibrium. By generalizing the Mattis–Bardeen conductivities to a quasithermal QP distribution and coupling them to a two-fluid wire model, the authors predict charge and flux noise, a $T_1$-limit in transmons, and a nearly white flux-noise background in flux qubits. Key findings show that asymmetric gap engineering (GE) dramatically suppresses QP tunneling across Josephson junctions, shifting residual loss to wire leads and producing flux noise compatible with experiments, while TLS mitigation remains essential to further improve coherence. The results offer practical design guidance for enhancing CPW resonator $Q$, extending $T_1$ and $T_2^*$ in GE qubits, and understanding the origins of flux noise in superconducting circuits.

Abstract

The quasiparticle density observed in low-temperature superconducting circuits is several orders of magnitude larger than the value expected at thermal equilibrium. The tunneling of this excess of quasiparticles across Josephson junctions is recognized as one of the main loss and decoherence mechanisms in superconducting qubits. Here, we present a unified impedance theory that accounts for quasiparticle energy loss in circuit regions both far and near (across) junctions. Our theory leverages the recent experimental demonstration that the excess quasiparticles are in \emph{quasiequilibrium} [T. Connolly et al., Phys. Rev. Lett. $\textbf{132}$, 217001 (2024)] and uses a generalized fluctuation-dissipation theorem to predict the amount of charge and flux noise generated by them. We compute the resulting energy relaxation time $T_1$ in transmon qubits with and without junction asymmetric gap engineering, and show that quasiparticles residing away from junctions can play a dominant role in the former case. They also may provide an upper limit for resonator quality factors if the density of amorphous two-level systems is reduced. In addition, we show that charge noise from quasiparticles leads to flux noise that is logarithmic-in-frequency, giving rise to a ``nearly white" contribution that is comparable to the flux noise observed in flux qubits. This contrasts with amorphous two-level systems, whose associated flux noise is shown to be superOhmic. We discuss how this quasiparticle flux noise can limit $T_2^{*}$ coherence times in flux-tunable qubits. The conclusion is that asymmetric gap engineering can greatly reduce noise and increase coherence times in superconducting qubits.

Figures (7)

• Figure 1: Circuit for a superconducting wire segment in the two-fluid model, where the nondissipative Cooper-pair current $I_{CP}$ is assumed to flow in parallel to the dissipative quasiparticle current $I_{QP}$. These are subjected to a kinetic inductor $L_{k}$ and a parallel resistor $R_{p}$, respectively. Both currents are subjected to the same geometric inductance $L_{g}$, as shown by $I=I_{CP}+I_{QP}$ in the circuit. An alternative approximate representation of the circuit assumes a quasiparticle resistor $R_s$ in series, satisfying $R_{s}\ll R_{p}$. A capacitor can be added to the circuit to describe a segment of a superconducting transmission line.
• Figure 2: Numerical calculation of $\sigma_1(\omega)$ assuming the quasithermal law for the QP distribution observed in experiments, $n(E)\propto e^{-E/k_BT}$. The plot is normalized by $\sigma_0=\sigma_N x_{{\rm QP}}\left(2\Delta/k_BT\right)^{3/2}$. In the low frequency range $\hbar \omega \lesssim k_BT$, $\sigma_1$ decreases logarithmically with increasing $\omega$; in the high frequency range it decreases as a power law. When $k_BT \lesssim 0.1 \Delta$ we find that the exact numerical result (red points) is well approximated by the analytical expression Eq. (\ref{['eq:sigma_1Approx']}) (shown as a solid line for comparison).
• Figure 3: Quality factor due to quasiequilibrium QPs (solid lines) in an aluminum CPW resonator for 3 values of $x_{\rm QP}^{\rm res}$ ranging from $10^{-9}$ (black) to $10^{-5}$ (orange). The length of the resonator is adjusted to match the resonance frequency $\Omega$ for the fundamental mode, $\ell=\frac{c \pi}{n \Omega}$. For comparison, we include the $Q$ due to dielectric loss with typical loss tangents in bulk and surface (dashed line). Other parameters in Table \ref{['table1']}.
• Figure 4: Circuit representation of (a) transmon, (b) flux qubit, and (c) split-junction transmon. Each wire includes the dissipative resistance due to Ohmic loss in addition to the inductive response as explained in Fig. \ref{['fig:Wire']}. In the transmon, each $Z_{\rm wire}$ includes a small lead connected to the junction and the large pad electrode forming the capacitor. In the flux qubit $Z_{\rm wire}$ represents the wire loop. For the split-junction transmon, each $Z_{\rm loop}$ represents one half of the wire loop with the pads considered separately in $Z_{\rm pad}$.
• Figure 5: Energy relaxation time $T_1$ due to quasiequilibrium QPs in a single-junction aluminum transmon. Solid line: $T_1$ for a non-gap engineered (NGE) transmon for $x_{\rm QP}^{\rm res}=10^{-5}$, which is dominated by QP tunneling across the junction. Dotted lines: $T_1$ for gap-engineered (GE) transmon for 3 values of $x_{\rm QP}^{\rm res}$. For comparison, the dashed-blue-line shows $T_1$ due to dielectric loss assuming typical loss tangents in bulk and surface.