Quasiparticle-induced decoherence of a driven superconducting qubit

Technical Summary: Quasiparticle-induced decoherence of a driven superconducting qubit

Problem Statement
Superconducting qubits operate at cryogenic temperatures ($T \sim 10$ mK) where the thermal population of quasiparticles (QPs) is theoretically negligible. However, non-equilibrium QPs persist due to environmental radiation and other sources, creating a channel for qubit decoherence. A primary mitigation strategy, "gap engineering," introduces a superconducting gap difference ($\delta\Delta$) across the Josephson junction (JJ) to act as a potential barrier, preventing QP tunneling when $\delta\Delta/\hbar > \omega_q$. While effective in stationary conditions, this approach does not eliminate QPs or prevent their generation. The central problem addressed is whether the microwave drives required for qubit operations (readout and gates) can reactivate QP-induced decoherence mechanisms that gap engineering was designed to suppress.

Methodology
The authors develop a perturbative theory for a driven flux-tunable transmon qubit. The system is modeled using a Hamiltonian comprising:

1. The quantized phase dynamics of the transmon ($\hat{H}_\phi$).
2. The quasiparticle Hamiltonian ($\hat{H}_{qp}$).
3. The QP tunneling term across the junctions in the presence of a microwave drive ($\hat{H}_T$).
4. A term describing Cooper pair breaking ($\hat{H}_{CP}$).

The microwave drive is treated as a time-dependent phase modulation $\phi_d(t) = a \cos(\omega_d t)$. The authors employ Fermi's Golden Rule to calculate transition rates, treating the tunneling and pair-breaking terms as perturbations. A key methodological feature is the inclusion of the drive's non-linear coupling to both the qubit phase and the tunneling matrix elements. This necessitates a diagrammatic approach to account for multiple interference channels (particle vs. hole channels) and processes involving the absorption of one or multiple drive photons. The Supplemental Materials provide a general formalism for calculating amplitudes involving an arbitrary number of photons using diagrammatic rules.

Key Contributions and Results
The paper identifies and quantifies two distinct mechanisms by which microwave drives induce QP-related decoherence in gap-engineered qubits:

1. Microwave-Assisted QP Tunneling (Existing QPs):

   • Mechanism: An existing QP tunnels across the JJ by simultaneously absorbing a qubit excitation ($\hbar\omega_q$) and one or more drive photons ($\hbar\omega_d$).
   • Threshold: For single-photon assistance, the process is allowed when $\omega_d + \omega_q > \delta\Delta/\hbar$. This effectively lowers the energy barrier that gap engineering creates.
   • Flux Dependence: The rate $\Gamma^{(1)}_{1\to0}$ depends on the flux bias $\Phi$. At $\Phi=0$, destructive interference between particle and hole tunneling channels suppresses the single-photon process. However, two-photon processes become dominant at $\Phi=0$ because the interference is constructive.
   • Excitation: The drive can also induce qubit excitation ($0 \to 1$) if $\omega_d > \delta\Delta/\hbar + \omega_q$, leading to leakage errors.

2. Multi-Photon Cooper Pair Breaking (New QP Generation):

   • Mechanism: The drive absorbs multiple photons ($n$) to break a Cooper pair at the junction, generating a new QP pair and inducing a qubit transition.
   • Threshold: This occurs when $n\hbar\omega_d + \hbar\omega_{fi} > \Delta_L + \Delta_R$.
   • Rate Behavior: The transition rates exhibit exponential dependence on $2\Delta/(\hbar\omega_d)$ and show distinct thresholds corresponding to the integer number of photons required. The rates oscillate based on the parity of $n$ due to the different threshold behaviors of the structure factors $\tilde{S}_+$ and $\tilde{S}_-$.

Quantitative Impact on Fidelity
The authors establish fundamental limits on the fidelity ($F$) of qubit operations:

• Readout: The fidelity loss is bounded by $1 - F \gtrsim \Gamma t_{RO}$. For state-of-the-art readout parameters, this scales as $1 - F \gtrsim 100 \cdot x_{QP}$. Following ionizing radiation events where $x_{QP}$ can exceed $10^{-4}$, fidelity losses of $\sim 1\%$ are predicted.
• Gates: Similarly, gate fidelity is limited by $1 - F \gtrsim \beta \cdot x_{QP} / (E_C t_{gate}/\hbar)$.
• High-Frequency Readout: The paper highlights a trade-off in high-frequency dispersive readout. While increasing $\omega_d$ improves signal-to-noise, it eventually crosses the threshold for multi-photon Cooper pair breaking (e.g., two-photon breaking at $\omega_d \approx 60$ GHz for aluminum), degrading fidelity.

Significance
The paper claims to establish a fundamental limitation on the performance of microwave qubit operations imposed by the presence and potential creation of QPs. It demonstrates that gap engineering, while effective in static conditions, does not fully protect qubits during active operation. The microwave drive can "reanimate" QP tunneling and generate new QPs, thereby negating the coherence benefits gained from gap engineering. These findings suggest that future improvements in qubit fidelity must account for drive-induced QP dynamics, potentially requiring hardware-level solutions such as junctions with larger gap differences or effective QP traps to isolate the qubit junctions. The theory provides a framework for optimizing drive parameters (power and frequency) to balance operation speed and fidelity against these decoherence channels.