Theory of quasiparticle generation by microwave drives in superconducting qubits

TL;DR

The paper addresses drive-induced quasiparticle generation in superconducting qubits by formulating a Floquet extended-space theory that captures multiphoton-assisted pair-breaking across charge- and flux-driven regimes. It derives a Floquet Fermi's golden rule for transitions between dressed qubit states, yielding rates that factorize into photon-structure factors and dressed matrix elements, with the total rate given by $\Gamma_{\alpha\beta}=\sum_n\Gamma_{\alpha\beta}^{(n)}$. Key findings reveal sharp stair-step thresholds at $\hbar\omega_d=2\Delta/n$, with even/odd $n$ dictating state-preserving vs state-changing processes, and demonstrate notable QP generation only under strong drives or high-frequency drives; flux driving can further enhance generation near specific amplitudes (e.g., $\varphi_{ac}^*$). The results provide practical guidelines for high-frequency readout and Floquet-engineered qubits and suggest material choices (larger $\Delta$, thinner films) to mitigate photon-assisted QP generation, shaping future designs of robust driven superconducting devices.

Abstract

Microwave drives play a central role in the control of superconducting quantum circuits, enabling qubit gates, readout, and parametric interactions. As the drive frequencies are typically an order of magnitude smaller than (twice) the superconducting gap, it is generally assumed that such drives do not disturb the BCS ground state. However, sufficiently strong drives can activate multiphoton pair-breaking processes that generate quasiparticles (QPs) and result in qubit errors. In this work, we present a theoretical framework for calculating the rates of multiphoton-assisted pair-breaking transitions induced by charge- or flux-coupled microwave drives. Through illustrative examples, we show that photon-assisted QP generation may affect novel high-frequency dispersive readout architectures, as well as Floquet-engineered superconducting circuits operating under strong driving.

Figures (5)

• Figure 1: Problem setup. (a) Multiphoton-induced pair-breaking across a JJ in a driven superconducting qubit, where the energy $n\hbar\omega_d$ from $n$ drive photons is absorbed to break a Cooper pair and excite a pair of QPs above the gap. Here, an example with $n = 3$ is shown. (b) The system is modeled as a driven qubit coupled to a bath of BCS QPs. Using Floquet theory, we promote the drive to a fictitious quantum degree of freedom, such that resulting Floquet Hamiltonian is time-independent. (c) The spectrum of the Floquet Hamiltonian consists of infinite replicas of the dressed qubit energies, separated by multiples of $\hbar\omega_d$. We compute transition rates in the Floquet extended Hilbert space between the states $\lvert\tilde{\Phi}_{\alpha, 0}\rangle$ and $\lvert\tilde{\Phi}_{\beta, -n}\rangle$ using a Floquet-basis version of Fermi's golden rule, representing transitions between qubit states $\alpha \to \beta$ accompanied by the absorption of $n$ drive photons. The actual transition rates $\Gamma_{\alpha\beta}$ in the original qubit Hilbert space are then obtained via a summation over the different photon-number contributions in Floquet space.
• Figure 2: Photon-assisted QP generation in transmon qubits. (a) Circuit of a transmon qubit subjected to a microwave drive. (b) Structure factors $\mathcal{S}_{\rm ph}^\pm$ [Eq. \ref{['eq:PAT_S_factors']}] for photon-assisted QP generation as a function of transition frequency. $\mathcal{S}_{\rm ph}^+$ and $\mathcal{S}_{\rm ph}^-$ have a threshold behavior at frequency $2\Delta/\hbar$. (c) Simulation of the total parity switching rate $\Gamma_g$ for the driven ground state of a transmon with $E_J/h = 3.025$ GHz, $E_C/h = 56$ MHz, and $n_g = 0$, as a function of the frequency $\omega_d$ and amplitude $\Omega$ of the drive. The rates exhibit sharp steps at $\hbar\omega_d = 2\Delta/n$, for integer $n$, reflecting the different $n$-photon processes required for pair-breaking. The red dashed line is a line of constant qubit ac-Stark shift $\delta_{\rm ac}$ (see the text). (d) Transition rates $\Gamma_{\alpha\beta}$ in the qubit $\{g, e\}$ manifold as a function of $2\Delta/\hbar\omega_d$ along the line of constant ac-Stark shift from panel (c). These rates display distinct steps, with alternating "sharpness" that depends on the even-odd parity of $n$, where $n = \lceil 2\Delta/\hbar\omega_d \rceil$ for state-preserving transitions ($\Gamma_{gg}, \Gamma_{ee}$) and $n = \lceil (2\Delta \pm\hbar\tilde{\omega}_q)/\hbar\omega_d \rceil$ for state-changing transitions ($\Gamma_{ge}$, $\Gamma_{eg}$) for Stark-shifted qubit frequency $\tilde{\omega}_q$. The inset shows an enlarged view of the behavior around the step $n = 4$.
• Figure 3: Pair-breaking in high-frequency readout. (a) We consider a readout resonator of frequency $\omega_r$ coupled to a transmon qubit with $E_J/h = 12.85$ GHz, $E_C/h = 218$ MHz, and $n_g = 0.1$, such that the bare qubit frequency is $\omega_q/2\pi = 4.5$ GHz. (b) We plot charge-parity switching lifetimes $T_g$ and $T_e$ starting from the ground and excited states of the qubit, respectively, as a function of resonator photon number $\bar{n}$, for $f_r = \omega_r/2\pi \in \{46, 34, 25, 20\}$ GHz. At each value of $\omega_r$, we adjust the qubit-resonator coupling $g$ so as to maintain a dispersive shift $\chi/2\pi = 1$ MHz. The horizontal black lines indicate a parity lifetime of 100 $\mu$s, and the secondary axis converts the parity lifetimes to an equivalent steady-state QP fraction, $x_{\rm qp}^\ast$ (see the text). In the inset, we plot $T_g$ versus $\bar{n}$ at $f_r = 25$ GHz for several values of the gate charge $n_g \in [0, 0.5]$.
• Figure 4: Pair-breaking in a flux-driven Floquet qubit. (a) Circuit schematic for a symmetric SQUID threaded by an external flux $\varphi_{\rm e}(t) = \varphi_{\rm ac}\sin(\omega_d t)$ [Eq. \ref{['eq:H_symmetric_SQUID']}]. Here, QP generation can occur across each of the junctions in the SQUID. (b) Top to bottom: Effective potentials $U_{\rm eff}$ [cf. Eq. \ref{['eq:H_eff_kapitza']}] at different drive amplitudes $\varphi_{\rm ac}/2\pi \in [0.5, 0.76, \varphi_{\rm ac}^\star/2\pi, 0.77, 0.9]$, where $\varphi_{\rm ac}^\star/2\pi \approx 0.76547$ is the drive amplitude that eliminates the $\cos(\hat{\varphi})$ component of $U_{\rm eff}$, leaving only the $\cos(2\hat{\varphi})$ component. Plotted potentials are normalized for clarity. (c) Parity lifetimes $T_\alpha$ for Floquet states that map onto $\lvert g_0\rangle$, $\lvert e_0\rangle$, $\lvert g_\pi\rangle$, and $\lvert e_\pi\rangle$ (ground and excited states in the wells at $\varphi=0$ and $\pi$, respectively). See the main text for circuit parameters. (d) Enlarged view of the data from panel (c) exhibiting a crossover at $\varphi_{\rm ac}^\star$ (black solid line), as the system's ground state changes from $\lvert g_0\rangle$ to $\lvert g_\pi\rangle$. The obtained lifetimes for $\lvert g_\pi/e_\pi\rangle$ when $\varphi_{\rm ac} < \varphi_{\rm ac}^\star$ and for $\lvert g_0/e_0\rangle$ when $\varphi_{\rm ac} > \varphi_{\rm ac}^\star$, respectively, exhibit large jumps due to the avoided crossings that these states undergo as the potential is adiabatically deformed. Bound states are present in both wells only for $0.76 \lesssim \varphi_{\rm ac}/2\pi \lesssim 0.77$. The insets depict the effective potential and phase-basis wavefunctions at flux drive amplitudes $\varphi_{\rm ac} = (1 \pm 0.0005)\varphi_{\rm ac}^\star$.
• Figure 5: Identifying labels for the Floquet eigenstates. State overlaps $|\ip{\lambda(\Omega)}{g, 0}|$ as a function of the drive amplitude $\Omega$. Here $\lvert\lambda(\Omega)\rangle$ are eigenstates obtained from numerical diagonalization of Eq. \ref{['eq:HF_dressed_mode']} for a driven transmon with parameters $E_J/h = 30$ GHz, $E_C/h = 0.15$ GHz, $n_g = 0$, and $\omega_d/2\pi = 5.059$ GHz, while $\lvert g, 0\rangle$ is the undriven qubit ground state with Floquet photon-number index $m = 0$. The colors in this plot indicate the indices of the eigenstates returned from numerical diagonalization and the black solid curve represents the identified labels for the driven eigenstates $\lvert\tilde{\Phi}_{g, 0}\rangle \equiv \lvert\widetilde{g, 0}\rangle$. The two features at $\Omega/2\pi = 0.5$ GHz and $\Omega/2\pi = 1.9$ GHz reflect multiphoton resonances in the Floquet spectrum, which we intentionally cross diabatically in our labeling procedure.