Excitation of Superconducting Qubits from Hot Nonequilibrium Quasiparticles

Superconducting qubits probe environmental defects such as nonequilibrium quasiparticles, an important source of decoherence. We show that “hot” nonequilibrium quasiparticles, with energies above the superconducting gap, affect qubits differently from quasiparticles at the gap, implying qubits can probe the dynamic quasiparticle energy distribution. For hot quasiparticles, we predict a non-negligible increase in the qubit excited state probability $P_e$. By injecting hot quasiparticles into a qubit, we experimentally measure an increase of $P_e$ in semiquantitative agreement with the model and rule out the typically assumed thermal distribution.

Figure 1

Qubit decay ${\Gamma }_{\downarrow }$ (blue) and excitation ${\Gamma }_{\uparrow }$ (red) mediated by tunneling quasiparticles. (a) Portion of the superconducting phase qubit potential energy diagram, showing the ${\Gamma }_{\downarrow }$ and ${\Gamma }_{\uparrow }$ transitions between the ground state $|g⟩$ and the excited state $|e⟩$, along with the qubit energy $E_{\mathrm{ge}}$. (b) Cold nonequilibrium quasiparticles, which have energies near the superconducting gap $\Delta$, can only absorb $E_{\mathrm{ge}}$, resulting in qubit ${\Gamma }_{\downarrow }$ decay. The density of states [$\rho (E)$, horizontal] on both sides of a Josephson junction (superconductor $S$-insulator $I$-superconductor $S$) is shown versus quasiparticle energy $E$ (vertical). The quasiparticle energy distribution $f(E)$ is shown by the shaded triangles. (c) Hot nonequilibrium quasiparticles with energy above $\Delta +E_{\mathrm{ge}}$ [portion of $f(E)$ with $E>\Delta +E_{\mathrm{ge}}$] not only can cause qubit ${\Gamma }_{\downarrow }$ transitions, but can also relax by causing qubit $|g⟩\to |e⟩$ transitions.

Figure 2

Calculated effects of hot quasiparticles. Plots are for three qubit frequencies $E_{\mathrm{ge}}/h$: 13.3 GHz (black), 6.7 GHz (cyan), and 3.3 GHz (red); lines are guides to the eye. Quasiparticle densities incorporate scattering and recombination. (a), (c) Quasiparticles injected at energy $E_{\mathrm{inj}}=1.8\Delta$ (solid lines) or $\Delta +0.94E_{\mathrm{ge}}$ (dashed line, 6.7 GHz), assuming $\Delta /k_B=2\mathrm{K}$ as for aluminum. (b), (d) Quasiparticle density is $n_{\mathrm{qp}}/n_{\mathrm{cp}}={10}^{-5}$. (a), (b) Qubit decay rate ${\Gamma }_{\downarrow }$ normalized by the cold rate ${\Gamma }_{\downarrow }^c$ from Eq. (2), showing that ${\Gamma }_{\downarrow }^c$ is a good approximation to ${\Gamma }_{\downarrow }$. (c) Qubit $|e⟩$ state probability $P_e={\Gamma }_{\uparrow }/({\Gamma }_{\downarrow }+{\Gamma }_{\uparrow })$ vs normalized quasiparticle density $n_{\mathrm{qp}}/n_{\mathrm{cp}}$. The $|e⟩$ state occupation increases linearly with the quasiparticle density for low densities. (d) Qubit probability $P_e$ vs injection energy $E_{\mathrm{inj}}$ expressed as fraction of $E_{\mathrm{ge}}$ above $\Delta$. Injecting low energy quasiparticles results in a greatly reduced $P_e$. For $E_{\mathrm{inj}}\gtrsim \Delta +1.7E_{\mathrm{ge}}$, $P_e$ is essentially constant.

Figure 3

Experimental apparatus and protocol. (a) Schematic of phase qubit, with voltage $V_{\mathrm{sq}}$ applied to readout SQUID and flux bias ${\Phi }_{\mathrm{bias}}$. Quasiparticles generated with the SQUID can only reach the qubit through the common ground. (b) Photo of qubit. Two paths for quasiparticles to diffuse to the qubit junction are indicated by dotted and solid arrows. (c) Pulse sequence. Quasiparticles are generated with a voltage pulse on the SQUID line of amplitude $V_{\mathrm{inj}}>2(\Delta +E_{\mathrm{ge}})/e$ and time $t_{\mathrm{inj}}$, which are varied to adjust the quasiparticle density. After a delay time $t_{\mathrm{dif}}$, $40\mu \mathrm{s}$ of which is at the operating bias, allowing quasiparticles to diffuse to the qubit, the qubit state is projected with a flux bias pulse and read out using a voltage pulse on the SQUID.

Figure 4

Experimental impacts of injected quasiparticles. Increases in (a), (b) qubit decay rate $\delta {\Gamma }_{\downarrow }$ and (c), (d) qubit excited state probability $\delta P_e$ are measured with respect to values without quasiparticle injection. The quasiparticle density was varied by changing (b), (d) the length $t_{\mathrm{inj}}$ of the injection pulse or (a), (c) the diffusion time $t_{\mathrm{dif}}$. Data are for two slightly different designs: 1 (Ref. 4) and 2 (Ref. 44); experimental parameters are in the Supplemental Material [45]. These increases directly demonstrate a nonequilibrium distribution.

Figure 5

Comparison of data and model. The experimental increase in qubit excited state probability $\delta P_e$ from Fig. 4 is parametrically plotted vs increases in qubit decay rate $\delta {\Gamma }_{\downarrow }$, equivalent to the quasiparticle density $\delta n_{\mathrm{qp}}/n_{\mathrm{cp}}$. Symbols are as in Fig. 4. The power law dependence of $\delta P_e$ on $\delta {\Gamma }_{\downarrow }$ is reduced to the left of $t_{\mathrm{inj}}+t_{\mathrm{dif}}\sim 350\mu \mathrm{s}$ (arrows). Theoretical predictions (solid lines) are for quasiparticles injected at $1.9\Delta$ using measured $E_{\mathrm{ge}}\simeq 0.15\Delta$. Dashed line is for a Fermi-Dirac quasiparticle distribution at quasiparticle temperatures selected for $\delta {\Gamma }_{\downarrow }$.