Energy Decay in Superconducting Josephson-Junction Qubits from Nonequilibrium Quasiparticle Excitations

We calculate the energy decay rate of Josephson qubits and superconducting resonators from nonequilibrium quasiparticles. The decay rates from experiments are shown to be consistent with predictions based on a prior measurement of the quasiparticle density $n_{\mathrm{qp}}=10/\mu {\mathrm{m}}^3$, which suggests that nonequilibrium quasiparticles are an important decoherence source for Josephson qubits. Calculations of the energy-decay and diffusion of quasiparticles also indicate that prior engineered gap and trap structures, which reduce the density of quasiparticles, should be redesigned to improve their efficacy. This model also explains a striking feature in Josephson qubits and resonators—a small reduction in decay rate with increasing temperature.

Figure 1

Plot of quasiparticle occupation versus energy for 3 phonon temperatures $T_p=140$, 70, and 0 mK (top to bottom). An injection rate was chosen to match the experimental quasiparticle density $10/\mu {\mathrm{m}}^3$, equivalent to $n_{\mathrm{qp}}/2D(E_F)\Delta =\int f\rho dE=1.8\times {10}^{-6}$. The state occupation at energies shown does not depend on the injection energies. The gray line shows the thermal occupation $f_T$ at 140 mK.

Figure 2

(a) Plot of calculated qubit energy relaxation rate versus phonon temperature $T_p$ for a quasiparticle density of $n_{\mathrm{qp}}/2D(E_F)\Delta =1.8\times {10}^{-6}$ and $E_{10}/h=6\mathrm{GHz}$. This shape of the curve is relatively insensitive to changes in $E_{10}$ and $n_{\mathrm{qp}}$. Note the small decrease in decay rate with increasing temperature up to 120 mK, and then a rapid increase due to thermally generated quasiparticles. (b) Experimental measurement of relaxation rate ${\Gamma }_1$ for a phase qubit. The relaxation decreases, and then increases with temperature, in qualitative agreement with theory. The slow change at $T_p\gtrsim 140\mathrm{mK}$ is probably due to nonidealities, such as the small occupation of the ground state (greater than 90% for $T_p<100\mathrm{mK}$, but only 77% at $T_p=130\mathrm{mK}$ and 55% at $T_p=160\mathrm{mK}$).

Figure 3

Plot of average quasiparticle diffusion length versus energy for both the electron-phonon $1/{\Gamma }^s$ ($e$-ph) and electron recombination $1/{\Gamma }^r$ ($e\mathrm{\text{−}}e$) decay times. The $e\mathrm{\text{−}}e$ interaction dominates for $E\lesssim 1.1\Delta$. Parameters are $D=60v_{\mathrm{qp}}{\mathrm{cm}}^2/\mathrm{s}$ (Al), $T_p=0$, and $n_{\mathrm{qp}}=10/\mu {\mathrm{m}}^3$.