Noise and microresonance of critical current in Josephson junction induced by Kondo trap states

We analyze the impact of trap states in the oxide layer of a superconducting tunnel junction on the fluctuation of the Josephson critical current, thus on coherence in superconducting qubits. Two mechanisms are usually considered: the current blockage due to repulsion at the occupied trap states and the noise from electrons hopping across a trap. We extend previous studies of noninteracting traps to the case where the traps have onsite electron repulsion inside one ballistic channel. The repulsion not only allows the appropriate temperature dependence of $1/f$ noise, but also the control of the coupling between the computational qubit and the spurious two-level systems inside the oxide dielectric. We use second-order perturbation theory, which allows to obtain analytical formulas for the interacting bound states and spectral weights, limited to small and intermediate repulsions. Remarkably, it still reproduces the main features of the model as identified from the numerical renormalization group. We present analytical formulations for the subgap bound state energies, the singlet-doublet phase boundary, and the spectral weights. We show that interactions can reverse the supercurrent across the trap. We finally work out the spectrum of junction resonators for qubits in the presence of onsite repulsive electrons and analyze its dependence on microscopic parameters that may be controlled by fabrication.

Figure 1

(a) Energy view on a Josephson junction sandwiched between two superconducting reservoirs. The two thick (red color) bands are the subgap bound states caused by the presence of magnetic impurity in the junction. (b) Cross section of the tunneling layer. Traps near the oxide surface (left) predominantly contribute to blocking noise—an occupied trap is charged and blocks off parts of the current. Traps in the center carry a contribution to the supercurrent that is influenced by interactions and can potentially lead to a $\pi$ junction.

Figure 2

Energy diagrams illustrating Cooper pair transport through a magnetic impurity. (a) Weak-Coulomb-interaction limit where transport occurs through a screened spinless level; denoted by screened arrow at the bound state $E_b$. The Cooper pair parity is conserved. (b) Strong-Coulomb-interaction limit where transport flips the impurity spin. The parity of the Cooper pair is reversed, which results in a reversal supercurrent flow.

Figure 3

Bound states at (a) zero-phase limit, $\phi =0$, due to the localized magnetic impurity inside a Josephson junction between two superconducting reservoirs, where $\Delta$ on the solid line is 0.06, dashed line 0.01, and dotted line 0.001, and (b) in variable phase $0<\phi <\pi$ for different $\frac{U}{\pi \Gamma }$, where $\Delta =0.06$. In both plots, $\pi \Gamma =0.01$. (c) The bound state in terms of $\Delta /\Gamma$ and $\Delta /U$ for the zero-phase case ($\phi =0$).

Figure 4

(a) The spectral diagonal weight $W\left(E_b\right)$ and (b) the off-diagonal weight $W\left(E_b\right)$ of the substates whose energies are given in Fig. 3. (c) The phase dependence of diagonal weight, $\Delta =0.01$. (d) The normal weight as a function of $\delta =\frac{\Delta }{U}$ and $\theta =\frac{\Delta }{2\pi \Gamma }$.

Figure 5

Phase diagram for singlet and doublet ground states. The critical Coulomb repulsion is denoted by $U_c$ to be compared with the doublet-singlet boundary at $T_K/\Delta =0.3$. The dashed vertical line is the doublet-singlet boundary at large $\Delta$ limit. The dash-dot line indicates the border between singlet and doublet states for very large gap superconductors. The off-diagonal weight $W_{\Delta }/\Delta$ is plotted in background color.

Figure 6

Supercurrent variation with respect to (a) phase difference in different Coulomb repulsions at $kT=0.1\Delta$ and (b) the Coulomb repulsion $U$ at different phases between 0 and $\pi$ at zero temperature; the reservoir-junction hybridization rate $\pi \Gamma =0.1$ and the gap $\Delta =0.01$, $I_c=2e\Delta /\hslash$.

Figure 7

Vertex correction in the second order.

Figure 8

The charge blockade noise spectrum for various Coulomb repulsions as listed in the legend.

Figure 9

The background colors indicate the spectral weight of the resonance noise as a function of ratio $\Delta /\Gamma$ and $U/\pi \Gamma$ for the phase difference $\phi =0$ (left) and $\phi =\pi /2$ (right). Dotted lines are the singlet-doublet boundary.