Microscopic model of critical current noise in Josephson-junction qubits: Subgap resonances and Andreev bound states

We propose a microscopic model of critical current noise in Josephson junctions based on individual trapping centers in the tunnel-barrier hybridized with electrons in the superconducting leads. We calculate the noise exactly in the limit of no on-site Coulomb repulsion. Our result reveals a noise spectrum that is dramatically different from the usual Lorentzian assumed in simple models. We show that the noise is dominated by sharp subgap resonances associated to the formation of pairs of Andreev bound states, thus providing a possible explanation for the spurious two-level systems (microresonators) observed in Josephson-junction qubits [R. W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004).]. Another implication of our model is that each trapping center will contribute a sharp dielectric resonance only in the superconducting phase, providing an effective way to validate our results experimentally. We derive an effective Hamiltonian for a qubit interacting with Andreev bound states, establishing a direct connection between phenomenological models and the microscopic parameters of a Fermionic bath.

Figure 1

(a) A current-biased Josephson junction adversely affected by the tunneling of electrons between one of the superconducting leads (sc) and a trapping-center defect in the insulating barrier (ins.). Josephson-junction critical current noise is directly related to fluctuations in trapping-center occupation due to modulation of the tunneling rate between superconducting leads. (b) A Cooper-pair box quantum bit affected by charge noise produced by a single trapping center in the barrier. (c) Proposed setup for measuring trap noise close to the superconducting transition. A single-electron transistor (SET) is weakly coupled to an artificial trap, e.g., a normal-state quantum dot or a nanotube.

Figure 2

Particlelike Andreev bound state as a function of hybridization $\gamma$ for different trap energies ${ϵ}_d$, all in units of the superconducting gap $\Delta$. The holelike Andreev bound state has the same energy but opposite sign.

Figure 3

(Color online) Depiction of the energy density as a function of energy for the TC plus superconductor model and the most important quasiparticle-quasihole excitations (denoted by arrows) determining the TC noise spectrum.

Figure 4

(Color online) Trapping-center noise spectrum near the superconducting transition temperature $T_c$, for ${ϵ}_d=0$. For $T>T_c$ the noise has the Lorentzian form characteristic of random telegraph noise (Ref. 17). As $T$ is lowered below $T_c$ a gap opens in the noise spectrum, and a sharp subgap resonance appears as a transition between two Andreev bound states (for convenience, we represent the subgap resonance with a phenomenological linewidth equal to $0.01\gamma$). This shows that TC noise is dramatically affected by superconductivity.

Figure 5

(Color online) Trapping-center noise for temperatures well below the superconducting transition. The parameters are $\gamma =0.5\Delta$, ${ϵ}_d=0$, $k_{B}T=0.1\Delta$, leading to Andreev energy $E_b=0.35\Delta$. About 60% of the noise power is due to one sharp subgap resonance represented here by a Lorentzian with linewidth $0.001\Delta$. The remaining 40% is dominated by processes involving the creation of a hole in the continuum and the excitation of an Andreev level at $+E_b$. This occurs only for $\omega >\Delta +E_b$.

Figure 6

(Color online) TC noise spectrum in the superconducting regime in a case where the Andreev bound states are very close to the gap edge $\left(E_b=0.981\Delta \right)$. Here the noise is dominated by continuum-continuum contributions (94% of the noise power) with the subgap resonance contributing only 0.3%.

Figure 7

(Color online) TC noise spectrum for the asymmetric case ${ϵ}_d=5\Delta$ with Andreev levels at $\pm E_b=\pm 0.999\Delta$. The noise spectra has a smooth peak close to $\hslash \omega =6\Delta$. This occurs because the spectral function peaks at ${ϵ}_d$. Similar to Fig. 6, the noise is dominated by continuum-continuum contributions (98%) with the subgap resonance contributing less than 0.1%.

Figure 8

(Color online) (a) Energy-level structure of the effective Hamiltonian for a superconducting qubit interacting with a pair of Andreev bound states (one holelike at energy $-E_b$ and one particlelike at $+E_b$). The Andreev-qubit coupling is assumed $\lambda =0.2E_b$. Anticrossing behavior occurs when the qubit energy splitting matches the subgap resonance, $\hslash \omega =2E_b$. (b) Energy differences between the ground state and the first and second excited states. The result is remarkably similar to spectroscopy measurements on phase (Ref. 6) and flux qubits (Ref. 8).