Anharmonicity and Charge-Noise Sensitivity of Fraunhofer Qubit

Abstract

We present a theory of a flux-tunable superconducting qubit, the "Fraunhofer qubit," based on the Fraunhofer interference in a wide ballistic Josephson junction. As magnetic flux threads the junction, the Josephson potential is effectively averaged over a phase window proportional to flux. For perfectly transmitting junctions, as flux approaches one flux quantum h/2e, the flux averaging transforms the potential near its minimum from a quadratic to a triangular shape, resulting in significantly enhanced anharmonicity. This enhancement persists for junctions with lower transparency conducting channels. Microscopic tight-binding simulations that include inhomogeneous electrostatic potential and disorder confirm the enhancement of anharmonicity. These results establish a framework for flux control in hybrid superconducting circuits, providing an operating point where anharmonicity and charge-noise protection can be optimally balanced.

Figures (5)

• Figure 1: Schematic of the Fraunhofer qubit. (a) Circuit diagram of the Fraunhofer qubit with zoom-in view of the Josephson junction, illustrating the external magnetic field applied in the normal region. (b) Illustration of the flux averaging window for potential, with black dashed lines for the triangular potential demonstration.
• Figure 2: Frequencies for $0\to1$ and $0\to 2$ qubit transitions as functions of magnetic flux, shown for different junction transparencies. In the low-flux regime, the perfect junction case exhibits an essentially flux-independent transition frequency, while junctions with lower transparencies show flux tunability at lower $\Phi$. In the intermediate-flux range (the middle panel), the qubit frequencies change faster with the flux for all values of transparency $T$. At high flux (the right panel), the transition frequencies are strongly suppressed, and the qubit gradually enters a charge-sensitive regime of a Cooper-pair box. Computations were performed for $E_C/h=200$MHz, $\Delta/h=50.7$ GHz and $N=20$ channels.
• Figure 3: Anharmonicity (Eq. \ref{['eq:anharmonicitydef']}) of the qubit as a function of magnetic flux for the same parameters as in Fig. 2 In the low-flux regime, the anharmonicity of a reflectionless junction is flux-independent and remains at $E_C/4$ as highlighted in the logarithmic-scale inset. The inset compares the anharmonicity versus flux with perturbative and triangular-well analytic predictions, with colors matched to those used in the main panel. In the intermediate-flux range, the reflectionless junction exhibits pronounced anharmonicity enhancement, which broadens as channel transparency decreases. At high flux, the potential well becomes increasingly shallow, and the qubit crosses over into a charge-sensitive regime.
• Figure 4: Tight-binding simulation. (a) Comparison of anharmonicity obtained from various approaches for $\Delta V = 0.3\mu$. The inset shows the transparency distribution extracted from the tight-binding model in the absence of magnetic flux. (b) Anharmonicity versus flux for different values of $\Delta V$. Solid lines represent results from the tight-binding model, while dashed lines correspond to calculations using the analytic potential, with tight-binding model transparencies. (c) Critical current and (d) qubit frequency as functions of flux for several disorder realizations.
• Figure 5: An example of a quasi-classical electron/hole trajectory which contributes to the bound state energy.