Enhanced Coherence in Superconducting Circuits via Band Engineering

In superconducting circuits interrupted by Josephson junctions, the dependence of the energy spectrum on offset charges on different islands is $2e$ periodic through the Aharonov-Casher effect and resembles a crystal band structure that reflects the symmetries of the Josephson potential. We show that higher-harmonic Josephson elements described by a $\cos (2\phi )$ energy-phase relation provide an increased freedom to tailor the shape of the Josephson potential and design spectra featuring multiplets of flat bands and Dirac points in the charge Brillouin zone. Flat bands provide noise-insensitive energy levels, and consequently, engineering band pairs with flat spectral gaps can help improve the coherence of the system. We discuss a modified version of a flux qubit that achieves, in principle, no decoherence from charge noise and introduce a flux qutrit that shows a spin-1 Dirac spectrum and is simultaneously quite robust to both charge and flux noise.

Figure 1

(a) Schematics of a charge-noise-insensitive flux qubit employing $\cos (2\phi )$ JJs, indicated as two parallel red lines. (b) Josephson potential of an ordinary FQ for ${\phi }_x=\pi$ and (c) Josephson potential Eq. (2) for the choice $\alpha =0.8$, $\beta =0.2$, and ${\phi }_x=\pi /2$. In (b) and (c) the unit cell is highlighted in white. (d) Exact spectrum showing the six lowest energy levels as a function of the two offset charges $q_1$ and $q_2$.

Figure 2

(a) Energy difference ${\omega }_{01}$ between the two qubit states as a function of the normalized offset charges. (b) Full spectrum highlighting the two lowest energy levels as a function of the applied flux $\delta {\phi }_x={\phi }_x-\pi /2$ and matrix elements $M_{\varphi }$ and $M_1$ determining the relaxation and pure decoherence rates.

Figure 3

(a) Schematics of circuit whose spectrum realizes a spin-1 Dirac model. (b) Josephson potential of Eq. (5) for the choice $\beta =0.3$ and $\gamma =0.4$, and ${\phi }_x=\pi$. The unit cell is highlighted in white. (c) Spectrum of the system showing the first three levels as a function of $q_1$ and $q_2$ for $\beta =0.3$, $\gamma =0.352$, and ${\phi }_x=\pi$.

Figure 4

(a) Matrix elements $M_1^{nm}$ and $M_{\varphi }^{nm}$ and spectrum of a flux qutrit as a function of the flux bias $\delta {\phi }_x={\phi }_x-\pi /2$. (b) Matrix elements $M_{\varphi }^{nm}$ and spectrum of a flux qutrit as a function of the offset charges $q_1=-q_2=q$. (c) Matrix elements $M_1^{nm}$ and spectrum as a function of $q$. In (a)–(c) thick lines indicate the three lowest energy eigenvalues.