Symmetries and model reduction in a $\cos 2\varphi$ transmon qubit

Abstract

Engineered symmetries in superconducting circuits can protect quantum information; much like selection rules in atomic physics, they forbid transitions between the qubit states. Protected circuits, however, combine several loops and islands and host multiple nonlinearly coupled modes, calling for reduced low-energy models that inherit the circuit symmetries and make the protection manifest. We develop such a reduction for the $\cos 2\varphi$ transmon qubit, recently realized experimentally with a two-Cooper-pair interferometer, whose circuit hosts three coupled modes. Exploiting the separation of energy scales between the heavy transmon mode and the light internal modes of the interferometer, we derive an effective single-band Born-Oppenheimer model, including its geometric correction, and extend it to several bands. We identify the symmetries of the full and reduced Hamiltonians and derive exact selection rules. Finally, we identify a parameter regime beyond the single-band description that suppresses flux-noise dephasing without degrading charge-noise protection.

The paper asks how the three-mode KITE-based $\cos 2\varphi$ transmon reduces to a protected-qubit model, which noise channels its symmetries forbid, and whether its charge/flux-noise tradeoff can be avoided. A Born–Oppenheimer (BO) reduction treats the shunted island phase as heavy and two internal fluxonium modes as light, with $E_{C_S}/E_{C_J}\ll1$. The single-band Hamiltonian includes a geometric potential omitted in earlier treatments and becomes $\pi$-periodic at half flux for identical arms. A multiband extension incorporates nonadiabatic couplings. Against full-circuit diagonalization at experimentally motivated parameters, three low-lying bands reproduce transitions across $0$–$4.5\,\mathrm{GHz}$ with maximum relative error $3\%$. Omitting the geometric correction overestimates the sweet-spot doublet splitting by roughly a factor of four. Reflection symmetries establish exact selection rules: symmetric charge and phase operators cannot connect the logical doublet, whereas antisymmetric internal operators can. Dephasing estimates give charge curvature $-\pi^2\bar\omega_q$ and flux curvature $\pi^2(E_L/\hbar)^2/\bar\omega_q$. A numerical parameter map identifies multiband trajectories that reduce $E_L$ while maintaining the doublet splitting and nominal plasmon/plasma frequencies, suggesting a route around the single-band tradeoff. This route is theoretical, not a demonstrated coherence improvement. Single-band and perturbative Schrieffer–Wolff descriptions fail when the internal phase-slip gap approaches $E_{C_S}$. Exact protections depend on the stated symmetry conditions; quasiparticle poisoning is neglected. A separate Blochnium limit suppresses remaining transition matrix elements but also collapses confinement, making finite-temperature operation impractical. Unmodelled decoherence, control, and readout remain open issues.