Bound states in the continuum in a fluxonium qutrit

The heavy fluxonium at zero external flux has a long-lived state when coupled capacitively to any other system. We analyze it by projecting all the fluxonium relevant operators into the qutrit subspace, as this long-lived configuration corresponds to the second excited fluxonium level. This state becomes a bound-state in the continuum (BIC) when the coupling occurs to an extended system supporting a continuum of modes. In the case without noise, we find BIC lifetimes that can be much larger than seconds $T_1\gg {\rm s}$ when the fluxonium is coupled to a superconducting waveguide, while typical device frequencies are in the order of ${\rm GHz}$. We have performed a detailed study of the different sources of decoherence in a realistic experiment, obtaining that upward transitions caused by a finite temperature in the waveguide and decay induced by $1/f$-flux noise are the most dangerous ones. Even in their presence, BICs decay times could reach the range of ${T_1\sim \rm 10^{-1} ms},$ while preparation times are of the order of $10^{2}$ns.