Demonstration of Protection of a Superconducting Qubit from Energy Decay

Long-lived transitions occur naturally in atomic systems due to the abundance of selection rules inhibiting spontaneous emission. By contrast, transitions of superconducting artificial atoms typically have large dipoles, and hence their lifetimes are determined by the dissipative environment of a macroscopic electrical circuit. We designed a multilevel fluxonium artificial atom such that the qubit’s transition dipole can be exponentially suppressed by flux tuning, while it continues to dispersively interact with a cavity mode by virtual transitions to the noncomputational states. Remarkably, energy decay time $T_1$ grew by 2 orders of magnitude, proportionally to the inverse square of the transition dipole, and exceeded the benchmark value of $T_1>2\mathrm{ms}$ (quality factor $Q_1>4\times {10}^7$) without showing signs of saturation. The dephasing time was limited by the first-order coupling to flux noise to about $4\mu \mathrm{s}$. Our circuit validated the general principle of hardware-level protection against bit-flip errors and can be upgraded to the $0-\pi$ circuit [P. Brooks, A. Kitaev, and J. Preskill, Phys. Rev. A 87, 052306 (2013)], adding protection against dephasing and certain gate errors.

Figure 1

(a) Images of a double-loop fluxonium device in a 3D cavity. The antenna is directly connected to the split junction. (b) Circuit models of a double-loop fluxonium atom (top) and its coupling to a 3D copper cavity (bottom). (c) Lowest four energy levels of the atom accurately positioned in the double-well potential profile $U(\varphi )$, along with their calculated wave functions. (d) The interwell fluxon transition 0–1 [magenta arrow in (c)] has a vanishing transition dipole $⟨0|\varphi |1⟩$ and hence is of the forbidden type. The intrawell plasmon transitions 0–2 and 1–3 [blue and red arrows in (c), respectively] by contrast have transition dipoles of the order of unity and are thus of the allowed type.

Figure 2

Transmission near cavity resonance (scale not shown) as a function of the coil current and spectroscopy tone frequency. Cavity resonance is off the scale at about 10 GHz. Dashed lines represent a fit to the circuit model in Fig. 1 (see the text). Left inset: An enlargement of the smallest fluxon-plasmon splitting region. Right inset: Measured values of the seven splittings visible in the plot vs the extracted value of $E_J^{1/2}$ in frequency units.

Figure 3

(a) The measured lifetime $T_1$ of the 0–1 transition (markers) and its frequency (solid line). Dotted lines are the dielectric loss prediction corresponding to the loss tangent values of $2\times {10}^{-5}$ and $2\times {10}^{-6}$. Measured dispersive shift (blue markers) and calculated dispersive shifts, taking into account all transitions to higher levels (solid line) and only states 0 and 1 (dashed line). Note that $\chi$ decreases much slower than $T_1$ grows. (b),(c) Example of time-domain data: the Rabi oscillation trace (red), $\pi$-pulse echo trace (green), and energy relaxation trace (blue) all measured simultaneously with an interleaved pulse sequence. The inset shows repetitive measurements of $T_1$ during a period of about one hour.

Figure 4

Measured energy decay times for transitions 1–0 (blue) and 2–1 (red) taken from a narrow range of frequencies plotted against the calculated values of the corresponding transition dipole squared. Dashed lines illustrate Fermi’s golden rule predictions for dielectric loss (a),(b) and quasiparticle loss (c),(d) using the expressions for their respective quality factors defined in Ref. [25].