Figure 1

Schematic of a macroscopic superconducting three-body system which consists of two Josephson junctions and an $LC$ resonator. Parameters obtained from fitting the spectra in Fig. 2 are $I_{c1}=21.388\pm 0.003\mu \mathrm{A}$, $I_{c2}=22.536\pm 0.003\mu \mathrm{A}$, $C=4.85\pm 0.05\mathrm{p}\mathrm{F}$, $L\equiv L_1+L_2=1.7\pm 0.05\mathrm{n}\mathrm{H}$, and $C_3=0.33\pm 0.01\mathrm{p}\mathrm{F}$. The two junctions J1 and J2 (in the left and right dashed boxes) have anharmonic potentials with unequal energy level spacings (shown below). The horizontal arrows represent quantum tunneling and the vertical arrows denote microwave-induced transitions. The $LC$ oscillator (in the center dashed box) has a harmonic well with equal energy level spacings.Reuse & Permissions

Figure 2

(a) Spectrum of the system when the bias current $I_{b1}$ for junction J1 is ramped and $I_{b2}=0$ for junction J2. Circles are measured microwave resonance positions; dashed black lines are from quantum mechanical calculations using Eq. (1) with parameters given in Fig. 1 and $I_{b2}=0$. The solid curves denote uncoupled $|0⟩$ to $|1⟩$ level spacings for J1 (black) and the resonator (gray), while that for J2 ($\sim 19\mathrm{G}\mathrm{H}\mathrm{z}$) is not shown in the plot. For low and for high bias, the energy level transitions are from the ground state $|000⟩$ to the indicated higher excited states, where the state notation is $|\mathrm{J}1,\mathrm{J}2,LC⟩$. At the degeneracy point $I_{b1}\approx 21.12\mu \mathrm{A}$ (shown in the lower box), the first two excited states are $(|001⟩\pm |100⟩)/\sqrt{2}$. The deviation of the fit for the third excited state is probably due to its large tunneling rates at high bias currents. (b) Spectrum of the system when J2 is ramped and J1 is zero biased ($I_{b1}=0$). Similar to (a), there is a degeneracy point at $I_{b2}=22.27\mu \mathrm{A}$ (shown in the lower box) where the two states are $(|001⟩\pm |010⟩)/\sqrt{2}$.Reuse & Permissions

Figure 3

Enhancement in the escape rate when the bias current $I_{b1}$ for junction J1 is ramped and junction J2 is biased at a constant current of $I_{b2}=22.330\mu \mathrm{A}$. Black corresponds to highest enhancement and light gray to zero enhancement. The white dashed lines are from quantum mechanical calculations using Eq. (1) with parameters given in Fig. 1 and $I_{b2}=22.330\mu \mathrm{A}$. The solid black lines indicate the uncoupled $|0⟩$ to $|1⟩$ level spacings for J1 (curved) and J2 (horizontal), while that for the $LC$ resonator ($\approx 7.1\mathrm{G}\mathrm{H}\mathrm{z}$) is not shown in the plot. At the triple degeneracy point $I_{b1}\approx 21.15\mu \mathrm{A}$ (shown in the lower box), the three lowest excited states are $(|100⟩-|010⟩-\sqrt{2}|001⟩)/2$, $(|100⟩+|010⟩)/\sqrt{2}$, and $(|100⟩-|010⟩+\sqrt{2}|001⟩)/2$. The higher energy states are superpositions of the multiply excited states $|200⟩$, $|020⟩$, $|002⟩$, $|110⟩$, $|101⟩$, and $|011⟩$.Reuse & Permissions