Reduced visibility of Rabi oscillations in superconducting qubits

Coherent Rabi oscillations between quantum states of superconducting microcircuits have been observed in a number of experiments, albeit with a visibility that is typically much smaller than unity. Here, we show theoretically that the coherent coupling of superconducting phase qubits to background charge fluctuators [R. W. Simmonds et al., Phys. Rev. Lett., 93, 77003 (2004)] leads to a significantly reduced visibility if the Rabi frequency is comparable to the coupling energy of microcircuit and fluctuator. For larger Rabi frequencies, transitions to the second excited state of the superconducting microcircuit become dominant in suppressing the Rabi oscillation visibility.

Figure 1

(Color online) Rabi oscillations for a squbit-fluctuator system. The probability $p_1\left(t\right)$ to find the squbit in state ∣1⟩ is obtained from numerical integration of Eq. (4) (solid line) and the analytical solution Eq. (5) (dashed line) which is valid for $b_x∕J⪢1$. For weak decoherence of the fluctuator, $\gamma ∕J<1$ [(a) and (c)], $p_1\left(t\right)$ shows damped beating. For $\gamma ∕J>1$, damped single-frequency oscillations are restored. The fluctuator leads to a reduction of the first maximum in $p_1\left(t\right)$ to $\sim 0.8$ [(a) and (b)] and $\sim 0.9$ [(c) and (d)], respectively. The parameters are (a) $b_x∕J=1.5$, $\gamma ∕J=0.5$; (b) $b_x∕J=1.5$, $\gamma ∕J=1.5$; (c) $b_x∕J=3$, $\gamma ∕J=0.5$; (d) $b_x∕J=3$, $\gamma ∕J=1.5$.

Figure 2

(Color online) (a) Level scheme for the squbit in Refs. 7, 10. (b) $p_1\left(t\right)$ obtained from numerical integration of the Schrödinger equation (solid line) in comparison with the analytical result in Eq. (10)(dashed line) for $b_x∕\hslash \Delta \omega =1∕3$.