Measurement scheme for the Lamb shift in a superconducting circuit with broadband environment

Motivated by recent experiments on quantum mechanical charge pumping in a Cooper pair sluice, we present a measurement scheme for observing shifts of transition frequencies in two-level quantum systems induced by broadband environmental fluctuations. In contrast to quantum optical and related setups based on cavities, the impact of a thermal phase reservoir is considered. A thorough analysis of Lamb and Stark shifts within weak-coupling master equations is complemented by nonperturbative results for the model of an exactly solvable harmonic system. The experimental protocol to measure the Lamb shift in experimentally feasible superconducting circuits is analyzed in detail and supported by numerical simulations.

Figure 1

Lamb shift according to Eq. (21) for a Drude spectral density as a function of the dimensionless inverse temperature ${\omega }_0\hslash \beta$. The cutoff frequencies increase from bottom to top with ${\omega }_c/{\omega }_0=10$ (dotted line), ${\omega }_c/{\omega }_0=40$ (dashed line), and ${\omega }_c/{\omega }_0=100$ (solid line). The coupling constant is $\gamma /{\omega }_0=0.05$. The shaded region and the dots denote the temperature range and the Lamb shifts accessible for the discussed experiment.

Figure 2

Circuit diagram for the Cooper pair sluice, the artificial environment (in the dashed box), and the coil used for exciting the system with external drive (on the right).

Figure 3

(a) Average net current in a microwave-driven Cooper pair sluice as a function of the frequency shift $\delta \Omega$. The minimum current is found at $\delta \Omega /{\omega }_0=0.063$ for $T=50$ mK (solid line) and at $0.033$ for $T=400$ mK (dashed line). Currents are given in units of the critical current $I_c=3.14$ nA yielding $\langle I_{\max }\rangle =0.93$ nA and $\langle I_{\min }\rangle =-0.06$ nA. Data are averaged over a total time $t_{\mathrm{fin}}=7.6$ ns with the system initialized to the ground state and the microwave drive turned on instantaneously at the beginning of the averaging period. (b) Same as in panel (a) but for various values of the phase bias ${\varphi }_0=\pi /2$ (solid curve) and ${\varphi }_0=0.5$ (dashed curve) at $T=50$ mK. The inset shows the normalized shape of the drop for the respective values of ${\varphi }_0$. Other parameters are $E_C/k_B=1$ K, $N_g=1/2$, $J_L=0.1E_C$, $J_R=0.05E_C$, and $\Delta =0.2$.