Time-Resolved Measurement of a Charge Qubit

We propose a scheme for monitoring coherent quantum dynamics with good time-resolution and low backaction, which relies on the response of the considered quantum system to high-frequency ac driving. An approximate analytical solution of the corresponding quantum master equation reveals that the phase of an outgoing signal, which can directly be measured in an experiment with lock-in technique, is proportional to the expectation value of a particular system observable. This result is corroborated by the numerical solution of the master equation for a charge qubit realized with a Cooper-pair box, where we focus on monitoring coherent oscillations.

Figure 1

CPB coupled to a transmission line with Ohmic effective impedance $Z_0$.

Figure 2

Decaying qubit oscillations with initial state $|\uparrow ⟩$ in a weakly probed CPB with 6 states for $\alpha =Z_0{e}^2/\hslash =0.08$, $A=0.1E_J/e$, $E_C=5.25E_J$ and $N_g=0.45$, so that $E_{\mathrm{el}}=2.1E_J$ and ${\omega }_{\mathrm{qb}}=2.3E_J/\hslash$. (a) Time evolution of the measured difference signal $⟨\dot{Q}⟩\propto ⟨{\xi }_{\mathrm{out}}⟩-⟨{\xi }_{\mathrm{in}}⟩$ (in units of $2e{E}_J/\hslash$) of the full CPB and its lock-in amplified phase ${\varphi }_{\mathrm{out}}$ (frequency window $\Delta \Omega =5E_J/\hslash$), compared to the estimated phase ${\varphi }_{\mathrm{hf}}^0\propto ⟨{\sigma }_x{⟩}_0$ in the qubit approximation. The inset resolves the underlying small rapid oscillations with frequency $\Omega =15E_J/\hslash$ in the long-time limit. (b) Power spectrum of $⟨\dot{Q}⟩$ for the full CPB Hamiltonian (solid) and for the two-level approximation (dashed).

Figure 3

(a) Fidelity defect $\delta F=1-F$ and (b) time-averaged trace distance between the driven and the undriven density operator of the CPB for various driving amplitudes as a function of the driving frequency. All other parameters are the same as in Fig. 2.