We theoretically study the quantum nondemolition measurements of a flux qubit coupled to a noisy superconduct- ing quantum interference device （SQUID）. The obtained analytical results indicate that the measurement probability is frequency-dependent in a short time scale and has a close relationship with the measurement-induced dephasing. Furthermore, when the detuning between the driven and bare resonator equals the coupling strength, we can obtain the maximum measurement rate that is determined by the character of the noise in the SQUID. Finally, we analysed the mixed effect caused by coupling between the non-diagonal term and the external variable. It is found that the initial information of the qubit is destroyed due to quantum tunneling between the qubit states.
We propose a universal analytical method of studying the dynamics of a multi-anticrossing system subjected to driving by a single large-amplitude triangle pulse, within a time scale smaller than the dephasing time. Our approach can explain the main features of the Landau-Zener St/ickelberg interference patterns recently observed in a tripartite system ]Nature Communications 1 51 （2010）]. In particular, we focus on the effect of the size of the anticrossing on interference and compare the calculated interference patterns with numerical simulations. In addition, a Fourier transform of the patterns can extract the information about the energy level spectrum.
