We consider an electrostatic qubit located near a Bose-Einstein condensate (BEC) of noninteracting bosons in a double-well potential, which is used for qubit measurements. Tracing out the BEC variables we obtain a simple analytical expression for the
qubits density-matrix. The qubits evolution exhibits a slow ($propto1/sqrt{t}$) damping of the qubits coherence term, which however turns to be a Gaussian one in the case of static qubit. This stays in contrast to the exponential damping produced by most classical detectors. The decoherence is, in general, incomplete and strongly depends on the initial state of the qubit.
We consider an electrostatic qubit, interacting with a fluctuating charge of single electron transistor (SET) in the framework of exactly solvable model. The SET plays a role of the fluctuating environment affecting the qubits parameters in a control
lable way. We derive the rate equations describing dynamics of the entire system for both weak and strong qubit-SET coupling. Solving these equation we obtain decoherence and relaxation rates of the qubit, as well as the spectral density of the fluctuating qubits parameters. We found that in the weak coupling regime the decoherence and relaxation rates are directly related to the spectral density taken at Rabi or at zero frequency, depending on what a particular qubits parameters is fluctuating. This relation holds also in the presence of weak back-action of the qubit on the fluctuating environment. In the case of strong back-action, such simple relationship no longer holds, even if the qubit-SET coupling is small. It does not hold either in the strong-coupling regime, even in the absence of the back-action. In addition, we found that our model predicts localization of the qubit in the strong-coupling regime, resembling that of the spin-boson model.
