  We discuss a deterministic model of detector coupled to a two-level system (a
qubit). The detector is a quasi-classical object whose dynamics is described by
the kicked rotator Hamiltonian. We show that in the regime of quantum chaos the
detector acts as a chaotic bath and induces decoherence of the qubit. We
discuss the dephasing and relaxation rates and demonstrate that several
features of Ohmic baths can be reproduced by our fully deterministic model.
Moreover, we show that, for strong enough qubit-detector coupling, the
dephasing rate is given by the rate of exponential instability of the
detector's dynamics, that is, by the Lyapunov exponent of classical motion.
Finally, we discuss the measurement in the regimes of strong and weak
qubit-detector coupling. For the case of strong coupling the detector performs
a measurement of the up/down state of the qubit. In the case of weak coupling,
due to chaos, the dynamical evolution of the detector is strongly sensitive to
the state of the qubit. However, in this case it is unclear how to extract a
signal from any measurement with a coarse-graining in the phase space on a size
much larger than the Planck cell.
