  A one-dimensional quantum oscillator is monitored by taking repeated position
measurements. As a first con- tribution, it is shown that, under a quantum
nondemolition measurement scheme applied to a system initially at the ground
state, (i) the observed sequence of measurements (quantum tracks) corresponding
to a single experiment converges to a limit point, and that (ii) the limit
point is random over the ensemble of the experiments, being distributed as a
zero-mean Gaussian random variable with a variance at most equal to the
ground-state variance. As a second contribution, the richer scenario where the
oscillator is coupled with a frozen (i.e., at the ground state) ensemble of
independent quantum oscillators is considered. A sharply different behavior
emerges: under the same measurement scheme, here we observe that the
measurement sequences are essentially divergent. Such a rigorous statistical
analysis of the sequential measurement process might be useful for
characterizing the main quantities that are currently used for inference,
manipulation, and monitoring of many quantum systems. Several interesting
properties of the quantum tracks evolution, as well as of the associated
(quantum) threshold crossing times, are discussed and the dependence upon the
main system parameters (e.g., the choice of the measurement sampling time, the
degree of interaction with the environment, the measurement device accuracy) is
elucidated. At a more fundamental level, it is seen that, as an application of
basic quantum mechanics principles, a sharp difference exists between the
intrinsic randomness unavoidably present in any quantum system, and the
extrinsic randomness arising from the environmental coupling, i.e., the
randomness induced by an external source of disturbance.
