Fluctuational and dynamical back action of the measuring device

During a continuous monitoring of a quantum object, the measuring device can act back on the object in general in two different ways: First, it can influence the evolution of the expectation value of the measured observables. This is called the “dynamical back action,” and it is a regular, predictable effect. (An example is the dynamical back action produced by the dynamical rigidity of a capacity sensor, which will be discussed in chapter X.) Second, the measuring device can perturb the observables in a random way, increasing their uncertainties, i.e. producing random deviations from their expectation values. A key feature of linear, continuous measurements, as discussed in the last chapter, is that for them these two types of back action are completely independent. The fluctuational back action is fundamental and irremovable, while the dynamical back action can be avoided entirely by a suitable construction of the measuring device.

For nonlinear systems the situation is completely different, as one can readily see from a typical example of a nonlinear, continuous measurement: a monitoring of an oscillator's energy. In accord with the uncertainty relation, the energy measurement produces perturbations of the oscillator's phase, and as the measurement proceeds continuously, these perturbations cause the phase to diffuse in a Brownian-motion type way.