Continuous time-dependent measurements: quantum anti-Zeno paradox with applications

Abstract

We derive differential equations for the modified Feynman propagator and for the density operator describing time-dependent measurements or histories continuous in time. We obtain an exact series solution and discuss its applications. Suppose the system is initially in a state with density operator ρ(0) and the projection operator E(t)=U(t) EU^†(t) is measured continuously from t=0 to T, where E is a projector obeying Eρ(0)E=ρ(0) and U(t) a unitary operator obeying U(0)=1 and some smoothness conditions in t. Then the probability of always finding E(t)=1 from t=0 to T is unity. Generically E(T)≠E and the watched system is sure to change its state, which is the anti-Zeno paradox noted by us recently. Our results valid for projectors of arbitrary rank generalize those obtained by Anandan and Aharonov for projectors of unit rank.