Jump time and passage time: the duration of a quantum transition

Under unitary evolution, systems move gradually from state to state. An unstable atom has amplitude in its original state after many lifetimes ($\tau_L$). But in the laboratory, transitions seem to go instantaneously, as suggested by the term "quantum jump." The problem studied here is whether the "jump" can be assigned a duration, in theory and in experiment. Two characteristic times are defined, jump time ($\tau_J$) and passage time ($\tau_P$). Both use Zeno time, $\tau_Z$, defined in terms of $H$ and its initial state as $\tau_Z \equiv \hbar/\sqrt{<\psi| (H-E_\psi)^2 |\psi>}$, with $E_\psi \equiv <\psi|H|\psi>$. $\tau_J$ is defined in terms of the time needed to slow (\`a la the quantum Zeno effect) the decay: $\tau_J \equiv \tau_Z^2/\tau_L$. It appears in several contexts. It is related to tunneling time in barrier penetration. Its inverse is the bandwidth of the Hamiltonian, in a time-energy uncertainty principle. $\tau_J$ is also an indicator of the duration of the quadratic decay regime in both experiment and in numerical calculations (cf. Fig.~2 of PRA 57,1509 (1998).) The passage time, $\tau_P$, arises from unitary evolution sans interpretation. It is based on a bound of Fleming (Nuov. Cim. 16 A, 232 (1973)): for any $H$ and $\psi$ a system cannot evolve to a state orthogonal to $\psi$ for $t< \tau_P \equiv \pi \tau_Z/2$. By including apparatus in $H$, $\tau_P$ limits the observation of decay according to the quantum measurement ideas proposed in "Time's Arrows and Quantum Measurement," Cambridge U. Press, 1997, thereby allowing an experimental test of these ideas.