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

Abstract

Under unitary evolution, systems move gradually from state to state. An unstable atom has amplitude in its original state after many lifetimes ($τ_L$). 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 ($τ_J$) and passage time ($τ_P$). Both use Zeno time, $τ_Z$, defined in terms of $H$ and its initial state as $τ_Z \equiv \hbar/\sqrt{<ψ| (H-E_ψ)^2 |ψ>}$, with $E_ψ\equiv <ψ|H|ψ>$. $τ_J$ is defined in terms of the time needed to slow (à la the quantum Zeno effect) the decay: $τ_J \equiv τ_Z^2/τ_L$. 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. $τ_J$ 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, $τ_P$, arises from unitary evolution sans interpretation. It is based on a bound of Fleming (Nuov. Cim. 16 A, 232 (1973)): for any $H$ and $ψ$ a system cannot evolve to a state orthogonal to $ψ$ for $t< τ_P \equiv πτ_Z/2$. By including apparatus in $H$, $τ_P$ 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.