dorsal/arxiv
View SchemaJump time and passage time: the duration of a quantum transition
| Authors | L. S. Schulman |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0103151 |
| URL | https://arxiv.org/abs/quant-ph/0103151 |
Abstract
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.
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"abstract": "Under unitary evolution, systems move gradually from state to state. An\nunstable atom has amplitude in its original state after many lifetimes\n($\\tau_L$). But in the laboratory, transitions seem to go instantaneously, as\nsuggested by the term \"quantum jump.\"\n The problem studied here is whether the \"jump\" can be assigned a duration, in\ntheory and in experiment. Two characteristic times are defined, jump time\n($\\tau_J$) and passage time ($\\tau_P$). Both use Zeno time, $\\tau_Z$, defined\nin terms of $H$ and its initial state as $\\tau_Z \\equiv \\hbar/\\sqrt{\u003c\\psi|\n(H-E_\\psi)^2 |\\psi\u003e}$, with $E_\\psi \\equiv \u003c\\psi|H|\\psi\u003e$.\n $\\tau_J$ is defined in terms of the time needed to slow (\\`a la the quantum\nZeno effect) the decay: $\\tau_J \\equiv \\tau_Z^2/\\tau_L$. It appears in several\ncontexts. It is related to tunneling time in barrier penetration. Its inverse\nis the bandwidth of the Hamiltonian, in a time-energy uncertainty principle.\n$\\tau_J$ is also an indicator of the duration of the quadratic decay regime in\nboth experiment and in numerical calculations (cf. Fig.~2 of PRA 57,1509\n(1998).)\n The passage time, $\\tau_P$, arises from unitary evolution sans\ninterpretation. It is based on a bound of Fleming (Nuov. Cim. 16 A, 232\n(1973)): for any $H$ and $\\psi$ a system cannot evolve to a state orthogonal to\n$\\psi$ for $t\u003c \\tau_P \\equiv \\pi \\tau_Z/2$. By including apparatus in $H$,\n$\\tau_P$ limits the observation of decay according to the quantum measurement\nideas proposed in \"Time\u0027s Arrows and Quantum Measurement,\" Cambridge U. Press,\n1997, thereby allowing an experimental test of these ideas.",
"arxiv_id": "quant-ph/0103151",
"authors": [
"L. S. Schulman"
],
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"quant-ph"
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"title": "Jump time and passage time: the duration of a quantum transition",
"url": "https://arxiv.org/abs/quant-ph/0103151"
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