A trapped ion sits in a trap whose energy levels are almost

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch Coherent states, Ehrenfest's theorem and the classical limit

Problem

A trapped ion sits in a trap whose energy levels are almost, but not exactly, evenly spaced: En=ℏωn+ℏδn2E_n = \hbar\omega n + \hbar\delta n^2, with ω=2π×8.00×105 rad s−1\omega = 2\pi\times 8.00\times10^{5}\,\mathrm{rad\,s^{-1}} (angular trap frequency 2π×800 kHz2\pi\times800\,\mathrm{kHz}) and a small anharmonic correction δ=5.00×10−4 ω\delta = 5.00\times10^{-4}\,\omega. The ion is prepared in a state whose number distribution has mean nˉ=9\bar n = 9 and spread Δn=nˉ=3\Delta n = \sqrt{\bar n} = 3, the same width a coherent state with that mean would have.Someone argues: "A coherent state stays exactly as narrow forever under harmonic-oscillator evolution, so any state that starts out with a narrow, Poisson-like spread in nn must stay narrow for all later times, whatever the potential looks like." ((a)) Explain what is wrong with this argument. ((b)) Find by how much the level spacing (En+1−En)/ℏ(E_{n+1}-E_n)/\hbar drifts per level. Then use the chapter's spreading estimate, tspread∼1/(∣δω∣ Δn)t_{\rm spread}\sim1/(|\delta\omega|\,\Delta n), to estimate the time at which the phase spread across the number distribution reaches about one radian. ((c)) Express tspreadt_{\rm spread} as a number of oscillation periods T=2π/ωT = 2\pi/\omega, and say what this shows about the quoted claim.

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