A single ion is held in a harmonic trap and prepared in the oscillator ground state…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations
The Structure of Quantum Mechanics, Ch Slow change, sudden change and the Berry phase
Problem
A single ion is held in a harmonic trap and prepared in the oscillator ground state ∣n=0⟩ at trap frequency ω1=2π×1.0MHz. The trapping laser intensity is then ramped so that the trap stiffens smoothly to twice the frequency, ω2=2ω1, over a time T. Recall that the n→n+1 energy spacing of a harmonic oscillator of frequency ω is ℏω, the same for every n.
((a)) Using the initial frequency ω1, compute the level-spacing time τ1=ℏ/(ℏω1)=1/ω1.
((b)) Decide, using τ1, whether the adiabatic condition or the sudden approximation applies (i) if the stiffening takes Ta=1.0ms, and (ii) if it takes Tb=1.0ns. Give the ratio T/τ1 in each case as your justification.
((c)) For case (i), state which number state the ion ends in immediately after the ramp, and why. For case (ii), state (without evaluating any overlap integral) what determines the probabilities of finding the ion in the various number states ∣n(ω2)⟩ of the new, stiffer trap.