A single trapped ion's motional state along the trap axis is prepared as the…

PHYSICS · Subjective · 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 single trapped ion's motional state along the trap axis is prepared as the superposition ∣Ψ⟩=N(∣α⟩+∣−α⟩)|\Psi\rangle = N\big(|\alpha\rangle+|-\alpha\rangle\big) of two coherent states of the trap oscillator, with α=1.2\alpha=1.2 real and positive and NN real and positive fixed by normalisation. Recall the number-state expansion ∣α⟩=e−∣α∣2/2∑n=0∞αnn!∣n⟩|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle and the general overlap of two coherent states, ⟨β∣α⟩=e−12∣α∣2−12∣β∣2+β∗α\langle\beta|\alpha\rangle=e^{-\frac12|\alpha|^2-\frac12|\beta|^2+\beta^*\alpha}, whose modulus squared is ∣⟨β∣α⟩∣2=e−∣α−β∣2|\langle\beta|\alpha\rangle|^2=e^{-|\alpha-\beta|^2}. (a) Find NN. (b) Show that ⟨n∣Ψ⟩\langle n|\Psi\rangle vanishes for every odd nn, and give the probability P(n)=∣⟨n∣Ψ⟩∣2P(n)=|\langle n|\Psi\rangle|^2 of finding n=0n=0, n=2n=2 and n=4n=4 quanta, to three significant figures. (c) Compute ∣⟨α∣−α⟩∣2|\langle\alpha|-\alpha\rangle|^2, and say whether a single measurement of position would tell the two component coherent states apart, and whether a single measurement of momentum would. (d) The odd-nn probabilities in (b) vanish because of interference between the ∣α⟩|\alpha\rangle and ∣−α⟩|-\alpha\rangle branches. Given the result of (c), explain in one or two sentences why that interference is still fully visible in the number distribution even though the two coherent states are almost distinguishable.

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