A single ion

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 171Yb+^{171}\mathrm{Yb}^+ ion, mass m=2.84×10−25 kgm=2.84\times10^{-25}\,\mathrm{kg}, is confined along one axis of a linear trap that acts as a harmonic potential V(x)=12mω2x2V(x)=\tfrac12m\omega^2x^2 with secular frequency ω=2π×1.5 MHz\omega=2\pi\times1.5\,\mathrm{MHz}. The motion is one-dimensional and undamped, and the trap is harmonic to the precision needed here. (a) Starting from the Heisenberg equation of motion, derive d⟨x^⟩dt=⟨p^⟩m\frac{d\langle\hat x\rangle}{dt}=\frac{\langle\hat p\rangle}{m} and d⟨p^⟩dt=−⟨V′(x^)⟩\frac{d\langle\hat p\rangle}{dt}=-\langle V'(\hat x)\rangle, and show that for this potential the second relation holds with no correction term for every state, giving ⟨x^⟩¨=−ω2⟨x^⟩\ddot{\langle\hat x\rangle}=-\omega^2\langle\hat x\rangle exactly. (b) At t=0t=0 the ion is prepared in the coherent state ∣α0⟩|\alpha_0\rangle, the eigenstate of a^\hat a with eigenvalue α0\alpha_0 real and positive, for which ⟨x^⟩0=x0=10 nm\langle\hat x\rangle_0=x_0=10\,\mathrm{nm} and ⟨p^⟩0=0\langle\hat p\rangle_0=0. Using ⟨α∣x^∣α⟩=2ℏ/mω Re α\langle\alpha|\hat x|\alpha\rangle=\sqrt{2\hbar/m\omega}\,\mathrm{Re}\,\alpha, find α0\alpha_0 numerically. (c) Solve the equation of part (a) with these initial conditions for ⟨x^⟩(t)\langle\hat x\rangle(t). (d) Starting instead from the number-state expansion of ∣α0⟩|\alpha_0\rangle and En=(n+12)ℏωE_n=(n+\tfrac12)\hbar\omega, show directly that U^(t)∣α0⟩=e−iωt/2∣α0e−iωt⟩\hat U(t)|\alpha_0\rangle=e^{-i\omega t/2}|\alpha_0e^{-i\omega t}\rangle, use this to recompute ⟨x^⟩(t)\langle\hat x\rangle(t), confirm it matches part (c), and evaluate ⟨x^⟩(t1)\langle\hat x\rangle(t_1) at t1=16×2πωt_1=\tfrac16\times\tfrac{2\pi}{\omega}.

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