A spin one-half particle precesses at Larmor frequency in a static field

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Spin one-half, SU(2) and two-level dynamics

Problem

A spin one-half particle precesses at Larmor frequency ω0=γB0\omega_0=\gamma B_0 in a static field B⃗0=B0z^\vec B_0=B_0\hat z, and is additionally driven by a field of amplitude B1B_1 rotating in the xyxy-plane at angular frequency ω\omega, so that the full Hamiltonian is H^(t)=−ℏω02σz−ℏω12(cos⁡ωt σx−sin⁡ωt σy),ω1≡γB1.\hat H(t) = -\frac{\hbar\omega_0}{2}\sigma_z - \frac{\hbar\omega_1}{2}\big(\cos\omega t\,\sigma_x - \sin\omega t\,\sigma_y\big),\qquad \omega_1\equiv\gamma B_1. ((a)) Transform to the frame rotating with the drive using ∣ψ′(t)⟩=U^R(t)∣ψ(t)⟩|\psi'(t)\rangle = \hat U_R(t)|\psi(t)\rangle with U^R(t)=e−iωtσz/2\hat U_R(t)=e^{-i\omega t\sigma_z/2}. Using iℏ d∣ψ′⟩/dt=H^′∣ψ′⟩i\hbar\,d|\psi'\rangle/dt = \hat H'|\psi'\rangle with H^′=U^RH^U^R†+iℏ(dU^R/dt)U^R†\hat H' = \hat U_R\hat H\hat U_R^\dagger + i\hbar(d\hat U_R/dt)\hat U_R^\dagger, show that H^′\hat H' is time-independent, H^′=−ℏδ2σz−ℏω12σx,δ≡ω0−ω.\hat H' = -\frac{\hbar\delta}{2}\sigma_z - \frac{\hbar\omega_1}{2}\sigma_x,\qquad \delta\equiv\omega_0-\omega. ((b)) Treating H^′\hat H' as a static-field Hamiltonian along the effective axis n^′=(ω1,0,δ)/ΩR\hat n' = (\omega_1,0,\delta)/\Omega_R with ΩR≡δ2+ω12\Omega_R\equiv\sqrt{\delta^2+\omega_1^2}, and noting that U^R\hat U_R is diagonal so it does not change the population of ∣↑⟩|{\uparrow}\rangle or ∣↓⟩|{\downarrow}\rangle, derive the probability of finding a spin that starts in ∣↑⟩|{\uparrow}\rangle in the state ∣↓⟩|{\downarrow}\rangle at time tt: P↓(t)=ω12ΩR2sin⁡2 ⁣(ΩRt2).P_\downarrow(t) = \frac{\omega_1^2}{\Omega_R^2}\sin^2\!\left(\frac{\Omega_Rt}{2}\right). ((c)) A spin with ω1=2π×2.0 MHz\omega_1 = 2\pi\times2.0\ \mathrm{MHz} is driven exactly on resonance (δ=0\delta=0). Find the shortest time at which P↓=1P_\downarrow=1 (a π\pi pulse).

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