An ensemble of two-level atoms is used to estimate an unknown angular frequency by…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Information & Technology · Physical realizations and quantum technologies
Quantum Information and Technology, Ch Quantum metrology and sensing
Problem
An ensemble of N two-level atoms is used to estimate an unknown angular frequency ω by Ramsey interferometry, either as N independent atoms or as a single N-atom GHZ state (∣0⟩⊗N+∣1⟩⊗N)/2. Each atom suffers independent dephasing with single-atom coherence time T2, so that after a free-precession time t a single atom's Ramsey fringe has visibility V(t)=e−t/T2, and the N-atom GHZ fringe (which oscillates as cos(Nωt)) has visibility e−Nt/T2. A total measurement time T is available, split into repeated shots of free-precession time t, giving T/t shots. For a two-outcome fringe of visibility V read at its optimal working point, the Fisher information about the phase actually read out, Θ, is F(Θ)=V2 per shot, and Fisher information adds over independent shots (and, for independent atoms, over atoms).
((a)) For N independent atoms, with Θ=ωt for each, show that the total Fisher information about ω accumulated over the whole time T is Findep(t)=NTte−2t/T2, and find the shot length t that maximises it.
((b)) For the N-atom GHZ state, with Θ=Nωt, show that the total Fisher information is FGHZ(t)=N2Tte−2Nt/T2, and find the shot length that maximises it.
((c)) Evaluate each maximised Fisher information and hence the best achievable Δω for both schemes, and show that the GHZ state gives no improvement over independent atoms once each scheme uses its own optimal shot length.
((d)) State, without further calculation, the largest factor by which any entangled state of the N atoms could beat independent atoms under this noise model.