- #1
David DCruz
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Homework Statement
J-coupling term between two spins is
HJ = ħJ/4 σz(1) σz(2)
In the measured magnetization spectrum of the spins, this leads to the splitting of the individual
spin lines by frequency J, which we’ll now derive. We can write the magnetization of spin 1 as:
<M1(t)> = tr(ρ(t)σ+(1)) = tr[ρ(t)σ+(1)⊗(e+(2)+e-(2))]
where e+(2) = matrix(1 0; 0 0 )
e-(2)) = matrix(0 0;0 1)
σ+ = σx + i σy
(1) refers to 1st qubit; (2) refers to 2nd qubit
Assume ρ(t) evolves according to U(t)=exp(-iHJt/ħ)
Show that
<M1(t)> = exp(iJt/2) tr[ρ(0)σ+(1)e+(2)] + exp(-iJt/2) tr[ρ(0)σ+(1)e-(2)]
Homework Equations
Mentioned above
The Attempt at a Solution
I expressed ρ(t) = U(t) ρ(0) U+(t)
Then I wrote <M1(t)> = tr[ρ(0)exp(iHJt/ħ)σ+(1)exp(-iHJt/ħ)⊗(e+(2)+e-(2))]
I expanded out the exponential hamiltonian to get
<M1(t)> = tr[ρ(0)exp(-iJtσz(2)/2) σ+(1)⊗(e+(2)+e-(2))]
I'm not sure how to proceed from here