Question

1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since this state is not an eigenstate of the Hamiltonian, it will evolve in time. Write down the Hamiltonian matrix for this system in the subspace spanned by the states 101) and 10). Calculate the time evolved state Jb(t)) b. Calculate the 4 × 4 density matrix ρ(t) corresponding to the state you found in part (a) c. Trace over the degrees of freedom of spin B to calculate the reduced density matrix describing spin A: A = Tre(ρ). d. Calculate the three components of the Bloch vector PA corresponding to A as a function of time. Again, the Bloch vector of an ensemble is defined from the following equation: 2 where σ are the usual Pauli matrices.

Answer 1

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