You are doing a Ramsey experiment as described in the lecture:
You prepare a beam of atoms in a state |1⟩, you do a π/2 pulse with a Rabi frequency of Ω0=(2π)⋅500 Hz and a detuning of δ=(2π)⋅3 Hz to the |1⟩→|2⟩ transition. Then there is a free flight for a time T = 0.5 s and a second π/2 pulse with the same Rabi frequency. The excited state |2⟩ has a decay rate of γ=0.5 s−1. At the end you measure the purity of the final state right after the second pulse.
What is the length of the Bloch vector after this experiment?
To find the length of the Bloch vector after the experiment, we need to first determine the density matrix of the final state. The density matrix describes the state of a quantum system in terms of its probabilities and quantum coherence.
In this case, we start with a beam of atoms in a state |1⟩. After the first π/2 pulse, the system is in a superposition state given by the following density matrix:
ρ₁ = 1/2 |1⟩⟨1| + 1/2 |2⟩⟨2| + 1/2 |1⟩⟨2| + 1/2 |2⟩⟨1|
During the free flight time T = 0.5s, the excited state |2⟩ can decay with a rate γ = 0.5s^(-1). This decay process can be described by the equation:
ρ(t) = e^(-γt) ρ₁
Where ρ(t) is the density matrix at time t.
After the second π/2 pulse, the final density matrix is given by:
ρ_f = e^(-γT) (1/2 |1⟩⟨1| + 1/2 |2⟩⟨2| + 1/2 |1⟩⟨2| + 1/2 |2⟩⟨1|)
Now, the length of the Bloch vector is given by the equation:
|B| = sqrt(tr(ρ_f²))
Where tr(ρ_f²) represents the trace of the square of the density matrix.
To calculate |B|, we need to express ρ_f² in terms of the basis states |1⟩ and |2⟩, and then calculate its trace.
ρ_f² can be written as:
ρ_f² = e^(-2γT) (1/4 |1⟩⟨1| + 1/4 |1⟩⟨2| + 1/4 |2⟩⟨1| + 1/4 |2⟩⟨2| + 1/4 |1⟩⟨2| + 1/4 |2⟩⟨1| + 1/4 |2⟩⟨2| + 1/4 |2⟩⟨1| + 1/4 |2⟩⟨2|)
Simplifying this expression, we have:
ρ_f² = e^(-2γT) (1/2 |1⟩⟨1| + 1/2 |2⟩⟨2| + 1/2 |1⟩⟨2| + 1/2 |2⟩⟨1|)
Taking the trace of ρ_f²:
tr(ρ_f²) = e^(-2γT)
So, the length of the Bloch vector |B| is equal to e^(-γT), which simplifies to:
|B| = e^(-γT) = e^(-0.5x0.5) ≈ 0.6065
Therefore, the length of the Bloch vector after the experiment is approximately 0.6065.