Electron Spin — Question 10

Question 10

Random phase noise reduces transverse polarization without changing z counts

Initially $|\chi\rangle=(|+z\rangle+|-z\rangle)/\sqrt2$. In each trial the negative-z amplitude acquires a random phase $\phi$ drawn from a zero-mean Gaussian of variance $v$. Use $\langle e^{i\phi}\rangle=e^{-v/2}$.

1. Find the ensemble density matrix after phase averaging.

2. Find the positive-x probability and purity for $v=2\ln2$.

Question 10 - Solution

Random phase noise reduces transverse polarization without changing z counts

(a) Method and calculation

$$\rho(\phi)=\tfrac12\begin{pmatrix}1&e^{-i\phi}\\e^{i\phi}&1\end{pmatrix}.$$$$\boxed{\bar\rho=\tfrac12\begin{pmatrix}1&e^{-v/2}\\e^{-v/2}&1\end{pmatrix}}.$$

(b) Interpretation and further calculation

$$v=2\ln2:\quad e^{-v/2}=1/2\Rightarrow\boxed{P_{+x}=(1+1/2)/2=3/4}.$$$$\boxed{\mathrm{Tr}(\bar\rho^2)=(1+e^{-v})/2=5/8}.$$

Physical check

Phase averaging preserves both z populations while suppressing off-diagonal terms. Purity falls because the unrecorded phases differ from trial to trial.