Checkpoint: Solve a Master Equation

The task

This checkpoint puts the whole module together. A single qubit is coupled to a zero-temperature bath through two channels you have now met:

• amplitude damping at rate $\gamma_1 = 1/T_1$ (jump operator $\sqrt{\gamma_1}\,\sigma_-$), and
• pure dephasing at rate $\gamma_\phi = 1/T_\phi$ (jump operator $\sqrt{\gamma_\phi/2}\,\sigma_z$).

You will solve the resulting master equation for the Bloch vector and report its transverse component after a fixed evolution time.

Bloch equations from the Lindblad equation

Writing $\rho = \tfrac12(\mathbb{1} + x\,\sigma_x + y\,\sigma_y + z\,\sigma_z)$ and substituting the two dissipators (with $H=0$, so no precession), the components decouple into simple exponential decays:

with the longitudinal fixed point $z_\infty = -1$ (the ground state at $T=0$) and the transverse rate set by

The longitudinal and transverse motions are independent: populations relax toward the ground state on the scale $T_1$, while coherences die on the scale $T_2$.

Solving for the transverse component

The transverse equation integrates immediately to

There is no constant offset because the transverse steady value is zero. So the entire problem reduces to forming $T_2$ from $T_1$ and $T_\phi$, then evaluating one exponential.

Try it

This is a numerical exercise — your code should return a number. With $T_1=50$, $T_\phi=75$, an initial equatorial state $\langle\sigma_x\rangle(0)=1$, and evolution time $t=30$ (consistent units), compute $T_2$ via $1/T_2 = 1/(2T_1)+1/T_\phi$ and return $\langle\sigma_x\rangle(t) = x_0\,e^{-t/T_2}$.

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