= Post-quench oscillations in a two-level system
{title2=$P_0(t)=1-\sin^2\theta\sin^2(\Delta_ft/(2\hbar))$}
After the <quantum quench>, hold $H_f$ fixed and evolve without an intervening <energy measurement>. The <unitary time evolution> is
$$
|\psi(t)\rangle=c\,e^{+i\Delta_ft/(2\hbar)}|g_f\rangle-s\,e^{-i\Delta_ft/(2\hbar)}|e_f\rangle.
$$
Projection onto the original $|0\rangle$ gives <probability amplitude> $c^2e^{+i\Delta_ft/(2\hbar)}+s^2e^{-i\Delta_ft/(2\hbar)}$. Squaring its <modulus> yields the <two-level quantum return probability>
$$
P_0(t)=1-\sin^2\theta\sin^2\left(\frac{\Delta_ft}{2\hbar}\right),\qquad
P_1(t)=\sin^2\theta\sin^2\left(\frac{\Delta_ft}{2\hbar}\right).
$$
Their sum is one. For $\theta=\pi/3$, the first minimum of $P_0$ is $1/4$ at $t=\pi\hbar/\Delta_f$, and the return is complete at $T=2\pi\hbar/\Delta_f$. At $\theta=0$ or $\pi$ there is no oscillation, because the initial <quantum state> is already a final <energy eigenstate>.
\Image[/two-level-post-quench-oscillations.png]
{title=Post-quench oscillations and constant final <energy> populations in a <two-level system>}
{description=Original ideal calculation at $\theta=\pi/3$ with final gap $\Delta$. Original-basis <Born rule> <probabilities> oscillate through interference of the evolving relative <quantum phase>. Final <energy measurement> <probabilities> remain $3/4$ and $1/4$, so $\langle H_f\rangle=-\Delta/4$ is constant after the driven switch. Curves describe independent preparations measured at different times.}
{source=two-level-post-quench-oscillations.py}
Changing measurement basis reveals the evolving relative <quantum phase>. Measuring the final <energy> first instead destroys this <quantum coherence> between final <energy eigenstates> in the unconditioned ensemble: its later original-basis return <probability> is $c^4+s^4=1-\tfrac12\sin^2\theta$, equal to $5/8$ for this example. That constant differs from the oscillation of the unmeasured <pure state>. The <work> protocol and the oscillation protocol therefore use separate preparations.
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