= Solution
For the <quantum harmonic oscillator>, the <canonical commutation relation> $[x,p]=i\hbar$ gives $[a,a^\dagger]=1$ and $H_0=\hbar\omega_0(a^\dagger a+1/2)$. Inverting the definition of the <annihilation operator> gives
$$
x=\sqrt{\frac{\hbar}{2m\omega_0}}(a+a^\dagger).
$$
With normalized <energy eigenstates> and consistent phases, $a|0\rangle=0$ and $a^\dagger|0\rangle=|1\rangle$. Their <orthogonality> therefore gives
$$
\boxed{|\langle1|x|0\rangle|=\sqrt{\frac{\hbar}{2m\omega_0}}\equiv x_{01}.}
$$
More generally, $\langle n|x|0\rangle=x_{01}\delta_{n1}$, the oscillator <electric-dipole selection rule> out of the <ground state>.
To derive the transition <amplitude>, insert the expansion in stationary <energy eigenstates> into the <Schrödinger equation> with $H=H_0+V_1(t)$. The derivatives of the explicit phases cancel the $H_0$ terms. Projection onto $\langle n|$ leaves the exact coefficient equations
$$
i\hbar\dot c_n(t)=\sum_r e^{i(E_n-E_r)t/\hbar}\langle n|V_1(t)|r\rangle c_r(t).
$$
The initial conditions are $c_0(-\infty)=1$ and $c_{n\ne0}(-\infty)=0$. To first order in the perturbation, replace the <amplitudes> on the right by these initial values. Since $E_n-E_0=n\hbar\omega_0$, integration gives <time-dependent perturbation theory> in the form
$$
\boxed{c_n^{(1)}(t)=\frac1{i\hbar}\int_{-\infty}^t e^{in\omega_0t'}\langle n|V_1(t')|0\rangle\,dt'\qquad(n\ne0).}
$$
This neglects depletion and higher-order transitions; it is justified when the total transition <probability> is small. Here $\langle0|x|0\rangle=0$, so there is also no first-order diagonal <amplitude> change from this particular perturbation.
For the exponential <electric field>, the <electric-dipole interaction> is $V_1=-qEx e^{t/\tau}$, with $\tau>0$. Only $n=1$ couples directly to the initial state, and
$$
c_1^{(1)}(0)=\frac{iqEx_{01}}\hbar\int_{-\infty}^0e^{(1/\tau+i\omega_0)t'}\,dt'.
$$
The <Born rule> gives
$$
\boxed{p_1(0)\simeq\frac{q^2E^2}{2\hbar m\omega_0}\left|\int_{-\infty}^0e^{i\omega_0t'}e^{t'/\tau}\,dt'\right|^2.}
$$
The real part $1/\tau>0$ makes the lower-limit contribution vanish. Thus
$$
\int_{-\infty}^0e^{(1/\tau+i\omega_0)t'}\,dt'=\frac1{1/\tau+i\omega_0},\qquad \boxed{p_1(0)\simeq\frac{q^2E^2\tau^2}{2\hbar m\omega_0[1+\omega_0^2\tau^2]}.}
$$
Set $u=\omega_0\tau$ and $\eta=qEx_{01}/(\hbar\omega_0)$. The dimensionless form is $p_1/\eta^2=u^2/(1+u^2)$, increasing monotonically from zero to one, with half the asymptotic value at $u=1$. A sufficient weak-field condition for the entire curve is $|\eta|\ll1$.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2018-oscillator-switch-on.png]
{title=Original-basis excitation approaches a finite limit during slow oscillator displacement}
For $\omega_0\tau\ll1$, the force has acted for too little time to appreciably change the state at the instant of measurement. The short-time <amplitude> is approximately $iqEx_{01}\tau/\hbar$, so
$$
\boxed{p_1(0)\simeq\frac{q^2E^2\tau^2}{2\hbar m\omega_0}\quad(\omega_0\tau\ll1).}
$$
For $\omega_0\tau\gg1$, instead,
$$
\boxed{p_1(0)\longrightarrow\frac{q^2E^2}{2\hbar m\omega_0^3}=\eta^2.}
$$
The nonzero slow-switch limit is a <probability> in the original $H_0$ basis. It does not contradict the <adiabatic theorem>. At $t=0$ the instantaneous oscillator is displaced by $d=qE/(m\omega_0^2)$:
$$
H(0)=\frac{p^2}{2m}+\frac12m\omega_0^2(x-d)^2-\frac{q^2E^2}{2m\omega_0^2}.
$$
Its <ground state> is $|0_E\rangle=|0\rangle+\eta|1\rangle+O(\eta^2)$ in the original basis. Adiabatic following therefore predicts exactly the finite original-basis limit found above.
The <exponential switch-on of a driven quantum oscillator> also permits an exact check. A linear force preserves a <coherent state>, with original-basis displacement $\alpha(0)=i\eta u/(1+iu)$. Thus the exact original-basis <probability> is $|\alpha|^2e^{-|\alpha|^2}$, whose weak-field limit is the expression above. In the instantaneous displaced basis the mean excitation is instead $|\alpha-\eta|^2=\eta^2/(1+u^2)$ and does tend to zero for slow switching. The two bases measure different notions of excitation.
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