= Harmonic-oscillator dipole response to switched radiation
For a localized <quantum harmonic oscillator> along $z$, let the transverse <magnetic vector potential> be $A_z(t)=2A_0\cos(\theta-\omega t)$ for $t>0$, with zero potential before the switch. The <Coulomb gauge> form of <minimal coupling> gives $H_1=eA_zp_z/m$. The <creation operator> <matrix element> $\langle1|p_z|0\rangle=i\sqrt{m\hbar\omega_0/2}$ means that only $|1\rangle$ is reached at first order from the <ground state>. Its <interaction picture> amplitude is
$$
b_1=eA_0\sqrt{\frac{\omega_0}{2m\hbar}}
\left[e^{i\theta}F_{\omega_0-\omega}(t)+e^{-i\theta}F_{\omega_0+\omega}(t)\right],\qquad
F_\nu(t)=\int_0^t e^{i\nu t'}dt'.
$$
In particular $F_0=t$, so resonant finite-time expressions are nonsingular. The induced <electric dipole moment> is $-2e\sqrt{\hbar/(2m\omega_0)}\operatorname{Re}(e^{-i\omega_0t}b_1)$. It contains a driven component with the <dynamic electric polarizability> and free-frequency transients required by the initial conditions. The discontinuity in $A_z$ gives an impulsive <electric field> and an initial mechanical velocity $eA_z(0^+)/m$ even though the initial canonical <momentum> has zero mean.
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