= Solution
For a free-space <electric dipole transition>, the <spontaneous emission> rate in SI units is
$$
\boxed{\Gamma=\frac{\omega_0^3|\mathbf d|^2}{3\pi\epsilon_0\hbar c^3}
\ \propto\ |d|^2\omega_0^3.}
$$
Here $|\mathbf d|^2$ includes the relevant dipole components; a specified single polarization requires its corresponding <matrix element>.
A complete scattering cycle returns the internal state to the <ground state> and changes the atomic <momentum> to $\mathbf P+\mathbf p-\mathbf p'$. Thus
$$
\boxed{\Delta E=\frac{|\mathbf P+\mathbf p-\mathbf p'|^2-P^2}{2M}
=\frac{\mathbf P\cdot(\mathbf p-\mathbf p')}{M}
+\frac{|\mathbf p-\mathbf p'|^2}{2M}.}
$$
For <photon-scattering recoil heating> with isotropic emission of fixed <momentum> magnitude, $\langle\mathbf p'\rangle=0$, giving
$$
\boxed{\langle\Delta E\rangle_{\rm em}
=\frac{\mathbf P\cdot\mathbf p}{M}+\frac{p^2}{2M}+\frac{p'^2}{2M}.}
$$
The first term is the velocity-dependent <work> of absorption and can cool an atom if the <photon> opposes its motion. The last two are positive absorption and emission <recoil> energies. If the actual absorbed directions are also isotropic and independent of $\mathbf P$, the first term averages to zero. For approximately equal incoming and outgoing <photon> momenta $\hbar k$,
$$
\boxed{\langle\Delta E\rangle=2E_R,\qquad E_R=\frac{\hbar^2k^2}{2M}.}
$$
Isotropic incident illumination alone would not guarantee isotropic absorption if Doppler-dependent excitation selected directions.
\b[Source qualification for the trapped calculation.] Set $m=M$ for the same atom. The printed Gaussian $\exp[-x^2/(2l_0)^2]$ has the wrong width for the specified <harmonic oscillator>: the printed ground and first-excited functions have squared norms $\sqrt2$ and $2\sqrt2$, respectively. The normalized oscillator states fixed by the given $l_0=\sqrt{\hbar/(m\omega_{\rm HO})}$ instead use
$$
\psi_0=\pi^{-1/4}l_0^{-1/2}e^{-x^2/(2l_0^2)},\qquad
\psi_1=\sqrt2(x/l_0)\psi_0.
$$
Also, one common complex phase multiplying both off-diagonal terms does not make the <Hamiltonian> a <Hermitian operator>. Use an absorption <matrix element> $(\hbar\Omega/2)e^{-ikx}$ and its <Hermitian conjugate> for emission; choosing the opposite propagation convention reverses the imaginary sideband phase but not a <transition probability>. The physical results below use these explicitly stated corrections. Parts (i) and (ii) also give the literal integrals of the printed functions.
\b[<Conditional motional heating and recoil averaging>.] With $\eta=kl_0/\sqrt2$, an oscillator <eigenstate> has $\langle K\rangle_n=\langle V\rangle_n=(n+1/2)\hbar\omega_{\rm HO}/2$. Therefore the absorption outcomes have
$$
\boxed{0\to0:\ \Delta K=\Delta V=0;\qquad
0\to1:\ \Delta K=\Delta V=\hbar\omega_{\rm HO}/2.}
$$
The same statements hold for the specified emission outcomes $|e,0\rangle\to|g,0\rangle$ and $|e,0\rangle\to|g,1\rangle$. Internal excitation <energy> goes into, or comes from, the <photon> and is distinct from this external <energy> change. Emission into the first motional state makes the emitted <photon> correspondingly less energetic.
A sudden <recoil> from $|0\rangle$ gives probabilities $P_n=e^{-\eta^2}\eta^{2n}/n!$, so $P_1\simeq\eta^2\ll1$ while $P_0\simeq1$. The rare first sideband transfers a whole trap quantum, much larger than $E_R$; the mean is nevertheless
$$
\boxed{\sum_n P_n\,n\hbar\omega_{\rm HO}
=\eta^2\hbar\omega_{\rm HO}=E_R.}
$$
The emission average is the same for a one-dimensional <recoil> of the same magnitude. Projecting isotropic three-dimensional emission onto one axis instead gives a factor $\langle\cos^2\theta\rangle=1/3$ for that axis.
A sudden unobserved kick initially raises <kinetic energy> by $E_R$ and leaves the position distribution, hence <potential energy>, unchanged. It creates a coherent <superposition> of motional states. The equal kinetic/potential increments above concern energy-eigenstate-conditioned outcomes, or time-averaged/dephased motion; these are different measurement conditions. There is no contradiction.
With $\Gamma>\omega_{\rm HO}$, motional sidebands are not cleanly resolved, and a short-lived excited atom has little time to move before decay. In the stronger sudden limit $\Gamma\gg\omega_{\rm HO}$ the recoil-kick picture is direct. For only the stated inequality, finite linewidth weights can change numerical factors; it does not fix an exact continuous-wave scattering rate. \b[The average heating remains of <recoil> order per absorption or emission], and of order $2E_R$ when both absorption and emission kicks are counted at full magnitude. For the axial component of isotropic three-dimensional emission, the emission contribution instead carries the factor $1/3$ noted above. Strong confinement suppresses the chance of changing $n$, but does not make its averaged <recoil energy> disappear. This agrees with free-space <recoil> heating while replacing small continuous <energy> changes by rare quantized motional changes.
Back to article page
