= Zeeman slower resonance tracking
{c}
{title2=$\delta_{\rm eff}=\omega_L+kv-\omega_0-\zeta B(z)$}
A <Zeeman slower> prepares a slow beam for capture; a narrow longitudinal <velocity> group is not by itself a cold three-dimensional cloud. The historical apparatus measured slowed-beam <velocities> with a separate <fluorescence> probe scanned in <frequency>, just after switching off the slowing beam. This avoided simultaneous-beam complications.<Nobel Lecture: Laser cooling and trapping of neutral atoms by William D. Phillips>[p. 724, Figs. 4–5]{ref}
For an original resonance-tracking calculation, let atoms travel along $+z$ with $v>0$ and the slowing beam along $-z$, so its atom-frame <angular frequency> is $\omega_L+kv$. Approximate the selected cycling transition by $\omega_{eg}(B)=\omega_0+\zeta B$ on one field branch; $\zeta$ is its signed <angular frequency> shift per unit <magnetic field>. A multilevel atom may require a nonlinear $\omega_{eg}(B)$ outside that branch. Maintaining a chosen effective <laser detuning> $\delta_s$ requires
$$
B(z)=\frac{\omega_L-\omega_0+kv(z)-\delta_s}{\zeta}.
$$
For chosen constant deceleration magnitude $a>0$, $v\,dv/dz=-a$ gives $v(z)=\sqrt{v_i^2-2az}$ and length $L=(v_i^2-v_f^2)/(2a)$. Substitution supplies a square-root <magnetic field> profile, plus a constant offset. The signs follow from the specified directions; reversing the selected transition's magnetic slope reverses the required field variation.
As a hypothetical design, take $v_i=300\ \mathrm{m\,s^{-1}}$, $v_f=30\ \mathrm{m\,s^{-1}}$ and $a=5.0\times10^4\ \mathrm{m\,s^{-2}}$. The required length is $0.891\ \mathrm m$ and flight time $(v_i-v_f)/a=5.4\ \mathrm{ms}$. For chosen $\lambda=589\ \mathrm{nm}$ and $m=3.82\times10^{-26}\ \mathrm{kg}$, <photon> recoil <velocity> $\hbar k/m\simeq0.0294\ \mathrm{m\,s^{-1}}$ implies roughly $9.17\times10^3$ directed absorption events. The mean required scattering rate is about $1.70\times10^6\ \mathrm{s^{-1}}$. This budget must lie below the available scattering rate, with margin for capture and intensity variation. <Optical pumping> into unwanted <hyperfine structure> or <magnetic sublevels> can interrupt the cycle. Longitudinal slowing, transverse recoil heating and final cloud <temperature> are separate quantities.
Back to article page
