= Doppler cooling limit
{c}
{title2=$T_D=\hbar\Gamma/(2k_B)$}
{wiki}
<Damping> removes random <kinetic energy>, but the discrete <photon> kicks cause <diffusion> in <momentum>. Their balance gives a nonzero <temperature>. Define the momentum diffusion coefficient $D_p$ by the noise contribution $d\langle p_x^2\rangle/dt=2D_p$. With atom <mass> $m$ and linear <damping> coefficient $\alpha$,
$$
\frac{d\langle p_x^2\rangle}{dt}=-\frac{2\alpha}{m}\langle p_x^2\rangle+2D_p,\qquad
k_BT=\frac{\langle p_x^2\rangle}{m}=\frac{D_p}{\alpha}
$$
at stationary balance, where $k_B$ is the <Boltzmann constant> and mean <momentum> is zero.
For ideal three-dimensional <optical molasses> with three identical independent beam pairs and isotropic <spontaneous emission>, the $x$-pair absorption rate at rest is $R_x=\Gamma s/(1+4d^2)$. Absorption from that pair contributes $(\hbar k)^2R_x$ to the noise growth of $\langle p_x^2\rangle$. Emission from all three pairs contributes the same amount: its total rate is $3R_x$ and each emitted <photon> has mean squared $x$-<momentum> $(\hbar k)^2/3$. Hence $D_p=(\hbar k)^2R_x$. Using the <Doppler cooling force> coefficient gives
$$
T(d)=\frac{\hbar\Gamma}{8k_B}\frac{1+4d^2}{-d},\qquad d<0.
$$
Put $a=-d>0$. Then $(1+4d^2)/(-d)=a^{-1}+4a\ge4$, with equality at $a=1/2$. The ideal weak-field <Doppler cooling limit> is therefore $T_D=\hbar\Gamma/(2k_B)$ at $\delta=-\Gamma/2$. It is the minimum of this specified diffusion-and-<damping> model, rather than a universal lower bound for <laser cooling>. The diffusion approximation also assumes <recoil energy> much smaller than $\hbar\Gamma$. In a one-pair geometry, emission patterns and transverse recoil change the diffusion bookkeeping. <Nobel Lecture: Laser cooling and trapping of neutral atoms by William D. Phillips>[William D. Phillips, Nobel Lecture: Laser cooling and trapping of neutral atoms, 1998, pp. 729–730, Eqs. (1)–(2): recoil heating, linewidth convention and ideal three-dimensional temperature limit]{ref}.
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