Recoil energy ($E_R=p_R^2/(2m)$)

Recoil energy is the kinetic energy gained by an initially stationary atom absorbing a photon of momentum $p_R$. In a standing-wave optical lattice of spacing $d$, the photon wavenumber is $\pi /d$, giving $E_R={\hslash }^2{\pi }^2/(2m{d}^2)$.

Absorption of momentum $\mathbf{p}$ followed by emission of ${\mathbf{p}}^{\prime }$ changes a free atom's momentum by $\mathbf{p}-{\mathbf{p}}^{\prime }$. For isotropic emission of fixed magnitude, its mean vector is zero. The energy change then separates into a directional work term and two positive recoil terms. If absorption directions are also isotropic and uncorrelated with the initial momentum, the work term averages away. Equal incoming and outgoing photon magnitudes give two recoil energies per cycle.