Optical lattice ($V(x)=s{E}_R{\sin }^2(\pi x/d)$)

An optical lattice is a periodic optical trap produced by interfering light fields. Its periodic potential gives atomic Bloch states and energy bands. For the displayed one-dimensional lattice, the recoil energy is $E_R={\hslash }^2{\pi }^2/(2m{d}^2)$ and $s$ measures the depth in recoil units.

Here $n$ is average number density and $\mu$ the ground-state chemical potential. This commonly used condensate response has units of inverse energy; thermodynamic isothermal compressibility is instead $\kappa /n$. In a weakly interacting repulsive condensate, the normalized one-cell ground state $f_s$, with average $|f_s{|}^2=1$, gives $\mu =E_0(s)+UnI(s)+\cdots$, $I(s)=d^{-1}{\int }_0^d|f_s{|}^{4}dx$. Localization raises $I$ and reduces $\kappa$ at fixed $n,U,d$. In deep harmonic wells $I\propto s^{1/4}$, hence $\kappa \propto s^{-1/4}$. This is a ground-band, repulsive-superfluid statement with the held variables specified, not a theorem for arbitrary signs of the interaction or arbitrary excited branches.

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.