Optical lattice

There are two important parameters of an optical lattice: the potential well depth and the periodicity.

A basic optical lattice is formed by the interference pattern of two counter-propagating laser beams. The trapping mechanism is via the Stark shift, where off-resonant light causes shifts to an atom's internal structure. The effect of the Stark shift is to create a potential proportional to the intensity. This is the same trapping mechanism as in optical dipole traps (ODTs), with the only major difference being that the intensity of an optical lattice has a much more dramatic spatial variation than a standard ODT.[1]

The energy shift to (and thus, the potential experienced by) an electronic ground state ${\displaystyle \vert g_{i}\rangle }$ is given by second-order time-independent perturbation theory, where the rapid time variation of the lattice potential at optical frequencies has been time-averaged.

$${\displaystyle U(\mathbf {r} )=\Delta E_{i}={\frac {3\pi c^{2}\Gamma }{2\omega _{0}^{3}}}I(\mathbf {r} )\times \sum _{j}{\frac {c_{ij}^{2}}{\Delta _{ij}}}}$$

where ${\textstyle \mu _{ij}=\langle e_{j}\vert \mu \vert g_{i}\rangle \equiv c_{ij}\|\mu \|}$ are the transition matrix elements for transitions from the ground state ${\textstyle \vert g_{i}\rangle }$ to the excited states ${\textstyle \vert e_{j}\rangle }$. For a two-level system, this simplifies to

$${\displaystyle U(\mathbf {r} )=\Delta E={\frac {3\pi c^{2}}{2\omega _{0}^{3}}}{\frac {\Gamma }{\Delta }}I(\mathbf {r} )}$$

where ${\displaystyle \Gamma }$ is the linewidth of the excited state transition.[1]

An alternative picture of the stimulated light forces due to the AC Stark effect is to view the process as a stimulated Raman process, where the atom redistributes photons between the counterpropagating laser beams which form the lattice. In this picture, it is clearer that the atoms can only acquire momentum from the lattice in units of ${\displaystyle \pm 2\hbar k}$, where ${\displaystyle \hbar k}$ is the momentum of a photon of one laser beam.[1]

The trapping potential experienced by atoms in an optical dipole trap is weak, generally below 1 mK. Thus atoms must be cooled significantly before loading them into the optical lattice. Cooling techniques used to this end include magneto-optical traps, Doppler cooling, polarization gradient cooling, Raman cooling, resolved sideband cooling, and evaporative cooling.[1]

Once cold atoms are loaded into the optical lattice, they will experience heating by various mechanisms such as spontaneous scattering of photons from the optical lattice lasers. These mechanisms generally limit the lifetime of optical lattice experiments.[1]

Once cooled and trapped in an optical lattice, they can be manipulated or left to evolve. Common manipulations involve the "shaking" of the optical lattice by varying the relative phase between the counterpropagating beams, or amplitude modulation of the lattice. After evolving in response to the lattice potential and any manipulations, the atoms can be imaged via absorption imaging.

A common observation technique is time of flight (TOF) imaging. TOF imaging works by first waiting some amount of time for the atoms to evolve in the lattice potential, then turning off the lattice potential (by switching off the laser power with an AOM). The atoms, now free, spread out at different rates according to their momenta. By controlling the amount of time the atoms are allowed to evolve, the distance travelled by atoms maps onto what their momentum state must have been when the lattice was turned off. Because the atoms in the lattice can only change in momentum by ${\displaystyle \pm 2\hbar k}$, a characteristic pattern in a TOF image of an optical-lattice system is a series of peaks along the lattice axis at momenta ${\displaystyle \pm 2n\hbar k}$, where ${\displaystyle n\in \mathbb {Z} }$. Using TOF imaging, the momentum distribution of atoms in the lattice can be determined. Combined with in-situ absorption images (taken with the lattice still on), this is enough to determine the phase space density of the trapped atoms, an important metric for diagnosing Bose–Einstein condensation (or more generally, the formation of quantum degenerate phases of matter).

• Bose–Hubbard model
• Ultracold atom
• List of laser articles
• Electromagnetically induced grating
• Magic wavelength

• More about optical lattices
• Introduction to optical lattices
• Optical lattice on arxiv.org