Radio Frequency Selective Addressing of Localized Atoms in a Periodic Potential

We study the localization and addressability of ultracold atoms in a combined parabolic and periodic potential. Such a potential supports the existence of localized stationary states and we show that applying a radio frequency field allows us to selectively address atoms in these states. This method is used to measure the energy and momentum distribution of the atoms in the localized states. We also discuss possible extensions of this scheme to address and manipulate atoms in single lattice sites.

Figure 1

(a) Spectrum of the Hamiltonian: representation of the 1D spectrum of the single particle Schrödinger equation for a combined parabolic and periodic potential with $s=3$. Each line represents one eigenstate of the system, which is plotted as density profile in gray scale. The vertical position of the profile corresponds to the energy of the eigenstate. (b) Cloud of atoms without rf field (upper image), and with applied rf field (lower image) after 1.5 ms time of flight. The parameters of the rf field are ${\nu }_{\mathrm{u}\mathrm{p}}=632\mathrm{k}\mathrm{H}z$, ${\nu }_{\mathrm{l}\mathrm{o}\mathrm{w}}=624\mathrm{k}\mathrm{H}z$. The substructure in the upper cloud is due to the finite resolution of our imaging system.

Figure 2

Energy distribution of localized states in a lattice with $s=9$. (a) Scan of the rf field through the cloud. The indicated frequencies are the upper frequency ${\nu }_{\mathrm{u}\mathrm{p}}$ of the rf field; the sweep range of the rf field is 3 kHz. (b) Atom number in the left cloud in dependence on ${\nu }_{\mathrm{u}\mathrm{p}}$. The solid line is a fit with an axial density of states that is proportional to $E^{-1/2}$ (see text).

Figure 3

Momentum distribution of the localized states. (a) Fourier transform of selected eigenstates of the spectrum shown in Fig. 1a. The first three states are delocalized and lie within the first band; the last state is the first state of the second band. The states in between are localized states. (b) Absorption images of a cloud of atoms after application of the rf field for $s=3$ and $s=9$. To reveal the momentum distribution, the time of flight was chosen to be 10 ms.

Figure 4

(a) Tunneling between the bands: density distribution of the 594th eigenstate for a potential with $s=0.3$. (b) Extension of the localized states in the first band in dependence on the lattice height for three different energies ($k_{\mathrm{B}}\times 100\mathrm{n}\mathrm{K}$, $k_{\mathrm{B}}\times 200\mathrm{n}\mathrm{K}$, and $k_{\mathrm{B}}\times 300\mathrm{n}\mathrm{K}$).