Microwave Control of Atomic Motion in Optical Lattices

We control the quantum mechanical motion of neutral atoms in an optical lattice by driving microwave transitions between spin states whose trapping potentials are spatially offset. Control of this offset with nanometer precision allows for adjustment of the coupling strength between different motional states, analogous to an adjustable effective Lamb-Dicke factor. This is used both for efficient one-dimensional sideband cooling of individual atoms to a vibrational ground state population of 97% and to drive coherent Rabi oscillation between arbitrary pairs of vibrational states. We further show that microwaves can drive well resolved transitions between motional states in maximally offset, shallow lattices, and thus in principle allow for coherent control of long-range quantum transport.

Figure 1

(a) Deep, slightly offset state-dependent optical lattices for the $|\downarrow ⟩$ and $|\uparrow ⟩$ hyperfine states. (b) The offset enables on site microwave transitions changing the vibrational quantum number. (c) Maximally offset, shallow lattices. (d) Microwave radiation can couple an initially localized population of $|\downarrow ⟩$ symmetrically to the neighboring potential wells and further throughout the lattice.

Figure 2

Microwave spectra for strong confining and slightly displaced ($\Delta x=24\mathrm{nm}$, $\vartheta =0.064\pi$) traps, for (a) molasses cooled and (b) sideband cooled atoms with a ground state population of 97%. The pulse area amounts to $\pi$ for both sidebands and to $1.6\pi$ for the carrier. (c) Spectrum for all vibrational states in the trap, starting from the axial ground state. The different spectra correspond to (from back to front) $\Delta x=\{0,43,111,176\}\mathrm{nm}$ ($\vartheta =\{0,0.112,0.280,0.420\}\pi$).

Figure 3

Rabi oscillations for the $|\uparrow ,0⟩\leftrightarrow |\downarrow ,1⟩$ (a) and $|\uparrow ,0⟩\leftrightarrow |\downarrow ,7⟩$ (b) transitions with Rabi frequencies of $2\pi \times 32\mathrm{kHz}$ and $2\pi \times 22\mathrm{kHz}$. Solid lines are sines damped by the empirically found function $\exp (-\sqrt{t/\tau })$ to account for radial dynamics. We find ${\tau }_{(a)}=(2.0\pm 0.1)\mathrm{ms}$, ${\tau }_{(b)}=(90\pm 10)\mu \mathrm{s}$. (c) Rabi oscillations on the carrier for a thermal initial state and (d) its Fourier transform. From the amplitudes we deduce $T_{\mathrm{ax}}=(8\pm 1)\mu \mathrm{K}$.

Figure 4

Measured Rabi frequencies of the carrier (○) and the red  (◇) and blue (□) sidebands, starting from state $|\downarrow ,1⟩$ for different polarization angles $\vartheta$. Solid lines are predictions of a full model. The offset $\Delta x$ (upper axis) depends nonlinearly on the polarization angle $\vartheta$. The inset shows measured and calculated Rabi frequencies in a shallow lattice, for $\vartheta \le \pi /2$.

Figure 5

Vibrational spectroscopy in a weakly confining lattice with $\Delta x={\lambda }_{\mathrm{lat}}/4$. (a) Calculated light shift of the $|\downarrow ,0⟩\leftrightarrow |\uparrow ,\{0,1\}⟩$ carrier and sideband transitions. The shaded areas indicate line broadening due to band curvature. The dashed line indicates the ${\omega }_{\mathrm{ax}}$ used in our experiment. (b) Microwave spectrum. Solid points are experimental data, the line a prediction from our full model. The various linewidths reflect the slope of the light shift curves in (a).