Robust calibration of an optical-lattice depth based on a phase shift

We report on a method to calibrate the depth of an optical lattice. It consists of triggering the intrasite dipole mode of the cloud by a sudden phase shift. The corresponding oscillatory motion is directly related to the interband frequencies on a large range of lattice depths. Remarkably, for a moderate displacement, a single frequency dominates the oscillation of the zeroth and first orders of the interference pattern observed after a sufficiently long time of flight. The method is robust against atom-atom interactions and the exact value of the extra weak external confinement superimposed to the optical lattice.

Figure 1

(a) Diffraction of a rubidium-87 Bose-Einstein condensate from a pulsed one-dimensional optical lattice for $s\simeq 2.7$ (see text). The different images correspond to different pulse durations separated by $3\mu \mathrm{s}$. (b) Orders $-2$, $-1$, 0, 1, and 2: Experimental data (dark points), Raman-Nath approximation (blue dotted lines), and Mathieu functions analysis (red solid lines).

Figure 2

(a) Time-of-flight pictures (expansion during 25 ms) obtained after a sudden release of an adiabatically loaded Bose-Einstein condensate into an optical lattice for different depths $U_0=1.65\pm 0.3$, $6.62\pm 0.31$, $10.92\pm 2.4E_L$. (b) Integrated profile corresponding to each image.

Figure 3

Oscillation in momentum space (obtained after a 25 ms time of flight) triggered by an initial phase shift ${\theta }_0={25}^{\circ }$ and for different depths $U_0=2\pm 0.05$ (a), $6.66\pm 0.10$ (b), and $11.38\pm 0.15E_L$ (c).

Figure 4

Populations ${\mathrm{\Pi }}_n\left(\nu \right)$ in momentum space vs frequency $\nu$, for the orders $n=0$ (a) and $n=\pm 1$ (b). The populations ${\mathrm{\Pi }}_n\left(\nu \right)$ are defined as the Fourier transform of the oscillation pattern resulting from the initial quench. The quench corresponds to an offset of the initial potential by an angle ${\theta }_0={10}^{\circ }$ ($a_1$ and $b_1$), ${\theta }_0={25}^{\circ }$ ($a_2$ and $b_2$), and ${\theta }_0={70}^{\circ }$ ($a_3$ and $b_3$). The grayscale corresponds to a decimal scale.

Figure 5

Population of the zeroth order of the diffraction pattern of Fig. 3 as a function of time: Experimental data (black squares), fit (solid line). We find a frequency $\nu =33.3$ kHz corresponding to a depth $U_0=6.66\pm 0.10E_L$.

Figure 6

Normalized populations ${\tilde{\mathrm{\Pi }}}_m$ in the ${\mathrm{m}}^{th}$ order of the diffraction pattern as a function of time for ${\theta }_0={70}^{\circ }$: (a) $m=-2$, (b) $m=-1$, and (c) $m=0$. The depth of the optical potential is $U_0=9.76\pm 0.2E_L$.