Direct Tunneling Delay Time Measurement in an Optical Lattice

We report on the measurement of the time required for a wave packet to tunnel through the potential barriers of an optical lattice. The experiment is carried out by loading adiabatically a Bose-Einstein condensate into a 1D optical lattice. A sudden displacement of the lattice by a few tens of nanometers excites the micromotion of the dipole mode. We then directly observe in momentum space the splitting of the wave packet at the turning points and measure the delay between the reflected and the tunneled packets for various initial displacements. Using this atomic beam splitter twice, we realize a chain of coherent micron-size Mach-Zehnder interferometers at the exit of which we get essentially a wave packet with a negative momentum, a result opposite to the prediction of classical physics.

Figure 1

(a) Time sequence showing the evolution of the wave packet in the lattice after a sudden displacement ($5\mu \mathrm{s}$ time interval between each picture). The data presented are averaged over two iterations. The images are taken after a 25 ms time of flight. (b) Sketch of the center-of-mass motion of the packets in each well. Their splitting by tunneling at the turning points is represented as dashed lines.

Figure 2

(a) Collective dipole mode period measured in dimensionless units as a function of the lattice depth (normalized to the lattice characteristic energy $E_L$) for various values of the external potential and the interactions: ${\tilde{\omega }}_{\text{ext}}=1/50$ (triangle), ${\tilde{\omega }}_{\text{ext}}=1/262$ (square), $\beta =0.1$ (open symbol), and $\beta =1$ (filled symbol). The proximity of the curves enables one to extract precisely the optical lattice depth (dashed line). The dotted curve corresponds to the classical oscillation period [34]. (b) Oscillation period as a function of the initial angle ${\theta }_0$ (which accounts for the initial sudden displacement of the lattice): experimental points (filled square), numerical simulation with the calibration presented in (a) (empty square), and classical prediction (dotted line).

Figure 3

(a) Experimental images: enlargements of $C$, $D_1$, and $D_2$ packets for various values of the initial offset angle of the lattice ${\theta }_0$. (b) Sketch of the tunneling process for different ${\theta }_0$ showing the thickness of the potential barrier at the first turning point (point $C$). (c) Quantitative tunneling time delay extracted from the experimental images (black disk), compared to the results of the numerical simulations without any adjustable parameter but including the uncertainty on the lattice depth (gray area). Inset: integrated density distribution of the reflected $D_1$ (black square) and tunneled $D_2$ (red triangles) packets for ${\theta }_0=40°$.