Klein Tunneling of a Quasirelativistic Bose-Einstein Condensate in an Optical Lattice

A proof-of-principle experiment simulating effects predicted by relativistic wave equations with ultracold atoms in a bichromatic optical lattice that allows for a tailoring of the dispersion relation is reported. We observe the analog of Klein tunneling, the penetration of relativistic particles through a potential barrier without the exponential damping that is characteristic for nonrelativistic quantum tunneling. Both linear (relativistic) and quadratic (nonrelativistic) dispersion relations are investigated, and significant barrier transmission is observed only for the relativistic case.

Figure 1

Band structure for an optical lattice potential for a relative phase of $\phi =0$ (left) and $\phi =\pi$ (right). The parameters for the lattice depths were $V_1=5E_r$ and $V_2=1.6E_r$. For $\phi =\pi$ the splitting between the first and the second excited band vanishes, and a Dirac point (marked by the solid blue box) is observed. The bands relevant for the experiment are shown by solid lines, and on the rescaled energy scale shown on the right-hand side, zero energy is chosen at the position of the band crossing.

Figure 2

Klein tunneling of atoms through a potential barrier. (a) Relevant part of the dispersion relation in the lattice for a relative phase between lattice harmonics of $\phi =0$. The large splitting between bands here suppresses a tunneling between bands. The reflection off the potential barrier leads to an oscillation between positive and negative values of the particle momentum (green arrows). (b) For $\phi =\pi$ the dispersion relation is linear and the two Bloch bands touch each other. Shown in the three diagrams is the variation of the atomic energy during passage of a potential barrier of height $V_b$. (c) Spatial distribution of the combined potential formed by gravitational and dipole trapping potential. Atoms can pass the potential well in the case of $\phi =\pi$ due to their possibility to drop to below the band crossing in the Bloch spectrum when loosing potential energy, see the middle graph in (b).

Figure 3

(a) Cuts (solid red) through the measured spatial atomic distribution (insets) for a 5 ms experiment time. Atoms proceeding from the center of the dipole trapping potential towards the potential barrier that detains against an outcoupling by the earths gravitational field for a relative phase between lattice harmonics of (top) $\phi =0$ and (bottom) $\phi =\pi$. The dotted black line indicates the potential $V_{\mathrm{slow}}(z)$. (b) Relative atomic population beyond the potential barrier versus phase $\phi$. The corresponding effective Compton wavelength ${\lambda }_{c,\mathrm{eff}}=2c_{\mathrm{eff}}h/\Delta E$ is shown on the top scale. The shown horizontal error bars refer to the latter scale and are dominated by the uncertainty in $\Delta E$, while the estimated uncertainty for the phase $\phi$ (lower scale) is below the drawing size of the dots. The solid line is the result of a numerical integration of the relativistic wave equation (see Eq. (1)). The only free fit parameters were amplitude and offset.

Figure 4

Fraction of unreflected atoms versus the height of the potential barrier $V_b$ for a phase shift (solid red) $\phi =0$ and (dashed blue) $\phi =\pi$. The solid lines show numerical simulations. The grey shaded region corresponds to barrier height values where the atomic velocity reaches the edge of the Brillouin zone at $|q|=\hslash k$. The desired dispersion relation is reached only in the white region. The experimental parameters are the same as in Fig. 3a except for the barrier height.