Quantum transport of ultracold atoms in an accelerating optical potential

We review our recent experiments on the motion of ultracold sodium atoms in an accelerating one-dimensional standing wave of light. Atoms are trapped in a far-detuned standing wave that is accelerated for a controlled duration. A small oscillatory component is added to the acceleration, and the fraction of trapped atoms is measured as a function of the oscillation frequency. Resonances are observed where the number of trapped atoms drops dramatically. The separation between resonances is found to be proportional to the acceleration, and they are identified as a Wannier–Stark ladder. At higher values of the acceleration, we observe an exponential decay in the number of atoms that remain trapped as a function of the interaction time. This loss is due to quantum tunneling, and we compare the decay rates with Landau– Zener theory. We also observe oscillations in the tunneling rate as a function of the acceleration; these are due to quantum interference effects. PACS: 03.75.-b; 32.80.Pj Quantum transport of particles in spatially tailored potentials has been a topic of active research in recent years, motivated by fundamental interest and by the prospect of controlling electron motion in microfabricated devices. In the regime of quantum transport, motion is dominated by tunneling and interference over macroscopic regions of phase space, and there are many basic questions that remain to be studied. The simplest case of a periodic potential was first studied theoretically in the 1930s by Bloch and Zener as a model of electron conduction in a crystalline lattice [1, 2]. A periodic potential in ∗ Present address: Conley, Rose and Tayon, P.C., 816 Congress Ave., Austin, TX 78701, USA ∗∗ Present address: Intel Corp., 5200 NE Elam Young Pkwy, Hillsboro, OR 97124, USA ∗∗∗ Present address: Hughes Aircraft Co., 2000 E. El Segundo Blvd., El Segundo CA 90245, USA ∗∗∗∗ Present address: Department of Mathematics, CSI-CUNY, Staten Island, NY 10314, USA ∗∗∗∗∗ Correspondence to: raizen@physics.utexas.edu one dimension leads to a quantized energy structure as shown in Fig. 1 for the case of a cosine potential [3]. The levels are broadened into bands due to resonant tunneling between adjacent wells. As the well depth is increased, tunneling in the low-lying bands is suppressed, and particle motion is dominated by single-well dynamics. Near the bottom of the well, the harmonic approximation is valid, further simplifying the analysis. This same band structure can also be displayed in the reciprocal lattice, as shown in Fig. 2. This plot is a dispersion relation between the energy E and the quasi-momentum hk (also referred to in textbooks as the crystal momentum). As a familiar point of reference, free particle motion would be represented in this picture as a parabola, due to the quadratic dependence of energy on quasi-momentum. The parabola is distorted when the periodic potential is turned on, opening band gaps. Note that the quasi-momentum is restricted here to the first Brillouin zone [3]. The natural basis set for this problem is composed of Bloch states that are spatially delocalized. An initial atomic wave packet that is spatially localized will then spread via resonant “Bloch” tunneling. −4π −2π 0 2π 4π φ 0.0 0.5 1.0 1.5 2.0