
Transport of BoseEinsteinCondensates
Klaus Richter, Peter Schlagheck,
Michael Hartung, and Timo Hartmann
The realization of BoseEinstein condensates with ultracold atomic gases [1] has lead to a number of fascinating new experiments on various condensed matter phenomena. Due to their extraordinary coherence properties, BoseEinstein condensates are ideal objects for probing macroscopic quantum phenomena such as interference effects [2] and vortex lattices [3]. Moreover, the flexibility of optical and magnetic tools for the manipulation of ultracold atoms as well as the high degree of precision and control that is generally available in the quantum optical context have lead to experiments that provide new insight into interactioninduced effects in manybody systems, such as the quantum phase transition from a superfluid to a Mott insulator state [4]. With the development of magnetic and optical waveguides for cold atoms, e.g. on "atom chips" [5], a natural link was opened to mesoscopic transport physics. Such waveguides generally consist of a harmonic confinement in two (transverse) spatial dimensions and permit free motion along the third (longitudinal) dimension. By engineering suitable scattering regions within such waveguides, one would be able to study the role of interaction in transmission processes from a completely new perspective, which would also shed new light on interactionaffected transport problems in the electronic context. In this context, we focus on two paradigmatic transport processes:

Transmission of a BoseEinstein condensate through atomic quantum dots
Such a quantum dot could, e.g., be realized by focusing a pair of bluedetuned laser beams onto the waveguide, which would induce two symmetric potential barriers for the atoms. The scattering process of the condensate in this system is calculated by the numerical integration of the nonlinear GrossPitaevskii equation, where an inhomogeneous source term is used to simulate the injection of atoms into the waveguide. Computing the transmission spectrum in this way, we encounter a bistability phenomenon, where the resonance peaks are strongly distorted due to the interactioninduced nonlinearity. Due to a blocking phenomenon which is similar to Coulomb blockade for electrons, resonant transport of the condensate would generally be suppressed in a straightforward propagation process. It can, however, be achieved if the waveguide potential is adiabatically varied during the propagation process [6].


Fig. 1: Propagation of a BoseEinstein condensate through a waveguide with a double barrier potential, acting as an atomic quantum dot. The two barriers can be realized by focusing bluedetuned laser beams onto the waveguide. 

Fig. 2: Transmission spectrum of a BoseEinstein condensate in presence of a double barrier potential. The distortion of the resonance peak arises due to the interactioninduced nonlinearity in the GrossPitaevskii equation. Only the lower branches of the resonance peak (black line) can be directly populated in a straightforward propagation experiment. 
Transport of a condensate through disorder potentials
Spatial disorder is naturally present on atom chips where it is induced by spatial inhomogeneities in the electric wires on the chip surface [7]. These inhomogeneities lead to smooth fluctuations of the magnetic field in the waveguide, which directly affect the free propagation of the atoms. The transmission process of the condensate through such a disorder potential is, as in the case of a double barrier potential, calculated by a numerical integration of the inhomogeneous GrossPitaevskii equation. For vanishing atomatom interaction, we find an exponential decrease of the average transmission with the length of the disordered waveguide, which is the typical signature of Anderson localization. At finite interactions, however, a crossover to an algebraic (Ohmlike) decrease of the transmission with the disorder length is encountered. This crossover is correlated with the appearance of permanently timedependent scattering processes of the condensate, even though the injection of matter waves onto the disorder region proceeds in a perfectly stationary way [8]. This indicates that singleatom excitations and the population of a thermal cloud around the condensate might play an important role in the transport through disorder.


Fig. 3: Propagation of a BoseEinstein condensate through a waveguide with smooth random fluctuations. On atom chips, such a disorder potential would be induced by spatial inhomogeneities in the underlying electric wires on the chip surface. 

Fig. 4: Transmission of a BoseEinstein condensate through a disorder potential. At nonvanishing interactions between the atoms, a crossover from an exponential (Andersonlike) to an algebraic (Ohmlike) decrease of the transmission with the length of the disorder sample is encountered. This crossover is correlated with the appearance of permanently timedependent scattering processes (red dots) which dominantly arise above a critical sample length (blue dots denote timeindependent scattering processes). 
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