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arXiv:1102.4047 · Physical Review A

Effective Dirac dynamics of ultracold atoms in bichromatic optical lattices

Dirk Witthaut, Tobias Salger, Sebastian Kling, Christopher Grossert, Martin Weitz

Published Sep 1, 2011• 47 citations
arXivHTMLPDF

Abstract

We study the dynamics of ultracold atoms in tailored bichromatic optical lattices. By tuning the lattice parameters, one can readily engineer the band structure and realize a Dirac point, i.e., a true crossing of two Bloch bands. The dynamics in the vicinity of such a crossing is described by the one-dimensional Dirac equation, which is rigorously shown beyond the tight-binding approximation. Within this framework we analyze the effects of an external potential and demonstrate numerically that it is possible to demonstrate Klein tunneling with current experimental setups.

DOI
10.1103/physreva.84.033601

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We study the dynamics of ultracold atoms in tailored bichromatic optical lattices. By tuning the lattice parameters, one can readily engineer the band structure and realize a Dirac point, i.e., a true crossing of two Bloch bands. The dynamics in the vicinity of such a crossing is described by the one-dimensional Dirac equation, which is rigorously shown beyond the tight-binding approximation. Within this framework we analyze the effects of an external potential and demonstrate numerically that it is possible to demonstrate Klein tunneling with current experimental setups.
We review the derivation of the effective Dirac equation for ultracold atoms in one-dimensional bichromatic optical lattices, following the proposal by Witthaut et al. [Phys. Rev. A 84, 033601 (2011)]. We discuss how such a derivation---based on a suitable rotation of the Bloch basis and on a coarse-graining approximation---is affected by the choice of the Wannier functions entering the coarsening procedure. We show that in general the Wannier functions obtained by rotating the maximally localized Wannier functions for the original Bloch bands can be sufficiently localized for justifying the coarse-graining approximation. We also comment on the relation between the rotation needed to achieve the Dirac form and the standard Foldy-Wouthuysen transformation. Our results provide a solid ground for the interpretation of the experimental results by Salger et al. [Phys. Rev. Lett. 107, 240401 (2011)] in terms of an effective Dirac dynamics.
The dispersion relation of ultracold atoms in variably shaped optical lattices can be tuned to resemble that of a relativistic particle, i.e. be linear instead of the usual nonrelativistic quadratic dispersion relation of a free atom. Cold atoms in such a lattice can be used to carry out quantum simulations of relativistic wave equation predictions. We begin this article by describing a Raman technique that allows to selectively load atoms into a desired Bloch band of the lattice near a band crossing. Subsequently, we review two recent experiments with quasirelativistic rubidium atoms in a bichromatic lattice, demonstrating the analogues of Klein tunnelling and Veselago lensing with ultracold atoms, respectively.
An experimental group in Beijing [Yueyang Zhai et al., Phys. Rev. A 87, 063638 (2013)] introduced the method of the standing-wave pulse sequence for efficiently preparing ultracold bosonic atoms into a specific excited band in a one-dimensional optical lattice. Here, we report on a theoretical extension of their work to the problem of one-dimensional bichromatic superlattices in order to understand a wave packet shaping process in higher bands. Varying the lattice parameters leads to the so-called Dirac point where a pair of excited bands crosses. This paper thus discusses simultaneously efficient excitation of the wave packet to the proximity of the Dirac point and its subsequent dynamics in the force field of a parabolic trap. We numerically explore optimal pulse-sequence parameters and find an optimized sequence which excites more than 99% of the atoms in an ideal situation to the first and second excited bands within 100 $\ensuremath{\mu}\mathrm{s}$. Our main finding is that the system permitting the Dirac point possesses a region of parameters where the excited energy bands become nearly parabolic, conducive to a robust wave packet with high coherence and isochronicity. We also provide an appropriate data set for future experimentation, including effects of the atom-atom interaction by way of the mean-field nonlinear term.
We simulate and analyze an experimental method of loading interacting ultracold atoms onto nontrivial quantum states such as nonlinear Bloch wave and soliton solutions in a one-dimensional bichromatic lattice. Of standard bands, inverted bands, and bands with Dirac-like points permitted by a bichromatic lattice, we consider the case of an inverted band and examine the loading process in terms of nonlinear Bloch waves formed by an aggregate of ultracold atoms described by the mean-field model. Specifically, we solved the Gross-Pitaevskii equation numerically and found an appropriate standing wave-pulse sequence for the inverted band, which sequence proved to be a suitable protocol for producing soliton solutions. In addition, we examined the effect of an external potential and dynamical instabilities for the postloading process. We also provide an appropriate data set for future experimental realization of our findings.
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Quantum simulation is a powerful tool to study a variety of problems in physics, ranging from high-energy physics to condensed-matter physics. In this article, we review the recent theoretical and experimental progress in quantum simulation of Dirac equation with tunable parameters by using ultracold neutral atoms trapped in optical lattices or subject to light-induced synthetic gauge fields. The effective theories for the quasiparticles become relativistic under certain conditions in these systems, making them ideal platforms for studying the exotic relativistic effects. We focus on the realization of one, two, and three dimensional Dirac equations as well as the detection of some relativistic effects, including particularly the well-known Zitterbewegung effect and Klein tunneling. The realization of quantum anomalous Hall effects is also briefly discussed.