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arXiv:cond-mat/0211187

Quantum Hall Effect in a Rotating Bose-Einstein Condensate: An Atomic Twin of the Electronic Brother?

Zeng‐Bing Chen, Bo Zhao, Zhang, Yong-De

Published Nov 10, 2002• 1 citation
arXivHTMLPDF

Abstract

We exploit the analogy with the quantum Hall (QH) effect for electrons to study the possible atomic QH states of a rapidly-rotating Bose-Einstein condensate. Actually, there is a nearly perfect map of the present problem in the QH regime to the QH physics for electrons. The profound map enables one to give a physically appealing definitions of the filling fraction and the "atomic Hall conductance" that is quantized for atomic Laughlin states. This quantization might imply an exotic fractionalization of atomic mass. We also briefly discuss an effective Chern-Simons theory for describing the atomic QH liquids where a gravitational-like field naturally emerges.

DOI
10.48550/arxiv.cond-mat/0211187

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We exploit the analogy with the quantum Hall (QH) effect for electrons to study the possible atomic QH states of a rapidly-rotating Bose-Einstein condensate. Actually, there is a nearly perfect map of the present problem in the QH regime to the QH physics for electrons. The profound map enables one to give a physically appealing definitions of the filling fraction and the "atomic Hall conductance" that is quantized for atomic Laughlin states. This quantization might imply an exotic fractionalization of atomic mass. We also briefly discuss an effective Chern-Simons theory for describing the atomic QH liquids where a gravitational-like field naturally emerges.
The close theoretical analogy between the physics of rapidly rotating atomic Bose condensates and the quantum Hall effect (i.e. a two-dimensional electron gas in a strong magnetic field) was first pointed out ten years ago. As a consequence of this analogy, a large number of strongly correlated quantum-Hall-type states have been predicted to occur in rotating Bose systems, and suggestions have been made for how to manipulate and observe their fractional quasiparticle excitations. Due to a very rapid development in experimental techniques over the past years, experiments on BEC now appear to be close to reaching the quantum Hall regime. This paper reviews the theoretical and experimental work done to date in exploring quantum Hall physics in cold bosonic gases. Future perspectives are discussed briefly, in particular the idea of exploiting some of these strongly correlated states in the context of topological quantum computing.
In this paper, we explore the composite particle description of the atomic quantum Hall (QH) effect. We further give the Chern–Simon–Gross–Pitaevskii (CSGP) effective theory for the atomic Hall liquid, which is the counterpart of Chern–Simon theory in electron Hall effect. What we obtained is equivalent to the Laughlin wavefunction approach. Our results show that in terms of composite particles, the atomic Hall effect is really the same as the electronic QH effect. The CSGP effective theory would shed new light on the atomic QH effect.
This chapter reviews the theoretical understanding of Bose and Fermi superfluids under rotation (or in an effective magnetic field) while placing this in context of the current experimental status. It focuses on theoretical predictions of the novel phases that can appear. One goal is to establish the connections with the physics of the fractional quantum Hall effect of electrons in semiconductors. As the chapter is intended to be tutorial in nature, it concentrates on the qualitative features and simple model calculations to present the essential ideas in as physically transparent a way as possible. It begins with a brief review of the properties of Bose–Einstein condensates subjected to rotation.
The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.
The long-term goal of our study is to perform a simulation on a rapidly rotating gas of ultracold bosonic atoms. When the rotation frequency is close to the harmonic trap frequency, the trapping force is compensated by the centrifugal force and only the Coriolis force remains, analogous to the Lorentz force. The quantum Hall effect regime is reached when the number of vortices is larger than the number of atoms. In this work, we demonstrate a variety of excitations of the fractional quantum Hall states of composite fermions and ultracold two-dimensional rapidly rotating bosonic atoms. Second, we can also check directly whether there is a relationship between these two phenomena by calculating the energy of N = 8 bosons in the ground state of the harmonic oscillator and a few-boson elementary excitations for the Bose–Laughlin state at the half filling factor.
After reviewing the ideal Bose-Einstein gas in a box and in a harmonic trap, the effect of interactions on the formation of a Bose-Einstein condensate are discussed, along with the dynamics of small-amplitude perturbations (the Bogoliubov equations). When the condensate rotates with angular velocity $\ensuremath{\Omega}$, one or several vortices nucleate, leading to many observable consequences. With more rapid rotation, the vortices form a dense triangular array, and the collective behavior of these vortices has additional experimental implications. For $\ensuremath{\Omega}$ near the radial trap frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$, the lowest-Landau-level approximation becomes applicable, providing a simple picture of such rapidly rotating condensates. Eventually, as $\ensuremath{\Omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{\ensuremath{\perp}}$, the rotating dilute gas is expected to undergo a quantum phase transition from a superfluid to various highly correlated (nonsuperfluid) states analogous to those familiar from the fractional quantum Hall effect for electrons in a strong perpendicular magnetic field.