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Fractional quantum Hall effect in ultracold rapidly rotating atomic gases

Nicolas Regnault

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

Ultracold atomic gases in rapidly rotating traps display a wealth of interesting physics. One of the most striking achievements in this field is the observation of th e Abrikosov lattice of vortices. When the rotation frequency is equal to the harmonic trap frequency, the atomic gases is similar to charged particles in a magnetic f ield. It has been predicted that for weak interactions and strong confinement along the rotation axis, this system will exhibit quantum phases equivalent to the one o f the fractional quantum Hall effect. The talk will be focused on bosonic atoms with only one hyperfine species (i.e spinless bosons). We will give numerical evidences of the fractional quantum Hall states in these systems. We will describe how they can be understood using the Laughlin wave function and the Jain's composite fermion theory. We will also discuss about more exotic quantum phases that have also been predicted in such regime. These quantum phases are described by the Read-Rezayi wave functions whose one of the most remarkable properties is that their excitations obey non-abelian statistics.

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Ultracold atomic gases in rapidly rotating traps display a wealth of interesting physics. One of the most striking achievements in this field is the observation of th e Abrikosov lattice of vortices. When the rotation frequency is equal to the harmonic trap frequency, the atomic gases is similar to charged particles in a magnetic f ield. It has been predicted that for weak interactions and strong confinement along the rotation axis, this system will exhibit quantum phases equivalent to the one o f the fractional quantum Hall effect. The talk will be focused on bosonic atoms with only one hyperfine species (i.e spinless bosons). We will give numerical evidences of the fractional quantum Hall states in these systems. We will describe how they can be understood using the Laughlin wave function and the Jain's composite fermion theory. We will also discuss about more exotic quantum phases that have also been predicted in such regime. These quantum phases are described by the Read-Rezayi wave functions whose one of the most remarkable properties is that their excitations obey non-abelian statistics.
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.
The non-perturbative effect of interaction can sometimes make interacting bosons behave as though they were free fermions. The system of neutral bosons in a rapidly rotating atomic trap is equivalent to charged bosons coupled to a magnetic field, which has opened up the possibility of fractional quantum Hall effect for bosons interacting with a short range interaction. Motivated by the composite fermion theory of the fractional Hall effect of electrons, we test the idea that the interacting bosons map into non-interacting spinless fermions carrying one vortex each, by comparing wave functions incorporating this physics with exact wave functions available for systems containing up to 12 bosons. We study here the analogy between interacting bosons at filling factors ν=n/(n+1)\nu=n/(n+1)ν=n/(n+1) with non-interacting fermions at ν∗=n\nu^*=nν∗=n for the ground state as well as the low-energy excited states and find that it provides a good account of the behavior for small nnn, but interactions between fermions become increasingly important with nnn. At ν=1\nu=1ν=1, which is obtained in the limit n→∞n\rightarrow \inftyn→∞, the fermionization appears to overcompensate for the repulsive interaction between bosons, producing an {\em attractive} interactions between fermions, as evidenced by a pairing of fermions here.
Cold atomic gases of interacting bosons subject to rapid rotation and confined in anharmonic traps can theoretically exhibit analogues of the fractional quantum Hall effect for electrons in strong magnetic fields. In this setting the Coriolis force due to the rotation mimics the Lorentz force on charged particles but artificial gauge fields can also be obtained by coupling the internal structure of the atoms to light fields. The chapter discusses mathematical aspects of transitions to different strongly correlated phases that appear when the parameters of a model Hamiltonian are varied.
The nonperturbative effect of interaction can sometimes make interacting bosons behave as though they were free fermions. The system of neutral bosons in a rapidly rotating atomic trap is equivalent to charged bosons coupled to a magnetic field, which has opened up the possibility of a fractional quantum Hall effect for bosons interacting with a short-range interaction. Motivated by the composite fermion theory of a the fractional Hall effect of electrons, we test the idea that the interacting bosons map onto noninteracting spinless fermions carrying one vortex each, by comparing wave functions incorporating this physics with exact wave functions available for systems containing up to 12 bosons. We study here the analogy between interacting bosons at filling factors $\ensuremath{\nu}=n∕(n+1)$ with noninteracting fermions at ${\ensuremath{\nu}}^{*}=n$ for the ground state as well as the low-energy excited states and find that it provides a good account of the behavior for small nnn, but interactions between fermions become increasingly important with nnn. At $\ensuremath{\nu}=1$, which is obtained in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the fermionization appears to overcompensate for the repulsive interaction between bosons, producing an attractive interaction between fermions, as evidenced by a pairing of fermions here.
The system of neutral bosons in rapidly rotating atomic trap is equivalent to charged bosons coupled to a magnetic field, which has opened up the possibility of fractional quantum Hall effect (FQHE) for bosons interacting with a short range interaction. We present numerical evidences for incompressible liquids at filling fractions ν = 1/2, 2/3, 3/4. Motivated by the composite fermion (CF) theory of the FQHE for electrons, we show how interacting bosons can be mapped into non-interacting spinless fermions carrying one vortex each. This provides an elegant explanation of the fractional quantum Hall states in this system. We use overlap calculations to test how the CF model fits with exact numerical results. At ν = 1, the fermionization appears to overcompensate for the repulsive interaction between bosons producing an attractive interaction between fermions, as evidence by a pairing of fermions. In that case, the Pfaffian state is a good candidate to describe the ground state of such system.
We use a Chern Simons Landau-Ginzburg (CSLG) framework related to hierarchies of composite bosons to describe 2D harmonically trapped fast rotating Bose gases in Fractional Quantum Hall Effect (FQHE) states. The predicted values for ννν (ratio of particle to vortex numbers) are ννν===pq{{p}\over{q}}qp​ (ppp, qqq are any integers) with even product pqpqpq, including numerically favored values previously found and predicting a richer set of values. We show that those values can be understood from a bosonic analog of the law of the corresponding states relevant to the electronic FQHE. A tentative global phase diagram for the bosonic system for ννν$