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arXiv:cond-mat/0203061 · Physical Review A

Fermionizing a small gas of ultracold bosons

B. Paredes, P. Zoller, J. I. Cirac

Published Sep 20, 2002• 42 citations
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Abstract

We study the physics of a rapidly rotating gas of ultracold atomic bosons, with an internal degree of freedom. We show that in the limit of rapid rotation of the trap the problem exactly maps onto that of noninteracting fermions with spin in the lowest Landau level. The spectrum of the real bosonic system is identical to the one of the effective fermions, with the same eigenvalues and the same density of states. When the ratio of the number of atoms to the spin degeneracy is an integer number, the ground state for the effective fermions is an integer quantum Hall state. The corresponding bosonic state is a fractional quantum Hall liquid whose filling factor ranges in the sequence $\ensuremath{\nu}=1/2,2/3,3/4,\dots{},$ as the spin degeneracy increases. Anyons with $1/2,1/3,1/4,\dots{}$ statistics can be created by inserting lasers with the appropriate polarizations. A special situation appears when the spin degeneracy equals the number of atoms in the gas. The ground state is then the product of a completely antisymmetric spin state and a $\ensuremath{\nu}=1$ Laughlin state. In this case the system exhibits fermionic excitations with fermionic statistics although the real components are bosonic atoms.

DOI
10.1103/physreva.66.033609

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We study the physics of a rapidly rotating gas of ultracold atomic bosons, with an internal degree of freedom. We show that in the limit of rapid rotation of the trap the problem exactly maps onto that of noninteracting fermions with spin in the lowest Landau level. The spectrum of the real bosonic system is identical to the one of the effective fermions, with the same eigenvalues and the same density of states. When the ratio of the number of atoms to the spin degeneracy is an integer number, the ground state for the effective fermions is an integer quantum Hall state. The corresponding bosonic state is a fractional quantum Hall liquid whose filling factor ranges in the sequence \ensuremathν=1/2,2/3,3/4,…,\ensuremath{\nu}=1/2,2/3,3/4,\dots{},\ensuremathν=1/2,2/3,3/4,…, as the spin degeneracy increases. Anyons with 1/2,1/3,1/4,…1/2,1/3,1/4,\dots{}1/2,1/3,1/4,… statistics can be created by inserting lasers with the appropriate polarizations. A special situation appears when the spin degeneracy equals the number of atoms in the gas. The ground state is then the product of a completely antisymmetric spin state and a \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 Laughlin state. In this case the system exhibits fermionic excitations with fermionic statistics although the real components are bosonic atoms.
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 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 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.
We study a rapidly rotating gas of unpolarized spin-1/2 ultracold fermions in the two-dimensional regime when all atoms reside in the lowest Landau level. Due to the presence of the spin degree of freedom both sss-wave and ppp-wave interactions are allowed at ultralow temperatures. We investigate the phase diagram of this system as a function of the filling factor in the lowest Landau level and in terms of the ratio between sss- and ppp-wave interaction strengths. We show that the presence of attractive interactions induces a wide regime of phase separation with formation of maximally compact droplets that are either fully polarized or composed of spin-singlets. In the regime with no phase separation, we give evidence for fractional quantum Hall states. Most notably, we find two distinct singlet states at the filling $\ensuremath{\nu}=2/3$ for different interactions. One of these states is accounted for by the composite fermion theory, while the other one is a paired state for which we identify two competing descriptions with different topological structures. This paired state may be an Abelian liquid of composite spin-singlet Bose molecules with Laughlin correlations. Alternatively, it may be a known non-Abelian paired state, indicated by good overlaps with the corresponding trial wave function. By fine tuning of the scattering lengths it is possible to create the non-Abelian critical Haldane-Rezayi state for $\ensuremath{\nu}=1/2$ and the permanent state of Moore and Read for $\ensuremath{\nu}=1$. For purely repulsive interactions, we also find evidence for a gapped Halperin state at $\ensuremath{\nu}=2/5$.
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.
In the limit of very fast rotation atomic Bose-Einstein condensates may entirely reside in the lowest two-dimensional Landau level (LLL). For small enough filling factor of the LLL, one may have formation of fractional quantum Hall states. We investigate the case of bosons with dipolar interactions as may be realized with chromium-52 atoms. We show that at a filling factor equal to unity the ground state is a Moore-Read (or Pfaffian) paired state as is the case of bosons with purely sss-wave scattering interactions. This Pfaffian state is destabilized when the interaction in the sss-wave channel is small enough and the ground state is a stripe phase with unidimensional density modulation. For filling factor 1/3, we show that there is formation of a Fermi sea of ``composite fermions.'' These composites are made of one boson bound with three vortices. This phase has a wide range of stability and the effective mass of the fermions essentially depends only on the scattering amplitude in momentum channels larger or equal to 2. The formation of such a Fermi sea opens up a new possible route to detection of the quantum Hall correlations.