Portal de Periódicos da CAPES

Rapidly rotating atomic gases

2008; Taylor & Francis; Volume: 57; Issue: 6 Linguagem: Inglês

10.1080/00018730802564122

1460-6976

Nigel R. Cooper,

Atomic and Subatomic Physics Research

Abstract In this article, we review developments in the theory of rapidly rotating degenerate atomic gases. The main focus is on the equilibrium properties of a single-component atomic Bose gas, which (at least at rest) forms a Bose–Einstein condensate. Rotation leads to the formation of quantized vortices which order into a vortex array, in close analogy with the behaviour of superfluid helium. Under conditions of rapid rotation, when the vortex density becomes large, atomic Bose gases offer the possibility to explore the physics of quantized vortices in novel parameter regimes. First, there is an interesting regime in which the vortices become sufficiently dense that their cores, as set by the healing length, start to overlap. In this regime, the theoretical description simplifies, allowing a reduction to single-particle states in the lowest Landau level. Second, one can envisage entering a regime of very high vortex density, when the number of vortices becomes comparable to the number of particles in the gas. In this regime, theory predicts the appearance of a series of strongly correlated phases, which can be viewed as bosonic versions of fractional quantum Hall states. In this article, we describe the equilibrium properties of rapidly rotating atomic Bose gases in both the mean-field and the strongly correlated regimes, and related theoretical developments for Bose gases in lattices, for multi-component Bose gases and for atomic Fermi gases. The current experimental situation and outlook for the future are discussed in light of these theoretical developments. Keywords: Bose–Einstein condensationsuperfluidityquantized vorticesfractional quantum Hall effect Acknowledgements I have benefitted very much from working closely with many people on this and related topics: Nick Read, Stavros Komineas, Ed Rezayi, Steve Simon, Kareljan Schoutens, Miguel Cazalilla, Duncan Haldane, Gunnar Möller and, in particular, Nicola Wilkin and Mike Gunn who introduced me to the subject and who have been continued sources of help and advice. I also acknowledge useful discussions with many others, including Misha Baranov, Gordon Baym, Eric Cornell, Jean Dalibard, Eugene Demler, David Feder, Sandy Fetter, Victor Gurarie, Jason Ho, Jainendra Jain, Thierry Jolicoeur, Wolfgang Ketterle, Maciej Lewenstein, Chris Pethick and Nicholas Regnault. Finally, I am grateful to Sandy Fetter, Gunnar Möller and Kareljan Schoutens for helpful comments on a draft version of this manuscript. Notes Notes 1. The connection between a rotating neutral fluid and a charged fluid in a magnetic field is clarified later. 2. Bose–Einstein condensation has been measured by neutron scattering, yielding a condensate fraction at low temperatures of about 9% (see Citation5). 3. For the strong inequalities in (Equation20) to hold, the factor of two in this expression is clearly unimportant. We leave this factor in this formula merely as a reminder that the lowest-energy single-particle excitation out of the lowest Landau level (LLL) has energy 2ℏω⊥. In practice, one expects the physics derived by theoretical studies in the regime (Equation20) to be at least qualitatively accurate even in experiments in which these inequalities are not well satisfied. 4. The validity of the restriction of single-particle states to the two-dimensional LLL has been explored in Citation37. The numerical results are interpreted to indicate disagreement with the condition (Equation20) for restriction to the LLL, for the case of incompressible states at fixed filling factor ν = N/N v, Equation (Equation66). However, the criterion applied, the shift of the rotation frequency Ω, appears to overlook the dependence of Ω on N v, e.g. (Equation92) for N v ≫ 1. An analysis based on (Equation92) together with the assumption that with a correction due to Landau-level mixing that is small (δμ ≪ μ) leads to the conclusion that the quantity g max defined in Citation37 should vary as g max∼N v²/N. Thus, g max ∝ 1/N at fixed N v and g max ∝ N at fixed ν, in rough agreement with the numerical results Citation37. 5. When the exact groundstate is known (as for the Laughlin state at L = N(N − 1)), discussions reduce to considerations of the properties of this state. 6. This is easily checked by constructing the Euler–Lagrange equations for (Equation70). 7. These are the guiding-centre coordinates of a particle in a single Landau level. 8. The exact relation N v = 2N − 2 should be interpreted as N v = N/ν − 𝒮 where 𝒮 is the 'shift' of the state on the sphere, see Appendix B. 9. As written the states (Equation79) are not eigenstates of total angular momentum. They form an over-complete basis for the zero-energy states with L > N(N − 1). 10. See Note 2.1 regarding the units in Citation100,Citation101. 11. Such geometries have been studied in the groups of E. Cornell and J. Dalibard. 12. Within mean-field theory the results are identical for the quasi-two-dimensional and three-dimensional LLL regimes. 13. The more general form is ν = 2k/(2kq + 3) with q an even/odd integer for bosons/fermions Citation182. 14. Since the components are different hyperfine states of the same atomic species, the mass M is the same for all components. 15. The more general forms are ν = 3k/(3kq + 4) for SU(4)| k and ν = k/(kq + 1) for SO(5)| k with q an even/odd integer for bosons/fermions Citation197. 16. One exception is the trivial statement that ν = 1 is a fully filled Landau level. 17. Note that, in the BCS limit, this is a much lower rotation rate than that required to put one flux quantum through one Cooper pair, which would be ℏΩ≳Δ. 18. This transition cannot be captured within the theory in the narrow resonance limit Citation220, as the bosonic molecules are non-interacting so the Laughlin state does not appear. 19. The relation νm = νa/4 follows from the fact that there are half as many molecules as atoms, and the vortex density for a molecule is twice that for an atom Citation119. 20. The lowest-energy gap is proportional to V 0 in the two-dimensional LLL, V 0 ≪ ℏω⊥. For V 0 ≫ ℏω⊥, there will be mixing of Landau levels and the lowest-energy gap will not grow beyond the cyclotron gap 2ℏω⊥.