Bose-Einstein condensates under rotation: The structures within

Transcription

1 Bose-Einstein condensates under rotation: The structures within Peter Mason Centre de Mathématiques Appliquées, Ecole Polytechnique Paris, France In collaboration with Amandine Aftalion and Thierry Jolicoeur: Project ANR VoLQuan and Natalia Berloff SISSA, April 21

2 Introduction to BEC s A Bose-Einstein Condensate (BEC) is the macroscopic occupation of a single particle state. - theoretically predicted to occur in atomic BEC s experimentally discovered in 1995 Requirements - need a dilute, weakly interacting Bose gas (a gas of bosons), e.g. 23 Na, 7 Li - temperatures below 1 5 K - λ db interparticle distance (i.e. wavefunctions overlap). Why Study? a) new form of matter b) applications in laser physics, optics, condensed matter etc.

3 Vortices are observed in experiments Quantised Vortices JILA 2 MIT 21 ψ classical, complex ψ = ρe is (S - phase) quantised vortices, ψ = distinct units of circulation, ħ/m ψ = There are other types of waves: Rarefaction waves are 2/3D structures with no vorticity and dark solitons are 1D structures.

4 Vortex Dynamics in homogeneous BEC s (1) Consider 2 vortices of opposite circulation (2D) U No external potential: GP becomes (non-dimensional) In a moving frame, x x Ut 2i ψ t = 2 ψ (1 ψ 2 )ψ 2iU ψ x = 2 ψ + (1 ψ 2 )ψ We seek a family of solutions for varying translational speed U.

5 Vortex Dynamics in homogeneous BEC s (2) E U p y E = 1 ψ (1 ψ 2 ) 2 dxdy p = i (ψ 1) ψ (ψ 1) ψ dxdy 2 and from ψ ψ + δψ we get U = E p = 1 2y same as classically Jones & Roberts, J. Phys. Math A (1982)

6 Single Component BEC s What do we observe when a condensate is placed under rotation? [Chevy & Dalibard, Europhysics News, 26] - Creation of vortices - Vortices arranged in a triangular lattice - Lowest Landau level analysis (fast rotation) or Thomas-Fermi analysis

7 Fast Rotation in Harmonic Traps The non-dimensional Gross-Pitaevskii energy is E[ψ] = ( ψ [H Ω ψ] + g 2 ψ 4) d 2 r for Hamiltonian H Ω = r2 2 ΩL z = 1 2 ( ia)2 + (1 Ω 2 ) r2 2 K.E. P.E. rotation energy Coroilis restoring centrifugal where L z = i(y x x y ) and A = Ω r with Ω = (,,Ω) and r = (x, y, ).

8 The Landau Levels A common eigenbasis of L z and H Ω is the Hermite functions φ j,k = e r2 /2 ( x + i y ) j ( x i y ) k (e r2 ). The eigenvalues for L z are j k, while for H Ω, they are E j,k = 1 + (1 Ω)j + (1 + Ω)k. Suppose Ω = 1. These are the Landau levels. If Ω 1 then two adjacent levels are separated by 2. However the distance between two adjacent states is 1 Ω 1.

9 The Lowest Landau Level We are interested in the lowest energy state: the lowest Landau level (LLL). This occurs when k =. Any function ψ of the LLL is a linear combination of the φ j, s and we can write ψ(r) = e r2 /2 P(u) = e r2 /2 n (u u j ) j=1 for P an analytic function of u = x + iy and u j the n complex zeros of P. Each u j is the position of a vortex!

10 Remember the GP energy is Minimisation of the GP Energy (1) E[ψ] = ( ψ [H Ω ψ] + g 2 ψ 4) d 2 r for which the wave function ψ minimising E[ψ] is H Ω ψ(r) + g ψ(r) 2 ψ(r) = µψ(r) where the chemical potential µ can be determined from the normalisation condition. We can then write the energy in the LLL as E LLL = Ω + ( (1 Ω)r 2 ψ 2 + g 2 ψ 4) d 2 r

11 Minimisation of the GP Energy (2) The minimisation of E LLL = Ω + ( (1 Ω)r 2 ψ 2 + g 2 ψ 4) d 2 r is equivalent to minimising E[ψ] = E LLL[ψ] Ω 1 Ω = r 2 ψ 2 + λ 2 ψ 4 d 2 r where λ = g 1 Ω. Minimisation of E LLL depends only on one parameter λ: a combination of g and Ω. We get that min[e] = λ π

12 Minimisation of the GP Energy (3) The density and radius of the disk are then ψ 2 = R = 2 (1 r2 πr 2 ) 1/4 ( 2λ π This is an inverted parabola! However it is not in the space of minimisation: to alleviate this we need to add vortices. Note R 2 ) The LLL analysis is valid for g(1 Ω) 1 and is obtained by balancing K.E. and P.E. terms. We have that the vortex size is of the same order as the intervortex distance (i.e. interaction between vortices becomes important). Vortices need not be small.

13 Two-Component BEC s What s different? Here are some experimental pictures: [Schweikhard et al. PRL, 24] [Hall et al. PRL, 1998] - Transition from triangular to square lattices - Creation of more unconventional topological defects; skyrmions, merons etc. - Two hyperfine states or two different atomic species that can be considered and confined concurrently.

14 The Gross-Pitaevskii Energy Functional The Gross-Pitaevskii energy functional for the two-component BEC in rotation is E[ψ 1, ψ 2 ] = k=1,2 ψ k for k = 1, 2 and where L z = i(y x x y ). ( V k (r) ΩL z + g ) k 2m k 2 ψ k 2 ψ k +g 12 ψ 1 2 ψ 2 2 Consider 2D condensate with rotation along the z-axis g 1 and g 2 represent the intra-component interation terms: g k = 4π 2 a k m k. g 12 represents the inter-component interaction term: g 12 = 2π 2 a 12 (m 1 +m 2 ) m 1 m 2. Take a harmonic trap; V k (r) = r 2 /2 and take g 1, g 2 and g 12 always positive (repulsive interactions).

15 Coreless Vortices or Singly quantised skyrmions Define Γ 12 = 1 g2 12 g 1 g 2. Lets have a look at Γ 12 =.6 and Ω = ψ 1 ψ Where one component has a vortex, the other component has a density peak at the same location. - This is the appearance of a coreless vortex.

16 Coreless vortices: Pseudospin Representation Define ψ k = ρ T χ, where ρ T = ρ 1 + ρ 2 is the total density. Then the spin density is defined as S = χσχ for Pauli matrices σ. Explicit expressions for S follow to be S x = χ 1χ 2 + χ 2χ 1 S y = i(χ 1χ 2 χ 2χ 1 ) S z = χ 1 2 χ 2 2. In this picture, ψ 1 has a vortex at the centre where ψ 1 vanishes. The pseudospin thus points down in the centre (from S z ). At the edge of the atmoic cloud, ψ 2 vanishes and the pseudospin points up. In between, the pseudospin changes continuously. We have then a coreless vortex.

17 Giant skyrmion or Multiply quantised skyrmion Lets have a look at Γ 12 =.6 and Ω = (a) 1 5 y y x (b) x - In the boundary layer between the two components, there is a topological defect. - This is the appearance of a giant skyrmion.