nigel cooper, nigel cooper and paul sutcliffe- stable skyrmions in two-component bose-einstein...

8/3/2019 Nigel Cooper, Nigel Cooper and Paul Sutcliffe- Stable Skyrmions in Two-Component Bose-Einstein Condensates

1/18

Stable Skyrmions in

Two-Component Bose-Einstein

Condensates

Nigel CooperT.C.M. Group, Cavendish Laboratory,

University of Cambridge

UMIST, 8 Feb 2002

Richard Battye (Cambridge/Manchester),Paul Sutcliffe (Kent)

[PRL 88, 080401 (2002)]


2/18

Overview

Atomic Bose-Einstein Condensates Multi-component condensates Topological Solitons

Stable Skyrmions Summary

1


3/18

Bose-Einstein Condensation

T

a

h2

m2T kBT a n1/3

T > a kBT


4/18

Atomic Bose-Einstein Condensates

kBTc h2m n2/3 100nK

87

Rb (JILA, 1995)7Li (Rice, 1995)23Na (MIT, 1995)1H (MIT, 1998)85Rb (JILA, 2000)4He (Orsay, 2001)

[Anderson et. al. [JILA], Science 269, 198 (1995).]

3


5/18


6/18

Multicomponent Condensates

Hyperfine interaction F = I + JTypically:

J = S = 1/2 (Alkali gases)I = 3/2 (87Rb, 23Na, 7Li)

|F = 2, m = 2,1, 0, 1, 2

|F = 1, m =

1, 0, 1

Magnetic trap87Rb |F = 2, m = 2 and |F = 1, m = 1

[Myatt et. al.[JILA], PRL78, 586 (1997)]

87Rb

|F = 2, m = 1

and

|F = 1, m =

1

[Hall et. al.[JILA], PRL81, 1539 (1998)]Optical trap

23Na |F = 1, m = 1, 0, 1[Stenger et. al.[MIT], Nature 396, 345 (1998)]

Different atoms / Isotopes85Rb and 87Rb [Bloch et. al.[Munich], PRA 64, 021402 (2001)]

5


7/18

Vortex in a two-component BEC[Matthews et. al. [JILA], PRL 83, 2498 (1998).]

|1 = |1,1

|2

=

|2, 1

6


8/18

Gross-Pitaevskii Mean-Field Theory

N Ni=1

(ri)

N =

d3r |(r)|2

Minimise the expectation value of the energy

E =

d3r

h2

2m||2 + V(r)||2 + 1

2U||4

at fixed N (chemical potential) U a

7


9/18

Multi-component case

N Ni=1

(ri)|i

N =

d3r |(r)|2

Energy (density)

h2

2m||2 + V(r)||2 + 1

2

,

U||2||2

U all mutual two-body scattering lengths N conserved separate chemical potentials

=

12

... topological solitons ?

8


10/18

Topological Textures

One-component condensate in one dimension.

(r) =

ei

x0 L

Topological invariant

Q =1

2

L0

d

dxdx

9


11/18

Two-component condensate in three dimensions.

12

=

cos(/2)ei1

sin(/2)ei2

Q =1

82 sin ij1k2 ijkd

3r

Skyrmion

[Al Kawaja and Stoof, Nature 411, 918 (2001);

Ruostekoski and Anglin, PRL 86, 3934 (2001)]

[Three components, textures vortices, monopoles...]

10


12/18

How to obtain stable Skyrmions

Large Trap (|r| ) = 0 10 Constrain N2 =

d3r |2|2

Regime of phase separation: U212 > U11U22

We study U11 U12 U22 s.t. (r) = 0.Find stable skyrmions of the form:

Imprint with lasers [Ruostekoski and Anglin, PRL 86, 3934 (2001)]

[cf. Cosmic vortons]

11


13/18

Mathematical Details

12

=

0

cos(/2)ei1

sin(/2)ei2

N2 = 0 sin2(/2)d3r

Q = 182

sin ij1k2 ijkd3r

Energy density

h20

2m 14||2 + cos2(/2)|1|2 + sin2(/2)|2|2+sin2

[ 18

20 (2U12 U11 U22)]

Lengthscales: h22m R2 = N20 1/3

E(, N2) =

h20R2

m

EQ()

R2

12


14/18

Q=1

13


15/18

Q=2

14


16/18

Axisymmetric Ansatz

(r, z), 1(r, z), 2 = m[(r,,z) are cylindrical polar co-ordinates]

Q = mn

0 10 20 30

0

100

200

300

m,n=1

m=7

m=6

m=5

m=4

m=3m=2

m=1

15


17/18

Moving Vortex Rings

Constrain the impulse (momentum)

Pi =h

2i

d3r[rjij rjji]

v =E

P

0 0.5 1 1.5 2

p

54

54.5

55

Q=1

67

67.5

68

Q=2

Q=1

Q=2

16


18/18

Summary

Atomic Bose-Einstein condensates offer thepossibility of studying interacting Bose gases with ahigh level of control (interaction strength,confinement, numbers of components).

In the regime of phase separation, two-componentBECs have stable textures with the topology ofSkyrmions (Q = 1, 2).

This regime is relevant for 2-component 87Rbsystems. We expect that textures imprinted by lasers

will relax to these stable Skyrmion configurations.

17

nigel cooper, nigel cooper and paul sutcliffe- stable skyrmions in two-component bose-einstein...

Documents