Kioloa, February 8, 2006

Chromium Bose-EinsteinCondensates

Luis SantosUniversität Stuttgart

Chromium BEC

BEC of Chromium atoms[Griesmaier et al., PRL 94, 160401 (2005)]

Also experiments on Chromium in Paris-Nord

It is NOT just another BEC, because …

Chromium has a large magneticmoment, µ=6µB

The ground state of 52Cr is 7S3

First available dipolar BEC

First available spin-3 BEC

Outline of the talk

Dipolar gases

Cr-BEC as spin-3 BEC

m=-3m=-2

m=-1m=0

m=+1m=+2

m=+3

B B

Dipolar gases

In typical experiments up to now the atomsinteract via short-range isotropic interactions

The interaction is given by the s-wave scattering length “a”

( ) )(4 2

rrm

arrV ′−≈′−rrhrr δπ

Dipolar gases

Recent experimental developments open a novel research area in cold gases: the analysis of dipolar gases

Atoms with large magnetic moment, as Cr [Griesmaier et al., PRL 94, 160401 (2005)]

Polar moleculesDirect cooling of polar molecules [Bethlem and Meijer, Int. Rev.

Phys. Chem. 22, 73 (2003)]

Photoassociation of polar molecules in optical lattices

[Jaksch et al., PRL 89, 040402 (2002); Damski et al., PRL 90, 110401 (2003);

Rom et al., PRL 93, 073200 (2004)]

Feshbach resonances in binary mixtures

[Stan et al. PRL 93, 143001 (2004);Inouye et al., PRL 93, 183201 (2004);

Petrov et al., cond-mat/0502010]

Laser induced dipole-dipole interaction [O‘Dell et al., PRL 84, 5687 (2000)]

Rydberg atoms [Jaksch et al., PRL 85, 2208 (2000)]

Dipolar gases

( )θ23

2

cos31)( −=rdrV r

Dipole-dipole interaction

Attractiveinteraction

Repulsive interaction

The interaction is anisotropic(partially attractive and

partially repulsive!)

Long-range interaction

Dipolar gases

At low temperatures the physics of a dipolar BEC isgiven by a nonlocal nonlinear Schrödinger equation

),(),()cos31(

),(),(2

),(2

3

2

222

trtr

rrrdg

trgtrVm

trt

id

r

rrr

r

rrh

rh ψ

ψθ

ψψ

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

′′−

−′+

+++∇−

=∂∂

∫

Nonlocal NLSE

ag ∝ 2dgd ∝

Nonlocal nonlinearity is also observed in other physical systemsPlasma physics

[Litvak et al., Sov. J. Plasma Phys. 1, 60 (1975)]Nematic Liquid Crystals[Peccianti et al., Nature 432, 733 (2004)]

Dipolar gases [Stuhler et al., PRL 95, 150406 (2005)]

The expansion dynamics of a dipolar BEC issignificantly affected by the dipolar interaction

[Góral and Santos, PRA 66, 023613 (2002); Yi and You, PRA 67, 045601 (2003); Giovanazzi et al., J. Opt. B 5, 208 (2003)]

00 22 44 66 88 1010 1212 141400

0.50.5

11

1.51.5

22

2.52.5

33

3.53.5

44

time of flight [ms]time of flight [ms]

asp

ect

ratio

asp

ect

ratio

//RRyyRRzz

zz

yy

�� BB��,, alongalong ::BB yy

��,, alongalong ::BB zz�� BB

BB First trace of

dipolar effects in BEC ever!!

Dipolar gases

Dipolar BEC presents many novel characteristic features

Trap-dependent stability[Yi and You, PRA 61, 041604 (2000); Góral et al., PRA 61, 051601 (2000);Santos et al., PRL 85, 1791 (2000)]

Roton-maxonexcitation spectrum

[O‘Dell et al., PRL 90, 110402 (2003); Santos et al., PRL 90, 250403 (2003)]

TrulyTruly--2D 2D motionmotion withoutwithout distortiondistortion!!!!

ε0ωh

ql0

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2

2.0

2.5

1.5

1.0

0.5

0.5 1.0 1.5 2.0

3.0

0

(b)

0

BOGOLIUBOV

Phonon

Single particle

Roton

2D solitons[Pedri and Santos,

PRL 95, 150406 (2005)]

Inelastic scattering

Outline of the talk

Dipolar gases

Cr-BEC as spin-3 BEC

m=-3m=-2

m=-1m=0

m=+1m=+2

m=+3

B B

Chromium BEC as a spinor BEC

Spinor BEC

Spin ½[Stenger et al.., Nature

396, 345 (1998) ]

1,1 −== FmF1,2 +== FmF

Magnetically trapped 87Rb[Hall et al., PRL 81, 1539 (1998)]

Spin 1 Optically trapped

[Schmalljohann et al., PRL 92, 040402 (2004);

Chang et al., PRL 93, 170402 (2004);

Higbie et al., PRL 95, 050401 (2005)]

0,1,1 ±== FmF

87RbNa

Spin 2 0,1,2,2 ±±== FmF

[Schmalljohann et al., PRL 92, 040402 (2004);Kuwamoto et al., PRA 69, 063604 (2004);

Widera et al., PRL 95, 190405 (2005)]

87Rb

Chromium BEC as a spinor BEC [Santos and Pfau, cond-mat/0510634]

The ground state of 52Cr is 7S3

m=-3m=-2

m=-1m=0

m=+1m=+2

m=+3

⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

=

+

+

+

−

−

−

3

2

1

0

1

2

3

ψψψψψψψ

ψrHence, unless we pump into one state (which is what is

done up to now), we will have to deal with a spinorwavefunction with 7 components !!!!

Spin-3 BEC

Chromium BEC as a spinor BEC

Hamiltonian

ddsr VVHH ˆˆˆˆ0 ++=

∫ ∑ ⎥⎦

⎤⎢⎣

⎡++∇

−= +

mmtrapm rpmrU

MrrdH )(ˆ)(

2)(ˆˆ 2

2

0rrhrr ψψ

Dipole-dipole interactionsSingle particle physics Short-range interactions

Zeeman effect Bgp Bµ=

(No quadratic Zeeman effect in Cr-BEC)

Chromium BEC as a spinor BEC

Hamiltonian

ddsr VVHH ˆˆˆˆ0 ++=

Dipole-dipole interactionsSingle particle physics Short-range interactions

Mag SS /4 2hπ=

Sa s-wave scattering length for total spin Sm m‘

n

n‘They conserve the total spin : S=m+m‘=n+n‘

Only even S is possible for Bosons: S=0,2,4,6

∫ ∑=S SSsr rPgrdV )(ˆ

21ˆ rr

[Ho, PRL 81, 742 (1998)]

This is similar as for S=2, but having S=3 will lead to new physics!!

Baa )20(72 −=?0 =a

Baa )14(1126 =Baa )6(584 =For 52Cr:

[Werner et al., PRL 94, 183201 (2005)]

Chromium BEC as a spinor BEC

Hamiltonian

ddsr VVHH ˆˆˆˆ0 ++=

Dipole-dipole interactionsSingle particle physics Short-range interactions

They do NOT conserve the total spin

∫ ∫ ′⎥⎦

⎤⎢⎣

⎡⋅⋅

−⋅′

′−′= ′

′′

′′+′

+ )(ˆ)(ˆ))((3

)(ˆ)(ˆ12

ˆ3 rr

eSeSSS

rrrr

rdrdcV nnnmmn

nmmnmm

ddd

rrrrrr

rr ψψψψ

This violation of the spin conservation means thatspin can be transferred into center-of-mass angular momentum!!

( )zyx SSSS ,,= ( ) '' rrrre rvrv −−=are the spin-3 matrices

Chromium BEC as a spinor BEC: Ground state

)()(ˆ rNr mmrr ψψ ≅Mean-field approximation

mm rr ψψ )()( rrΦ=Single-mode approximation

Magnetic field in z-direction

{ }[ ] ⎟⎠⎞

⎜⎝⎛ ++−+Θ++≅ ++

2222223

22

21 2

2112

23

74~~

zzzzzm SSSSSccScSpE ψ

∑ −−=Θm mm

m ψψ)1(21

6261 65.018/)( gggc ≈−=

6064202 374.033/)52755( ggggggc +≈−+−+= ( )∫ Φ= 42~ rrdNpp rr

66423 002.0198/77/126/ ggggc −≈+−=

Chromium BEC as a spinor BEC: Ground state

In order to get a feeling, let‘s compare with spin-2 BEC

22

21 5

4~ Θ++≅ cScSpE zzThe same but without

the c3 term !![Ciobanu et al., PRA 61, 033607 (2000);Koashi and Ueda, PRL 84, 1066 (2000)]

Ferromagnetic

Polar

m=-2m=-1 m=0 m=+1m=+2

m=-2m=-1 m=0 m=+1m=+2

Cyclic

87Rb

0=Θ

0=Θ2=zS

0≠Θ

2≠zS

4/~1 pc ≤

4/~20/21 pcc ≤−

02 <c4/~20/21 pcc >−

02 >c4/~

1 pc >Relative phases important

For spin-1: only ferromagnetic (87Rb) or polar (Sodium) [Ho, PRL 81, 742 (1998)]

Chromium BEC as a spinor BEC: Ground state

g /g0 6

g6

p~0.2

0.3

0.1

0

0.4

−0.1 0 0.1 0.2 0.3

CY−3,2

CY−3,2 CY−1,2OR

CY−3,−1,1,3

S−2,0,2

−3,−1,1,3SOR

FERROMAGNETIC

POLAR

{ }[ ] ⎟⎠⎞

⎜⎝⎛ ++−+Θ++≅ ++

2222223

22

21 2

2112

23

74~~

zzzzzm SSSSSccScSpE ψ

0=Θ 02 =+S1

~2/~ cpSz −=

12 ~2/~6 cpSz +=

0=Θ 02 =+S1

~2/~ cpSz −=

12 ~2/~6 cpSz += 1

~2/~2 cp−

1~6c

0=Θ 02 ≠+S0≠Θ 02 ≠+S

0=+ zSS

B<0.1mG

or

Biaxial nematics[Diener and Ho, cond-mat/0511751] [Madsen et al., PRL 92, 45505 (2004);

Acharya et al., PRL 92, 145506 (2004)]

Chromium BEC as a spinor BEC: Dynamics

The dipole-dipole interaction plays a little role in the ground-state properties

But it can play a crucial role in the dynamics!

Only short-range part With dipolar interaction

m=-3m=-2

m=-1m=0

m=+1m=+2

m=+3

Spin-relaxation due to the dipolar interaction!

m=-3m=-2

m=-1m=0

m=+1m=+2

m=+3

Forbidden byspin-conservation

The loss in spin, must be gained by the center of mass angular momentum

Chromium BEC as a spinor BEC: Dynamics

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

-10 -5 0 5 10x

-10

-5

0

5

10

y

(b)

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009

-10 -5 0 5 10x

-10

-5

0

5

10y

(a)

Density for the state m=-2

The state m=-2 starts to rotate!

It resembles the Einstein-de Haas effect

[Kawaguchi et al., cond-mat/0511052]

The coherent EH-effect is destroyed if ωµ h>>Bg B

Chromium BEC as a spinor BEC: Dynamics

Also the c3 terms have a significant role int he spinor dynamicsand may lead to a rapid transfer from m=0 to m=+3 and m=-3

|+2> |+1> |0> |−1> |−2>|+2> |+1> |0> |−1> |−2>

(a)

(c)

(b)

(d)

200ms

40ms

10ms 50ms

150ms

500ms

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

"m=0""m=1""m=2""m=3"

tω

Popu

latio

n m=0

m=3

[Schmalljohann et al., PRL 92, 040402 (2004]

87Rb

For F=2 a sequential population is observed

Summary

Nonlocal nonlinearity. Multidimensional solitons

Properties of Dipolar BECs

Cr-BEC as spinor BEC

m=-3m=-2

m=-1m=0

m=+1m=+2

m=+3

B B

1D Dipolar gases[Sinha and Santos, in preparation)]

Dipolar Lattice gases[Góral et al., PRL 88, 170406 (2002)]

[Baranov et al., PRL 92, 250403 (2004)]Fermionic dipolar gasesRotating dipolar gases [Baranov et al., PRL 94, 070404 (2005); Rezayi et al., cond-mat/0507064]

[Brennen et al., PRL 82, 1060 (1999); Jaksch et al., PRL 85, 2208 (2000); DeMille et al., PRL 88, 067901 (2002)]

Quantum information

People

P. PedriS. Sinha

S. GiovanazziR. Nath

L. Santos

J. StuhlerA. Griesmaier

T. KochM. FattoriT. Pfau

M. Baranov, B. Damski, K. Góral, M. Lewenstein, E. Tiemann, G. V. Shlyapnikov, S. Kotochigova

P. Julienne, P. Zoller