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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
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