44th EGAS
Göteborg – July 11, 2012
The anisotropic excitation spectrum of a
chromium Bose-Einstein Condensate
Olivier GORCEIX
Laboratoire de Physique des Lasers
Université Sorbonne Paris Cité
Villetaneuse - France
Interactions within a BEC
Van der Waals / contact interactions :
isotropic and short ranged
Effective potentiel proportionnal to aS d(R), with aS = scattering length,
aS adjustable thanks to Feshbach resonances
Dipole-dipole interactions: anisotropic and long-ranged
highly magnetic atoms Cr, Er, Dy, dipolar molecules; Rydberg atoms
Chromium atoms carry a permanent magnetic dipole of 6µB
MDDI are 36 times bigger than in alkali BECs but still
edd (Cr)=0.159 while edd (Rb)=0.0044
where edd quantifies the ratio dipolar / contact interactions
Chrome (S=3): both contact AND dipolar interactions
Dipole-dipole interaction potential
Vdd 
0 2
1
2
S  g J  B  1  3cos 2 ( )  3
4
R
Anisotropy
Mean field becomes non
local and anisotropic

Spin and rotation are
coupled
R
is equal to 0.16
First reported effects of DDIs on BECs with Cr BECs
B
Striction of the BEC
(non local effect)
DDIs
e dd  0.16
Vdd adds a non local
anisotropic mean-field
Eberlein et al, PRL 92, 250401 (2004)
Anisotropy in the
BEC expansion
Pfau et al,PRL 95, 150406 (2005)
The effects of DDIs
are experimentally
evidenced by
differential
measurements,
for two orthogonal
orientations of the B field
Bismut et al., PRL 105,
040404 (2010)
DDIs change in
the few % range
the physics of
a ground state BEC
repulsion
attraction
1st PART
COLLECTIVE OSCILLATIONS
Impact of the dipolar interactions on their frequencies
A small correction induced by dipolar interactions
Collective excitations of a dipolar BEC
MDDI are anisotropic, they impact on the q-pole excitation
eigenfrequencies of a trapped BEC
In this mode, oscillations along y and z are in opposition
Parametric excitation
We repeat the experiment for
two orthogonal orientations
of B
15 000 atoms
Aspect ratio
1.2
1.0
0.8
0.6
5
t (ms )
10
15
20
Trap geometry dependence of the measured frequency shift
e dd  0.16
Shift of the
quadrupole
mode
frequency (%)
BEC always stretches along B
Shift of the
aspect ratio
(%)
While the sign of the
quadrupole shift depends on
the trap geometry
Theory PRL 92, 250401 (2004)
Trap anisotropy
This exp:
Bismut et al., PRL 105, 040404 (2010)
Good agreement with ThomasFermi predictions
e
2nd PART
RAMAN-BRAGG SPECTROSCOPY OF A DIPOLAR BEC
Sound velocity anisotropy induced by dipolar interactions
A more pronounced dipolar effect
Excitation spectrum of a BEC with pure contact interactions
c is the sound velocity
c is also the critical velocity
for superfluidity in the Landau model
Rev. Mod. Phys. 77, 187 (2005)
Bogoliubov spectrum
 2k 2
Ek 
2 m
k  1
k  1
when
e k  Ek ( Ek  2n0 gc ) where gc= 4ħ2a /m
Ek  2n0 gc  k  1 / 
Quasi-particles, phonons
free particles
healing length
e k  c k
e k  Ek
Excitation spectrum of the BEC with DDIs
4 d 2
V (k ) 
(3cos 2  k  1)
3
B
e k  Ek ( Ek  2n0 gc )
k
becomes:
e k  Ek Ek  2n0 gc 1  e dd 3 cos 2  k  1
if
 k  0 , c  c// and if
c// / c 

k
e dd  0.16
 k   / 2 , c  c
1  2e dd
 1.2
1  e dd
A 20% effect expected on the speed of sound !
Much larger than the (~3%) effects for striction and collective excitations
absorption image after TOF of 5 ms
or B┴
frequencies W and Ww
Angular separation a
Profile at resonance ie when ħ
w  e(q)
Raman-Bragg spectroscopy of a BEC
Moving lattice upon the BEC
Energy
a=14°
Phonon regime
W
a
k  0.8
Ww
1
2
k
Lattice beams crossing with an angle and a detuned
causes resonant momentum exchange when
k  2 kL sin(a / 2)
From the excitation spectrum
we infer
the speed of sound
Fraction of excited atoms
Anisotropic speed of sound

0.15
0.10

0.05
0.00
0
1000
2000
Frequency difference (Hz)
3000
c┴ ≠ c 
Width of resonance curve: finite size effects (inhomogeneous broadening)
Speed of sound depends on the relative angle between spin and excitation wavevector
Anisotropic speed of sound
Bismut el al, arXiv :1205.6305
Good agreement between
theory and experiment:
Theo
Exp
Parallel
3.6 mm/s
3.4 mm/s
Perpendicular
3 mm/s
2.8 mm/s
Magnetism and thermodynamics - spinor physics
At ultralow T ≈ 300nK and B ≈ 40 nT the chemical potential becomes greater than
the Zeeman splitting -> NEW PHYSICS
TOF + Stern-Gerlach
3
2
1
0
-1
-2 
-3
Above and
Below threshold
Spontaneous demagnetization
Spin 3 phase diagram
Pasquiou et al, PRL 106, 255303 and 108, 045307
Conclusion
Dipolar interactions induce anisotropy in the BEC excitation spectra
- collective modes;
- phonon dispersion law : sound velocity
Magnetism and thermodynamics of a spin 3 BEC with free magnetisation
Perspectives
Extension of our work on magnetism to strongly correlated states in 3D optical latttices
Einstein-de-Haas effect: spin-rotation coupling
Improved spatial resolution for in situ imaging of magnetization domains
Extension to a Fermi sea of 53Cr atoms
The chromium BEC crew
www-lpl.univ-paris13.fr:8082
E.Maréchal, OG, P. Pedri, Q. Beaufils (PhD), B. Laburthe, L. Vernac, B. Pasquiou (PhD),
G. Bismut (PhD)
Thank you for your attention
… PhD students welcomed in our group…