Collective excitations in a dipolar Bose-Einstein Collective excitations in a dipolar Bose-Einstein CondensateCondensate

Laboratoire de Physique des LasersUniversité Paris NordVilletaneuse - France

Former PhD students and post-docs: Q. Beaufils, T. Zanon, R. Chicireanu, A. Pouderous

Former members of the group: J. C. Keller, R. Barbé

B. Pasquiou

O. Gorceix P. Pedri

B. Laburthe

L. Vernac

E. Maréchal

G. Bismut

Why are dipolar gases interesting?

Strongly anisotropic Magnetic Dipole-DipoleInteractions (MDDI)

repulsive interactions attractive interactions

3

2220 )(cos31

4 RgSV BJdd

Angle between dipoles

Long range radial dependence

Great interest in ultracold gazes of dipolar molecules

What’s so special about Chromium?

6 valence electrons (S=3): strong magnetic dipole of

•Dimensionless quantity: strength of MDDI relative to s-wave scattering

gdmdd 3/20

B6

Large dipole-dipole interactions: 36 times larger than for alcali atoms.

Magnetic dipole of Cr52

maS /4 2

Only two groups have a Chromium BEC: in Stuttgart and Villetaneuse

How to make a Chromium BEC in 14s and one slide ?How to make a Chromium BEC in 14s and one slide ?

425 nm

427 nm

650 nm

7S3

5S,D

7P3

7P4

An atom: 52Cr

N = 4.106

T=120 μK

750700650600550500

600

550

500

450

(1) (2)

Z

An oven

A small MOT

A dipole trap

A crossed dipole trap

All optical evaporation

A BEC

(Rb=109 or 10)

(Rb=780 nm)

Oven at 1350 °C(Rb 150 °C)

A Zeeman slower

Q. Beaufils et al., PRA 77, 061601 (2008)

Outline

I) Hydrodynamics of a Dipolar BEC

II) Experimental results for collective excitations

III) How to measure the systematic effects

Similar results in Stuttgart

PRL 95, 150406 (2005)

I) 1 - One first effect of dipole dipole interactions: Modification of the BEC aspect ratio

Thomas Fermi profileStriction of BEC(non local effect)

Parabolic ansatz is still a good ansatz

B

B

The magnetic field is turned of 90°

Shift of the aspect ratio σ

))2/()0(/())2/()0((2

x

y

z

y

z

x

I) 2 - Dynamic properties of interactions in a BEC

•2 quadrupole modesLowest modes

•1 monopole modeHighest mode

Out of equilibrium: 3 collective modes

I) 2 - Dynamic properties of interactions in a BEC

•2 quadrupole modesLowest modes

•1 monopole modeHighest mode

Out of equilibrium: 3 collective modes

I) 2 - Dynamic properties of interactions in a BEC

•2 quadrupole modesLowest modes

•1 monopole modeHighest mode

Out of equilibrium: 3 collective modes

Theory: Superfluid hydrodynamics of a BEC in the Thomas-Fermi regime

Continuity equation

Euler Equation

)v.(nt

n

)( 2ddext gnVmv

t

vm

I) 3 - Introducing a dipolar mean field

Theory: Non local mean-field

The frequencies of the collective modes depend on the orientation of the magnetic field relative to the trap axis.

)r(n)rr(Vrd)r(dd

3

dd

dependent on the orientation of the magnetic dipoles

B

B

)0(Qfrequency )2/(Qfrequency

))2/()0(/())2/()0((2

exp

QQQQ

B

We measure a relative shift

•Frequency shift proportional to dd

II) 1 - How to excite one collective mode of the BEC

15ms modulation of the IR power with a 20% amplitude at a frequency ω close to the intermediate collective mode resonance.

The cloud then oscillates freely for a variable time

Imaging process with TOF of 5ms

Aλ/2 plate controls the trap geometry : angle Φ

Parametric excitations:Modulation of the « stiffness » of the trap

by modulating its depth

II) 2- Oscillations of the aspect ratio of the BEC after parametric excitations

•Trap geometry close to cylindrical symmetry•Very low (3%) noise on the TF radii

•High damping due to the large anharmonicity of the trap

Change between two directions of the magnetic

field

We measure exp

27 OOHzQ 7 %5,2exp

II) 3 - Trap geometry dependence of the measured frequency shift

Large sensitivity of the collective mode to trap geometry at the vicinity of spherical symmetry, unlike the striction of the BEC

Good agreementWith theoretical

predictions

•Related to the trap anisotropy

Relative shift of the quadrupole

mode frequency

Relative shift of the aspect ratio

II 1 - Influence of the BEC atom number

smaller number of atoms

Gaussian anzatz in order to take the quantum kinetic energy into account.

In our experiment, it is not negligible compared to the mean-field due to MDDI.

Hz25mR/2

TF

2

Large number of atoms (>10000)

Thomas Fermi Regime

Parabolic density profile

No more in the Thomas Fermi Regime

Parabolic anzatz is not valid

Results of simulations with the Gaussian anzatz:It takes three times more atoms for the frequency shift of the collective

mode to reach the TF predictions than for the striction of the BEC

Simulations with Gaussian anzatz

Blue and Red

Two different trap geometries

III) 1 - Measurement of the trap frequencies

parametric oscillations of the trap depth

+Potential gradient

Excitation of center of mass motion

Center of mass motion only depends on external

potential

Direct measurement of the trap frequencies

A good way of measuring systematic shifts of trap frequencies

III) 2 - Origins of the systematic shifts on the trap frequencies

In a Gaussian trap: magnetic gradient induced frequency shift

=> Trap geometry dependent Shift

Light shift of Cr is slightly dependent on the laser polarization orientation with respect to the static magnetic field.Relative associated shift independent of the trap geometry.

422

3

wm

g

Acceleration due to magnetic potential gradient

Waist of the trap along the gradient

III) 3 - Experimental results for the systematic shifts of the trap frequencies

)2(cos/ 4 ba

Fit byExcitation of

center of mass motion

Measurement of the trap frequencies

The magnetic field is turned of

90°

Measurement of relative systematic

shiftD

Summary Characterization of the effect of MDDI on a collective mode

of a Cr BEC. Good agreement with TF predictions for a large enough

number of Atoms.

Large sensitivity to trap geometry.

Useful tool to characterize a BEC beyond the TF regime, for lower numbers of atoms.

First measurement of the tensorial light shift of Chromium.

Have left: Q. Beaufils, J. C. Keller, T. Zanon, R. Barbé, A. Pouderous, R. Chicireanu

Collaboration: Anne Crubellier (Laboratoire Aimé Cotton)

B. Pas

quio

u

O. G

orce

ix

Q. B

eauf

ils

Paolo

Ped

ri

B. Lab

urth

e

L. V

erna

c

J. C

. Kel

ler

E. Mar

écha

l

G. B

ismut

Trap geometry (aspect ratio) dependent shifts

Theoretical results with a parabolic anzatz

Eberlein, PRL 92, 250401 (2004)with assumed cylindrical symmetryof the trap

See also:Pfau, PRA 75, 015604 (2007)for non axis-symmetric traps

Collective excitations of a BEC

Collisionless hydrodynamics of a BEC in the Thomas-Fermi regime

Continuity equation

Euler Equation

Time evolution of the BEC

Scaling law

Superfluid velocity

)v.(nt

n

)gnVmv(t

vm

ext

2

)t(R

z

)t(R

y

)t(R

x1

)t(R)t(R)t(R8

N15)t,r(n 2

z

2

2

y

2

2

x

2

zyx

2

z

2

y

2

xz)t(y)t(x)t(

2

1)t,r(v

with)t(R

)t(R)t(

j

j

j

Equation of Motion

From the s-wave pseudopotential with a being the s-wave scattering lenght.

Three solution for the linearized equation:

Two « quadrupole » modesIn our case the two lowest modes

One « monopole » modeIn our case the highest mode

)t(R),t(R),t(Ru)t(Rzyxjj

with shouuu

and

zyx2

2

sRRR

naN15

u