Lecture IV Bose-Einstein condensate Superfluidity New trends

Lecture IV

Bose-Einstein condensateSuperfluidityNew trends

The Hamiltonian:2

1

( ) ( )2

Ni

i i ji i j

pH V r W r r

m

Confiningpotential

Interactionsbetween atoms

At low temperature, we can replace the real potential by :( )i jW r r

( )i jW r r

( )i jg r r

, a : scattering legnth

Hartree approximation: 1 2 1 2( , ,..., ) ( ) ( )... ( )N Nr r r r r r

Gross-Pitaevski equation (or non-linear Schrödinger’s equation) :2

2( ) ( ) ( ) ( )

2V r Ng r r r

m

24 ag

m

Theoretical description of the condensate

The scattering length can be modified: a ( B ) Feshbach’s resonances

a > 0 : Repulsive interactionsa = 0 : Ideal gasa < 0 : Attractive interaction

a = 0 a > 0

Gaussian Parabolic

a < 0, 3D

N < Nc

« Collapse »

a < 0, 1D

Soliton

Different regime of interactions

-0,5 0,0 0,5

0,0

0,2

0,4

0,6

0,8

1,0

int.

opt.

dens

[arb

.uni

ts]

axial direction [mm]

2 m s

3 m s

4 m s

5 m s

6 m s

7 m s

8 m s

8 ms

7 ms

6 ms2 ms

0 1 2 3 4 5 6 7 8 90

25

50

75

100

125

150

[

m]

Temps [ms]0 1 2 3 4 5 6 7 8 9

0

25

50

75

100

125

150

[

m]

Temps [ms]

Experimental realization

Science 296, 1290 (2002)

Time-dependent Gross-Pitaevski equationHydrodynamic equations Review of Modern Physics 71, 463 (1999)

with the normalization

Phase-modulus formulation

evolve according to a set of hydrodynamic equations (exact formulation):

continuity

euler

Thomas Fermi approximation in a trap

Appl. Phys. B 69, 257 (1999)

Thomas Fermi energy point of view

Kinetic energy Potential energy Interaction energy

87 Rb : a = 5 nmN = 105

R = 1 m

Scaling solutions

Equation of continuity

Scaling ansatzScaling parametersTime dependent

Normalization

Euler equation

Scaling solutions: Applications

Quadrupole modeMonopole mode

• Time-of-fligth: microscope effect

• Coupling between monopole and quadrupole mode in anisotropic harmonic traps

1 m

100 m

Bogoliubov spectrum

Equilibrium statein a box

uniform

Linearization ofthe hydrodynamicequations

We obtain

speed of sound

At low momentum, the collective excitations have a linear dispersion relation:

P*

E(P*)

Microscopic probe-particle:

Conclusion : For the probe cannot deposit energy in the fluid. Superfluidity is a consequence of interactions.

For a macroscopic probe: it also exists a threshold velocity, PRL 91, 090407 (2003)

Landau argument for superfluidity

before collision and after collision

A solution can exist if and only if

HD equations: Rotating Frame, Thomas Fermi regime

velocity in the laboratory frame

position in the rotating frame

Stationnary solution

We find a shape which is the inverse of a parabola

But with modified frequencies

Introducing the irrotational ansatz

PRL 86, 377 (2001)

Determination of

Equation of continuity gives

From which we deduce the equation for

We introduce the anisotropy parameter

Determination of

dashed line: non-interacting gas

Solutions which break the symmetry of the hamiltonianIt is caused by two-body interactions

Center of mass unstable

Velocity field: condensate versus classical

Condensate Classical gas

Moment of inertia

The expression for the angular momentum is

It gives the value of the moment of inertia, we find

where

Strong dependence with anisotropy !

PRL 76, 1405 (1996)

Scissors Mode

PRL 83, 4452 (1999)

Scissors Mode: Qualitative picture (1)

Kinetic energy for rotation

For classical gas

For condensate

Extra potential energy due to anisotropy

MomentofInertia

classical

condensate

We infer the existence of a low frequency mode for the classical gas, but not for the Bose-Einstein condensate

Scissors Mode: Qualitative picture (2)

Scissors Mode: Quantitative analysis

Classical gas: Moment method for <XY>

Two modes and

One mode

Bose-Einstein condensate in the Thomas-Fermi regime

One modeLinearization of HD equations

Experiment (Oxford)

PRL 84, 2056 (2001)

Experimentl proof of reduced moment of inertiaassociated as a superfluid behaviour

Vortices in a rotating quantum fluid

In a condensate

the velocity is such that

( )( ) ( ) iS rr r e

v Sm

.nh

v drm

Liquid superfluid helium

Below a critical rotation c, no motion at all

Above c, apparition of singular lines on which the density is zero and around which the circulation of the velocity is quantized

Onsager - Feynman

incompatible with rigid body rotation v r

Preparation of a condensate with vortices

1. Preparation of a quasi-pure condensate (20 seconds)

Laser+evaporative cooling of 87Rb atoms in a magnetic trap

2 2 2 2 21 1

2 2 zm x y m z

/ 2 200 Hz

/ 2 10 Hzz

105 to 4 105 atoms

T < 100 nK

120 m

6 m

2. Stirring using a laser beam (0.5 seconds)

16 m

Y t

X

2 2 21U( )

2 X Yr m X Y X=0.03 , Y=0.09

controlled with acousto-optic modulators

From single to multiple vortices

Just belowthe critical frequency

Just abovethe critical frequency

Notably abovethe critical frequency

For large numbersof atoms:Abrikosov lattice

PRL 84, 806 (2000)

It is a real quantum vortexangular momentum h PRL 85, 2223 (2000)

also at MIT, Boulder, Oxford

Stable branch

Dynamicallyunstable branch

Dynamics of nucleation

PRL 86, 4443 (2001)