XI Escola de Vero Jorge Andr Swieca deptica Quntica e ptica No Linear11 a 22 de fevereiro de 2008, IFUSP, So Paulo, SP

- TUTORIAL-Fundaments of

Bose-Einstein condensationV.S.Bagnato

ISFC/USP

T > Tc T < Tc T

Three lectures:

1) How to make a BEC: principles and experimental

details, optical detection, characteristics features, etc

2) How to calculate effects related to BEC:

calculation of Tc, No, interactions, etc

3) The thermodynamics of a quantum degenerate gas

4) Coherent modes : similarities with quantum optics

5) Recent experiments on this matter in S. Carlos: final

temperature, coherent modes, vortices and quantum

turbulence,.

EXCITATION OF COHERENT MODES

INTERACTIONSINTERACTIONS

WITH:

Gross-Pitaevskii equation

(a = SCATTERING LENGTH)

Collisions or interaction are responsible for all nice properties

Example: excitation of the coherent modes

FIRST: CALCULATION OF GROUND AND EXCITED STATES

HARMONIC TRAP

UNITS ON NATURAL

VARABLES

INTERACTION

PARAMETER

External pumping

Solutions:

Population of levels

-> INTERACTION

TRANSITION

AMPLITUDE

EQUATIONS

FOR THE TWO

MAIN STATES

ANALYTICAL

SOLUTION FOR

THE TWO

POPULATIONS

RABI TYPE

FREQUENCY

NORMALIZED

VARIABLES

EQUATIONS

FOR THE TWO

MAIN STATES

ANALYTICAL

SOLUTION FOR

THE TWO

POPULATIONS

RABI TYPE

FREQUENCY

NORMALIZED

VARIABLES

Ramsey Pulseb=0.4

V(r)

tt1 t2

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5 Rabi toff = t1

np

Ramsey Pulseb=0.4

V(r)

tt1 t2

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5 Rabi toff = t1 Rabi t

off = 2t

1

np

Ramsey Pulseb=0.4

V(r)

tt1 t2

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5 Rabi toff = t1 Rabi t

off = 2t

1

Ramsey = 3t1

np

Ramsey Pulseb=0.4

V(r)

tt1 t2

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

= t1

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

= 2t1

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

np

np

np

np

= 3t1

-5 -4 -3 -2 -1 0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

= 8t1

EVIDENCIAS DAS PRIMEIRAS OSCILAES TIPO RABI

0 10 20 30 40 500

5

10

15

20

MIS

TU

RA

DE

ES

TA

DO

S

TEMPO DE EXCITAO

B

PRESENA DE SUPERPOSIAO DE ESTADOS COM TEMPO DE EXCITAO

OSCILAES??????

Vortices in a stirred condensate

0 c

Time of flightanalysis (25 ms)

imaging

beam

x 20

centrifugal limit

Cylindrical trap +

stirringENS, Boulder, MIT, Oxford

ENS 2000: Chevy,Madison,Rosenbusch, Bretin

The single vortex caseAfter time-of-flightexpansion:

The intermediate rotation regimeThe number of vortices is notably larger than 1.

MIT

Uniform surface density of vortices nv with

Coarse-grain average for the velocity field

However one keeps the rotation frequency notably below core size

VORTICES

excitation 20ms - 250mVpp (twice the previous)

excitation 20ms - 250mVpp (same as previous)

Again the zoom.

excitation 40ms - 250mVpp

For the same conditions of the previous images we also see the cloud in this way: with the long axis changed, more elongated than the usual and with some structure inside it.

The axis is always bent to the same side. Are we rotating the whole cloud?

A zoom of the previous image shows clearly 3 holes.

excitation 40ms (twice the time of the previous) - 250mVpp (same amplitude)

TANGLED VORTICES

V0 V1 V2 V3 V4 -- VC0

5

10

15

20

25

30

Y A

xis

Titl

e

X Axis Title

B

QUANTUM TURBULENCE

EXCITATION IN AT LEAST TWO PLANS ENOUGH AMPLITUDE TO TRANSFER

MANY UNITS OF ANGULAR MOMENT

Thermodynamics of cold trapped atoms

IntensiveXExtensiveT ),,( =

Can one make an analysis of Pressure-Volume for trapped atoms?

VOLUME PRESSURE

Particles interact everywhere with the confining potential, not Particles interact everywhere with the confining potential, not only at only at the walls as in regular thermodynamics!!!the walls as in regular thermodynamics!!!

)( Measuring? PevaluateHow rnr

Harmonic Trap

( )2222222

1zyxmU zyx ++=

Quadrupolar Trap

[ ] 21222 )()()( zAyAxAU zyx ++=

rd)r(U)r(n3

2P 3

3

0 =

zyx

1V

=

zyx AAA

1V =

rd)r(U)r(n3

AP 3

3

=

V

V

CT

U =

( )dT

d NP

( )2

2

dT

d NP

V

2

2

T

=

T

PT

V

CV

T

Indicative of BEC Indicative of BEC phasephase--transition by transition by CCvv!!!!!!