Post on 02-Jan-2019
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!!!!!!