18th international iupap conference on few-body problems in physics santos – sp – brasil -...
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18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Global variables to describe the Global variables to describe the thermodynamics of Bose-Einstein thermodynamics of Bose-Einstein
condensatescondensates
Emanuel A. L. HennEmanuel A. L. Henn
Kilvia M. F. Magalhães Kilvia M. F. Magalhães
Victor Romero-Rochin*Victor Romero-Rochin*
Gabriela B. SecoGabriela B. Seco
Luis G. Marcassa Luis G. Marcassa
Vanderlei S. BagnatoVanderlei S. Bagnato
Instituto de Física de São Carlos – USPInstituto de Física de São Carlos – USP
*Universidade Nacional Autónoma do México*Universidade Nacional Autónoma do México
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Summary
Introduction
Definition of global thermodynamical variables
Measurements in magnetically trapped cold atoms
Measurements in the route to BEC
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Introduction
• Rotating degenerated gases
• Mixtures Boson – Boson / Boson - Fermion
• Optical Lattices / Condensed Matter
• New species / Dipolar Gases
• Feshbach ressonances / Molecules / BEC - BCS
• Thermodynamics? Equation of state of a cold gas?
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Advantages of defining and measuring the EOS of a cold gas
• Definition of thermodinamical properties of the gas: compressibility, heat capacity, entropy, etc.
• For non-ideal gas: magnitude of interactions, differences from the ideal gas curve, etc
• For phase transitions: observation of discontinuities of macroscopic thermodinamical quantities across the transition.
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
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 only at Particles interact everywhere with the confining potential, not only at the walls as in regular thermodynamics!!!the walls as in regular thermodynamics!!!
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
For N noninteracting bosons
)(),,( 4
3
gkT
kTT
kT
)(4 g Bose function ;Bose function ;
N, E and S are extensiveN, E and S are extensive
T and T and are intensive are intensive
3
1
is extensive!!!is extensive!!!
)(3
3
gkT
N
)(3 4
3
gkT
kTE
)()(4 34
3
ggkT
kS
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
)()(
43
3
,
zgkT
kTPV
PT
3
1
V
)(
)(
3
4
zg
zgNkTPV
In a trap, for a given T, the volume occupied In a trap, for a given T, the volume occupied by most particles is of the order of by most particles is of the order of
Defining “harmonic volume”Defining “harmonic volume”
We obtain the intensive We obtain the intensive variable conjugate to variable conjugate to harmonic volume: harmonic volume:
harmonic pressure Pharmonic pressure P
NkTPV Classical limitClassical limit
Equation of state of a cold Equation of state of a cold trapped trapped noninteractingnoninteracting gas gas
PVT ),,(
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
If we include interactions:
ext
H
VUKH
TrekTTNF
int
)ln(),,( Helmholtz free energyHelmholtz free energy
wherewhere
3
1
V
It can be shown that the generalized volume can be defined again as:It can be shown that the generalized volume can be defined again as:
)()(3
2
3
2
33
3
,
rVrrndP
VPV
FP
ext
extT
The generalized pressure becomes: The generalized pressure becomes:
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
)( Measuring? PevaluateHow rn
Harmonic Trap
222222
2
1zyxmU zyx
Quadrupolar Trap
21222 )()()( zAyAxAU zyx
rd)r(U)r(n3
2P 3
30
zyx
1V
zyx AAA
1V
rd)r(U)r(n3
AP 3
3
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Experimental system and procedure
• Na23 system designed for BEC
• Thermal beam decelerated by Zeeman tuning technique
• 109 collected in a Dark-MOT
• Magnetic trapping: quadrupole trap (linear potential) and QUIC trap (harmonic potential)
• Rf evaporative cooling
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Measurements in magnetically trapped cold atoms
Quadrupole trap
In-trap fluorescence image
Measurements for 5 different compressions (“volumes”)
TOF measurement for determination of temperature for each compression: ~ 200 K (isothermic compression)
Imaging processing for correcting fluorescence distorted by magnetic field
Integration of the intensity profile gives “pressure”
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
ResultsResults
Distortion from the ideal gas curve! Distortion from the ideal gas curve! Interactions are more important as the gas is more compressed!Interactions are more important as the gas is more compressed!
Classical Virial expansion of the equation of state
PV = NkT [ 1+ B(T)N/V + …..]
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Classical Virial expansion of the equation of state
PV = NkT [ 1+ B(T)N/V + …..]
B(T) = 1/2 (b2/8)[ 1/8π(kT)3]
Hard sphere: b2 = -4π/3 (2R)3
R ~ 10-6 m
Need to take into account the interaction potential of two sodium atoms for a better value!
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
0 1 2 3 4 50
1
2
3
4
5 Data: Data1_BModel: user2 Chi^2/DoF = 0.03702R^2 = 0.98654 P1 0.5841 ±0.08042P2 0.83429 ±0.0635
Com
pres
sibi
lity
( cm
3 /erg
4 ) 10
67
Pressure ( erg4/cm3) 10-67
k= 0,5/P0,8
Compressibility
k = - 1/V [ dV/dP]
k=1/P ( for ideal gas)
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Measurements in the route to BEC
Harmonic TrapHarmonic Trap
Isochoric curve – constant Isochoric curve – constant volumevolume
In-situ absorption imagesIn-situ absorption images
Integration along beam pathIntegration along beam path
Symmetry considerations to Symmetry considerations to evaluate pressureevaluate pressure
1 experimental point after BEC1 experimental point after BEC
Finite pressure even at T ~0Finite pressure even at T ~0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
P/N
Temp. microK
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
VV
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 phase-transition by Cphase-transition by Cvv!!!!!!
18th International IUPAP Conference on Few-Body Problems in Physics
Santos – SP – Brasil - Agosto - 2006
Some conclusions and next steps
• Global variables seen to be a powerful tool to study cold gases, in classical and quantum regime.
• Possibility of quantifying interactions through new methods
• Measurements of these quantities in more detail in the new Rb system