Global variables to describe the thermodynamics of Bose-Einstein condensates

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



Summary

Introduction

Definition of global thermodynamical variables

Measurements in magnetically trapped cold atoms

Measurements in the route to BEC



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?



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.



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!!!



For N noninteracting bosons

)(),,( 4

3

gkTkTT

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

gkTN

)(3 4

3

gkTkTE

)()(4 34

3

ggkTkS



)()(43

3

,

zgkTkTPV

PT

3

1

V

)()(

3

4

zgzgNkTPV

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 ),,(



If we include interactions:

ext

H

VUKHTrekTTNF

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:

)()(32

32

33

3

,

rVrrndP

VPVFP

ext

extT

The generalized pressure becomes: The generalized pressure becomes:



)( Measuring? PevaluateHow rn

Harmonic Trap

222222

21 zyxmU zyx

Quadrupolar Trap

21222 )()()( zAyAxAU zyx

rd)r(U)r(n3

2P 330

zyx

1V

zyx AAA

1V

rd)r(U)r(n3AP 3

3



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





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”



ResultsResultsDistortion 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 + …..]



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!



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)



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 ~00.00 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



VV

CTU

dT

d NP

2

2

dT

d NP

V2

2

T

TPT

VCV

T

Indicative of BEC Indicative of BEC phase-transition by Cphase-transition by Cvv!!!!!!



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

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Global variables to describe the thermodynamics of Bose-Einstein condensates

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