Global variables to describe the thermodynamics of Bose-Einstein condensates

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Global variables to describe the thermodynamics
of Bose-Einstein condensates
Emanuel A. L. Henn Kilvia M. F. Magalhães Victor
Romero-Rochin Gabriela B. Seco Luis G. Marcassa
Vanderlei S. Bagnato Instituto de Física de São
Carlos USP Universidade Nacional Autónoma do
México
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Summary

• Introduction
• Definition of global thermodynamical variables
• Measurements in magnetically trapped cold atoms
• Measurements in the route to BEC

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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?

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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.

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Thermodynamics of cold trapped atoms
Can one make an analysis of Pressure-Volume for
trapped atoms?
VOLUME PRESSURE
Particles interact everywhere with the confining
potential, not only at the walls as in regular
thermodynamics!!!
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For N noninteracting bosons
Bose function
N, E and S are extensive
T and ? are intensive
is extensive!!!
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In a trap, for a given T, the volume occupied by
most particles is of the order of
Defining harmonic volume
We obtain the intensive variable conjugate to
harmonic volume harmonic pressure P
Classical limit
Equation of state of a cold trapped
noninteracting gas
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• If we include interactions

Helmholtz free energy
where
It can be shown that the generalized volume can
be defined again as
The generalized pressure becomes
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Harmonic Trap
Quadrupolar Trap
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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

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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

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Results
Distortion from the ideal gas curve!
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 ..
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Classical Virial expansion of the equation of
state PV NkT 1 B(T)N/V .. B(T) 1/2
(b2/8) 1/8p(kT)3 Hard sphere b2 -4p/3
(2R)3 R 10-6 m Need to take into account the
interaction potential of two sodium atoms for a
better value!
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Compressibility k - 1/V dV/dP k1/P ( for
ideal gas)
k 0,5/P0,8
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Measurements in the route to BEC

• Harmonic Trap
• Isochoric curve constant volume
• In-situ absorption images
• Integration along beam path
• Symmetry considerations to evaluate pressure
• 1 experimental point after BEC
• Finite pressure even at T 0

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T
Indicative of BEC phase-transition by Cv!!!
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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