Global variables to describe the thermodynamics of Bose-Einstein condensates

1. 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

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

3. 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?

4. 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.

5. 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!!!

6. For N noninteracting bosons Bose function ; N, E and S are extensive T and  are intensive is extensive!!!

7. 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

8. Helmholtz free energy If we include interactions: where It can be shown that the generalized volume can be defined again as: The generalized pressure becomes:

9. Harmonic Trap Quadrupolar Trap

10. 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

12. 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”

13. 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 + …..]

14. 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!

15. Compressibility k = - 1/V [ dV/dP] k=1/P ( for ideal gas) k= 0,5/P0,8

16. 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

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

18. 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