18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto 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

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto 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 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 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 Thermodynamics of cold trapped atoms Can one make an analysis of Pressure-Volume for trapped atoms? VOLUMEPRESSURE Particles interact everywhere with the confining potential, not only at the walls as in regular thermodynamics!!!

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto For N noninteracting bosons Bose function ; N, E and S are extensive T and  are intensive is extensive!!!

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto 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

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto 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:

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto Harmonic Trap Quadrupolar Trap

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto Experimental system and procedure Na 23 system designed for BEC Thermal beam decelerated by Zeeman tuning technique 10 9 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

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto 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 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 + …..]

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto Classical Virial expansion of the equation of state PV = NkT [ 1+ B(T)N/V + …..] B(T) = 1/2 (b 2 /8)[ 1/8π(kT) 3 ] Hard sphere: b 2 = -4π/3 (2R) 3 R ~ 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 k= 0,5/P 0,8 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 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

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto T Indicative of BEC phase-transition by C v !!!

18th International IUPAP Conference on Few-Body Problems in Physics Santos – SP – Brasil - Agosto 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