1. VARIATIONAL APPROACH FOR THE TWO-DIMENSIONAL TRAPPED BOSE GAS L. Pricoupenko Trento, 12-14 June 2003 LABORATOIRE DE PHYSIQUE THEORIQUE DES LIQUIDES Université Pierre et Marie Curie (Paris)

2. Motivations 2D experiments in the degenerate regime:  Innsbrück (Rudy Grimm)  Firenze (Massimo Inguscio)  Villetaneuse (Vincent Lorent)  MIT (Wolgang Ketterle) Why trapped 2D Bose gas interesting ?  Thermal fluctuations  Interplay between KT and BEC  Non trivial interaction induced by the geometry  Beyond mean-field effects

3. Summary Brief review of the actual experimental settings Back to the two-body problem Contact condition versus Pseudo-potential Variational Formulation of Hartree-Fock-Bogolubov (HFB) Numerical Results

4. The actual experimental settings MIT Firenze Innsbrück Villetaneuse Reach the 2D regime by decreasing N in an anisotropic trap A. Görlitz and al. Phys. Rev. Lett 87, 130402 (2001) Use a 1D optical lattice  Slices of 2D condensates S. Burger and al. Europhys. Lett., 57, pp. 1-6 (2002) Evanescent-wave trapping S. Jochim and al. Phys. Rev. Lett., 90, 173001 (2003) Evanescent-wave trapping Anisotropy parameter

5. Atoms trapped in a planar wave guide Two-body problem: Zero range approach: Eigenvalue problem defined by the contact conditions : The “2D induced” scattering length Maxim Olshanii (private communication) Dima Petrov and Gora Shlyapnikov, Phys. Rev. A 64, 100503 (2001)

6. The pseudo-potential approach Motivation : Hamiltonian formulation of the problem Example : the Fermi-Huang potential in 3D Construct a potential which leads to the contact condition of the 2-body problem Zero range potentialRegularizing operator The «  potential » in the 2D world 2-body t-matrix at energy

7. Many-body problem for trapped atoms 1)Contact conditions 2)Pseudo-potential Validity of the zero range approach Validity of the mean-field approach Two possibilities Constraints on the mean density

8. Summary of the zero-range approach Mean inter-particle spacing Possible description of a molecular phase  freedom a 2D >0 can be tuned via a 3D (Feshbach resonance) highly anisotropic traps Observables do not depend on the particular value of  Possible study of a highly correlated dilute system

9. Condensate/Quasi-condensate T=0K + Thomas-Fermi Near T=Tc Almost BEC Phase in near future experiments 2D character Actual experiments

10. The ingredients of HFB U(1) symmetry breaking approach (Phase of the condensate fixed : T<<T  ) Gaussian Variational ansatz (Number of atoms fluctuates) Use the 2D zero range pseudo-potential  A Dangerous game ! ! ! The atomic Bose gas is not the ground state of the system BEC Phase

11. HFB Equations Generalized Gross-Pitaevskii equation “Static spectrum” Implicit Born approximation Pairing field (satisfies the contact condition)

12. The gap spectrum “disaster”  Change the phase of  cost no energy  Anomalous mode solution of the linearized time dependent equations (RPA)  (  *  NOT SOLUTION (in general) of the static HFB equations Parameters of the Gaussian ansatz for the density operator « static spectrum » Eigen-energies of the RPA equations « dynamic spectrum » Spurious energy scale in the thermodynamical properties

13. Gapless HFB Search   such that Impose that the anomalous mode is solution of the static HFB equations

14. Link with the usual regularizing procedure Standard approach : At the Born level UV-div …for the next order systematic determination of  beyond the LDA procedure Variational approach

15. 2D Equation Of State (T=0) HFB EOS Popov’s EOS Schick’s EOS (For Hydrogen : ) Possible to probe the EOS using a Feshbach resonance ! (Example: K=100)

16. Thomas-Fermi Limit Trap parameters : Comparison between … LDA +Popov EOS ….and the full variational scheme

17. Velocity effects on the coupling constant 2-body scattering theory with  * determined by the mode amplitudes (Large distance behavior) Effective coupling constant Expect velocity dependence at the mean field level

18. The anomalous mode of the vortex Understanding the tragic fate of a single vortex The unexpected stabilization of the core at finite temperature D.S. Rokhsar, Phys. Rev. Lett 79, 2164 (1997) T. Isoshima and K. Machida, Phys. Rev. A 59, 2203 (1999) Usual self-consistent equation Effective “pining potential” for the vortex Anomalous mode Vortex core

19. Restoration of the instability L ocal D ensity A ppoximation for the t-matrix Full variational approach function of the local chemical potentialdepends on the configuration Calculate the “static spectrum” without thermalizing the anomalous mode

20. Conclusions and perspectives >>1 is necessary for observing 2D many-body properties Closed Formalism from the 2 body problem which includes velocity effects at the mean-field level  beyond LDA Collective modes : T ime D ependent HFB  RPA  a possible way to probe the EOS ? Variational description of the quasi-condensate phase

21. Appendix 1) Minimizing the Grand-potential with respect to h,  3) An equivalent condition for searching    4) Numerical procedure  2) The “gap equation