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VARIATIONAL APPROACH FOR THE TWO-DIMENSIONAL TRAPPED BOSE GAS

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). Motivations. 2D experiments in the degenerate regime: Innsbrück (Rudy Grimm)

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VARIATIONAL APPROACH FOR THE TWO-DIMENSIONAL TRAPPED BOSE GAS

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  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 Anisotropy parameter Reach the 2D regime by decreasing N in an anisotropic trap A. Görlitz and al. Phys. Rev. Lett 87, 130402 (2001) MIT Firenze Innsbrück Villetaneuse 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

  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 Construct a potential which leads to the contact condition of the 2-body problem Motivation : Hamiltonian formulation of the problem Example : the Fermi-Huang potential in 3D Zero range potential Regularizing operator The « L-potential » in the 2Dworld 2-body t-matrix at energy

  7. Many-body problem for trapped atoms • Contact conditions • Pseudo-potential Twopossibilities Constraints on the mean density Validity of the zero range approach Validity of the mean-field approach

  8. Summary of the zero-range approach highly anisotropic traps Mean inter-particle spacing Possible description of a molecular phase L freedom a2D>0 can be tuned via a3D (Feshbach resonance) Observables do not depend on the particular value of L Possible study of a highly correlated dilute system

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

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

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

  12. The gap spectrum “disaster” • Change the phase of f cost no energy • Anomalous mode solution of the linearized time dependent equations (RPA) • (F,-F*) NOT SOLUTION (in general) of the static HFB equations Parameters of the Gaussian ansatzfor the density operator « static spectrum » Eigen-energies of the RPA equations « dynamic spectrum » Spurious energy scale in the thermodynamical properties

  13. Gapless HFB Impose that the anomalous mode is solution of the static HFB equations Search L* such that

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

  15. 2D Equation Of State (T=0) Popov’s EOS HFB 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 (Large distance behavior) Effective coupling constant with L* determined by the mode amplitudes Expect velocity dependence at the mean field level

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

  19. Restoration of the instability Calculate the “static spectrum” without thermalizing the anomalous mode

  20. Conclusions and perspectives • l>>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 : Time Dependent 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,D,F : 2) The “gap equation 3) An equivalent condition for searching L* : 4) Numerical procedure :

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