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

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

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

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, (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, (2003) Evanescent-wave trapping Anisotropy parameter

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, (2001)

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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