1. QUANTUM DEGENERATE BOSE SYSTEMS IN LOW DIMENSIONS G. Astrakharchik S. Giorgini Istituto Nazionale per la Fisica della Materia Research and Development Center on Bose-Einstein Condensation Dipartimento di Fisica – Università di Trento Trento, 14 March 2003

2. Bose – Einstein condensates of alkali atoms dilute systems na 3 <<1 3D mean-field theory works low-D role of fluctuations is enhanced 2D thermal fluctuations 1D quantum fluctuations beyond mean-field effects many-body correlations

3. Summary General overview Homogeneous systems Systems in harmonic traps Beyond mean-field effects in 1D Future perspectives

4. BEC in low-D: homogeneous systems Textbook exercise: Non-interacting Bose gas in a box Thermodynamic limit Normalization condition momentum distribution fixed density

5. D=3 converges D=3 if D  2 for any T >0 If  =0 infrared divergence in n k D  2 diverges   0 chemical potential

6. Interacting case T  0 Hohenberg theorem (1967) Bogoliubov 1/k 2 theorem “per absurdum argumentatio” If Rules out BEC in 2D and 1D at finite temperature Thermal fluctuations destroy BEC in 2D and 1D quantum fluctuations?

7. T=0 Uncertainty principle (Stringari-Pitaevskii 1991) If But fluctuations of particle operator fluctuations of density operator static structure factor sum rules result Rules out BEC in 1D systems even at T=0 Quantum fluctuations destroy BEC in 1D (Gavoret – Nozieres 1964 ---- Reatto – Chester 1967)

8. Are 2D and 1D Bose systems trivial as they enter the quantum degenerate regime ? Thermal wave-length

9. One-body density matrix : central quantity to investigate the coherence properties of the system condensate density liquid 4 He at equilibrium density long-range order

10. 2D Something happens at intermediate temperatures low-T from hydrodynamic theory (Kane – Kadanoff 1967) high-T classical gas

11. Berezinskii-Kosterlitz-Thouless transition temperature T BKT (Berezinskii 1971 --- Kosterlitz – Thouless 1972) Universal jump (Nelson – Kosterlitz 1977) Dilute gas in 2D: Monte Carlo calculation (Prokof’ev et al. 2001) T<T BKT system is superfluid T>T BKT system is normal Thermally excited vortices destroy superfluidity Defect-mediated phase transition

12. Torsional oscillator experiment on 2D 4 He films (Bishop – Reppy 1978) Dynamic theory by Ambegaokar et al. 1980

13. 1D From hydrodynamic theory (Reatto – Chester 1967) T=0 T  0 4 He adsorbed in carbon nanotubes Cylindrical graphitic tubes: 1 nm diameter 10 3 nm long Yano et al. 1998superfluid behavior Teizer et al. 19991D behavior of binding energy degeneracy temperature in 1D

14. BEC in low-D: trapped systems a) ) anisotropy parameter motion is frozen along z kinematically the gas is 2D motion is frozen in the x,y plane kinematically the gas is 1D

15. Goerlitz et al. 2001 3D  2D 3D  1D

16. b)Finite size of the system cut-off for long-range fluctuations fluctuations are strongly quenched BEC in 2D (Bagnato – Kleppner 1991) Thermodynamic limit

17. But density of thermal atoms Perturbation expansion in terms of g 2D n breaks down Evidence of 2D behavior in T c (Burger et al. 2002) BKT-type transition ? Crossover from standard BEC to BKT ?

18. 1D systems No BEC in the thermodynamic limit N  For finite N macroscopic occupation of lowest single-particle state If (Ketterle – van Druten 1996) 2-step condensation

19. Effects of interaction (Petrov - Holzmann – Shlyapnikov 2000) (Petrov – Shlyapnikov – Walraven 2000) Characteristic radius of phase fluctuations 2D 1D true condensate (quasi-condensate) condensate with fluctuating phase

20. Dettmer et al. 2001 Richard et al. 2003

21. Beyond mean-field effects in 1D at T=0 Lieb-Liniger Hamiltonian Exactly solvable model with repulsive zero-range force Girardeau 1960 --- Lieb – Liniger 1963 --- Yang – Yang 1969 at T=0 one parameter n|a 1D | a 1D scattering length

22. Tonks-Girardeau fermionization Equation of state mean-field

23. One-body density matrix Quantum Monte-Carlo (Astrakharchik – Giorgini 2002)

24. Momentum distribution

25. Lieb-Liniger + harmonic confinement Exactly solvable in the TG regime (Girardeau - Wright - Triscari 2001) Local density approximation (LDA) (Dunjko - Lorent - Olshanii 2001) If 1D behavior is assumed from the beginning

26. 3D-1D crossover Quantum Monte-Carlo (Blume 2002 --- Astrakharchik – Giorgini 2002) Harmonic confinement Interatomic potential (a s-wave scattering length) highly anistropic traps hard-sphere modelsoft-sphere model (R=5a)

27. Compare DMC results with Mean-field – Gross-Pitaevskii equation 1D Lieb-Liniger (Olshanii 1998) with

31. Possible experimental evidences of TG regime size of the cloud (Dunjko-Lorent-Olshanii 2001) collective compressional mode (Menotti-Stringari 2002) momentum distribution (Bragg scattering – TOF)

32. Infrared behavior k >1/R z

33. Future perspectives Low-D and optical lattices –many-body correlations  superfluid – Mott insulator quantum phase transition (in 3D Greiner et al. 2002) –Thermal and quantum fluctuations  low-D effects Investigate coherence and superfluid properties

34. Tight confinement and Feshbach resonances (Astrakharchik-Blume-Giorgini) Quasi-1D system confinement induced resonance (Olshanii 1998 - Bergeman et al. 2003)