QUANTUM DEGENERATE BOSE SYSTEMS IN LOW DIMENSIONS

1. QUANTUM DEGENERATEBOSE SYSTEMSIN LOW DIMENSIONS G. Astrakharchik S. Giorgini Istituto Nazionale per la Fisica della MateriaResearch and Development Center onBose-Einstein Condensation Dipartimento di Fisica – Università di Trento Trento, 14 March 2003

2. Bose – Einstein condensates of alkali atoms • dilute systems na3<<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 fixed density momentum distribution

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

6. Interacting case T0Hohenberg theorem (1967)Bogoliubov 1/k2 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. fluctuations of particle operator fluctuations of density operator T=0 Uncertainty principle (Stringari-Pitaevskii 1991) If But 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 long-range order liquid 4He at equilibrium density

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

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

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

13. 1D From hydrodynamic theory (Reatto – Chester 1967) T=0 T0 4He adsorbed in carbon nanotubes Cylindrical graphitic tubes: 1 nm diameter 103 nm long Yano et al. 1998 superfluid behavior Teizer et al. 1999 1D behavior of binding energy degeneracy temperature in 1D

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

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

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

17. But density of thermal atoms Perturbation expansion in terms of g2D n breaks down Evidence of 2D behavior in Tc (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|a1D| a1Dscattering length

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

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 (as-wave scattering length) highly anistropic traps hard-sphere model soft-sphere model (R=5a)

27. Compare DMC results with • Mean-field – Gross-Pitaevskii equation • 1D Lieb-Liniger (Olshanii 1998) with 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 behaviork<<1/ --- Finite-size cutoffk>>1/Rz

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)