Optically Trapped Low-Dimensional Bose Gases in Random Environment

1. Optically Trapped Low-Dimensional Bose Gasesin Random Environment Zhao-xin Liang (梁兆新) Institute of Metal Research, Chinese Academy of Sciences (中国科学院金属研究所) Collaborators Ying Hu (胡颖) (HKBU), Ke-zhao Zhou (周可召)(IMR) 03. 08. 2010

2. Outline • Two Basic Concepts Revisited • Superfluid Density vs. Condensate Fraction • Disorder • Why? • Bose-Einstein Condensate +Optical Lattice+ Disorder • Optically-trapped low-dimensional Bose gases in random environment • Summary

3. Outline • Two Basic Concepts Revisited • Superfluid Density vs. Condensate Fraction • Disorder • Why? • Bose-Einstein Condensate + Disorder • Two recent work: PRA80_043629(2009),81_053621(2010) • Conclusion

4. Condensate Fraction Bose-Einstein condensation refers to the macroscopic occupation of a single quantum state

5. Superfluidity Superfluidity refers to a set of fascinating hydrodynamic phenomena, notably persistent flow. • Two-fluid model (finite temperature) - Tisza (1940), Landau (1941) - • Superfluid density Andronikashvili, J. Phys. USSR 10, 201 (1946) N. R. Cooper & Z. Hadzibabic PRL 104, 030401 (2010) I. Carusotto, Physics 3, 5 (2010) T. L. Ho and Q. Zhou, Nature Phys. 6, 131 (2010)

6. Method I: superfluidity and linear response theory Longitudinal perturbation Transverse perturbation - Only normal fluid is draggedby transverse perturbation - Both normal fluid and superfluid are pushed along z-direction K.Huang and H.F.Meng, Phys. Rev. Lett. 69 644 (1962) G.Baym, in Mathematical Methods in Solid State and Superfluid Theory

7. Current-current response function • Average momentum density <g(r,t)> induced by the external perturbation v(r) • Static current-current response function Superfluidity is a kinetic property of a system and the superfluid density is a transport coefficient rather than an equilibrium property. D.Pines and P.Nozieres, The Theory of Quantum Liquids

8. How to calculate superfluid density Longitudinal response Transverse response Total density Normal fluid density D.Pines and P.Nozieres, The Theory of Quantum Liquids

9. Method II: construct wave functions displaying condensate motion • Basic idea: In such wave-functions, the particles are no longer condensed in the state k=0, but in a state with a non-uniform wave-function describing a non-zero velocity of the condensate. The superfluid is thus characterized by a change in some suitably defined ‘condensate wave function’. • Definition: supposing that a linear phase is imposed on the originally static bosonic field which gives rise to a superfluid velocity . In response, the thermodynamic potential of the system is changed by • Comment: It must be realized that the response function method is less general than that of explicit wave function construction. By starting with the unperturbed wave-functions, one misses a large class of wave function, which can not be obtained by treating the probe as a small perturbation. D.Pines and P.Nozieres, The Theory of Quantum Liquids

10. Bose-Einstein Condensation vs. Superfluidity Connection VS. • The identification of the superfluid velocity and the gradient of the phase of the order parameter represents a key relationship between Bose-Einstein condensation and superfluidity. L. Pitaevskii and S. Stringari, Bose-Einstein condensation

11. Bose-Einstein condensation vs. Superfluidity Contrast 4 Generally, the two concepts of superfluid density and condensate density cannot be confused with each other. A typical illustration is provided by weakly interactingBose gases in the presence of disorder at zero temperature.

12. Outline 1, Basic Concepts Symmetry Breaking, Order Parameter, Condensation and Superfluidity 2,Disorder Why? 3, Bose-Einstein Condensate + Disorder Two recent work: PRA80_043629(2009),81_053621(2010) 4, Conclusion

13. Ultracold atoms in disordered potential • Why disorder? • - Disorder is a key ingredient of the microscopic (macroscopic) world. • - Fundamental element for the physics of conduction. • -Pronounced contrast between BEC and superfluidity in the presence of • disorder even at zero temperature • Why cold atoms • - Ultra-cold atoms are a versatile tool to study disorder-related phenomena. • - Allow precise control on the type and amount of disorder in the system. • Interplay between disorder and interaction • - Bose glasses (strongly interacting systems). • - Anderson localization (weakly interacting systems).

14. Different ways to produce disorder • Optical potential Speckle fields or multi-chromatic lattices • B. Damski et al., PRL91_080403(2003) • R. Roth & K. Burnett, PRA68_023604(2003) • Collision-induced disorder • Interaction with randomly distributed impurities • U.Gavish &Y.Castin, PRL95_020401(2005) • Magnetic potential • H. Gimperlein et al., PRL95_170401(2005)

15. Anderson localization in a BEC Nature 453_895 (2008)

16. BEC+Disorder

18. Outline 1, Basic Concepts Symmetry Breaking, Order Parameter, Condensation and Superfluidity 2, Disorder Why? 3, Bose-Einstein Condensate + Optical lattice + Disorder 4, Conclusion

19. Optically Trapped Low-Dimensional BEC BEC trapped in a 1D optical lattice BEC trapped in a 2D optical lattice Quasi-1D BEC Quasi-2D BEC Quasi-low-dimensional BEC: • Kinematically, the gas is 2D or 1D; • The difference from purely 2D or 1D gases is only related to the value of the inter-particle interactionwhich now depends on the tight confinement. Quasi-low-dimensional BEC: • Kinematically, the gas is 2D or 1D; • The difference from purely 2D or 1D gases is only related to the value of the inter-particle interactionwhich now depends on the tight confinement.

20. BEC in presence of a 1D (2D) optical lattice and disorder Action functional within the grand-canonical ensemble Disorder Effective two-body coupling constant Order parameter field 1D (2D) optical lattice Quasi-low-dimensional Bose gases: • Kinematically, the gas is 2D or 1D; • The difference from purely 2D or 1D gases is only related to the value of the inter-particle interactionwhich now depends on the tight confinement.

21. Effective two-body coupling constant Chemical potential: Pseudopotential: Dimensionality of g 3D 1D 2D Density-dependent

22. 3D limit( ) • 2D limit: ( ) Effective coupling constant tuned by a 1D tight optical lattice Tight-binding approximation • Quasi-2D limit ( ) D.S.Petrov et al., PRL84_2551 (2000); G.Orso and G. V. Shlyapnikov, PRL95_260402 (2005)

23. Treatment of disorder Affected by optical lattice Disorder is produced by random potential associated with quenched impurities • Small concentration of disorder: • Two basic statistical properties K. Huang and Hsin-Fei Meng, PRL69_644 (1992)

24. Bose gases trapped in a 2D optical lattice and random potential Beyond-mean-field ground state energy

25. Lieb-Liniger solution of 1D model expanded in the weak coupling regime in the presence of disorder Dimensional crossover from 3D to 1D

26. In the 3D regime, h(x) and K(x) decay Quantum depletion The first term diverges NO BEC

27. Bose gases trapped in a 1D optical lattice and random potential Beyond-mean-field Ground state energy

28. Dimensional crossover from 3D to 2D In the asymptotic 3D limit: In the 2D limit:

29. Ground state energy in 2D

30. Quantum depletion • 3D limit: • 2D limit:

31. Disorder-induced superfluid depletion in 2D 3D limit: 2D limit: Contrast between superfluidity and BEC becomes More pronounced in low dimensions.

32. Conclusion • Within Bogoliubov’s approximation, quantum fluctuations and superfliud density of a BEC trapped in 1D and 2D optical lattice with quenched disorder are investigated in details. • Dimensional crossover from 3D to 1D (2D) is studied in random environment. • Such lattice-controlled dimensional crossover presents an effective way to investigate the properties of low-dimensional Bose gases. • Reference: • 1, Y. Hu, Z. X. Liang and B. Hu, Phys. Rev. A 80, 043629 (2009). • 2, Y. Hu, Z. X. Liang and B. Hu, Phys. Rev. A 81, 053621 (2010). • 3, K. Z. Zhou, Y. Hu, Z. X. Liang and Z. D. Zhang, submitted into PRA.

33. Thank you!