1. Optically Trapped Low-Dimensional Bose Gases in 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)

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 Disorder Why?
Superfluid Density vs. Condensate Fraction
Disorder
Why?
Bose-Einstein Condensate + Disorder
Two recent work: PRA80_043629(2009),81_ (2010)
Conclusion 3

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, (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
Transverse perturbation
Longitudinal perturbation
- Only normal fluid is dragged by transverse perturbation
- Both normal fluid and superfluid are pushed along z-direction
K.Huang and H.F.Meng, Phys. Rev. Lett (1962)
G.Baym, in Mathematical Methods in Solid State and Superfluid Theory 6

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 7

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 8

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 9

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
Condensate density
Superfluid density
Noninteracting Bose gas
~100%
Weakly interacting Bose gas
Liquid He
~10%
~90%
2D Bose gas (in special case)
nonzero 4
Generally, the two concepts of superfluid density and condensate density cannot be confused with each other. A typical illustration is provided by weakly interacting Bose gases in the presence of disorder at zero temperature.

12. Outline 1, Basic Concepts 2, Disorder Why?
Symmetry Breaking, Order Parameter, Condensation and Superfluidity
2, Disorder
Why?
3, Bose-Einstein Condensate + Disorder
Two recent work: PRA80_043629(2009),81_ (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 2, Disorder Why?
Symmetry Breaking, Order Parameter, Condensation and Superfluidity
2, Disorder
Why?
3, Bose-Einstein Condensate + Optical lattice + Disorder
4, Conclusion 18

19. Optically Trapped Low-Dimensional BEC
BEC trapped in a 2D optical lattice
BEC trapped in a 1D 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 interaction which 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 interaction which now depends on the tight confinement. 19

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 interaction which now depends on the tight confinement.

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

22. Effective coupling constant tuned by a 1D tight optical lattice
Tight-binding approximation
3D limit ( )
Quasi-2D limit ( )
2D limit: ( )
D.S.Petrov et al., PRL84_2551 (2000); G.Orso and G. V. Shlyapnikov, PRL95_ (2005)

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

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

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

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

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

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

29. Ground state energy in 2D29

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

31. Disorder-induced superfluid depletion in 2D
3D limit:
Disorder induced superfluid depletion
Disorder induced condensate depletion 3D
4 / 3 2D 2
2D limit:
Contrast between superfluidity and BEC becomes
More pronounced in low dimensions. 31

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, (2009).
2, Y. Hu, Z. X. Liang and B. Hu, Phys. Rev. A 81, (2010).
3, K. Z. Zhou, Y. Hu, Z. X. Liang and Z. D. Zhang, submitted into PRA. 32

33. Thank you!