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

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

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

Condensate Fraction

Bose-Einstein condensation refers to the macroscopic occupation of a single quantum stateBose-Einstein condensation refers to the macroscopic occupation of a single quantum state

Superfluidity

Superfluidity refers to a set of fascinating hydrodynamic phenomena, notably persistent flow.

Two-fluid model (Two-fluid model (finite temperature)finite temperature)

- Tisza (1940), Landau (1941)

Andronikashvili, J. Phys. USSR 10, 201 (1946)

Superfluid densitySuperfluid density

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)

-

Method I: superfluidity and linear response theory

- Only normal fluid is draggedOnly normal fluid is dragged by transverse perturbation

- BothBoth normal fluid and superfluid are pushed along z-direction

TransverseTransverse perturbation perturbation LongitudinalLongitudinal perturbation perturbation

nn = ρ

j=nv

n = ρ

K.Huang and H.F.Meng, Phys. Rev. Lett. 69 644 (1962)G.Baym, in Mathematical Methods in Solid State and Superfluid Theory

Current-current response function

Average momentum density <g(r,t)> induced by the external perturbation v(r)

Static current-current response functionStatic current-current response function

Superfluidity is a Superfluidity is a kinetic propertykinetic property of a system and the superfluid density is a of a system and the superfluid density is a transport coefficienttransport coefficient rather than an equilibrium property. rather than an equilibrium property.

D.Pines and P.Nozieres, The Theory of Quantum Liquids

How to calculate superfluid density

Normal fluid densityTotal density

Longitudinal response Transverse response

D.Pines and P.Nozieres, The Theory of Quantum Liquids

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-non-uniform wave-function describing a non-zero velocity of the condensatezero 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 the response function method is less general than that of explicit wave function constructionthan 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

Bose-Einstein Condensation vs. Superfluidity

VS.

The identification of the superfluid velocity and the gradient of the phase of The identification of the superfluid velocity and the gradient of the phase of the order parameter represents a key relationship between Bose-Einstein the order parameter represents a key relationship between Bose-Einstein condensation and superfluidity.condensation and superfluidity.

ConnectionConnection

L. Pitaevskii and S. Stringari, Bose-Einstein condensation

Bose-Einstein condensation vs. Superfluidity

ContrastContrast Condensate density Superfluid density

Noninteracting Bose gas ~100% 0

Weakly interacting Bose gas

~100% ~100%

Liquid He ~10% ~90%

2D Bose gas (in special case)

0 nonzero

4

Generally, the two concepts of superfluid density and condensate density Generally, the two concepts of superfluid density and condensate density cannot cannot be confusedbe confused with each other. A typical illustration is provided by with each other. A typical illustration is provided by weakly interactingweakly interacting Bose gases in the presence of disorder at zero temperatureBose gases in the presence of disorder at zero temperature..

Outline1, 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

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 Pronounced contrast between BEC and superfluidity in the presence of disorder even at zero temperaturedisorder 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).

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)

Anderson localization in a BEC

Nature 453_895 (2008)

BEC+Disorder

Outline1, Basic Concepts

Symmetry Breaking, Order Parameter, Condensation and Superfluidity

2, Disorder Why?

3, Bose-Einstein Condensate + Optical lattice + Disorder

4, Conclusion

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 BECQuasi-low-dimensional BEC: KinematicallyKinematically, the gas is 2D or 1D; The difference from purely 2D or 1D gases is only related to the value of the value of the

inter-particle interactioninter-particle interaction which now depends on the tight confinementwhich now depends on the tight confinement.

Quasi-low-dimensional BECQuasi-low-dimensional BEC: KinematicallyKinematically, the gas is 2D or 1D; The difference from purely 2D or 1D gases is only related to the value of the value of the

inter-particle interactioninter-particle interaction which now depends on the tight confinementwhich now depends on the tight confinement.

BEC in presence of a 1D (2D) optical lattice and disorder

Action functional within the grand-canonical ensemble

Order parameter field1D (2D) optical lattice

Disorder

Effective two-body coupling constant

Quasi-low-dimensional Bose gasesQuasi-low-dimensional Bose gases: KinematicallyKinematically, the gas is 2D or 1D; The difference from purely 2D or 1D gases is only related to the value of the value of the

inter-particle interactioninter-particle interaction which now depends on the tight confinementwhich now depends on the tight confinement.

Effective two-body coupling constant

Chemical potential:

Pseudopotential:

Dimensionality of g

3D

1D

2D Density-dependentDensity-dependent

Effective coupling constant tuned by a 1D tight optical lattice

3D limit ( )

Quasi-2D limit ( )

Tight-binding approximationTight-binding approximation

2D limit: ( )

D.S.Petrov et al., PRL84_2551 (2000); G.Orso and G. V. Shlyapnikov, PRL95_260402 (2005)

Treatment of disorder

Disorder is produced by random potential associated with Disorder is produced by random potential associated with quenched impuritiesquenched impurities

Small concentration of disorder:

Two basic statistical properties

Affected by optical lattice

K. Huang and Hsin-Fei Meng, PRL69_644 (1992)

Bose gases trapped in a 2D optical lattice and random potential

Beyond-mean-field ground state energy

Dimensional crossover from 3D to 1D

Lieb-Liniger solution of 1D model expanded in the weak coupling regime in the presence of disorder

Quantum depletion

The first term diverges NO BEC

In the 3D regime, h(x) and K(x) decay

Bose gases trapped in a 1D optical lattice and random potential

Beyond-mean-field Ground state energy

Dimensional crossover from 3D to 2D

In the asymptotic 3D limit:

In the 2D limit:

Ground state energy in 2D

Quantum depletion

3D limit:

2D limit:

Disorder-induced superfluid depletion in 2D

3D limit:

2D limit:

Contrast between superfluidity and BEC becomesMore pronounced in low dimensions.

Disorder induced superfluid depletion

Disorder induced condensate depletion

3D 4 / 3

2D 2

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.

Thank you!