Commensurate-Incommensurate phase transition in cold ... · Commensurate-Incommensurate phase...

Commensurate-Incommensurate phase transition in cold atomic gases

Commensurate-Incommensurate phase transition in cold atomic gases

05/03/2010SISSA, Trieste

Marcello Dalmonte,Dipartimento di Fisica, Alma Mater Studiorum-

Università di Bologna e Sezione INFN, Bologna

Journal club on...Journal club on...

And complementary information and experiments:●W. Zwerger, arXiv:cond-mat/0211314v2;●T. Stöferle et al., Phys. Rev. Lett. 92, 130403 (2004);●D. Clément et al., Phys. Rev. Lett. 102, 155301 (2009).

Cold atoms in 1 DCold atoms in 1 D

1D gas of bosons with contact interactions

with dimensionless interaction strength:

+optical lattice

Cold atoms and optical lattices: the deep lattice case

Cold atoms and optical lattices: the deep lattice case

D. Jaksch et al., PRL98D. Jaksch et al., PRL98

When , atoms occupy the lowest vibrational level in each well

Bose-Hubbard model

SF

MI

Phase transition point:

Weak optical lattice: what happens?Weak optical lattice: what happens?

Treat the lattice as a perturbation if the system is strongly interacting!

Lieb-Liniger model + perturbation

+ optical lattice

Lieb-Liniger modelLieb-Liniger model

T. Kinoshita et al., Science(2004); B. Paredes et al., Nature(2004).T. Kinoshita et al., Science(2004); B. Paredes et al., Nature(2004).

Quasi-condensate

Tonks gas,Fermionic limit

More sensitive to perturbations!

Effective hydrodynamic approachEffective hydrodynamic approach

F. D. M. Haldane, PRL 81 F. D. M. Haldane, PRL 81

Density and phase felds with commutation relation:

Depends on interactions, canbe determined via Bethe ansatz

Luttinger liquid pictureLuttinger liquid picture

Recasting the previous Hamiltonian in the Luttinger-liquid form:

Only K determines the long-distance decay of correlation functions:

Weak lattice as a perturbationWeak lattice as a perturbation

+

=

Close to one particle per site

Physical meaning of QPhysical meaning of Q

Describes competition between two-lenghtscales!

Inverse inter-particle distance Inverse lattice spacing

Strong repulsion wants to maximise interparticle distance

Background potential wants to pin particles at potential mimima

Sine-Gordon physics: Q=0Sine-Gordon physics: Q=0

Commensurate case: the only relevant length-scale is the lattice spacing:

BKT equations to evaluate a phase transition between superfuid and insulator

Sine-Gordon physics: Q=0Sine-Gordon physics: Q=0

If , a vanishingly small optical lattice can pin all particles!

Non-commensurate case: Pokrovsky-Talapov model Non-commensurate case: Pokrovsky-Talapov model

Rescaling the density:

Leads to:

Effective chemical potential for density

excitations!

Energy competitionEnergy competition

Cosine term: any excitation over the “commensurate” confguration

cost an energy contribution proportional to the gap

Chemical potential: excitations are favored for large enough Q

Effect of a trapping potentialEffect of a trapping potential

True potential is inhomogeneous due to the trap

Coexistence of insulator and superfuid regions!

Density profle:

Add-ons: experimental setupAdd-ons: experimental setup

E. Haller et al., arXiv:1002.3795E. Haller et al., arXiv:1002.3795

1)Mott insulator of Cs atoms: the optical lattice is deep and 3D

2)Lower down the lattice in one direction

3)Tune the interaction between atoms with a resonance

Add-ons: experimental resultsAdd-ons: experimental results

E. Haller et al., in preparationE. Haller et al., in preparation

In excellent agreementwith BKT predictions

even at V=2

Confrms BH numerical predictions

Add-ons: spectrum and gap scalingAdd-ons: spectrum and gap scaling

lattice depth0.7 Er1.0 Er1.4 Er2.1 Er

Very good accordancewith Zamolodchikov

formula for V<1

Open questions and suggestionsOpen questions and suggestions

What happens when both models break down?

Gap scaling away from weak coupling?

●Dynamical properties of the sine-Gordon and PT models●Thermalization properties●Commensurate-incommensurate transition in 2D and 3D●Finite-size effects?

LabLab

ElmarHaller

Johann Danzl

RussellHart

Manfred Mark

Mattias Gustavsson

Hanns-Christoph Nägerl Lukas

Reichsöllner

OliverKriegelsteiner

AndreasKlinger

Guido Pupillo