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