Cold atoms in 2D optical lattices under staggered rotation - - PowerPoint PPT Presentation

Cold atoms in 2D optical lattices under staggered rotation

Cristiane MORAIS SMITH Institute for Theoretical Physics, Utrecht University, The Netherlands

• INSTANS 2008 – p.1/28

Collaborators

Lih-King Lim and Andreas Hemmerich

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Outline

Low-D systems: observation of quantum effects 2DES in a

: several interesting quantum phases electron-liquid phases: Laughlin, Moore-Read, Read-Rezayi electron-solid phases: Wigner crystals, bubbles, stripes nematic phases, BEC of excitons in bilayers, etc...

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2D Systems: cond-mat

Uniform GaAs, Si-MOSFETs

B I

L H

U U

2D electron gas _ _ _ _ _ _ + + + + + +

Lattice graphene

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2D Systems: cold atoms

Uniform rotating BECs

Laser Gaspacho BEC

Lattice rotating BECs in

• ptical lattices

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Cold atoms in optical lattices

Sorensen, Demler, Lukin, PRL 94, 086803 (2005) quadrupolar potential + tunneling for bosons FQHE Laughlin state: 95% overlap larger gap than in harmonically trapped BECs because in- teraction energy is larger

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Cold atoms in optical lattices

3D: M. Greiner et al., Nature 419, 51 (2002) 2D: Phillips group, PRL 100, 120404 (2008)

Superfluid-Mott insulator transition

Theoretical description: Bose-Hubbard model

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Cold atoms in optical lattices

simulators of cond-mat systems

• full control of lattice parameters
• load with bosons or fermions
• control of interactions (Feshbach resonance)
• no disorder

generate NEW situations

• alternating magnetic fields

Haldane 1988: graphene with zero net magnetic field per plaquette QHE: no uniform B, but break time-reversal symmetry

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Cold atoms in optical lattices

• A. Hemmerich and C.M.S., PRL 99, 113002 (2007)

Bosons and fermions Staggered rotational field Novel phases

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Staggered Rotational Field

• A. Hemmerich and C.M.S., PRL 99, 113002 (2007)

How to realize it experimentally? Linearly polarized bichromatic light-field

We assume

,

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Staggered Rotational Field

• A. Hemmerich and C.M.S., PRL 99, 113002 (2007)
• ✓✕✔

Stationary term

• ✄

Time-dependent term

• ✆

• INSTANS 2008 – p.10/28

Optical Setup

PZT M BS cold # atoms Laser AOM

Two nested Michelson interferometers PZT: piezoelectric transducer M: mirror BS: beam splitter AOM: acousto-optic frequency shifter

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Time-Dependent Bose-Hubbard Model

B A A B B A B A A

λ

• ✓

• ☛

• ☛

H.c.

• ✒

• ✒

• ✒

where

anisotropic time-varying hopping

time-varying energy offset

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Effective Hamiltonian

Hamiltonian is periodic:

• ✓

• ✓

with

• ✂

Dyson Series:

• ✑

• ✂

• ✑

• ✂

• ✓

• ✓

• ✓

Effective Hamiltonian

• eff

• ☛

• ☛

H.c.

• ✒

• ✒

.

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Single-particle spectrum

Write

• in

space

• ✄

• ✞

• ✆

• ☛

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Bosons: Mean Field Theory

• ✁
• ✂

• ✁

• ✠✡

• ✁

• ✠

• ✁

• ✔

eff

Mott regime

Hubbard-Stratonovich field

to decouple the hopping term Integrate out the boson fields

• ✁

• ✖

Effective action (quadratic order in

,

)

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Mean Field Theory - Effective Action

• ✁

Real frequencies (

),

Quasi-particle (hole) energy dispersion

where

: energy for creating a quasiparticle-quasihole pair.

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Phase Diagram

L.-K. Lim, C.M.S., and A. Hemmerich, PRL 100, 130402 (2008)

• (

)-plane: dashed white line

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Phase Diagram

Bosons: BEC at lowest single-particle state

• ✁

: min at

GS is uniform SF

• ☎

: min at

GS is finite

SF

θ = Ar ct an[ W / J] U / 4( J2+W 2)

1/ 2

π/ 2 2 6

2nd or der

M ot tI nsul at

• r

π/ 4 st agger ed vor t ex super f l ui d

1stor der

uni f

• r

m super f l ui d 5. 83

2 n d

• r

d e r

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Superfluid Phases

Variational mean-field ansatz for the ground state:

• ✟

• ✞

• ✞

• ✁

:

• ☎

:

Order parameter

changes discontinuously by

at

• ☎

Finite momentum SF: analogies with Abrikosov lattice and “FFLO states”

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Experimental Detection

Momentum distribution:

• ✁

• (a) Uniform SF

(b) Staggered vortex SF

: Fourier transform of Wannier function

: structure factor (B: Bravais lattice, P: plaquette)

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Fermions in optical lattices

At half-filling: anisotropic Dirac cones Graphene under uniaxial pressure At

• ☎

: staggered-

flux phase (HTSC)

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Simulating Graphene

2 ineq. points:

• ☎

Long-wavelength expansion around and

• ✄

• ✌☎✄

• ☎

• ✠

• ✌☎✄

• ☎

• ✠

• ✞

Tight-binding model for graphene (

• ☎

)

• ✞

• ✞

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Graphene under uniaxial pressure

Cheol-Hwan Park et al., Nature Physics 4, 213 (2008) P . Dietl, F . Piechon, G. Montambaux, cond-mat/0707.0219

A B t t t’ a1 a2

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Fermions and Bosons

• ✝✟✞

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

Fermions: same terms, replace

• by

, neglect

terms (spin polarized fermions: neglect

• wave collisions at

low-

)

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Fermions and Bosons

• ✝

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

• ✕

Bosons condense at

: integrate them out New “phonon-mediated” interaction which couples fermions in

• sublattices

: Yukawa-like interaction MF decoupling: generate Mass term for fermions See Kekulé distortion for graphene

• C. Mudry

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Conclusions

Cold atoms in optical lattices under staggered rotation: Superfluid / Mott-insulator transition Staggered flux phase (bosons) finite momentum condensate Simulate graphene under pressure (fermions) Simulate staggered-

flux phase for high-Tc superconductors at

• Simulate Kekulé distortion in graphene (fermions and

bosons) Novel Phases? QHE, bilayers, supersolids, etc...

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Perspectives

Hofstadter butterfly : flux/plaquette cond-mat:

• T

uniform

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Perspectives

Staggered

• ✠

field role of interactions ring-exchange and longer-range int. triangular, hexago- nal geometries

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