Cold atoms in zig-zag optical lattices [0.7cm] …€¦ · zig-zag lattices: realize Haldane...

Cold atoms in zig-zag optical lattices

Sebastian Greschner (ITP Hannover)

Goslar, 29. November 2012

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RTG seminar - Cold atoms in zig-zag optical lattices

Experiment: 2D classical spins

J. Struck, et al. 2011

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Bosons in zig-zag optical lattices

Optical lattices

triangular lattice and incoherentsuperposition with additional lattice

Bose Hubbard model

H =−∑〈ij〉

Jij a†i aj +

∑i

µni +

+∑

i

U

2ni (ni − 1)

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Lattice shaking

Fast shaking of optical latticewith periodic orbit (e.g.sinusoidal)

inertial force F = −mx isdescribed by potentialmodulations vi (t) = −ri · F

H(t) = −J∑〈ij〉

a†i aj +∑

i

vi (t)ni + Hon−site

Time averaging fast oscillationover a shaking period leads toan effective renormalizedhopping J → Jeff = J J0

(K~ω)

H = −Jeff∑〈ij〉

a†i aj + Hon−site

H. Lignier, et al. 2007

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Bosons in zig-zag optical lattices

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For U = 0 the Hamiltonian is diagonalized withthe dispersion

ε(k) = |t|(cos k + j cos 2k)

where the frustration parameter j ≡ t ′/t.

-Π Πk

-0.5

0.5

1ΕHkL

-Π Πk

-0.5

0.5

1

ΕHkL

-Π Πk

-0.5

0.5

1

ΕHkL two nonequivalentdegenerate minima at

± arccos(− 1

4j

)At the Lifshitz point j = 1/4, the single particle dispersion relationsplits due to frustration from one minimum at k = π into twononequivalent degenerate minima at k = ± arccos[−1/4j ].

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2 component vs chiral superfluiddue to interactions different groundstates are possible

2 component SF chiral SF

-π 0 πk

-π 0 πk

-π 0 πk

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2 component vs chiral superfluiddue to interactions different groundstates are possible

2 component SF chiral SF

J. Struck, et al. 2011 6 / 18


Chiral Superfluid

non-vanishing local current, orchirality

κi =i

2(b†i bi+1 − bib

†i+1).

appearance or vanishing of(sharp) peaks at Q or −Q inthe momentum distribution

-π 0 πk

-π 0 πk

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Spontaneous symmetry breaking: 2SF vs. CSF

Symmetry breaking does not necessarily haveto take place

0 0.5j

0

5

10

U

ρ = 0.2

ρ = 0.1

ρ = 0.05

SF

2SFCSF

1/√81/4

-Π Πk

-0.5

0.5

1ΕHkL

j < 1/4

-Π Πk

-0.5

0.5

1

ΕHkL

j > 1/4

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Properties of 2SF

0 0.2 0.4 0.6 0.8j

0

0.5

1

1.5

2

2.5

3

centr

al c

har

ge

c

0

0.04

chir

alit

y κ

2

SF 2 SF CSF

N=200, ρ=0.1, U=10

50 150

site i

1.2

1.7

Sv

N

0 50

site i

0.001

0.01

<k

0k

i>

Central charge c = 2 for two component superfluid

chirality correlator 〈κiκj〉

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Commensurate fillings: The Mott regime

0 0.2 0.4 0.6 0.8 1

j

0

1

2

U /

t

MI

CSF

SFCMI

(a.)-Π Π

k

-0.5

0.5

1

ΕHkL

Lifshitz point j = 1/4

commensuratefilling ρ = 1

Parity orderexp[iπ

∑i<l<j δnl ]

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Chiral Mott-Insulator

0 0.2 0.4 0.6 0.8 1

j

0

1

2

U /

t

MI

CSF

SFCMI

(a.)

Intermediate phase between MI and CSF phase

Insulating phase with excitation gap andchirality

0.66 0.68 0.7 0.72 0.74j

0

2

4

κ2 L

1/4

L=100L=150L=200L=300

-20 0 20

(j-jc) L

0

6

κ2 L

1/4

first: phase transition to chiral phase(Ising type)

0.66 0.68 0.7 0.72 0.74j

0.2

0.8

n(k

max

) L

-3/4

L = 100L = 150L = 200L = 300

π / 2 πk

n(k

)

j=0.2

j=0.4

j=0.6

j=0.7

0.2 0.4 0.6 0.8j

π / 2

πk

max

(c.)(b.)

(a.)

then: phase transition to gapless CSFphase (BKT type)

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Full phase diagramm

0 0.5 1

j

0

10

20

30

40

µ

0 0.5 1 1.5

j

-2

0

2

CSF

SF

ρ = 1

ρ = 4

ρ = 2

ρ = 3

ρ = 5

ρ = 6

ρ = 7

MI

CSFSF

MI ρ = 1

ρ = 0

Dimer (ρ = 1/2)

TLL2

(a) (b)

Mott phase for unit filling

Dimerized phase for half filling

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Haldane Insulator Phase

X. Deng, L. Santos, 2012

topological gapped phase

Bose-Hubbard model withdipolar interaction

“dilute antiferromangnet“+00− 0 +−000 + 0−String orderO2

S ≡ δni exp[iπ∑

i<l<j δnl ]δnj

zig-zag lattices: realize Haldane insulator phase without polarinteractions

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3-body hardcore constraint

inelastic 3 atom collision

molecule + atom ejected from lattice

ubiquitous, but typically undesirable

However

dynamic suppression of triple onsite occupation(analogous Quantum Zeno Effect)

strong losses create an effective 3-bodyhardcore constraint (not more than twoparticles (n = 0, 1, 2) on one site)

stabilize bosonic system with attractiveinteractions

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Zig-zag ladder with 3-body hardcore constraintIn order to understand the influence of the 3-body-constraint, we slowy ”turnon”U3

H → HBH + U3(b†i )3(bi )3

0 0.2 0.4 0.6 0.8 1

j

0

U3 /

t

CSF

HI

SF

CHI

(b.)

-0.2 0 0.2 0.4 0.6 0.8

j

-12

-8

-4

0

4

U /

t

CSF

PSF

DWPSF

SFHI

MI

Bosons with 3-body hard-core constraint, U3 →∞Haldane-insulator phase in the absence of polar (long-range) interaction

String order O2S ≡ δni exp[iπ

∑i<l<j δnl ]δnj

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Properties of polar bosons

H = HBH +∑

ij

V

|i − j |3ninj

Long range interactions

Devil’s staircase structure

Double Haldane insulator phase possible

0 0.5j

-2

0

µ

0 0.3 0.6 0.9j

0

1

V

ρ = 1/3

ρ = 1/2

SF

DW

CSF

Dimer

-4 -2 0 2 4U / t

-1

0

1

2

V /

t

DDW

CSF MI

F

DH

PSF

j=1

unit filling, 3-body hard-core constraint

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Summary

Using cold atom systems as quantumsimulators to study (quantum)frustrated magnetism

Ingredient I: Ultracold (atomic gases) inspecial optical lattice geometries

Ingredient II: shaking techniques

Explore rich physics, e.g. chiral phases,chiral insulator...

spontaneous symmetry breaking, chiralphases,

exotic gapped chiral phases

Haldane insulator phase even withoutpolar long range interaction

0 0.2 0.4 0.6 0.8 1

j

0

1

2

U /

t

MI

CSF

SFCMI

(a.)

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Thank you for the attention!

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Cold atoms in zig-zag optical lattices [0.7cm] …€¦ · zig-zag lattices: realize Haldane...

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