Cold atoms in zig-zag optical lattices [0.7cm] …€¦ · zig-zag lattices: realize Haldane...
Transcript of 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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
2 component vs chiral superfluiddue to interactions different groundstates are possible
2 component SF chiral SF
J. Struck, et al. 2011 6 / 18
RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
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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RTG seminar - Cold atoms in zig-zag optical lattices
Thank you for the attention!
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