Adiabatic preparation of a Heisenberg antiferromagnet using an ...
Adiabatic preparation of a Heisenberg antiferromagnet using an optical superlattice Michael Lubasch, Valentin Murg, Ulrich Schneider, J. Ignacio Cirac and Mari-Carmen Bañuls Max Planck Institute of Quantum Optics, Garching, Germany University of Vienna, Austria Ludwig-Maximilians-University Munich, Germany
Main ideas
Experimental proposal Adiabaticity conditions for the total lattice for a sublattice
Effect of holes and harmonic trap
Motivation
Motivation recent experimental realization of fermionic Hubbard model in optical lattice [Schneider et al., Science’08; Jördens et al., Nature’08]
U
ˆ = H
x
t
t
X
hl,mi,
x † (cl,
x cm, +
† cm,
x cl, ) + U
x X l
n ˆ l," n ˆ l,#
Motivation
limit of strong interactions U
x ˆ = H
t
X
hl,mi,
t † (˜ cl,
t : t-J model
x
x c˜m, +
c˜†m,
c˜l, ) + J
x
2
J := 4t /U
J
X
hl,mi
x
~l · S ~m (S
n ˆln ˆm ) 4
Motivation experimental realization of various phases:
ˆ +V H
X l
taken from [Schneider et al., Science’08]
(l
l0 ) 2 n ˆl
Experimental proposal
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
1. band insulating ground state |BI> [Schneider et al., Science'08]
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
2. dimerized ground state [Trotzky et al., PRL'10]
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
3. quantum Heisenberg antiferromagnet |AF M i
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Advantage over direct preparation of Mott insulator: Band insulator has less entropy!
Adiabaticity conditions for the total lattice
Adiabaticity on total lattice experimental observable: squared staggered magnetization 2 Mstag
1 = 2 N
1
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
1D
m
2
0,8 N=22 N=42 N=62
0,6 0,4 0
10
20
1D
30
40
ramping time T
50
Adiabaticity on total lattice 1D
experimental observable: squared staggered magnetization 1 = 2 N
1
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
T(m =0.85)
2 Mstag
N X
1D
m
2
0,8
0,4 0
10
20
30
40
ramping time T
2
N=22 N=42 N=62
0,6
50
Landau-Zener formula: T / 1/
50 40 40 20 30 0 2000 20 2 N 10 0 10 20 30
40
lattice size N
2
1D: gap closes at end of protocol [Matsumoto et al., PRB’01]
/ 1/N
50
!
T /N
2
60
Adiabaticity on total lattice experimental observable: squared staggered magnetization
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM 1 0,8 0,6 0,4 0,2
2D
m
1 = 2 N
2
2 Mstag
N X
2D
N=4x4 N=6x6 N=10x10 2
4
6
ramping time T
8
Adiabaticity on total lattice experimental observable: squared staggered magnetization 1 = 2 N
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM 1 0,8 0,6 0,4 0,2
2D
m
2
2 Mstag
N X
2D
N=4x4 N=6x6 N=10x10 2
4
6
ramping time T
8
high magnetization in short ramping time
Adiabaticity conditions for a sublattice
Adiabaticity on sublattice 1D
experimental observable: squared staggered magnetization L X 1 2 ~l · S ~m i : m2 (T ) := M 2 (T )/M 2 Mstag = 2 ( 1)l+m hS stag stag,AFM L l,m=1
1
m
2
0,8 L=22 L=42 L=62
0,6 0,4 0
10
20
30
40
ramping time T
50
Adiabaticity on sublattice 1D
experimental observable: squared staggered magnetization L X 1 2 ~l · S ~m i : m2 (T ) := M 2 (T )/M 2 Mstag = 2 ( 1)l+m hS stag stag,AFM L l,m=1
T(m =0.85)
1
m
2
0,8
0,4 0
10
20
30
40
ramping time T
effective local gap:
2
L=22 L=42 L=62
0,6
50
/ 1/L
50 40 40 20 30 0 20 10 0 10
2000
L 20
2
30
40
sublattice size L
!
T / L2
50
Adiabaticity on sublattice 1D
experimental observable: squared staggered magnetization L X 1 2 ~l · S ~m i : m2 (T ) := M 2 (T )/M 2 Mstag = 2 ( 1)l+m hS stag stag,AFM L l,m=1
T(m =0.85)
1
m
2
0,8
0,4 0
10
20
30
40
ramping time T
effective local gap:
2
L=22 L=42 L=62
0,6
50
/ 1/L
50 40 40 20 30 0 20 10 0 10
2000
L 20
2
30
40
sublattice size L
!
T / L2
high magnetization in short ramping time on small part
50
Effect of holes
Effect of holes † (˜ cl,
c˜m, +
† c˜m,
ˆ spin c˜l, ) + H
=2
=1
=0
hl,mi,
=3
t
=4
ˆ = H
X
1 0,5 0 0,1
one hole one particle
0 0,08 0,04 0 0,16 0,08 0 0,08 0,04
lattice site l
1D
Effect of holes t
† (˜ cl,
c˜m, +
† c˜m,
ˆ spin c˜l, ) + H
=3
=2
=1
=0
hl,mi,
=4
ˆ = H
X
1 0,5 0 0,1
one hole one particle
0 0,08 0,04 0 0,16 0,08 0 0,08 0,04
lattice site l
hole spreads as free particle: velocity v = 2t
1D
Effect of holes ˆ = H
t
X
† (˜ cl,
c˜m, +
† c˜m,
ˆ spin c˜l, ) + H
hl,mi, -16
energy increase:
Espin
-16,5
~l · S ~l+1 i| Espin ⇡ |hS
-17 -17,5 -18
0
2
4
6
8
10
6
8
10
time 0,09
M
2
0,08 0,07 0,06 0,05
0
2
4
time
1D
Effect of holes ˆ = H
t
X
† (˜ cl,
c˜m, +
† c˜m,
hl,mi,
J
Espin
~l · S ~l+1 i| Espin ⇡ |hS
-17 -17,5 0
2
4
6
8
10
6
8
10
time 0,09 0,08
M
2
~l · S ~m S
energy increase:
-16,5
0,07 0,06 0,05
X
hl,mi
-16
-18
ˆ spin c˜l, ) + H
0
2
4
time
a)
b)
1D
Effect of holes ˆ = H
t
X
† (˜ cl,
c˜m, +
† c˜m,
hl,mi,
J
~l · S ~m S
energy increase:
-16,5
Espin
X
hl,mi
-16
~l · S ~l+1 i| Espin ⇡ |hS
-17 -17,5 -18
ˆ spin c˜l, ) + H
0
2
4
6
8
10
a)
b)
time 0,09
M
2
0,08
drastic magnetization reduction
0,07 0,06 0,05
0
2
4
time
6
8
10
1D
Effect of holes experimental observable: squared staggered magnetization 1 = 2 N
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,8 0,6 0,4
2
m (L=42)
2 Mstag
N X
0,2 0
1D
2 holes on 84 sites 4 holes on 86 sites 10 20 30 40 50
ramping time T
Effect of holes 1D
experimental observable: squared staggered magnetization
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,8 0,6
2
0,4 0,2 0
20
T(m =0.85)
1 = 2 N
2
m (L=42)
2 Mstag
N X
2 holes on 84 sites 4 holes on 86 sites 10 20 30 40 50
ramping time T
hole arrival
15 10 5 0
10
15
20
25
sublattice size L
30
Effect of holes 1D
experimental observable: squared staggered magnetization
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM 20
0,8 0,6
2
0,4 0,2 0
T(m =0.85)
1 = 2 N
2
m (L=42)
2 Mstag
N X
2 holes on 84 sites 4 holes on 86 sites 10 20 30 40 50
ramping time T
hole arrival
15 10 5 0
10
15
20
25
sublattice size L
drastic magnetization reduction
30
Harmonic trap
Harmonic trap ˆ =H ˆ kin + H ˆ spin + V H
X
(l
2
l0 ) n ˆl
l
=2
=1
=0
=3 =4 =5 =6 =7 =8
0 0,1
=9
=6 =7
0 0,1
=10
0 0,1
=8
0 0,2
0 0,1
=10
=1 =2 =3 =4
0 0,2
=9
=5
0 0,2
0 0,1
1 0,5 0 0,4 0,2 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1
=0
1 0,5 0 0,4 0,2 0 0,2
lattice site l
lattice site l
no trap
weak trap
1D
Harmonic trap ˆ =H ˆ kin + H ˆ spin + V H
X
(l
2
l0 ) n ˆl
l
=2
=1
=0
=3 =4 =5 =6 =7 =8
0 0,1
=9
=6 =7
0 0,1
=10
0 0,1
=8
0 0,2
0 0,1
=10
=1 =2 =3 =4
0 0,2
=9
=5
0 0,2
0 0,1
1 0,5 0 0,4 0,2 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1
=0
1 0,5 0 0,4 0,2 0 0,2
lattice site l
lattice site l
no trap
weak trap
1D
Harmonic trap ˆ =H ˆ kin + H ˆ spin + V H
X
(l
2
l0 ) n ˆl
l
=2
=1
=0
=3 =4 =5 =6 =7 =8
0 0,1
=9
=6 =7
0 0,1
=10
0 0,1
=8
0 0,2
0 0,1
=10
=1 =2 =3 =4
0 0,2
=9
=5
0 0,2
0 0,1
1 0,5 0 0,4 0,2 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1
=0
1 0,5 0 0,4 0,2 0 0,2
lattice site l
lattice site l
no trap
weak trap
1D
Harmonic trap experimental observable: squared staggered magnetization
2
m (L=82)
2 Mstag
1 = 2 N
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,6 0,4 0,2 0
10
20
1D
V=0.004J V=0.006J V=0.02J 30 40 50
ramping time T
Harmonic trap 1D
experimental observable: squared staggered magnetization 2 Mstag
1 = 2 N
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,4 0,2 0
10
20
V=0.004J V=0.006J V=0.02J 30 40 50
ramping time T
2
0,6
T(m =0.85)
2
m (L=82)
50 40 30 20 10 0
10
20
30
40
sublattice size L
50
Harmonic trap 1D
experimental observable: squared staggered magnetization 2 Mstag
1 = 2 N
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,4 0,2 0
10
20
V=0.004J V=0.006J V=0.02J 30 40 50
nl(T=50)
ramping time T
2
0,6
T(m =0.85)
2
m (L=82)
50 40 30 20 10 0
10
20
30
40
sublattice size L
50
1 0,5
strong trap: hole-free case lattice site l
Conclusions
Conclusions
feasible timescales
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Conclusions
feasible timescales sublattice adiabaticity: governed only by its size
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Conclusions
feasible timescales sublattice adiabaticity: governed only by its size
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
destructive effect of holes: controlled by harmonic trap
3.
|AFM> V2
Conclusions
feasible timescales sublattice adiabaticity: governed only by its size
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
destructive effect of holes: controlled by harmonic trap
3.
|AFM>
details: [M. Lubasch, V. Murg, U. Schneider, J. I. Cirac, M.-C. Bañuls, PRL 107, 165301 (2011)]
V2
Main ideas
Experimental proposal Adiabaticity conditions for the total lattice for a sublattice
Effect of holes and harmonic trap
Motivation
Motivation recent experimental realization of fermionic Hubbard model in optical lattice [Schneider et al., Science’08; Jördens et al., Nature’08]
U
ˆ = H
x
t
t
X
hl,mi,
x † (cl,
x cm, +
† cm,
x cl, ) + U
x X l
n ˆ l," n ˆ l,#
Motivation
limit of strong interactions U
x ˆ = H
t
X
hl,mi,
t † (˜ cl,
t : t-J model
x
x c˜m, +
c˜†m,
c˜l, ) + J
x
2
J := 4t /U
J
X
hl,mi
x
~l · S ~m (S
n ˆln ˆm ) 4
Motivation experimental realization of various phases:
ˆ +V H
X l
taken from [Schneider et al., Science’08]
(l
l0 ) 2 n ˆl
Experimental proposal
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
1. band insulating ground state |BI> [Schneider et al., Science'08]
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
2. dimerized ground state [Trotzky et al., PRL'10]
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
3. quantum Heisenberg antiferromagnet |AF M i
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Experimental proposal 1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Advantage over direct preparation of Mott insulator: Band insulator has less entropy!
Adiabaticity conditions for the total lattice
Adiabaticity on total lattice experimental observable: squared staggered magnetization 2 Mstag
1 = 2 N
1
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
1D
m
2
0,8 N=22 N=42 N=62
0,6 0,4 0
10
20
1D
30
40
ramping time T
50
Adiabaticity on total lattice 1D
experimental observable: squared staggered magnetization 1 = 2 N
1
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
T(m =0.85)
2 Mstag
N X
1D
m
2
0,8
0,4 0
10
20
30
40
ramping time T
2
N=22 N=42 N=62
0,6
50
Landau-Zener formula: T / 1/
50 40 40 20 30 0 2000 20 2 N 10 0 10 20 30
40
lattice size N
2
1D: gap closes at end of protocol [Matsumoto et al., PRB’01]
/ 1/N
50
!
T /N
2
60
Adiabaticity on total lattice experimental observable: squared staggered magnetization
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM 1 0,8 0,6 0,4 0,2
2D
m
1 = 2 N
2
2 Mstag
N X
2D
N=4x4 N=6x6 N=10x10 2
4
6
ramping time T
8
Adiabaticity on total lattice experimental observable: squared staggered magnetization 1 = 2 N
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM 1 0,8 0,6 0,4 0,2
2D
m
2
2 Mstag
N X
2D
N=4x4 N=6x6 N=10x10 2
4
6
ramping time T
8
high magnetization in short ramping time
Adiabaticity conditions for a sublattice
Adiabaticity on sublattice 1D
experimental observable: squared staggered magnetization L X 1 2 ~l · S ~m i : m2 (T ) := M 2 (T )/M 2 Mstag = 2 ( 1)l+m hS stag stag,AFM L l,m=1
1
m
2
0,8 L=22 L=42 L=62
0,6 0,4 0
10
20
30
40
ramping time T
50
Adiabaticity on sublattice 1D
experimental observable: squared staggered magnetization L X 1 2 ~l · S ~m i : m2 (T ) := M 2 (T )/M 2 Mstag = 2 ( 1)l+m hS stag stag,AFM L l,m=1
T(m =0.85)
1
m
2
0,8
0,4 0
10
20
30
40
ramping time T
effective local gap:
2
L=22 L=42 L=62
0,6
50
/ 1/L
50 40 40 20 30 0 20 10 0 10
2000
L 20
2
30
40
sublattice size L
!
T / L2
50
Adiabaticity on sublattice 1D
experimental observable: squared staggered magnetization L X 1 2 ~l · S ~m i : m2 (T ) := M 2 (T )/M 2 Mstag = 2 ( 1)l+m hS stag stag,AFM L l,m=1
T(m =0.85)
1
m
2
0,8
0,4 0
10
20
30
40
ramping time T
effective local gap:
2
L=22 L=42 L=62
0,6
50
/ 1/L
50 40 40 20 30 0 20 10 0 10
2000
L 20
2
30
40
sublattice size L
!
T / L2
high magnetization in short ramping time on small part
50
Effect of holes
Effect of holes † (˜ cl,
c˜m, +
† c˜m,
ˆ spin c˜l, ) + H
=2
=1
=0
hl,mi,
=3
t
=4
ˆ = H
X
1 0,5 0 0,1
one hole one particle
0 0,08 0,04 0 0,16 0,08 0 0,08 0,04
lattice site l
1D
Effect of holes t
† (˜ cl,
c˜m, +
† c˜m,
ˆ spin c˜l, ) + H
=3
=2
=1
=0
hl,mi,
=4
ˆ = H
X
1 0,5 0 0,1
one hole one particle
0 0,08 0,04 0 0,16 0,08 0 0,08 0,04
lattice site l
hole spreads as free particle: velocity v = 2t
1D
Effect of holes ˆ = H
t
X
† (˜ cl,
c˜m, +
† c˜m,
ˆ spin c˜l, ) + H
hl,mi, -16
energy increase:
Espin
-16,5
~l · S ~l+1 i| Espin ⇡ |hS
-17 -17,5 -18
0
2
4
6
8
10
6
8
10
time 0,09
M
2
0,08 0,07 0,06 0,05
0
2
4
time
1D
Effect of holes ˆ = H
t
X
† (˜ cl,
c˜m, +
† c˜m,
hl,mi,
J
Espin
~l · S ~l+1 i| Espin ⇡ |hS
-17 -17,5 0
2
4
6
8
10
6
8
10
time 0,09 0,08
M
2
~l · S ~m S
energy increase:
-16,5
0,07 0,06 0,05
X
hl,mi
-16
-18
ˆ spin c˜l, ) + H
0
2
4
time
a)
b)
1D
Effect of holes ˆ = H
t
X
† (˜ cl,
c˜m, +
† c˜m,
hl,mi,
J
~l · S ~m S
energy increase:
-16,5
Espin
X
hl,mi
-16
~l · S ~l+1 i| Espin ⇡ |hS
-17 -17,5 -18
ˆ spin c˜l, ) + H
0
2
4
6
8
10
a)
b)
time 0,09
M
2
0,08
drastic magnetization reduction
0,07 0,06 0,05
0
2
4
time
6
8
10
1D
Effect of holes experimental observable: squared staggered magnetization 1 = 2 N
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,8 0,6 0,4
2
m (L=42)
2 Mstag
N X
0,2 0
1D
2 holes on 84 sites 4 holes on 86 sites 10 20 30 40 50
ramping time T
Effect of holes 1D
experimental observable: squared staggered magnetization
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,8 0,6
2
0,4 0,2 0
20
T(m =0.85)
1 = 2 N
2
m (L=42)
2 Mstag
N X
2 holes on 84 sites 4 holes on 86 sites 10 20 30 40 50
ramping time T
hole arrival
15 10 5 0
10
15
20
25
sublattice size L
30
Effect of holes 1D
experimental observable: squared staggered magnetization
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM 20
0,8 0,6
2
0,4 0,2 0
T(m =0.85)
1 = 2 N
2
m (L=42)
2 Mstag
N X
2 holes on 84 sites 4 holes on 86 sites 10 20 30 40 50
ramping time T
hole arrival
15 10 5 0
10
15
20
25
sublattice size L
drastic magnetization reduction
30
Harmonic trap
Harmonic trap ˆ =H ˆ kin + H ˆ spin + V H
X
(l
2
l0 ) n ˆl
l
=2
=1
=0
=3 =4 =5 =6 =7 =8
0 0,1
=9
=6 =7
0 0,1
=10
0 0,1
=8
0 0,2
0 0,1
=10
=1 =2 =3 =4
0 0,2
=9
=5
0 0,2
0 0,1
1 0,5 0 0,4 0,2 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1
=0
1 0,5 0 0,4 0,2 0 0,2
lattice site l
lattice site l
no trap
weak trap
1D
Harmonic trap ˆ =H ˆ kin + H ˆ spin + V H
X
(l
2
l0 ) n ˆl
l
=2
=1
=0
=3 =4 =5 =6 =7 =8
0 0,1
=9
=6 =7
0 0,1
=10
0 0,1
=8
0 0,2
0 0,1
=10
=1 =2 =3 =4
0 0,2
=9
=5
0 0,2
0 0,1
1 0,5 0 0,4 0,2 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1
=0
1 0,5 0 0,4 0,2 0 0,2
lattice site l
lattice site l
no trap
weak trap
1D
Harmonic trap ˆ =H ˆ kin + H ˆ spin + V H
X
(l
2
l0 ) n ˆl
l
=2
=1
=0
=3 =4 =5 =6 =7 =8
0 0,1
=9
=6 =7
0 0,1
=10
0 0,1
=8
0 0,2
0 0,1
=10
=1 =2 =3 =4
0 0,2
=9
=5
0 0,2
0 0,1
1 0,5 0 0,4 0,2 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1 0 0,2 0,1
=0
1 0,5 0 0,4 0,2 0 0,2
lattice site l
lattice site l
no trap
weak trap
1D
Harmonic trap experimental observable: squared staggered magnetization
2
m (L=82)
2 Mstag
1 = 2 N
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,6 0,4 0,2 0
10
20
1D
V=0.004J V=0.006J V=0.02J 30 40 50
ramping time T
Harmonic trap 1D
experimental observable: squared staggered magnetization 2 Mstag
1 = 2 N
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,4 0,2 0
10
20
V=0.004J V=0.006J V=0.02J 30 40 50
ramping time T
2
0,6
T(m =0.85)
2
m (L=82)
50 40 30 20 10 0
10
20
30
40
sublattice size L
50
Harmonic trap 1D
experimental observable: squared staggered magnetization 2 Mstag
1 = 2 N
N X
l,m=1
~l · S ~m i : m2 (T ) := M 2 (T )/M 2 ( 1)l+m hS stag stag,AFM
0,4 0,2 0
10
20
V=0.004J V=0.006J V=0.02J 30 40 50
nl(T=50)
ramping time T
2
0,6
T(m =0.85)
2
m (L=82)
50 40 30 20 10 0
10
20
30
40
sublattice size L
50
1 0,5
strong trap: hole-free case lattice site l
Conclusions
Conclusions
feasible timescales
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Conclusions
feasible timescales sublattice adiabaticity: governed only by its size
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
3.
|AFM> V2
Conclusions
feasible timescales sublattice adiabaticity: governed only by its size
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
destructive effect of holes: controlled by harmonic trap
3.
|AFM> V2
Conclusions
feasible timescales sublattice adiabaticity: governed only by its size
1.
|BI> V1
2. | >
| >
| >
| >
| > V2
destructive effect of holes: controlled by harmonic trap
3.
|AFM>
details: [M. Lubasch, V. Murg, U. Schneider, J. I. Cirac, M.-C. Bañuls, PRL 107, 165301 (2011)]
V2
