This document discusses quantum correlations and entanglement in far-from-equilibrium spin systems. It summarizes research on long-range interacting spin models, including exact results showing the time evolution of correlations. It also discusses light cones and Lieb-Robinson bounds, as well as current experimental work on simulating long-range interacting spin Hamiltonians using trapped ions or Rydberg atoms. The key takeaway is that relaxation in long-range systems may involve long-lived quasi-stationary states, information can propagate instantaneously over long ranges, and different types of entangled states can be created.

1. 1 / 11
Quantum correlations and entanglement in
far-from-equilibrium spin systems
Mauritz van den Worm
National Institute of Theoretical Physics
Stellenbosch University
NITheP Bursars Workshop

2. | Introductory words 2 / 11

3. | Introductory words 2 / 11

4. | Introductory words 2 / 11

5. | Introductory words 2 / 11

6. | Introductory words 2 / 11

7. | Introductory words 2 / 11

8. | Introductory words 2 / 11

9. | Introductory words 2 / 11
What do we use to study this?
lim
t→∞
1
t
t
0
A (τ)dτ = lim
t→∞
1
t
t
0
e−iHt
AeiHt
dτ

10. | Introductory words 2 / 11
What do we use to study this?
lim
t→∞
1
t
t
0
A (τ)dτ = lim
t→∞
1
t
t
0
e−iHt
AeiHt
dτ
A system is said to thermalize if
lim
t→∞
1
t
t
0
A (τ)dτ =
1
Z
Tr Ae−βH

11. | Introductory words 2 / 11
Interaction satisfies:
Ji,j ∝ r−α
0 < α < dim(System)
Gravitating Masses Coulomb Interactions (no screening)

12. | Exact analytic results 3 / 11

13. | Exact analytic results 4 / 11
Long-Range Ising: Time evolution of expectation values

14. | Exact analytic results 4 / 11
Long-Range Ising: Time evolution of expectation values
Ingredients
D dimensional lattice Λ
H = j∈Λ C2
j
Ji,j = |i − j|−α
Long-range Ising Hamiltonian
H = −
(i,j)∈Λ×Λ
Ji,j σz
i σz
j − B
i∈Λ
σz
i

15. | Exact analytic results 4 / 11
Long-Range Ising: Time evolution of expectation values
Orthogonal Initial States
ρ(0) =
i1,··· ,i|Λ|
∈Λ
a1,··· ,a|Λ|
∈{0,x,y}
R
a1,··· ,a|Λ|
i1,··· ,i|Λ|
σa1
i1
· · · σ
a|Λ|
i|Λ|

16. | Exact analytic results 4 / 11
Long-Range Ising: Time evolution of expectation valuesGraphical Representation of Correlation Functions
Σ0
x
t
Σ 1
x
Σ1
x
t
Σ 1
y
Σ1
y
t
Σ 1
y
Σ1
z
t
Α 0.4
0.01 0.1 1 10
t
0.2
0.4
0.6
0.8
1.0
Σi
a
Σj
b
t
Figure : Time evolution of the normalized spin-spin correlators. The respective
graphs were calculated for N = 102
, 103
and 104
. Notice the presence of the
pre-thermalization plateaus of the two spin correlators.

17. | Exact analytic results 5 / 11
Light-Cones and Lieb-Robinson Bounds

18. | Exact analytic results 5 / 11
Light-Cones and Lieb-Robinson Bounds
[OA(t), OB (0)] ≤ K exp
v|t| − d(A, B)
ξ
v t
x
t
Short Range

19. | Exact analytic results 5 / 11
Light-Cones and Lieb-Robinson Bounds
[OA(t), OB (0)] ≤ K
ev|t|
− 1
[d(A, B) + 1]D−α
ln x
x
t
Long Range

20. | Exact analytic results 5 / 11
Light-Cones and Lieb-Robinson Bounds
Ρ ΠB
Tr B e iHt
UA ΡUA
†
eiHt
Tr B e itH
ΡeiHt
0 t
Tt
Nt

21. | Exact analytic results 5 / 11
Light-Cones and Lieb-Robinson Bounds
Product Initial State
pt ≥ 1 − exp



−
4t2
5
j∈B
[1 + d (A, j)]−2α



Entangled Initial State
pt ≥ 1 −
1
2



1 + cos

t
j∈B
[1 + d (A, j)]−α






22. | Exact analytic results 5 / 11
Light-Cones and Lieb-Robinson BoundsExact results for Ising
H =
1
2
i=j
1
|i − j|α
σz
i σz
j , σx
i σx
j (t) − σx
i (t) σx
j (t)
α = 1/4 α = 3/4 α = 3/2
0 50 100 150
0.00
0.02
0.04
0.06
0.08
0.10
∆
t
0 50 100 150
0.00
0.05
0.10
0.15
0.20
∆
20 40 60 80
0.0
0.1
0.2
0.3
0.4
∆
Figure : Density contour plots of the connected correlator σx
0 σx
δ c (t) in the
(δ, t)-plane for long-range Ising chains with |Λ| = 1001 sites and three different
values of α. Dark colors indicate small values, and initial correlations at t = 0
are vanishing.

23. | Exact analytic results 5 / 11
Light-Cones and Lieb-Robinson BoundstDMRG results for XXZ
H =
1
2
i=j
1
|i − j|α
Jz
σz
i σz
j +
J⊥
2
σ+
i σ−
j + σ+
j σ−
i , σz
0 σz
δ c (t)
α = 3/4 α = 3/2 α = 3
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
∆
t
0 5 10 15 20
0.0
0.5
1.0
1.5
2.0
∆
t
0 5 10 15 20
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
∆
t
0.0 0.5 1.0 1.5 2.0 2.5
5
4
3
2
1
0
ln ∆
lnt
0.0 0.5 1.0 1.5 2.0 2.5
5
4
3
2
1
0
ln ∆
0.0 0.5 1.0 1.5 2.0 2.5
5
4
3
2
1
0
1
ln ∆

24. | What is being done experimentally? 6 / 11

25. | What is being done experimentally? 6 / 11
Current state of the art
H =
1
2
i=j
Jx
ij Sx
i Sx
j + Jy
ij Sy
j Sy
i + Jz
ij Sz
i Sz
j
[To appear in PRA, Arxiv:1406.0937 - Kaden Hazzard, MVDW, Michael Foss-Feig, et al.]

26. | What is being done experimentally? 6 / 11
Long-range Ising Hamiltonian
H =
i<j
Ji,j σz
i σz
j − Bµ ·
i
σi

27. | What is being done experimentally? 6 / 11
Graphical Representation of Correlation Functions
Σi
x
Σi
y
Σj
z
Σi
y
Σj
y
Σi
x
Σj
x
Α 0.25
0.01 0.1 1 10
t
0.2
0.4
0.6
0.8
1.0
Σi
x
Σi
y
Σj
z
Σi
y
Σj
y
Σi
x
Σj
x
Α 1.5
0.01 0.1 1 10
t
0.2
0.4
0.6
0.8
1.0
(a) (b)
Figure : Time evolution of the normalized spin-spin correlations. Curves of the
same color correspond to different side lengths L = 4, 8, 16 and 32 (from right
to left) of the hexagonal patches of lattices. In figure (a) α = 1/4, results are
similar for all 0 ≤ α < ν/2. In figure (b) α = 3/2, with similar results for all
α > ν/2.

28. | What is being done experimentally? 6 / 11
Ising XXZ

29. | Exact analytic results 7 / 11

30. | Exact analytic results 8 / 11
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0) =
j∈Λ
cos
θj
2
eiφj /2
| ↑ j + sin
θj
2
e−iφj /2
| ↓ j

31. | Exact analytic results 8 / 11
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0) =
j∈Λ
cos
θj
2
eiφj /2
| ↑ j + sin
θj
2
e−iφj /2
| ↓ j

32. | Exact analytic results 8 / 11
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0) =
j∈Λ
cos
θj
2
eiφj /2
| ↑ j + sin
θj
2
e−iφj /2
| ↓ j

33. | Exact analytic results 8 / 11
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0) =
j∈Λ
cos
θj
2
eiφj /2
| ↑ j + sin
θj
2
e−iφj /2
| ↓ j

34. | Exact analytic results 8 / 11
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0) =
j∈Λ
cos
θj
2
eiφj /2
| ↑ j + sin
θj
2
e−iφj /2
| ↓ j

35. | Exact analytic results 9 / 11
dB spin squeezing entanglement entropy concurrence
ϕ=π/2
a
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
Α
t
b
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
Α
t
c
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
Α
t
α=3/4
d
0 Π
4
Π
2
3 Π
4
Π
0
2
4
6
8
t
e
0 Π
4
Π
2
3 Π
4
Π
0
2
4
6
8
t
f
0 Π
4
Π
2
3 Π
4
Π
0
2
4
6
8
t

36. | Take home message 10 / 11
Take home message
In long-range systems...
Relaxation process might include long-lived
quasi-stationary states
Information can propagate instantaneously if
interaction range is long-enough
Lieb-Robinson bounds greatly overestimate
maximum group velocities
Different types of entangled states can be created

37. | Collaborators 11 / 11
Collaborators
Michael Kastner
Supervisor
John Bollinger
NIST
Boulder, Colorado
Brian Sawyer
NIST
Boulder, Colorado
Emanuele Dalla Torre
Bar Ilan University
Tel Aviv, Isreal
Tilman Pfau
Universit¨at Stuttgart
Stuttgart, Germany
Ana Maria Rey
JILA
Boulder, Colorado
Kaden Hazzard
JILA
Boulder, Colorado
Michael Foss-Feig
JQI
Gaithersburg, Maryland
Salvatorre Manmana
University of G¨ottingen
G¨ottingen, Germanay
Jens Eisert
Freie Universit¨at
Berlin, Germany