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.
Quantum correlations and entanglement in far-from-equilibrium spin systems
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
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
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.
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
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