Research talk presented at QICP2. I talk about equilibration and thermalization of closed quantum systems, the different timescales of two-point correlations functions and the evolution of different measures of entanglement for a quantum lattice system with long-range interactions.

1. 1 / 10
Relaxation timescales, decay of correlattions and
multipartite entanglement in a long-range interacting
quantum simulator
Mauritz van den Worm
National Institute of Theoretical Physics
Stellenbosch University
QICP2

2. | Introductory words 2 / 10

3. | Introductory words 2 / 10

4. | Introductory words 2 / 10

5. | Introductory words 2 / 10

6. | Introductory words 2 / 10

7. | Introductory words 2 / 10

8. | Introductory words 2 / 10

9. | Introductory words 2 / 10
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 / 10
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. | Exact analytic results 3 / 10

12. | Exact analytic results 4 / 10
Long-Range Ising: Time evolution of expectation values

13. | Exact analytic results 4 / 10
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

14. | Exact analytic results 4 / 10
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|

15. | Exact analytic results 4 / 10
Long-Range Ising: Time evolution of expectation values
σx
i (t) = σx
i (0)
j=i
cos
2t
|i − j|α
σy
i σz
j (t) = σx
i (0) sin (2tJi,j )
k=i,j
cos (2tJk,i )
σx
i σx
j (t) = P−
i,j + P+
i,j
σy
i σy
j (t) = P−
i,j − P+
i,j
P±
i,j =
1
2
σx
i σx
j (0)
k=i,j
cos 2t
1
|i − k|α
±
1
|j − k|α

16. | Exact analytic results 4 / 10
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. | What is being done experimentally? 5 / 10
Trapped Ion Experiments
Long-range Ising Hamiltonian
H = −
i<j
Ji,j σz
i σz
j − Bµ ·
i
σi

18. | What is being done experimentally? 5 / 10
Trapped Ion Experiments
Long-range Ising Hamiltonian
H = −
i<j
Ji,j σz
i σz
j − Bµ ·
i
σi
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.

19. | Exact analytic results 6 / 10

20. | Exact analytic results 7 / 10
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

21. | Exact analytic results 7 / 10
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

22. | Exact analytic results 7 / 10
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

23. | Exact analytic results 7 / 10
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

24. | Exact analytic results 7 / 10
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

25. | Exact analytic results 8 / 10

26. | Exact analytic results 8 / 10
S [ρi (t)] = −Tr [ρi (t) log2 ρi (t)]

27. | Exact analytic results 8 / 10
α = 0.75 θ = π/2
0
Π
4
Π
2
3 Π
4
Π
0.0
0.2
0.4
0.6
0.8
1.0
Θ
t
0.2
0.6
1
1.4
1.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Α
t
0.2
0.6
1
1.4
1.8
Figure: Left: von Neumann entanglement in the (θ, t)-plane. Notice different
saturation levels for different tipping angles. Right: von Neumann entanglement
in the (α, t)-plane. Notice the three distinct regions.

28. | Exact analytic results 9 / 10

29. | Exact analytic results 9 / 10

30. | Exact analytic results 9 / 10
Squeezing Parameter
ξ(t) =
√
N min
ψ
∆ S · ˆnψ
S (t)
, S =
i∈Λ
σx
i , σy
i , σz
i

31. | Exact analytic results 9 / 10
α = 0.75 θ = π/2
0
Π
4
Π
2
3 Π
4
Π
0.0
0.1
0.2
0.3
0.4
0.5
Θ
t
0.25
0.75
1.25
1.75
2.25
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.1
0.2
0.3
0.4
0.5
Α
t
0.5
1.5
2.5
3.5
Figure: Left: Decibell spin squeezing −10 log10 ξ(t) in the (θ, t)-plane. Right:
Decibell spin Squeezing in the (α, t)-plane

32. | Collaborators 10 / 10
Collaborators
Michael Kastner
Supervisor
John Bollinger
NIST
Boulder, Colorado
Brian Sawyer
NIST
Boulder, Colorado
Ana Maria Rey
JILA
Boulder, Colorado
Kaden Hazzard
JILA
Boulder, Colorado
Michael Foss-Feig
JQI
Gaithersburg, Maryland