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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.
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1 / 10
Relaxation timescales, decay of correlattions andmultipartite entanglement in a long-range interacting
quantum simulator
Mauritz van den Worm
National Institute of Theoretical Physics
Stellenbosch University
QICP2
| Introductory words 2 / 10
| Introductory words 2 / 10
| Introductory words 2 / 10
| Introductory words 2 / 10
| Introductory words 2 / 10
| Introductory words 2 / 10
| Introductory words 2 / 10
| Introductory words 2 / 10
What do we use to study this?
limt→∞
1
t
∫ t
0〈A〉 (τ)dτ = lim
t→∞
1
t
∫ t
0
⟨e−iHtAe iHt
⟩dτ
| Introductory words 2 / 10
What do we use to study this?
limt→∞
1
t
∫ t
0〈A〉 (τ)dτ = lim
t→∞
1
t
∫ t
0
⟨e−iHtAe iHt
⟩dτ
A system is said to thermalize if
limt→∞
1
t
∫ t
0〈A〉 (τ)dτ =
1
ZTr[Ae−βH
]
| Exact analytic results 3 / 10
| Exact analytic results 4 / 10
Long-Range Ising: Time evolution of expectation values
Graphical Representation of Correlation Functions
XΣ0x\HtL
XΣ-1x Σ1
x\HtLXΣ-1
yΣ1
y\HtLXΣ-1
yΣ1
z \HtL
Α = 0.4
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
XΣiaΣ j
b\HtL
Figure: Time evolution of the normalized spin-spin correlators. The respectivegraphs were calculated for N = 102, 103 and 104. Notice the presence of thepre-thermalization plateaus of the two spin correlators.
| Exact analytic results 4 / 10
Long-Range Ising: Time evolution of expectation values
Ingredients
D dimensional lattice Λ
H =⊗
j∈Λ C2j
Ji ,j = |i − j |−α
Long-range Ising Hamiltonian
H = −∑
(i ,j)∈Λ×Λ
Ji ,jσzi σ
zj − B
∑i∈Λ
σzi
Graphical Representation of Correlation Functions
XΣ0x\HtL
XΣ-1x Σ1
x\HtLXΣ-1
yΣ1
y\HtLXΣ-1
yΣ1
z \HtL
Α = 0.4
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
XΣiaΣ j
b\HtL
Figure: Time evolution of the normalized spin-spin correlators. The respectivegraphs were calculated for N = 102, 103 and 104. Notice the presence of thepre-thermalization plateaus of the two spin correlators.
| 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}
Ra1,··· ,a|Λ|i1,··· ,i|Λ| σ
a1i1· · ·σa|Λ|i|Λ|
Graphical Representation of Correlation Functions
XΣ0x\HtL
XΣ-1x Σ1
x\HtLXΣ-1
yΣ1
y\HtLXΣ-1
yΣ1
z \HtL
Α = 0.4
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
XΣiaΣ j
b\HtL
Figure: Time evolution of the normalized spin-spin correlators. The respectivegraphs were calculated for N = 102, 103 and 104. Notice the presence of thepre-thermalization plateaus of the two spin correlators.
| Exact analytic results 4 / 10
Long-Range Ising: Time evolution of expectation values
〈σxi 〉(t) = 〈σxi 〉(0)∏j 6=i
cos
(2t
|i − j |α
)〈σyi σ
zj 〉(t) = 〈σxi 〉(0) sin (2tJi ,j)
∏k 6=i ,j
cos (2tJk,i )
〈σxi σxj 〉(t) = P−i ,j + P+i ,j
〈σyi σyj 〉(t) = P−i ,j − P+
i ,j
P±i ,j =1
2〈σxi σxj 〉(0)
∏k 6=i ,j
cos
[2t
(1
|i − k |α± 1
|j − k|α
)]
Graphical Representation of Correlation Functions
XΣ0x\HtL
XΣ-1x Σ1
x\HtLXΣ-1
yΣ1
y\HtLXΣ-1
yΣ1
z \HtL
Α = 0.4
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
XΣiaΣ j
b\HtL
Figure: Time evolution of the normalized spin-spin correlators. The respectivegraphs were calculated for N = 102, 103 and 104. Notice the presence of thepre-thermalization plateaus of the two spin correlators.
| Exact analytic results 4 / 10
Long-Range Ising: Time evolution of expectation valuesGraphical Representation of Correlation Functions
XΣ0x\HtL
XΣ-1x Σ1
x\HtLXΣ-1
yΣ1
y\HtLXΣ-1
yΣ1
z \HtL
Α = 0.4
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
XΣiaΣ j
b\HtL
Figure: Time evolution of the normalized spin-spin correlators. The respectivegraphs were calculated for N = 102, 103 and 104. Notice the presence of thepre-thermalization plateaus of the two spin correlators.
| What is being done experimentally? 5 / 10
Trapped Ion Experiments
Long-range Ising Hamiltonian
H = −∑i<j
Ji ,jσzi σ
zj − Bµ ·
∑i
σi
Graphical Representation of Correlation Functions
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 0.25
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 1.5
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
(a) (b)
Figure: Time evolution of the normalized spin-spin correlations. Curves of thesame color correspond to different side lengths L = 4, 8, 16 and 32 (from rightto left) of the hexagonal patches of lattices. In figure (a) α = 1/4, results aresimilar for all 0 ≤ α < ν/2. In figure (b) α = 3/2, with similar results for allα > ν/2.
| What is being done experimentally? 5 / 10
Trapped Ion Experiments
Long-range Ising Hamiltonian
H = −∑i<j
Ji ,jσzi σ
zj − Bµ ·
∑i
σi
Graphical Representation of Correlation Functions
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 0.25
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
YΣix]
YΣiy
Σ jz]
YΣiy
Σ jy]
YΣix
Σ jx]
Α = 1.5
0.01 0.1 1 10t
0.2
0.4
0.6
0.8
1.0
(a) (b)
Figure: Time evolution of the normalized spin-spin correlations. Curves of thesame color correspond to different side lengths L = 4, 8, 16 and 32 (from rightto left) of the hexagonal patches of lattices. In figure (a) α = 1/4, results aresimilar for all 0 ≤ α < ν/2. In figure (b) α = 3/2, with similar results for allα > ν/2.
| Exact analytic results 6 / 10
| Exact analytic results 7 / 10
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0)〉 =⊗j∈Λ
[cos
(θj2
)e iφj/2| ↑〉j + sin
(θj2e−iφj/2
)| ↓〉j
]
| Exact analytic results 7 / 10
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0)〉 =⊗j∈Λ
[cos
(θj2
)e iφj/2| ↑〉j + sin
(θj2e−iφj/2
)| ↓〉j
]
| Exact analytic results 7 / 10
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0)〉 =⊗j∈Λ
[cos
(θj2
)e iφj/2| ↑〉j + sin
(θj2e−iφj/2
)| ↓〉j
]
| Exact analytic results 7 / 10
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0)〉 =⊗j∈Λ
[cos
(θj2
)e iφj/2| ↑〉j + sin
(θj2e−iφj/2
)| ↓〉j
]
| Exact analytic results 7 / 10
Long-Range Ising: Time evolution of expectation values
Product Initial States
|ψ(0)〉 =⊗j∈Λ
[cos
(θj2
)e iφj/2| ↑〉j + sin
(θj2e−iφj/2
)| ↓〉j
]
| Exact analytic results 8 / 10
α = 0.75 θ = π/2
0Π
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 differentsaturation levels for different tipping angles. Right: von Neumann entanglementin the (α, t)-plane. Notice the three distinct regions.
| Exact analytic results 8 / 10
S [ρi (t)] = −Tr [ρi (t) log2 ρi (t)]
α = 0.75 θ = π/2
0Π
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 differentsaturation levels for different tipping angles. Right: von Neumann entanglementin the (α, t)-plane. Notice the three distinct regions.
| Exact analytic results 8 / 10
α = 0.75 θ = π/2
0Π
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 differentsaturation levels for different tipping angles. Right: von Neumann entanglementin the (α, t)-plane. Notice the three distinct regions.
| Exact analytic results 9 / 10
α = 0.75 θ = π/2
0Π
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
| Exact analytic results 9 / 10
α = 0.75 θ = π/2
0Π
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
| Exact analytic results 9 / 10
Squeezing Parameter
ξ(t) =√N min
ψ
∆(S · n̂ψ
)∣∣〈S〉(t)∣∣ , S =
∑i∈Λ
(σxi , σ
yi , σ
zi
)
α = 0.75 θ = π/2
0Π
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
| Exact analytic results 9 / 10
α = 0.75 θ = π/2
0Π
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
| Collaborators 10 / 10
Collaborators
Michael KastnerSupervisor
John BollingerNIST
Boulder, Colorado
Brian SawyerNIST
Boulder, Colorado
Ana Maria ReyJILA
Boulder, Colorado
Kaden HazzardJILA
Boulder, Colorado
Michael Foss-FeigJQI
Gaithersburg, Maryland
