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

1 / 11

Quantum correlations and entanglement infar-from-equilibrium spin systems

Mauritz van den Worm

National Institute of Theoretical Physics

Stellenbosch University

NITheP Bursars Workshop

| Introductory words 2 / 11








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


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

]


Interaction satisfies:

Ji ,j ∝ r−α

0 < α < dim(System)

Gravitating Masses Coulomb Interactions (no screening)

| Exact analytic results 3 / 11


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



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|


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.


Light-Cones and Lieb-Robinson Bounds



‖ [OA(t),OB (0)] ‖ ≤ K exp

[v|t| − d(A, B)

ξ

] ~v t

x

t

Short-Range



‖ [OA(t),OB (0)] ‖ ≤ Kev|t| − 1

[d(A, B) + 1]D−α

~ ln x

x

t

Long-Range



Ρ Π B

TrL� B @e- iHt UA ΡUA†

e iHt D

TrL� B @e- itH ΡeiHt D

0 t

Tt

N t



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


Light-Cones and Lieb-Robinson BoundsExact results for Ising

H =1

2

∑i 6=j

1

|i − j |α σzi σ

zj , 〈σx

i σxj 〉(t)− 〈σx

i 〉(t)〈σxj 〉(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 〈σx0σ

xδ 〉c(t) in the

(δ, t)-plane for long-range Ising chains with |Λ| = 1001 sites and three differentvalues of α. Dark colors indicate small values, and initial correlations at t = 0are vanishing.


Light-Cones and Lieb-Robinson BoundstDMRG results for XXZ

H =1

2

∑i 6=j

1

|i − j |α

[Jzσz

i σzj +

J⊥

2

(σ+i σ−j + σ+

j σ−i

)], 〈σz

0σzδ〉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 ∆

| What is being done experimentally? 6 / 11


Current state of the art

H =1

2

∑i 6=j

[JxijS

xi S

xj + Jy

ij Syj S

yi + Jz

ijSzi S

zj

][To appear in PRA, Arxiv:1406.0937 - Kaden Hazzard, MVDW, Michael Foss-Feig, et al.]


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.


Ising XXZ




Product Initial States

|ψ(0)〉 =⊗j∈Λ

[cos

(θj2

)e iφj/2| ↑〉j + sin

(θj2e−iφj/2

)| ↓〉j

]




|ψ(0)〉 =⊗j∈Λ

[cos

(θj2


(θj2e−iφj/2

)| ↓〉j

]




|ψ(0)〉 =⊗j∈Λ

[cos

(θj2


(θj2e−iφj/2

)| ↓〉j

]




|ψ(0)〉 =⊗j∈Λ

[cos

(θj2


(θj2e−iφj/2

)| ↓〉j

]




|ψ(0)〉 =⊗j∈Λ

[cos

(θj2


(θj2e−iφj/2

)| ↓〉j

]


dB spin squeezing entanglement entropy concurrenceϕ

=π/2

a

0.0 0.5 1.0 1.5 2.00

2

4

6

8

Α

t

b

0.0 0.5 1.0 1.5 2.00

2

4

6

8

Α

t

c

0.0 0.5 1.0 1.5 2.00

2

4

6

8

Α

t

α=

3/4

d

0Π

4

Π

2

3 Π

4Π

0

2

4

6

8

j

t

e

0Π

4

Π

2

3 Π

4Π

0

2

4

6

8

j

tf

0Π

4

Π

2

3 Π

4Π

0

2

4

6

8

jt

| Take home message 10 / 11

Take home message

In long-range systems...

Relaxation process might include long-livedquasi-stationary states

Information can propagate instantaneously ifinteraction range is long-enough

Lieb-Robinson bounds greatly overestimatemaximum group velocities

Different types of entangled states can be created

| Collaborators 11 / 11

Collaborators

Michael KastnerSupervisor

John BollingerNIST

Boulder, Colorado

Brian SawyerNIST

Boulder, Colorado

Emanuele Dalla TorreBar Ilan UniversityTel Aviv, Isreal

Tilman PfauUniversitat StuttgartStuttgart, Germany

Ana Maria ReyJILA

Boulder, Colorado

Kaden HazzardJILA

Boulder, Colorado

Michael Foss-FeigJQI

Gaithersburg, Maryland

Salvatorre ManmanaUniversity of GottingenGottingen, Germanay

Jens EisertFreie UniversitatBerlin, Germany

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

Science

t x j t

x j t x

j j i

j b t figure

j jz z i z j j

j z i z j b i z

b v t x t short range

cos t jb