Relaxation Timescales and Decay of Correlations in a Long-Range Interacting Quantum SimulatorMauritz van den Worm1, Brian C. Sawyer2, John J. Bollinger 2 and Michael Kastner 1,3

1Institute of Theoretical PhysicsStellenbosch University

2US National Institute of Standards and TechnologyTime and Frequency Division, Boulder, Colorado

3National Institute for Theoretical PhysicsStellenbosch

Long-Range Ising Model

• Lattice Λ, dim(Λ) = ν <∞• C2

i attached at each i ∈ Λ• Dynamics occurs on H = ⊗

i∈Λ C2i

H = Nα

∑(i,j)∈Λ×Λ

Ji,jσzi σ

zj − h

∑i∈Λ

σzi ,

• coupling constant Ji,j := 1|i−j|α, where α ≥ 0

Equilibration of Long-Range Ising Model

〈A〉(t) = Tr[αΛt (A) ρ(0)

], A =

∑i

aiσxi

αΛt (A) := eiHtAe−iHt

ρ(0) = initial density matrixSimplest method: make use of

σ±i = 12

(σxi ± iσyi )

Time Evolution

Contribution from magnetic terms:

exp

−ih∑j∈Λ

σzj

σ±i exp

ih∑k∈Λ

σzk

= σ±i exp [∓2iht]

Magnetic terms play no role in the equilibration.Contribution from interacting terms:

exp

itNα ∑(j,k)∈Λ×Λ

Jj,kσzjσ

zk

σ±i exp

−itNα ∑(l,m)∈Λ×Λ

Jl,mσzl σ

zm

= σ±i exp

2iNαt∑k∈Λ\i

Jk,iσzk

Class of Orthogonal Initial States

Define the tripleA := (A1, A2, A3)

where the Ai’s are mutually disjoint finite subsets of the lat-tice Λ. Define the operator

σA :=

∏i∈A1

σxi

∏j∈A2

σyj

∏k∈A3

σzk

.We say that ρ(0) is in the class of orthogonal initial states iffor any triple A = (A1, A2, A3) with A3 6= ∅ we have

Tr[σAρ(0)

]= 0.

Analytic Results [2, 3] (σx-Polarized)

〈σxi 〉(t) = 〈σxi 〉(0) cos (2ht)∏k∈Λ\i

cos (2NαtJk,i)

〈σyi 〉(t) = 〈σxi 〉(0) sin (2ht)∏k∈Λ\i

cos (2NαtJk,i)

〈σxi σxj 〉(t) = P−i,j + cos (4ht)P+i,j

〈σyi σyj 〉(t) = P−i,j − cos (4ht)P+

i,j

〈σxi σyj 〉(t) = − sin (4ht)P+

i,j

〈σxi σzj 〉(t) = sin (2ht)P zi,j

〈σyi σzj 〉(t) = cos (2ht)P zi,j

Where we have definedP±i,j :=1

2〈σxi σ

xj 〉(0)

∏k∈Λ\{i,j}i,j

cos[2Nαt

(Ji,k ± Jj,k

)]P zi,j :=− 〈σxi 〉(0) sin

(2NαJi,j

) ∏k∈Λ\{i,j}

cos(2NαtJi,k

).

Britton et al. [1]

(a) (b) (c)

H = −∑i<j

Ji,jσzi σ

zj −Bµ ·

∑i

σi

• Expressed i.t.o. the transverse phonon eigenfunctions• Numerical evaluation shows Ji,j ∝ D−αi,j• Tune 0 ≤ α ≤ 3

Application to Trapped-Ion QuantumSimulator

After upgrades to the current system the group at NISTshould be able to handle:•N = 217 ions• optical dipole force generated from two 10 mW beamscrossing with an angular separation of 35◦

• frequency difference tuned to generate an α = 1/2 powerlaw interaction

Timescales:• Relaxation timescales will be

P+i,j ≈ 30 µs, P−i,j ≈ 430 µs

• Spontaneous emission time is ∼ 4 ms• Timescales short compared to the T2 & 50 ms coherencetime of the Be+ valence electron spin qubits.

Correlators on Finite Hexagonal Patch of Triangular Lattice

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

Figure: Time evolution of the normalized spin-spin correlations 〈σxi σxj 〉(t)/〈σxi σxj 〉(0) (blue), 〈σyi σyj 〉(t)/〈σ

xi σ

xj 〉(0) (purple), 〈σyi σzj 〉(t)/〈σxi 〉(0) (red) and

〈σxi 〉(t)/〈σxi 〉(0) (green). The lattice sites i and j are chosen one lattice site to the right, respectively left, of the center of the hexagonal patch, as indicated by

die blue dots in figure 1. The various curves of the same color in each plot correspond to different side lengths L = 4, 8, 16 and 32 (from right to left) of thehexagonal patches of lattices. Figure (a) is for power law interactions with an exponent α = 1/4, but results are qualitatively similar for all α between zero andν/2, where ν is the dimension of the lattice. Figure (b) is for α = 3/2, with qualitatively similar results for all α > ν/2.

Γ2 HtL

Γ1 HtL

Α = 0.25

0.01 0.1 1 10t

0.2

0.4

0.6

0.8

1.0

ΓnHtL Mechanism Responsible forPrethermalization PlateausDefine the n−spin purity

γn(t) := Tr [ρn(t)]where ρn is the n−spin reduced density matrix.Figure: The various curves of the same colour show the single and spin-spinpurities corresponding to side lengths L = 4, 8 and 16 (from right to left) ofthe hexagonal patches of triangular lattices. Each relaxation step inspinâĂŞspin correlations is accompanied by a drop in γn(t). This suggeststhat dephasing is responsible for all relaxation steps.

Formation of Entanglement

S [ρn(t)] = −Tr [ρn(t) log2 ρn(t)]

N=16

N=8

N=4

Α = 0.25

0.01 0.1 1 10t

0.2

0.4

0.6

0.8

1.0

SHF1L

N=16

N=8

N=4

Α = 0.25

0.01 0.1 1 10t

0.5

1.0

1.5

2.0

SHF2L

Figure: The figures respectively show the von Neumann entropies of the single and two spin reduced density matrices of the long-range Ising model on a finitehexagonal patch of a triangular lattice. Both figures show that if we start from an initially unentangled state entropy is created in time. The formation ofentanglement occurs on a faster timescale for lattices with increasing side-lengths N .

References

[1] Britton et al, Nature 484 (2012), no. 7395, 489–92.[2] Michael Kastner, PRL 106 (2011), no. 13.[3] Mauritz van den Worm et al, arXiv:1209:3697v2, To

appear in NJP - August 2013.

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