Accurate Rydberg quantum simulations of spin-1/2 models · 2017. 11. 22. · Accurate Rydberg quantum simulations of spin-1/2 models ICQSIM — Paris — November 16, 2017 Theory

Accurate Rydberg quantum simulations of spin-1/2 models

ICQSIM — Paris — November 16, 2017

Theory Sebastian Weber, Hans Peter Büchler (University of Stuttgart)

Experiment Sylvain De Léséleuc, Vincent Lienhard, Daniel Barredo, Thierry Lahaye, Antoine Browaeys (Université Paris-Saclay)

Sebastian Weber Accurate Rydberg quantum simulations

Rydberg quantum simulations of spin Hamiltonians

1

• Strong interactions

I) dipole-dipole ~ n4/R3 II) van der Waals ~ n11/R6, anisotropy possible III) cutoff potentials via Rydberg dressing

• Long radiative lifetime ~ n3

• Defect-free arbitrary atom arrays experimentally realized

• Requirement: accurate mapping of multilevel Rydberg atoms to spins with just a few levels

n ≫ 1

Basis size for 49 spin-1/2 particles: 249 ≈ 6 ⋅ 1014

➡direct numerical simulations intractable

H. K

im e

t al.,

N

at. C

omm

un. 7

, 13

317

(201

6)

D. B

arre

do e

t al.,

Sc

ienc

e 35

4,

6315

(201

6)

M. Endres et al., Science 354, 6315 (2016)


Example: Ising-like system of spin-1/2 particles

2

We want to map the Rydberg atoms to spin-1/2 particles:

• ,

• effective interaction between and :

➡ with ,

Ising-like model with transverse field

(a) (b)

(c)

Interatomic distance R

Energ

y

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

small electric field, left over after compensation of stray electric fields

requires less laser power for excitation from ground state as nS states, anisotropic van der Waals interaction

|ri ! | "i|gi ! | #i

H =X

i

~⌦2

�i

x

+1

2

X

i 6=j

Uij

ni

nj

Uij

�i

x

= |ri hg|i

+ |gi hr|i

ni = |ri hr|i

|rii |rij


Example: Ising-like system of spin-1/2 particles

3

Ideal interaction potential How reality looks like for generic

experimental parameters

➡ Describing atoms as two-level systems is an approximation that can be difficult to fulfill

➡ Determination of suitable parameters requires calculation of Rydberg pair interaction potentials

RInteratomic distance

Ener

gy


Ener

gy

(a) (b)

(c)


Energ

y

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

Non-perturbative calculation of Rydberg pair interaction potentials

J. Phys. B 50, 133001 (2017)


Pair potential calculation Step 1: set up the Hamiltonian

4

• Born–Oppenheimer approximation

• R > Le Roy radius ➡multipole approximation, no exchange

interaction

• R < wavelength of Rydberg-Rydberg transitions ➡no retardation effects

energies of unperturbed Rydberg states

H = (Hatom1

+Hatom1-fields

)⌦ 1 + 1 ⌦ (Hatom2

+Hatom2-fields

) +Hmultipole interaction

interaction with static E and B-fields

interaction between the two Rydberg atoms

x

y z, quantization axis

✓R

atom 1

atom 2

BE


Pair potential calculation Step 2: define the basis

5

Full basis set:

Set of all pair states

Restricted basis set:

If interested in the pair potential of the pair state , restrict basis to states with ... • similar energy as • similar principal and momentum quantum number as • same symmetry as

| i| i

| i

| i

Rotation: mj1+mj2 conserved Reflection

xInversion

Permutation (if no interaction of higher order than dipole-dipole)

|n1, l1, j1,mj1i ⌦ |n2, l2, j2,mj2i


Pair potential calculation Step 3: diagonalize the Hamiltonian within the basis

6

Diagonalize the Hamiltonian matrix

➡eigen energies plotted as a function of the interatomic distance make up the pair potentials

Calculate matrix elements

• decompose into single atom matrix elements

• radial and angular part separate

- radial matrix elements: radial Schrödinger equation

- angular matrix elements: Wigner-Eckart theorem

hbi|H|bji

Our Pairinteraction software is available as open-source: https://pairinteraction.github.io

Similar open-source project by Nikola Šibalić et al.: https://github.com/nikolasibalic/ARC-Alkali-Rydberg-Calculator

where are basis elements

0

B@hb1|H|b1i hb1|H|b2i · · ·hb2|H|b1i hb2|H|b2i · · ·

......

. . .

1

CA

|bii

https://pairinteraction.github.io

https://github.com/nikolasibalic/ARC-Alkali-Rydberg-Calculator

Usage of pair interaction potentials for the determination of optimal parameters

arXiv:1710.06156 (2017)

Ideal interaction potentialReality for generic parameters


Ener

gy


Ener

gy

(a) (b)

(c)


Energ

y

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr


Influence of B-fields on pair potentials

7

(a) (b)

(c)

Interatomic distance REn

ergy

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

(a) (b)

(c)


Energ

y

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

(a) (b)

(c)


ergy

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

78° : 61D3/2 mj = 3/2

Ener

gy (M

Hz)

10

0

-10

B = 0 G

Interatomic distance (µm)R

6 8 10 12 14

B = -6.9 G


6 8 10 12 14

B = 6.9 G


6 8 10 12 14

0.01 10.1Overlap with |rri

|ri

Degenerate Zeeman levels T. G. Walker and M. Saffman, Phys. Rev. A 77, 032723 (2008)

Crossing Zeeman levels B. Vermersch et al., Phys. Rev. A 91, 023411 (2015)

Reasonable Rydberg blockade


Influence of an additional small E-field

8

Ener

gy (M

Hz)

10

0

-10

E = 0 mV/cm


E = 20 mV/cm

0.01 10.1Overlap with |rri

B =

6.9 G

Ener

gy (M

Hz)

10

0

-10

B =

3.5 G

6 8 10 12 14Interatomic distance (µm)R

6 8 10 12 14

(a) (b)

(c)


ergy

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

(a) (b)

(c)


Energ

y

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

(a) (b)

(c)


ergy

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

78° : 61D3/2 mj = 3/2|ri


Experimental test of the predictions

9

E = 0 mV/cm

Ωτ/2π

E = 20 mV/cmB

= 6.9 G

Prob

abili

ty fo

r tw

o Ry

dber

g ex

cita

tions

0.4

0.2

0.0

B =

3.5 G

0 1 2

(a) (b)

(c)


ergy

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

(a) (b)

(c)


Energ

y

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

(a) (b)

(c)


ergy

EB|g

nD3/2

5P1/2

5S1/2

|rΩb

ΩrR

θ≈ Ω

|g

|r


Energ

y C6(θ)/R6

z

|rr

78° : 61D3/2 mj = 3/2

Ω = 2π⋅1.2MHz

|ri

P rr

Ωτ/2π Ωτ/2π

E = 0 mV/cm E = 20 mV/cmB =

6.9 GB

= 3.5 G0.2

0.0

0.4

(a) (b)

(c) (d)

0 1 2 0 1 2

0.2

0.0

0.4SimulationExperiment

Spin-1/2 model

P rr

0.4

0.2

0.0

Predicted breakdown of Rydberg blockade experimentally seen as increasing probability for two Rydberg excitations

Ωτ/2π0 1 2

6.5µm


Influence of geometry

10

Mag

netic

fiel

d B z

(G)

5

0

-5

0° 15° 30° 45° 60°Angle between the z-axis and the interatomic axis✓

5

0

-5 0.0

0.1

0.2

0.3

0.4

0.5

Long

-tim

e pr

obab

ility

for t

wo

Rydb

erg

exci

tatio

ns

75° 90°

E = 0 mV/cm

0 mV/cm < E < 20 mV/cm


Quantum simulation of a 8-atom ring

11

Prior to the experiment, stray electric field compensated better than 5 mV/cm

Revisit of the experiment reported in H. Labuhn et al., Nature 534, 667 (2016)

B = 6.9 G (E-field sensitivity) B = 3.5 G (ideal regime)

Frac

tion

of R

ydbe

rg a

tom

s

0.4

0.2

0.0

0.6

0.8

1.0

Ωτ/2π0.0 0.5 1.51.0

Ωτ/2π0.0 0.5 1.51.0

z


Quantum simulation of a defect-free 7x7 lattice

12

Prior to the experiment, stray electric field compensated better than 5 mV/cm

B = 6.9 G (E-field sensitivity) B = 3.5 G (ideal regime)

Frac

tion

of R

ydbe

rg a

tom

s

0.4

0.2

0.0

0.6

0.8

1.0

Ωτ/2π0.0 0.5 1.51.0

Ωτ/2π0.0 0.5 1.51.0

z


• E-field sensitivity of Rydberg blockade for D states, resolved by tuning the B-field

• Predictions verified in 2-atom blockade experiments

• Increased accuracy of Rydberg quantum simulations

Conclusion

13


Outlook

14

• Optimize further parameters

• Use optimization procedure for other experiments

• Investigate mapping to dipolar interacting spin-1 particles, e.g. useful for realizing topological bands

Details on ...

• pair potential calculations: J. Phys. B 50, 133001 (2017) • optimization of mapping to spin-1/2 particle: arXiv:1710.06156 (2017)

Accurate Rydberg quantum simulations of spin-1/2 models · 2017. 11. 22. · Accurate Rydberg quantum simulations of spin-1/2 models ICQSIM — Paris — November 16, 2017 Theory

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