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)
Sebastian Weber Accurate Rydberg quantum simulations
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
Interatomic distance 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
Sebastian Weber Accurate Rydberg quantum simulations
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
RInteratomic distance
Ener
gy
(a) (b)
(c)
Interatomic distance R
Energ
y
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
Non-perturbative calculation of Rydberg pair interaction potentials
J. Phys. B 50, 133001 (2017)
Sebastian Weber Accurate Rydberg quantum simulations
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
Sebastian Weber Accurate Rydberg quantum simulations
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
Sebastian Weber Accurate Rydberg quantum simulations
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
Usage of pair interaction potentials for the determination of optimal parameters
arXiv:1710.06156 (2017)
Ideal interaction potentialReality for generic parameters
RInteratomic distance
Ener
gy
RInteratomic distance
Ener
gy
(a) (b)
(c)
Interatomic distance R
Energ
y
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
Sebastian Weber Accurate Rydberg quantum simulations
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
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
(a) (b)
(c)
Interatomic distance R
Energ
y
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
(a) (b)
(c)
Interatomic distance REn
ergy
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance 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
Interatomic distance (µm)R
6 8 10 12 14
B = 6.9 G
Interatomic distance (µm)R
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
Sebastian Weber Accurate Rydberg quantum simulations
Influence of an additional small E-field
8
Ener
gy (M
Hz)
10
0
-10
E = 0 mV/cm
Interatomic distance (µm)R
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)
Interatomic distance REn
ergy
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
(a) (b)
(c)
Interatomic distance R
Energ
y
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
(a) (b)
(c)
Interatomic distance REn
ergy
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
78° : 61D3/2 mj = 3/2|ri
Sebastian Weber Accurate Rydberg quantum simulations
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)
Interatomic distance REn
ergy
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
(a) (b)
(c)
Interatomic distance R
Energ
y
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance R
Energ
y C6(θ)/R6
z
|rr
(a) (b)
(c)
Interatomic distance REn
ergy
EB|g
nD3/2
5P1/2
5S1/2
|rΩb
ΩrR
θ≈ Ω
|g
|r
Interatomic distance 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
Sebastian Weber Accurate Rydberg quantum simulations
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
Sebastian Weber Accurate Rydberg quantum simulations
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
Sebastian Weber Accurate Rydberg quantum simulations
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
Sebastian Weber Accurate Rydberg quantum simulations
• 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
Sebastian Weber Accurate Rydberg quantum simulations
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)