Quantum Simulations with Yb + crystal ~5 m Trapped Atomic Ions.

Quantum Simulations with

Yb+ crystal

~5 mm

Trapped Atomic Ions

Ramanbeatnotes:

wHF ± m

ki

tik

tik

ki

kixi

kk eaeabxkH,

)()(0

)( ][ˆ †

uppersidebands

frequencywHF+m

carrierlower

sidebands

wHF -m

global spin-dependent oscillating force

ki

tik

tik

ki

kixi

kk eaeabxkH,

)()(0

)( ][ˆ †

k

kkik

kii aa ])()([)(ˆ *

)sincos()(22

,

ki

k

ikiki ie

ik

†

phonons

k kk

k

kk

k

k

kjkijiji

bb

m

k

2

2sin

)(

)sin(

)(

)sin(

2

)()(

22

,,2

,

interaction between qubits (entangling gates etc..)

ji

jx

ixji

i

ixi iU

,

)()(,

)( )()(ˆexp)(

evolution operator

...)]](),([),([

6)](),([

2

1)(exp)(

232

0

1231

0

2

0

3

0

121

0

2

0

ttt

tHtHtHdtdtdti

tHtHdtdttdtHiU

“Adiabatically eliminate” phonons: | - m wk| >> hW0 “SLOW MOLMER”

1)sincos()( ,22

,

k

ikik

i

k

ikiki

iie

ik

)()(, ˆˆ j

xi

xji

jieff JH

k k

kj

kiji

ji

bb

m

kJ

22

2

, 2

)(

General effective Hamiltonian theory:D. F. James, Canadian J. Phys. 85, 625 (2007)

uppersidebands

frequency

carrierlower

sidebands

m

wk

sidebandlinewidth= iki ,

Ramanbeatnote:

mwHF ± m

uppersidebands

frequencywHF+m

carrierlower

sidebands

wHF -m

)()(, ˆˆ j

xi

xji

jieff JH

m m

mj

mi

jiji

bb

m

kJ

22

2

, 2

i

iyB )(̂

wHF ( = /2Df p ) wHF

control

normal modeeigenvectors(ion i mode m)

IsingModel

global spin-dependent oscillating force

Quantum Simulation: What is it?

Hdt

di

Y Describes N interacting systems, each system having D degrees of freedom

DN coupled differential equations

Hdt

di

PhysicalSystem

Y

TrialH

Hdt

di

PhysicalSystem

Y

ChooseH

Two approaches

(1)

(2)

i

iy

jx

ix

jijieff BJH )()()(

, ˆˆˆ

Quantum simulations with trapped ions

Porras and Cirac, PRL 92, 207901 (2004)Deng, Porras, Cirac, PRA 72, 063407 (2005)Taylor and Calarco, PRA 78, 062331 (2008)

A. Friedenauer, et al., Nature Phys. 4, 757 (2008)K. Kim et al., Phys. Rev. Lett. 102, 250502 (2009)K. Kim et al., Nature 465, 590 (2010)E. Edwards et al., Phys. Rev. B 82, 060412 (2010)J. Barreiro et al., Nature 470 , 486-491 (2011)R. Islam, et al., Nature Comm. 2, 377 (2011)B. Lanyon et al., Science 334, 57 (2011)J. Britton et al., Nature 484, 489 (2012)A. Khromova et al., PRL 108, 220502 (2012) R. Islam, et al., Science 340, 583 (2013)P. Richerme, et al., ArXiv 1303.6983 (2013) P. Richerme, et al., ArXiv 1305.2253 (2013)

Frustration and Entanglement ?AFM

AFM

AFM

Spin Liquids

1936: Giauque and Stout, “The Entropy of Water and the Third Law of Thermodynamics. Heat Capacity of Ice from 15 to 273°K”

Zero-point entropy in 'spin ice’, A. P. Ramirez, A. Hayashi, R. J. Cava, R. Siddharthan and B. S. Shastry, Nature 399, 333 (1999)(pyrochloric “spin ice” Dy2Ti2O7)

1945: L. Pauling The Nature of the Chemical Bond (Cornell Univ. Press), pp. 301-4

Ice

Control Range of Interaction!

Theory

ji

1

~

Pow

er L

aw E

xpon

ent a

COM

uppersidebands

frequencywHF

carrierlower

sidebands

Dw

-m wCOM

tunelaserhere

tunelaserhere

k k

kj

ki

ji

bb

m

kJ

22

22

, 2

)(

Initialization

CoolingOptical PumpingSpins along y(or against y)

DetectionMeasure each spin along x)()(

, ˆˆ jx

ix

jijiJ

i

iyB )(̂

time

Adiabatic Quantum Simulation

i

iy

jx

ix

jijieff tBJH )()()(

, ˆ)(ˆˆ

A. Friedenauer, et al., Nature Phys. 4, 757 (2008)

N=2

B/Jrms 0.210

0.50

0.25

0.000 100 200 300

0.50

0.25

0.000 100 200 300

t (ms)

B/Jrms 0.210

t (ms)

Exact Ground StateMeasured Populations

J12=J13=J23 < 0

Initialization

CoolingOptical PumpingSpins along y


, ˆˆ jx

ix

jijiJ

i

iyB )(̂

time

P↓↓↓

P↓↓↑

P↓↑↓

P↓↑↑

P↑↓↓

P↑↓↑

P↑↑↓

P↑↑↑

P↓↓↓, P↑↑↑

E. Edwards, et al., Phys. Rev. B 82, 060412 (2010)

N=3

B/Jrms

0.50

0.25

0.000 100 200 300

0.50

0.25

0.000 100 200 300

t (ms)

B/Jrms

t (ms)

J12=J13=J23 > 0

0.21010

P↓↓↓

P↓↓↑

P↓↑↓

P↓↑↑

P↑↓↓

P↑↓↑

P↑↑↓

P↑↑↑ P↓↓↓, P↑↑↑

Initialization

CoolingOptical PumpingSpins along y


, ˆˆ jx

ix

jijiJ

i

iyB )(̂

time

0.2

E. Edwards, et al., Phys. Rev. B 82, 060412 (2010)

Exact Ground StateMeasured Populations

N=3

FM

Ferromagnetic couplingsFM FM

J12=J13=J23 < 0

K. Kim, et al., Nature 465, 590 (2010)

|Y = |+|

ground state is entangled

P0 P1 P2 P3

Bx=0

|Y2 = |

no entanglement

|Y1 = |

no entanglement

Bx0

P0 P1 P2 P3

symmetrybreaking field Bx

N=3

Competing AFM: spin frustration

AFM

AFM AFM

J12=J13=J23 > 0

?

K. Kim, et al., Nature 465, 590 (2010)

|Y = | +|+| +| +|+|

ground state is entangled Bx=0

P0 P1 P2 P3

|Y1 = | +|+|

still entangled!

symmetrybreaking field Bx

|Y2 = | +|+|

still entangled!

Bx0

P0 P1 P2 P3

Frustration Entanglement

N=3

Emergence of ferromagnetism vs. # spins N(all FM couplings: Jij<0)

|mx|

R. Islam et al., Nature Communications 2, 377 (2011)

t(ms)

0

0.25

0.5

B/|J|

N

4J

B1

J

B 3.0J

B05.0

J

B12

J

B

Ion index, j

Time/τ0 52.5

0

12

6

B

Long Range Antiferromagnetism (N=10)

i

iy

jx

ix

jijieff BJH )()()(

, ˆˆˆ

)()1()()1( jxx

jxx

pair correlation

G1,j =

Ji,j 1

|i-j|1.1

Frustration and energy gaps

Short range: exponent 1.5

Long range: exponent 0.5

Ground state Neel ordered:

Abandoning adiabaticity probes frustration

Low-lying energy states in antiferromagnetic model

StructureFunction

||

||),(1

ji

jiikejiGN

Spatial frequency k (2p)

Short range

Long range

R. Islam et al., Science340, 583 (2013)

Frustration of Magnetic Order (N=10)

Antiferromagnetic Néel order of N=10 spinsAll in state

2600 runs, a=1.12

AFM ground state order 222 events

441 events out of 2600 = 17% Prob of any state at random =2 x (1/210) = 0.2%

219 events


All in state

First Excited States(Pop. ~2% each)

Second Excited States(Pop. ~1% each)

Distribution of all 210 = 1024 states

Prob

abili

ty

0 341 682 1023

NominalAFMstate

B << J0

0101010101 1010101010

Prob

abili

ty

0.10

0.08

0.06

0.04

0.02

Initialparamagnetic

state

B >> J0


Distribution of states ordered by energy (N=10)

Energy/J0R. Islam et al., Science

340, 583 (2013)

a = 1.12a = 0.86

ji

JJ ji

0

,Thermalization??

Cum

uliti

ve P

rob

AFM order of N=14 spins (16,384 configurations)

i

ixx

i

iyy

jx

ix

ji

jix BtBJH )()()()(, ˆˆ)(ˆˆ

==

At By = 0:

AFM Ising with AXIAL field

010010


010010

AFM Ground States

2-Bright Ground State

1-Bright Ground States


P. Richerme, et al., ArXiv 1303.6983 (2013)







5-Bright (AFM) Ground States

System exhibits a completedevil's staircase for N → ∞

P. Bak and R. Bruinsma, PRL 49, 249 (1982) P. Richerme, et al., ArXiv 1303.6983 (2013)


Modulate transverse B field to drive transitions

between ground and excited states

i

iyy

jx

ix

ji

jixeff tBJH )()()(, ˆ)(ˆˆ

timeBy

Jxi,j

ampl

itude

Dynamics: many-body spectroscopy

C. Senko et. al., in preparation

timeBy

Jxi,j

ampl

itude

Start from

Drive to

N = 6 Dynamics: many-body spectroscopy


Start from

Drive to

N = 5

timeBy

Jxi,j

ampl

itude



Start from

Drive to

N = 5 Dynamics: many-body spectroscopy


N = 5

Modulation frequency (kHz)

Measurement

Theory

Spin states in order of energy


Complete spectrum of 5 spins


111111111110011111111111

111111111101101111111111

111111111011110111111111

111111110111111011111111

111111101111111101111111

111111011111111110111111


Dynamics: many-body spectroscopyN = 12

Modulation frequency (kHz)

Measurement

Theory

111111111110011111111111

111111111101101111111111

111111110111111011111111

111111101111111101111111


Dynamics: many-body spectroscopyN = 12

FM

Po

pu

lati

on

+ +

+ +

+Y =

Drive system with all frequencies simultaneously(and control relative phases)

Create equal superposition of single-spin flip states

(W state)

Dynamics: quantum engineering (FM: N=4)


)sin()( 11 tAtBy )sin( 22 tA

timeBy

Jxi,j

ampl

itude

+ ++Y =

2222

21 zyxxbipartite JJ

NJJNW

entangled

f = 340°

f = 160°

Dynamics: quantum engineering (FM: N=4)


ji

JJ ji

0

,

“Ising Quench”(a) Prepare (↓+↑)N “kT = ”(b) Meaure correlations Cmidpoint, j (t)

Dynamics: “light cone” of interaction propagationwith long range interactions

Theory: Z. Gong and A. Gorshkov (JQI)

a=2.5N=41 N=41 N=41 a=1.5 a=0.5

N=11 J0=0.5kHz a=0.81

shorter rangelonger range

N=11 J0=0.5kHz a=1.3

neutrals (nearest-neighbor interactions): M. Cheneau et al., Nature 481, 484 (2012)

E.H. Lieb and D.W. Robinson, “The finite group velocity of quantum spin systems,” Commun. Math. Phys. 28, 251–257 (1972).

Dynamics: L-R bounds with long range interactions

PreliminaryData

N=11 spins

• Formation of localized defects: nonequilibrium dynamicsM. Knap, E. Demler, I. Bloch (in preparation)

• XY model

• Spin-1: topological excitations

• Programmable fully connected spin network

Up next…

N beams, each with N spectral components

:ˆˆ )()(, jx

ix

ji

jiJH

2

)1( NNinteractions

S. Korenblit, et al., New. J. Phys. 14, 095024 (2012)

i

i

ji

jy

iy

jiy

jx

ix

jixeff BJJH )()()(,)()(, ˆˆˆˆ

Example: programming a 2D kagome lattice with a linear chain of 36 ions

atom # spectral component

Theory

J. Garcia-Ripoll et al., Phys. Rev. A 71, 062309 (2005)S. Korenblit et al., ArXiv 12010776 (2012)

• GET MORE SPINS!!

B/J = 0.01B/J = 5

16 spin AFM simulation 18 spin FM simulation

mx = total spin along x

Prob

abili

ty

N=16 ()

N=1

N=0 ()

Photon count histograms for N=16 ions

# photons

Global Spin Detection: 16 ions

N=8

Quantum simulation with N=16 ions

B>>J

B ~ J

B << J

Ferro couplings

Quantum Phase

Transition

Decreasing B/J

FM/AFM order

paramagnetic polarization

Anti-ferro couplings

G(1

,j)G

(1,j)

Distance from 1st ion, j

B ~ J

B << J

N=16 ()

N=0 ()

Theoretical photon count histograms

# photons

N=8

B>>J

Ferro couplings

~few 100Be+ ions ina Penning Trap

J. Britton et al., Nature 484, 489 (2012)

QuantumHard-drive?

Going Cold: N>50

GaTech Res. Inst.Al/Si/SiO2

Maryland/LPSGaAs/AlGaAs

Sandia Nat’l Lab: Si/SiO2

NIST-BoulderAu/Quartz

a (C.O.M.)

b (stretch)

c (Egyptian)

d (stretch-2)

Mode competition – example: axial modes, N = 4 ions

Fluo

resc

ence

cou

nts

Raman Detuning dR (MHz)

-15 -10 -5 0 5 10 15

20

40

60

a b

c

d

a

bcd

2a

c-a

b-a

2b,a

+c b+

c

a+b

2a

c-a

b-a

2b,a

+c

b+c

a+b

carrier

axial modes only

modeamplitudes

cooling beam

D. Kielpinski, CM, D. Wineland, Nature 417, 709 (2002)

Large scale vision (103 – 106 atomic qubits?)

• New hierarchical and modular quantum computer architecture• Different model for circuit optimization• Error correction thresholds exist! (R. Raussendorf)

C.M., et al., ArXiv 1208.0391 (2012).

0.001 Hz then, ~1 Hz now, ~1 kHz soon

ENIAC (1946) Solid-state transistor (1947)

right idea, wrong platform

Quantum Simulations with Yb + crystal ~5 m Trapped Atomic Ions.

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