John BollingerNIST-Boulder

Ion storage group

Justin Bohnet (Honeywell), Kevin Gilmore, Elena Jordan, Brian Sawyer (GTRI) ,

Joe Britton (ARL)

theory –Rey goup (JILA/NIST)Freericks group (Georgetown)

Dan Dubin (UCSD)

Quantum sensing and simulation with single plane crystals of trapped ions

B+V

+V

• motional amplitude sensing

• quantum simulation – measure quantum dynamics with OTOC

NIST ion storage group

Kevin GilmoreJustin Bohnet

Brian SawyerGTRI

Joe BrittonARL

Elena Jordan

Outline:

● sensing small COM (center-of-mass) motion

- spin-dependent forces

● Quantum simulation with ion crystals in a Penning trap- engineering Ising interactions with spin-dependent forces

- Loschmidt echo and out-of-time order correlation functions

ji

z

j

z

ijiJN

H ,Ising

1

- high field qubit, modes● Penning trap features

Penning trap: many particle confinement with static fields

● radial confinement due to rotation –ion plasma rotates v = r r due to ExB fields

in rotating frame, Lorentz force is directed radially inward

2

2

22

222

2

1

2

1),(

22

1),(

rzmzr

rzmzr

z

rcrzrot

ztrap

rotatingframe

c m

Den

sity

no

m c / 2

Rotation frequency r

nB

9Be+, B0 = 4.5 T

Ω𝑐

2𝜋~ 7.6 MHz,

𝜔𝑧

2𝜋~ 1.6 MHz,

𝜔𝑚

2𝜋~ 160 kHz

Ion crystals form as a result of minimizing Coulomb potential energy

T→ 0.4 mK (Doppler laser cooling) ⇒ ൗ𝑞2𝑎𝑊𝑆 ≫ 𝑘𝐵𝑇, 2𝑎𝑊𝑆~ ion spacing

B

0.5 mm

single planes

c mD

ensi

ty n

o

m c / 2

Rotation frequency r

nBtype of crystal, nearest neighbor

ion spacing depend on 𝜔𝑟

bcc crystals with N>100 k

observed with:Bragg scatteringion fluorescence imaging

14 μm

Mitchell et.al., Science (1998)

𝑉𝑠𝑒𝑐𝑡𝑜𝑟 = 𝑉𝑊𝑎𝑙𝑙 𝑠𝑖𝑛 𝜔𝑑𝑟𝑖𝑣𝑒𝑡 + 𝜙

𝜙 =0o

270o

180o

90o

270o

90o

180o

360o

𝜔𝑤𝑎𝑙𝑙 = 𝜔𝑑𝑟𝑖𝑣𝑒/2

𝜔𝑟

𝜔𝑟

𝜔𝑤𝑎𝑙𝑙

torque drives 𝜔𝑟 = 𝜔𝑤𝑎𝑙𝑙

𝜔𝑤𝑎𝑙𝑙

Precise 𝝎𝒓 control with a rotating electric field

rotatingwall

electrodes

𝑉𝑠𝑒𝑐𝑡𝑜𝑟 = 𝑉𝑊𝑎𝑙𝑙 𝑠𝑖𝑛 𝜔𝑑𝑟𝑖𝑣𝑒𝑡 + 𝜙

𝜙 =0o

270o

180o

90o

270o

90o

180o

360o

𝜔𝑤𝑎𝑙𝑙 = 𝜔𝑑𝑟𝑖𝑣𝑒/2

𝜔𝑟

𝜔𝑟

𝜔𝑤𝑎𝑙𝑙

torque drives 𝜔𝑟 = 𝜔𝑤𝑎𝑙𝑙

B

𝜔𝑤𝑎𝑙𝑙

Precise 𝝎𝒓 control with a rotating electric field

rotatingwall

electrodes

Be+ high magnetic field qubit

+1/2 = | ↑ ›

-1/2 = | ↓ ›

2s 2S1/2 124 GHz

2p 2P1/2

2p 2P3/2

-1/2

+1/2

-3/2

-1/2

+1/2

+3/2

~ 40 GHz

Doppler

cooling

repump

~ 80 GHz

mJ

9Be+ , B ~ 4.5 T, o /2 ~124.1 GHz

B

cooling

repump

kHz 1510

,ˆ

B

BHi

x

iW

Transverse (drumhead) modes𝐸 × 𝐵 modes transverse modes cyclotron modes

𝜔𝑧𝜔𝑚 Ω𝑐

Transverse (drumhead) modes𝐸 × 𝐵 modes transverse modes cyclotron modes

𝜔𝑧𝜔𝑚 Ω𝑐

Freericks group, PRA (2013)Baltrush, Negretti, Taylor,

Calarco, PRA (2011)Dubin, UCSD

Modes characterized by eigenfrequency 𝜔𝑚and eigenvector 𝑏𝑖,𝑚

Freq

ue

ncy

𝜔𝑚

⟶

Transverse (drumhead) modes𝐸 × 𝐵 modes transverse modes cyclotron modes

𝜔𝑧𝜔𝑚 Ω𝑐

Spin-dependent force frequency 𝜇 (kHz)

Spin

pre

cess

ion

COM modetilt modeMeasure mode spectrum with spin-dependent force

𝜔𝑧

Outline:

● sensing small COM (center-of-mass) motion

- spin-dependent forces

● Quantum simulation with ion crystals in a Penning trap- engineering Ising interactions with spin-dependent forces

- Loschmidt echo and out-of-time order correlation functions

ji

z

j

z

ijiJN

H ,Ising

1

- high field qubit, modes● Penning trap features

Motional amplitude sensing or

Trapped ions as sensitive 𝑬-field and force detectorsMaiwald, et al., Nature Physics 2009 – 1 yN Hz-1/2

Hempel et al., Nature Photonics 2013 – detect single photon recoilShaniv, Ozeri, Nature Communications, 2017 – high sensitivity (~28 zN Hz-1/2) at low frequencies

⋮Biercuk et al., Nature Nanotechnology, 2010 – 100-ion crystal (400 yN Hz-1/2 )

Basic idea: map motional amplitude onto spin precession

Single ion

⟺

N ion crystal

N+1levels⟺

N ion crystal• Less projection noise

• Smaller zero-point motion, 𝑧𝑧𝑝𝑡 ≈ 2 nm

for N=100

~1

𝑁

Sensing small center-of-mass motion

Ƹ𝑧

ො𝑥 ො𝑦

𝛿𝑘

900 nm

20° 𝐻𝐼 =

𝑖

𝐹0 cos(𝜇𝑡) Ƹ𝑧𝑖 ො𝜎𝑖𝑧

Implement classical COM oscillation: Ƹ𝑧𝑖 → Ƹ𝑧𝑖 + 𝑍𝑐 𝑐𝑜𝑠 𝜔𝑡 + 𝜙

𝐻𝐼 ≅ 𝐹0 ∙ 𝑍𝑐 𝑐𝑜𝑠 𝜔 − 𝜇 𝑡 + 𝜙 σ𝑖ෝ𝜎𝑖𝑧

2

= 𝐹0 ∙ 𝑍𝑐 𝑐𝑜𝑠 𝜔 − 𝜇 𝑡 + 𝜙 መ𝑆𝑧

For 𝜇 = 𝜔, produces spin precession with rate ∝ 𝐹0 ⋅ 𝑍𝑐cos(𝜙)

Measuring spin precession

𝜋

2ቚ𝑦

Cool & Prepare

DetectODF𝜋

2ቚ𝑦

𝜃

𝜃

Probability of measuring spin up:

𝑃↑ =1

21 − 𝑒−Γ𝜏 cos 𝜃

=1

21 − 𝑒−Γ𝜏𝐽0

𝐹0ℏ𝑍𝑐𝜏

Precession 𝜃,

𝜃 =𝐹0ℏ𝑍𝑐𝜏 cos 𝜙

−𝐹0ℏ𝑍𝑐𝜏 < 𝜃 <

𝐹0ℏ𝑍𝑐𝜏

𝜏

Measuring spin precession

ODF Difference Frequency 𝜇 (kHz)

𝑃↑

(%)

COM mode

ODF Difference Frequency 𝜇 (kHz)

𝑃↑

(%)

COM mode

Ƹ𝑧

ො𝑥 ො𝑦 𝝎

𝝎

𝜔𝑧

tilt mode

Sensitivity limits/ signal-to-noise

50 pm – smallest detected amplitude

SNR limited due to noise from fluctuations in 𝜙

Small signal limits due to:projection noisespontaneous emission

Sin

gle

tria

l

Gilmore et al., PRL 2017

Sensing small center-of-mass motion

Future:

• Fixed phase sensing off-resonance (i.e. fixed 𝜙 in 𝑍𝑐𝑐𝑜𝑠 𝜔𝑡 + 𝜙 )

- 74 pm in single experimental trial

- 18 pm/ 𝐻𝑧

- Exploit spins: squeezed states

• On-resonance with COM mode

- Enhance force and electric field sensitivities by 𝑄~106

- Protocols for evading zero-point fluctuations, backaction ??- 20 pm amplitude from a resonant 100 ms coherent drive

• force/ion of 5 × 10−5 yN• electric field of 0.35 nV/m

Potential for dark matter search(axions and hidden photons)

20 pm amplitude from a resonant 100 ms coherent drive• force/ion of 5 × 10−5 yN• electric field of 0.35 nV/m

log10𝜖

GHzMHzkHz THz

log10𝑚 [𝑒𝑉]

𝜖 =𝐸

3.3nVm

∗ 10−12

S. Chaudhuri, et al., Phys. Rev. D (2015).

Technical improvement: EIT coolingMorigi PRA 67 (2003); exp results with smaller ion numbers: Innsbruck, NIST

Technical improvement: EIT coolingMorigi PRA 67 (2003); exp results with smaller ion numbers: Innsbruck, NIST

Outline:

● sensing small COM (center-of-mass) motion

- spin-dependent forces

● Quantum simulation with ion crystals in a Penning trap- engineering Ising interactions with spin-dependent forces

- Loschmidt echo and out-of-time order correlation functions

ji

z

j

z

ijiJN

H ,Ising

1

- high field qubit, modes● Penning trap features

Sensing small center-of-mass motion

Ƹ𝑧

ො𝑥 ො𝑦

𝛿𝑘

900 nm

20° 𝐻𝐼 =

𝑖

𝐹0 cos(𝜇𝑡) Ƹ𝑧𝑖 ො𝜎𝑖𝑧

Implement classical COM oscillation: Ƹ𝑧𝑖 → Ƹ𝑧𝑖 + 𝑍𝑐 𝑐𝑜𝑠 𝜔𝑡 + 𝜙

𝐻𝐼 ≅ 𝐹0 ∙ 𝑍𝑐 𝑐𝑜𝑠 𝜔 − 𝜇 𝑡 + 𝜙 σ𝑖ෝ𝜎𝑖𝑧

2

= 𝐹0 ∙ 𝑍𝑐 𝑐𝑜𝑠 𝜔 − 𝜇 𝑡 + 𝜙 መ𝑆𝑧

Engineering quantum magnetic couplings

N

j

z

jjODF ztFtH1

0ˆˆcosˆ

ˆˆ21

†

N

m

ti

m

ti

m

m

jmmm eaea

Mb

N drumhead eigenvalues 𝜔𝑚 and

eigenvector 𝑏𝑚

frequency⟶(kHz)

16001400

higher frequency drumhead modes

𝜇

detuning from COM

dipole-dipole coupling

Infinite range ⟹Single axis twisting

𝐻𝐼𝑠𝑖𝑛𝑔 =𝐽

𝑁σ𝑖<𝑗 𝜎𝑖

𝑧𝜎𝑗𝑧 =

2𝐽

𝑁𝑆𝑧2

where 𝑆𝑧 = σ𝑖𝜎𝑖𝑧

2

generates a “cat state” 1

2| ۧ↑↑↑∙∙∙↑ 𝑥 + | ۧ↓↓↓∙∙∙↓ 𝑥

at long times 𝜏, such that 2𝐽

𝑁𝜏 =

𝜋

2

Ƹ𝑧

ො𝑥 ො𝑦

Benchmarking quantum dynamics

● employ infinite range interactions 𝐻𝐼𝑠𝑖𝑛𝑔 ≈2𝐽

𝑁𝑆𝑧2, 𝑆𝑧 ≡ σ𝑖 𝜎𝑖

𝑧 /2

Ising

µWave

ODFtime

Cool Detect

● prepare eigenstate of , turn on i

x

iBH ̂ IsingH

measure global spin

polarization መ𝑆𝑧 ,

variance Δ𝑆𝑧2 = መ𝑆𝑧 − መ𝑆𝑧

2

general rotation

𝑈𝑅𝜏

N=85Ƹ𝑧

ො𝑥 ො𝑦

Bohnet et al., Science 352, 1297 (2016)

•Measurements of Ramsey squeezing parameter ⇒prove entanglement for 25 < 𝑁 < 220

•Largest inferred squeezing: -6.0 dB

Benchmarking quantum dynamics

Ƹ𝑧

ො𝑥

Bohnet et al., Science 352, 1297 (2016)

N=85

Benchmarking quantum dynamics

Out-of-time-order correlation functions

𝐹(𝑡) ≡ 𝜓 𝑊 𝑡 †𝑉†𝑊 𝑡 𝑉 𝜓 where W t = e𝑖𝐻𝑡𝑊 0 𝑒−𝑖𝐻𝑡 ,

𝑉,𝑊(0) = 0

𝑅𝑒 𝐹(𝑡) = 1 − 𝑊 𝑡 , 𝑉 2 /2

⇒ measures failure of initially commuting operators to commute at later times

⇒ quantifies spread or scrambling of quantum information across a system’s degrees of freedom

Swingle et al., arXiv:1602.06271; Shenker et al., arXiv:1306.0622; Kitaev (2014)

Difficult to measure ⟺ requires time-reversal of dynamics

time reversal is possible in many quantum simulators!

Time reversal of the Ising dynamics

𝐻𝐼𝑠𝑖𝑛𝑔 =𝐽

𝑁σ𝑖<𝑗 ො𝜎𝑖

𝑧 ො𝜎𝑗𝑧,

𝐽

𝑁≅

𝐹02

ℏ4𝑚𝜔𝑧∙

1

𝜇−𝜔𝑧

Change 𝜇 = 𝜔𝑧 + 𝛿 (antiferromagnetic)to 𝜇 = 𝜔𝑧 − 𝛿 (ferromagnetic)

Multiple quantum coherence protocol• Probe higher-order coherences and correlations (Pines group, 1985)

𝑆𝑥 , 𝜓0|𝜓𝑓2

𝐻𝐼𝑠𝑖𝑛𝑔 −𝐻𝐼𝑠𝑖𝑛𝑔

prepare measure

| ۧ𝜓0

Multiple quantum coherence protocol

𝐻𝐼𝑠𝑖𝑛𝑔 −𝐻𝐼𝑠𝑖𝑛𝑔

prepare measure

| ۧ𝜓0 𝑆𝑥

𝑆𝑥 = 𝛹0| 𝑒𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏 𝑒𝑖𝜙𝑆𝑥 𝑒−𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏𝑆𝑥 𝑒

𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏 𝑒−𝑖𝜙𝑆𝑥 𝑒−𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏|𝛹0

=2

𝑁𝛹0| 𝑒

𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏 𝑊† 𝑒−𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏𝑉† 𝑒𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏 𝑊 𝑒−𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏𝑉|𝛹0

𝑊†(𝑡) 𝑉†(0) 𝑊(𝑡) V(0)

Out-of-time-order correlation (OTOC) function⇒ quantifies spread or scrambling of quantum

information across a system’s degrees of freedom

Swingle et al., arXiv:1602.06271; Shenker et al., arXiv:1306.0622; Kitaev (2014)

Multiple quantum coherence protocol

𝐻𝐼𝑠𝑖𝑛𝑔 −𝐻𝐼𝑠𝑖𝑛𝑔

prepare measure

| ۧ𝜓0 𝑆𝑥

𝑆𝑥 = 𝛹0| 𝑒𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏 𝑒𝑖𝜙𝑆𝑥 𝑒−𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏𝑆𝑥 𝑒

𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏 𝑒−𝑖𝜙𝑆𝑥 𝑒−𝑖𝐻𝐼𝑠𝑖𝑛𝑔𝜏|𝛹0

=

𝑚

𝛹|𝐶𝑚|𝛹 𝑒𝑖𝜙𝑚 𝐶𝑚 =𝜎1𝑧𝜎4

𝑦…𝜎𝑘

𝑧

At least m terms

𝑚𝑡ℎ order Fourier coefficient 𝛹|𝐶𝑚|𝛹 indicates | ۧ𝛹 has correlations of at least order 𝑚

≡ | ۧ𝛹

MQC protocol – 𝑺𝒙 measurement

𝐻𝐼𝑠𝑖𝑛𝑔 −𝐻𝐼𝑠𝑖𝑛𝑔

prepare measure

| ۧ𝜓0 𝑆𝑥

33

𝐻𝐼𝑠𝑖𝑛𝑔 = 𝐽/𝑁

𝑖<𝑗

𝜎𝑖𝑧𝜎𝑗

𝑧

𝐽 ≲ 5𝑘𝐻𝑧

𝑁 = 111

Γ = 93𝐻𝑧

[Gärttner, Bohnet et al. Nature Physics 2017]

Fourier transform of magnetization

Martin Gärttner - DAMOP 2017

• Measure build-up of 8-body correlations

• Only global spin measurement

• Illustrates howOTOCs measure spread of quantum information

[Gärttner, Bohnet et al. Nature Physics 2017]

Summary:

Future directions:

•transverse field, variable range interaction, longitudinal fields

• spin-phonon models (Dicke model)

−𝛿𝑎†𝑎 −𝑔0

𝑁𝑎 + 𝑎† 𝑆𝑧 + 𝐵⊥𝑆𝑥 arXiv:1711.07392

• mitigate decoherence, improve single ion readout

• 3-dimensional crystals with thousands of ions?

• trapped ion crystals – motional amplitude sensing below

the zero-point fluctuations

• employed spin-squeezing, OTOCs to benchmarked quantum dynamics with long range Ising interactions

𝑖<𝑗

𝐽𝑖,𝑗𝜎𝑖𝑧 𝜎𝑗

𝑧 + 𝐵⊥

𝑖

𝜎𝑖𝑥 +

𝑖

ℎ𝑖𝜎𝑖𝑧

Lab selfie ∼ 𝟐𝟎𝟏𝟒

Joe BrittonARL

Justin BohnetHoneywell

Brian SawyerGTRI

Theory

Ana Maria Rey

Martin Gärttner

Michael Wall ArghavanSafavi-Naini

MichaelFoss-Feig (ARL)

Kevin GilmoreCU grad student

Elena JordanLeopoldina PD

𝟐𝟎𝟏𝟕

Time dependence of squeezed and anti-squeezed variance

N = 85

Bohnet et al., Science352 (2016)

Benchmarking quantum dynamics and entanglement

Writing a spin gradientmethod: generate Stark shift gradient in the rotating frame

𝝁W𝜋/2 𝜋/2

o

err 1.0~

ODF, beat note = crystal rot 𝝎𝒓

z

j

j

jrjerro

ODF ttRkk

FH ˆcossincos

𝜇 = 𝜔𝑟 produces static Stark shift in the rotating frame

z

j

j

j

j

z

jjjerro hRkk

F ˆˆ)sin(sinJ1

jlabx ,

Random field Ising model j

z

jj

ji

z

j

z

iji hJN

ˆˆˆ1

,

In-plane modes

Lowest frequency ExB modes Lowest frequency cyclotron modes

Freericks group, PRA 87 (2013)

𝐸 × 𝐵 modes transverse modes cyclotron modes

𝜔𝑧𝜔𝑚 Ω𝑐