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
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