Ana Maria Rey

Computers

Clocks

QIP Tutorials Boulder, Colorado, January 12th, 2019 Quantum materials Quantum Supremacy

Trapped Ions

Quantum Gravity Black holes Highest accuracy Dark matter • Penning trap: 2D triangular crystals of 20-300 ions

• Two hyperfine states used as spin ½ system

124 GHz

Single qubit gates with 99.9 fidelity • Spin–spin interactions generated by lasers

+ ODF 𝝎𝝎𝑶𝑶𝑶𝑶𝑶𝑶 𝝎𝝎𝑶𝑶𝑶𝑶𝑶𝑶 𝛍𝛍

• Spin dependent force N ˆ z HODF (t) = −F0 cos(µt) ∑ zˆ j ⋅σˆ j j=1

z Pauli matrix σˆ j on spin j N ˆ z HODF (t) = −F0 cos(µt) ∑ zˆ j ⋅σˆ j j=1 Written in terms of drumhead eigenmodes

N  † iωmt −iωmt ∑bjm (aˆme + aˆme ) m=1 2Mωm N drumhead eigenvalues and eigenvector • Ions are not independent: form a crystal due to Coulomb interactions 𝜔𝜔𝑚𝑚 𝑏𝑏𝑚𝑚 CM

• Far detuning > | |: phonons can be adiabatically eliminated

• Effective Hamiltonian:𝜹𝜹 𝝁𝝁 − Ising𝝎𝝎𝒎𝒎 spin model 1 = ~ 2 𝑧𝑧 𝑧𝑧 𝐹𝐹0 1 𝑆𝑆𝑆𝑆 𝑖𝑖𝑖𝑖depends𝑖𝑖 𝑗𝑗 on eigenmodes and ODF 2detuning2 ( ) 𝐻𝐻 � ,𝐽𝐽 𝑡𝑡 𝜎𝜎 𝜎𝜎 𝐽𝐽𝑖𝑖𝑖𝑖 ℏ ∑𝜇𝜇 𝑏𝑏𝜇𝜇𝑖𝑖 𝑏𝑏𝜇𝜇𝑗𝑗 𝜇𝜇 −𝜔𝜔𝜇𝜇 𝑁𝑁 𝑖𝑖<𝑗𝑗 𝐽𝐽𝑖𝑖 𝑗𝑗 𝜇𝜇 |↑ z z Initial |↓↓↓↓ ⟩ Rotate:θ ⟩

Wait τ/2 x x y echo

Wait τ/2 |↓ N. Ramsey. Nobel prize Read ⟩ 1989

Verify spin Create strong Explore regime Goals: intractable to model correlations theory 1 mean field cos = = = 𝑁𝑁 limit 𝑧𝑧 𝑧𝑧precession 𝑧𝑧 𝜃𝜃 𝐻𝐻𝑆𝑆𝑆𝑆 � 𝐽𝐽𝑖𝑖𝑖𝑖 𝜎𝜎𝑖𝑖 𝜎𝜎𝑗𝑗 𝐻𝐻𝑀𝑀𝑀𝑀 � 𝐵𝐵𝑗𝑗𝜎𝜎𝑗𝑗 𝐵𝐵𝑗𝑗 � 𝐽𝐽𝑖𝑖𝑗𝑗 𝑁𝑁Short𝑖𝑖<𝑗𝑗 time used to𝑗𝑗 measure = 𝑁𝑁 /𝑖𝑖≠𝑗𝑗 � 𝑗𝑗 𝑗𝑗 𝐽𝐽𝐽𝐽 ≪ 𝑁𝑁 100 𝐵𝐵 ∑ 𝐵𝐵 𝑁𝑁 � 𝐵𝐵

10

1 0 5 10 15 20 25 30 35 40 No mean field dynamics at θ=π/2 [Khz] Not the full story J. W. Britton𝜇𝜇et− al 𝜔𝜔 Nature1 484, 489 (2012) = Detuning

𝜔𝜔1 𝛿𝛿 𝜇𝜇 − CM

µ

= ( ) 𝐽𝐽 2 Collective Ising𝑆𝑆̂𝑧𝑧 Model 𝐻𝐻�zz 𝑁𝑁 Prepare Evolve: Entangled uncorrelated state Interaction state | Hamiltonian Trapped Ions: NIST,.. → ⋯ →⟩ = ( ) Cavity QEDs: MIT, Stanford,.. 𝐽𝐽 Bose Einstein Condensates: One-Axis Twisting2 𝑯𝑯� 𝑶𝑶𝑶𝑶𝑶𝑶 𝑁𝑁 𝑆𝑆̂𝑧𝑧 Heidelberg, Georgia Tech, Max-Planck-Institute, Hannover,… How entanglement builds up? ↑ , ↓ Spin S=N/2 = , = Spin ½ 2 𝑁𝑁 �𝑆𝑆 𝑆𝑆𝑧𝑧 𝑀𝑀� Dimension 2 Dimension: N+1 = { , … . , } 2 2 𝑁𝑁 𝑁𝑁 ψ = ∑CM M 𝑀𝑀 − M = , , ⃗̂ ̂𝑥𝑥 ̂𝑦𝑦 ̂𝑧𝑧 = , , 𝑆𝑆 𝑆𝑆 𝑆𝑆 𝑆𝑆 θ ̂ 𝑆𝑆⃗ 𝑺𝑺�𝒙𝒙 𝑺𝑺�𝒚𝒚 𝑺𝑺�𝒛𝒛 Evolution

2 𝜋𝜋 𝜑𝜑 Projects coherence onto 𝜏𝜏 the spin population

Contrast or transverse magnetization

= ( ) = ( ) + ( ) = ( ) 2 2 + 𝒞𝒞 𝑆𝑆̂ 𝜏𝜏 𝑆𝑆�𝑥𝑥 𝜏𝜏 𝑆𝑆�𝑦𝑦 𝜏𝜏 𝑆𝑆�𝑥𝑥 𝜏𝜏 ( ) = ( ) =0

𝑆𝑆�𝑦𝑦 𝜏𝜏 𝑆𝑆�𝑧𝑧 𝜏𝜏

At the mean field limit is constant. But that is not the case when quantum correlations are included 𝒞𝒞 ● Coherent spin demagnetization: Bloch vector length | | vs time

𝑆𝑆𝑥𝑥 = / 𝑵𝑵 𝑵𝑵−𝟏𝟏 𝒄𝒄𝒄𝒄𝒄𝒄 𝑱𝑱𝑱𝑱 𝑵𝑵 𝑺𝑺�𝒙𝒙 𝟐𝟐 Decoherence

Lines: Theory Symbols: Experiment

Bohnet et al., Science, 352, 1297(2016). ?? Heisenberg Uncertainty Relations Quantum Mechanics Sets the Size of the “Uncertainty Blob”

| /2| Quantum Fuzziness zΔ𝑆𝑆̂𝑦𝑦 Δ𝑆𝑆̂𝑧𝑧 ≥ ℏ 𝑆𝑆̂𝑥𝑥 Coherent Spin states | + | y x ↑⟩ ↓⟩ Uncorrelated 2

=N/2 , =0 ̂𝑦𝑦 𝑧𝑧 𝑆𝑆̂𝑥𝑥 𝑆𝑆 =0 , = 𝑁𝑁 ∆𝑆𝑆̂𝑦𝑦 𝑧𝑧 4 ∆𝑆𝑆̂𝑥𝑥 ~

𝐽𝐽 𝑆𝑆̂𝑧𝑧 𝐻𝐻�zz ⟨𝑆𝑆̂𝑧𝑧⟩ Spin Squeezing𝑁𝑁 Parameter

z < 1 y 𝜓𝜓 x 2 𝜉𝜉 A. Sørensen et al Nature 409, 63 (2001) • Entanglement witness • Enhanced sensitivity • Useful only for Gaussian states ● largest inferred squeezing: -6.0 dB

Anti-squeezed variance N = 86 Theory

Squeezed variance Theory

Bohnet et al., Science, 352, 1297(2016). • Disappearance of squeezing at longer time does not mean no entanglement • Squeezing is only useful for Gaussian states • How can we quantify entanglement? : Density Matrix of the close system Tr( ) = 1 Purity 𝜌𝜌�Tr( ) = 1 A2 B =𝜌𝜌�Tr 𝜌𝜌�

Reduce𝜌𝜌�𝐴𝐴 density𝐵𝐵 𝜌𝜌� Matrix of subsystem A Renyi entropy: Purity of the A subsystem

= ln Tr( ) 2 Do not lose Product𝑆𝑆𝐴𝐴 − state 𝜌𝜌 𝐴𝐴 = = 0 information when I cut 𝜌𝜌� ⨂𝑖𝑖𝜌𝜌�𝑖𝑖 𝑆𝑆𝐴𝐴 Entangled state > 0 Lose information when I cut 𝑆𝑆𝐴𝐴 A = / 𝑵𝑵 𝑵𝑵−𝟏𝟏 𝒄𝒄𝒄𝒄𝒄𝒄 𝑱𝑱𝑱𝑱 𝑵𝑵 =1 𝑺𝑺�𝒙𝒙 𝟐𝟐 𝐿𝐿𝐴𝐴 = log 1 + 1 2 2 𝑆𝑆1 − 2 𝑁𝑁 𝑆𝑆̂𝑥𝑥 Magnetization /N x 1 𝑆𝑆 2S

=1

𝐿𝐿𝐴𝐴 2Jτ/Nπ 2Jτ/Nπ / / / 𝒎𝒎𝒎𝒎𝒎𝒎 𝑺𝑺𝑵𝑵 𝟐𝟐 𝑺𝑺𝑵𝑵 𝟐𝟐 2Jt/N π N= 40, 60 Jτ 𝑐𝑐𝑐𝑐𝑐𝑐 =N 2

𝑆𝑆1 2J τ /N π Boring Quantum Thermalization in Entanglement closed systems

(0) = Product state

= 0 𝜌𝜌� (0) ⨂𝑖𝑖𝜌𝜌�𝑖𝑖 t=0 A

𝑆𝑆𝐴𝐴 𝜌𝜌�𝐴𝐴 Apparent loss of information in local observables

T>0 > 0 ( ) A

𝑆𝑆𝐴𝐴 𝜌𝜌�𝐴𝐴 𝑡𝑡 Black Hole Entangled state Information D’Alessio et al, Adv. in Phys.(2016) paradox Information not loss but Scrambled Spread over many-body degrees of freedom, becoming inaccessible to local measurements

Kaufman et al, Science(2016) Greiner group: quantum gas microscope

 Single site addressing  Only in small systems L=6

But entanglement entropy is hard to measure in large systems Brydges,…., P. Zoller, R. Blatt, C. F. Roos, arXiv:1806.05747 Out-of-time-Order-Correlators OTOCs

= 0 ( ) (0) † † 𝐹𝐹 𝑡𝑡 𝑊𝑊� 𝑡𝑡 𝑉𝑉� 𝑊𝑊� 𝑡𝑡 𝑉𝑉� =1-C(t) C(t) = | [ , 0 ] | 𝟐𝟐 � � 𝐹𝐹 𝑡𝑡 𝑊𝑊 𝑡𝑡 𝑉𝑉 Measurement of the degree of non-commutativity of (0) and the time evolved version of ( ) 𝑉𝑉� � [Hayden-Preskill, Sekin𝑊𝑊o-Susskind𝑡𝑡 , Shenker-Stanford ’13, Kitaev ‘14] • Black holes scramble quantum information as fast as possible • Fast scramblers: C(t)~eλt

• Bound of growth of quantum chaos: λ Lyapunov exponent [Maldacena-Shenker-Stanford][Martinis’16]

Can we access OTOCS? [Swingle et al 16] [Yao et al16] [Zhu et al16] Measure |Prepare † 𝑦𝑦 𝑦𝑦 𝑅𝑅� 𝑅𝑅� 𝑧𝑧 � � 0 ↓ ⋯ ↓⟩ −𝒊𝒊𝒕𝒕t𝑯𝑯 𝒊𝒊𝒕𝒕𝑯𝑯t 𝜌𝜌 𝑽𝑽� | 𝒆𝒆 |𝑾𝑾� 𝒆𝒆 | 𝜳𝜳𝟎𝟎⟩ 𝒕𝒕⟩ 𝜳𝜳𝒇𝒇� and Two commuting𝜳𝜳 operators � � = |𝑾𝑾 𝑽𝑽 | | =| 𝒊𝒊𝒕𝒕𝐻𝐻� † −𝒊𝒊𝒕𝒕𝐻𝐻� † 𝒊𝒊𝒕𝒕𝐻𝐻� −𝒊𝒊𝒕𝒕𝐻𝐻� 𝟎𝟎 � � � 𝟎𝟎 𝐹𝐹 𝑡𝑡 = 𝜳𝜳 | 𝒆𝒆 𝑊𝑊 𝒆𝒆 𝑉𝑉 𝒆𝒆 𝑊𝑊 𝒆𝒆 𝜳𝜳| 𝑉𝑉� 𝜳𝜳𝟎𝟎⟩ 𝜳𝜳𝟎𝟎⟩ 𝒊𝒊𝒕𝒕𝐻𝐻� † −𝒊𝒊𝒕𝒕𝐻𝐻� † 𝒊𝒊𝒕𝒕𝐻𝐻� −𝒊𝒊𝒕𝒕𝐻𝐻� 𝜳𝜳𝟎𝟎 𝒆𝒆 𝑊𝑊� 𝒆𝒆 𝑉𝑉� 𝒆𝒆 𝑊𝑊� 𝒆𝒆 𝑉𝑉� 𝜳𝜳𝟎𝟎 † = 𝑊𝑊�𝒕𝒕 𝑊𝑊�𝒕𝒕 OTOCS † † Garttner et 𝑊𝑊al� Nat.𝒕𝒕 𝑉𝑉�Physics𝑊𝑊�𝒕𝒕𝑉𝑉 �(2017) Garttner et al PRL (2018) M. Munowitz and M. Mehring , Sol. St. Com., 64, 605 (1987)

𝝆𝝆�𝟎𝟎 | Prepare Measure 𝑧𝑧 � � 0 ↓ ⋯ ↓⟩ −𝒊𝒊𝒕𝒕t𝑯𝑯 𝒊𝒊𝒕𝒕𝑯𝑯t 𝜌𝜌 𝑽𝑽� 𝒆𝒆 𝑾𝑾� 𝒆𝒆 = In NMR states are highly mixed = (1+ε ) �𝒛𝒛 = 𝒊𝒊𝝓𝝓𝑺𝑺 Fourier transform gives the Multi-Quantum𝝆𝝆� spectrum𝟎𝟎 𝑺𝑺�𝒛𝒛. 𝑾𝑾� 𝒆𝒆 𝟎𝟎 Multi-quantum 𝑽𝑽� 𝝆𝝆�= 𝑁𝑁 −𝑖𝑖𝑖𝑖𝜙𝜙 intensities ℑ𝝓𝝓 𝜏𝜏 � 𝐼𝐼𝑚𝑚 𝑒𝑒 𝑚𝑚=−𝑁𝑁 = [ ( ) ( )]

𝑰𝑰𝒎𝒎 𝐓𝐓𝐓𝐓 𝜌𝜌�−𝒎𝒎 :𝒕𝒕 all𝜌𝜌� matrix𝒎𝒎 𝒕𝒕 elements with = coherences between states 𝑚𝑚 𝜌𝜌differing� in Sz by m 𝜌𝜌� � 𝜌𝜌�𝑚𝑚 𝑚𝑚 MQC Detailed structure of the state …

= [ ] 𝟐𝟐 𝑰𝑰𝟎𝟎 𝐓𝐓𝐓𝐓 𝝆𝝆𝟎𝟎

| .

→ ⋯ →⟩ 𝐼𝐼𝑚𝑚 Requirements: 1) Invert many-body time evolution. = ( ) µ µ 𝐽𝐽 2 𝑆𝑆̂𝑧𝑧 𝐻𝐻�zz 𝑁𝑁 ~ ~ 0 𝐽𝐽0 𝐽𝐽 𝐽𝐽 𝛿𝛿 𝐽𝐽 − 𝛿𝛿

2) Measure initial state.

Fidelity: Probability of all down = | | 𝑧𝑧 𝜌𝜌�0 ↓ ⋯ ↓⟩⟨↓ ⋯ ↓ N=48 Solid lines: decoherence + phonons

6.5 %τcat~1ms

Light scattering ( ) = −𝜞𝜞𝑵𝑵𝑵𝑵 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 0 ( ) = ( + 0) 𝐼𝐼 𝜏𝜏 𝒆𝒆 𝐼𝐼 −𝑁𝑁Γ𝑡𝑡 𝒑𝒑𝒑𝒑𝒑𝒑𝒑𝒑 𝟐𝟐 𝟐𝟐 −𝟏𝟏 𝑒𝑒 𝟎𝟎Garttner et al Nature Physics, 𝑰𝑰 doi:10.1038𝝉𝝉 /nphys𝟏𝟏 4119𝑱𝑱 𝝉𝝉 Fidelity measurements decays too fast due to decoherence Measure magnetization instead of fidelity = = ( ) = 𝑽𝑽 𝝆𝝆�𝟎𝟎 𝑽𝑽 𝑆𝑆̂𝒙𝒙 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 Less sensitive to− 𝜞𝜞decoherence𝝉𝝉 . 𝐴𝐴𝑚𝑚 𝜏𝜏 𝒆𝒆 𝐴𝐴𝑚𝑚 𝐼𝐼𝑚𝑚 𝐴𝐴𝑚𝑚 Also an OTOC but what information gives us?

𝐴𝐴𝑚𝑚 A non-zero Am >0 implies >0 𝛼𝛼1 𝛼𝛼2 𝛼𝛼𝑚𝑚 𝜎𝜎1 𝜎𝜎2 ⋯ 𝜎𝜎𝑚𝑚 Signal buildup of at least m-body correlations. 7.3 % τcat=1 ms • Successful N=111 benchmark • Decoherence under control • Access features of Hamiltonian and prepared states

Up to m=8 significant correlations!!

Garttner et al Nature Physics, doi:10.1038/nphys4119 Scrambling in 4 nuclear spins in NMR, China Probing localization with OTOCs MIT

Chaos in a kicked BEC, U. of 7 Ions: Verified Information Illinois Scrambling, JQI

arXiv:1806.02807 arXiv:1705.06714v1 Can we simulate • Add Transverse Field fast scrambling and • Involve Phonons analogs of black holes with trapped ions? = ( ) = + 2 † 𝑔𝑔 † † 𝐽𝐽 𝒂𝒂� +𝒂𝒂� 𝑺𝑺�𝒛𝒛 𝑺𝑺�𝒛𝒛 𝑯𝑯� −𝜹𝜹𝒂𝒂� 𝒂𝒂� − 𝑁𝑁 −𝜹𝜹𝒃𝒃� 𝒃𝒃� 𝑁𝑁 ( + )/ N : CM phonons = creation† operator † 𝑔𝑔 𝒊𝒊 𝒂𝒂� � ̂𝑧𝑧 𝒛𝒛� ∝, , 𝒂𝒂� 𝒂𝒂� √ , , 𝑏𝑏 𝑎𝑎� − 𝑆𝑆 = Collective spin: S=N/2 𝛿𝛿= 𝑁𝑁 2 𝒙𝒙 𝒚𝒚 𝒛𝒛 𝒙𝒙 𝒚𝒚 𝒛𝒛 𝟏𝟏 𝑔𝑔 𝑺𝑺� � 𝝈𝝈�𝒊𝒊 Displaced Harmonic𝐽𝐽 Oscillator� 𝟐𝟐 𝒊𝒊 𝛿𝛿

V At = 2 Decoupling points ↑ V↓

𝑡𝑡𝛿𝛿 =𝜋𝜋𝜋𝜋 ( ) 𝐽𝐽 2 Collective Ising𝑆𝑆̂𝑧𝑧 Model 𝐻𝐻�zz 𝑁𝑁 = ( ) † 𝑔𝑔 † Phonons Spin𝑎𝑎-�Phonon+𝑎𝑎� 𝑆𝑆̂𝑧𝑧 −Transverse𝐵𝐵𝑆𝑆̂𝑥𝑥 Field 𝐻𝐻� −𝛿𝛿𝑎𝑎� 𝑎𝑎� − 𝑁𝑁 ˆ ˆ x No decoupling points [HODF , S ] ≠ 0 We can NOT eliminate the phonons

= ( ) 𝐽𝐽 2 𝑆𝑆̂𝑧𝑧 𝐻𝐻� 𝑁𝑁 − 𝐵𝐵𝑆𝑆̂𝑥𝑥 = + ( ) : CM phonons creation† operator † 2𝑔𝑔 † 𝒂𝒂� Phonons Spin-𝑎𝑎�Phonon+𝑎𝑎� 𝑆𝑆̂𝑧𝑧 −Transverse𝐵𝐵𝑆𝑆̂𝑥𝑥 Field 𝐻𝐻� 𝛿𝛿𝑎𝑎� 𝑎𝑎� 𝑁𝑁

Safavi-Naini et al (arXiv:1807.03178)PRL 2018 Renew interest in cold-atoms

T. Esslinger group 2010: Effective Dicke Model in a BEC (self-organization) = ( ) ( ) † 𝑔𝑔 † δ<0 𝑎𝑎� +𝑎𝑎� 𝑆𝑆̂𝑧𝑧−𝐵𝐵 𝑡𝑡 𝑆𝑆̂𝑥𝑥 � Phonons Spin𝑁𝑁 -Phonon Transverse Field Normal to superradiant𝐻𝐻 −𝛿𝛿𝑎𝑎� 𝑎𝑎�second− order phase transition at =

= + ( ) ( ) = = 𝑩𝑩𝒄𝒄 𝑱𝑱 2 𝐽𝐽 2 𝑔𝑔 𝑔𝑔 𝑧𝑧 † 𝑆𝑆̂𝑧𝑧 −𝐵𝐵 𝑡𝑡 𝑆𝑆̂𝑥𝑥 𝐽𝐽 � 𝑏𝑏� 𝑎𝑎� − 𝑆𝑆̂ 𝐻𝐻� −𝛿𝛿𝑏𝑏� 𝑏𝑏� 𝑁𝑁 𝛿𝛿 𝛿𝛿 𝑁𝑁 :

𝑩𝑩 ≫ 𝑩𝑩𝒄𝒄 𝐍𝐍𝐍𝐍B𝐍𝐍x 𝐍𝐍𝐍𝐍𝐍𝐍 𝐁𝐁 ≫ 𝐁𝐁𝐜𝐜 Energy

Paramagnetic, 0 No phonons Decoupled spin/phonon

𝑆𝑆̂𝑧𝑧 = ( ) ( ) † 𝑔𝑔 † δ<0 Phonons Spin𝑎𝑎�-Phonon+𝑎𝑎� 𝑆𝑆̂𝑧𝑧−𝐵𝐵Transverse𝑡𝑡 𝑆𝑆̂𝑥𝑥 Field 𝐻𝐻� −𝛿𝛿𝑎𝑎� 𝑎𝑎� − 𝑁𝑁 Normal to superradiant second order phase transition at = 𝟐𝟐 𝒈𝒈 𝒄𝒄 �𝜹𝜹 = + ( ) ( ) = = 𝑩𝑩 2 𝑔𝑔 † 𝐽𝐽 2 𝑔𝑔 � � ̂𝑧𝑧 𝑆𝑆̂𝑧𝑧 −𝐵𝐵 𝑡𝑡 𝑆𝑆̂𝑥𝑥 𝐽𝐽 𝛿𝛿 𝑏𝑏: 𝑎𝑎� − 𝑆𝑆 𝐻𝐻� −𝛿𝛿𝑏𝑏� 𝑏𝑏� 𝑁𝑁 𝛿𝛿 𝑁𝑁 𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒𝐒 𝑩𝑩 ≪ 𝑩𝑩𝒄𝒄 ± 𝒄𝒄 𝑩𝑩𝑩𝑩≪≪𝑩𝑩𝑩𝑩𝒄𝒄 Energy Ferromagnetic, Spin/phonon Entanglement Macroscopic phonon population N

0 ∝

𝑆𝑆̂𝑧𝑧 N=70 Thermodynamic limit the gap closes at Bc

Paramagnetic, Ferromagnetic, Spin/phonon Entanglement decoupled spin/phonon Macroscopic phonon population N ∝  Initial state (large B): Spin aligned | ( ) = | | along x, thermal phonon state ~6 ⟩ 𝚿𝚿 𝟎𝟎 ⟩ ← ⋯ ←⟩⨂ 𝑛𝑛̅  Optimal slow ramps to follow 𝑛𝑛� adiabatic state  Trade-off: adiabaticity vs decoherence (light scattering) Ramp limited to ~2 ms ramps Experiment Theory

𝑡𝑡 𝐵𝐵 𝑡𝑡 =𝐵𝐵0 1− 𝜏𝜏ramp

− 𝑡𝑡 𝐵𝐵 𝑡𝑡 =𝐵𝐵0𝑒𝑒 𝜏𝜏

• Measure full spin distributions (global fluorescence) • Benchmark the quantum simulator |

c I II E

Normal

Energy/| III

𝐵𝐵𝑐𝑐⁄𝐵𝐵 • Excited State Phase Transition at Ec =-NB/2 and B>Bc Singularity in the energy level structure

• B>Bc Poissonian Integrable Emary&Brandes, PRE (2003) Brandes,PRE (2013) • E>Ec: Wignger-Dyson: Chaos Altland & Haake, PRL (2012) E

0 = ( ) = (0𝑥𝑥)⃗ 𝑆𝑆̂𝒙𝒙 𝑆𝑆̂𝒚𝒚 𝑆𝑆̂𝒛𝒛 𝑎𝑎� 𝑹𝑹 𝑎𝑎� 𝑰𝑰𝑰𝑰 𝜆𝜆𝐿𝐿𝑡𝑡 Butterfly effect 𝑥𝑥⃗ 𝑡𝑡: Lyapunov− 𝑥𝑥⃗ ExponentΔ𝑥𝑥 𝑡𝑡 Δ𝑥𝑥 𝑒𝑒 Strogatz Book > 0: Signature of classical chaos 𝐿𝐿 • State𝜆𝜆 | = | | = Maximal Exponent 𝜆𝜆𝐿𝐿 𝒄𝒄 𝒄𝒄 𝒄𝒄 𝜳𝜳𝟎𝟎⟩ → ⋯ →⟩⨂ 𝟎𝟎⟩ 𝛹𝛹𝟎𝟎 𝐻𝐻� 𝛹𝛹𝟎𝟎 𝐸𝐸𝑐𝑐

𝜆𝜆𝐿𝐿 = | |

𝑽𝑽� 𝜳𝜳𝟎𝟎⟩⟨𝜳𝜳𝟎𝟎 Chaos Entanglement

Scrambling

Liapunov Volume Exponents Law Thermalization Air: Arcimboldo 1566 Water: Arcimboldo 1566 = = | | <<1 Small𝒊𝒊𝜹𝜹 Perturbation𝜹𝜹𝑮𝑮� � 𝑾𝑾 𝒆𝒆 𝑽𝑽� 𝜳𝜳𝟎𝟎⟩⟨𝜳𝜳𝟎𝟎 𝛿𝛿𝛿𝛿 , 1 1 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝟐𝟐 2 𝐹𝐹Great𝑮𝑮 𝛿𝛿𝛿𝛿 Insight:𝑡𝑡 ≈ − 𝛿𝛿𝛿𝛿 𝜳𝜳𝒕𝒕 𝐺𝐺� 𝜳𝜳𝒕𝒕 − 𝜳𝜳𝒕𝒕 𝐺𝐺� 𝜳𝜳𝒕𝒕 ≡ − 𝛿𝛿𝛿𝛿 𝛥𝛥 𝐺𝐺  Provide a semi-classical picture of the scrambling dynamics  Variance can be computed by phase space methods  Compute large systems intractable with numerical methods  Connect classical and quantum Liapunov exponents ~ 1 , ~( ) = Classical𝝀𝝀𝑳𝑳𝒕𝒕 Quantum 𝟐𝟐 𝝀𝝀𝑸𝑸𝒕𝒕 𝟐𝟐 𝟐𝟐𝝀𝝀𝑳𝑳𝒕𝒕 𝜹𝜹𝜹𝜹 𝒄𝒄 𝒆𝒆 − 𝐹𝐹𝑮𝑮 𝛿𝛿𝛿𝛿 𝑡𝑡 𝛿𝛿𝛿𝛿 𝒆𝒆 ≡ 𝛿𝛿𝛿𝛿 𝒆𝒆 𝑸𝑸 𝑳𝑳 Schmitt et al arXiv:1802.06796 𝝀𝝀 𝟐𝟐𝝀𝝀 Rozenbaum et al, PRL (2017) = = ( + ) | = | |0 𝟏𝟏 † � � 𝒄𝒄 • Fast scrambling𝐺𝐺 𝑋𝑋 in Dicke𝒂𝒂� M.𝒂𝒂� 𝟎𝟎 𝟐𝟐 𝜳𝜳 ⟩ → ⋯ →⟩⨂ ⟩ • Scrambling time ~ log 𝟐𝟐

𝛿𝛿𝛿𝛿 𝑄𝑄 ∗ / 𝝀𝝀𝑸𝑸𝒕𝒕 𝜆𝜆 𝑡𝑡 𝑁𝑁 ] 𝑡𝑡∗

𝑡𝑡 𝒆𝒆 , 10 10 10 10 N | 2 𝑥𝑥 5 7 𝑄𝑄 𝛿𝛿𝛿𝛿 3 𝜆𝜆 =𝒄𝒄 𝑥𝑥 𝑮𝑮 𝜳𝜳𝟎𝟎⟩ 𝐿𝐿 𝐹𝐹 𝜆𝜆 𝒄𝒄

− 𝐄𝐄 𝑬𝑬 1 [ | 𝑦𝑦 = 2 1 2 𝟎𝟎⟩𝒚𝒚 𝜆𝜆𝑄𝑄 t* 3 𝜳𝜳 𝑦𝑦 t(ms) 𝐄𝐄 𝟎𝟎 𝜆𝜆𝐿𝐿 At the critical energy scrambling is faster = B/(2π)(kHz) 𝝀𝝀𝑸𝑸 𝟐𝟐𝟐𝟐𝑳𝑳 A B : Density Matrix Tr( ) = 1 𝜌𝜌� 2 = Tr 𝜌𝜌� Reduce density Matrix of A 𝜌𝜌�𝐴𝐴 𝐵𝐵 𝜌𝜌� Renyi entropy: Purity of = log ( ) 2 𝐴𝐴 𝐴𝐴 𝐴𝐴 = Tr 𝜌𝜌� 𝑆𝑆 − 𝑇𝑇𝑇𝑇 𝜌𝜌 −𝑺𝑺𝑨𝑨 † −𝜷𝜷𝑯𝑯 † −𝜷𝜷𝑯𝑯 † 𝒆𝒆 � 𝑾𝑾𝒕𝒕 𝑶𝑶� 𝒆𝒆 𝑶𝑶�H. Zhai𝑾𝑾𝒕𝒕:𝑶𝑶 �Science𝒆𝒆 Bulletin(𝑶𝑶� 2017) =𝑊𝑊𝜖𝜖𝜖𝜖 B. Yoshida: JHEP02 (2016) −𝜷𝜷𝑯𝑯 † Sum over𝑽𝑽 complete𝑶𝑶� 𝒆𝒆 𝑶𝑶�set of operators acting on B :Exponential 4^B terms At = | | = | | FOTOC † Probing𝜷𝜷 → ∞entanglement𝑽𝑽 𝑶𝑶� 𝜳𝜳𝒈𝒈 �entropy�𝜳𝜳𝒈𝒈 𝑶𝑶� via 𝜳𝜳randomized𝟎𝟎⟩⟨𝜳𝜳𝟎𝟎 measurements: Up to N=20 P. Zoller, R. Blatt, C. F. Roos, arXiv:1806.05747 = , | | | | ′ , Spins 𝑛𝑛 𝑛𝑛 Phonons ′ | 𝜌𝜌�= ( � 𝜚𝜚)𝑚𝑚| 𝑚𝑚 𝑚𝑚=′⟩⟨𝑚𝑚 |⊗ 𝑛𝑛�⟩⟨𝑛𝑛 | = | = | † 𝒓𝒓 ̂ 𝑮𝑮�Spin𝑆𝑆 𝑚𝑚 𝒓𝒓Renyi⟩ Entropy:𝒆𝒆𝒓𝒓 � 𝑆𝑆⃗ 𝑚𝑚𝒓𝒓 � =𝑚𝑚𝒓𝒓 𝑚𝑚log𝒓𝒓⟩ Tr𝑮𝑮�[𝒏𝒏 𝒏𝒏]⟩ : Tracing𝒂𝒂� 𝒂𝒂� 𝒏𝒏 over⟩ 𝒏𝒏phonons𝒏𝒏⟩ 2 2 𝑆𝑆 , , 𝑆𝑆 Tr[ ]= +𝑆𝑆 𝜌𝜌� − + 𝜌𝜌� 2 𝑆𝑆̂𝒓𝒓 𝒏𝒏� 𝑆𝑆̂𝒓𝒓 𝑛𝑛� 𝑆𝑆̂𝒓𝒓 𝑛𝑛� 𝜌𝜌�𝑠𝑠 𝐼𝐼0 𝑰𝑰𝟎𝟎 − 𝐷𝐷diagMulti-Quantum𝐶𝐶off Intensities. = | |

( ) = , ( ) = , 0 0 𝟐𝟐𝝅𝝅 𝟐𝟐𝝅𝝅 𝑉𝑉� 𝛹𝛹 ⟩⟨𝛹𝛹 𝑆𝑆̂𝒓𝒓 𝑛𝑛� = 𝟏𝟏 𝑆𝑆 𝟏𝟏 𝐼𝐼0 𝒕𝒕 � 𝐹𝐹𝑮𝑮 𝒓𝒓 𝜙𝜙 𝑡𝑡 𝑑𝑑𝜙𝜙 0 𝑮𝑮𝑛𝑛 � 𝟎𝟎 𝐼𝐼 , 𝒕𝒕 ,�𝟎𝟎 𝐹𝐹 𝜙𝜙 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑖𝑖𝜙𝜙𝐺𝐺 Purely-diagonal𝟐𝟐𝝅𝝅 elements: = (𝟐𝟐𝝅𝝅 , ) ~1/( ) 𝑊𝑊�𝐺𝐺 𝑒𝑒 𝑆𝑆̂𝒓𝒓 𝑛𝑛� 𝑛𝑛 𝑛𝑛 2 Off-diagonal elements: 𝐷𝐷diag � 𝜚𝜚𝑚𝑚 𝑚𝑚 𝑁𝑁𝑛𝑛𝑝𝑝ℎ

, , , Dephase for < ~ = = , , , ′ , , , 0 ̂𝒓𝒓 ′ ′ ′ −1 𝑆𝑆 𝑛𝑛� 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 ′ ′ m , ′ ′ 𝑐𝑐 𝑄𝑄 𝐶𝐶off m� 𝜚𝜚 𝑚𝑚, 𝑚𝑚 𝜚𝜚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 For > :𝑡𝑡 randomize𝑡𝑡 𝜆𝜆 � 𝜚𝜚 ′ 𝜚𝜚 ⟶ ′ ≠ 𝑚𝑚 𝑛𝑛 ≠ 𝑛𝑛′ ≠ 𝑚𝑚 𝑛𝑛 ≠ 𝑛𝑛′ 𝑡𝑡 𝑡𝑡𝑐𝑐 FOTOCS can give use access to Renyi entropy | = | |0 N=40 𝑐𝑐 0 𝛹𝛹 � → ⋯ →⟩⨂ ⟩ BBc : Integrable case , = log = log + 𝑆𝑆̂𝒓𝒓 𝑛𝑛� 𝑆𝑆̂𝒓𝒓 ̂𝒙𝒙 ̂𝒙𝒙 𝑛𝑛� 𝑆𝑆 𝑆𝑆 , 𝐹𝐹 0 0 𝐹𝐹 0 | 𝑆𝑆( < −)| 0 𝐼𝐼Initial 𝐼𝐼condition 𝑆𝑆 , − 𝐼𝐼 𝑆𝑆̂𝒙𝒙 𝑛𝑛� | | 0 , , ( > −1)| 0 |𝐶𝐶off 𝑡𝑡 𝜆𝜆𝑄𝑄 ≪ Scrambling 𝑆𝑆̂𝒙𝒙 𝑛𝑛� 𝑆𝑆̂𝒓𝒓 𝑛𝑛� 𝑆𝑆̂𝒓𝒓 𝑛𝑛� 𝑛𝑛� , −1 𝐶𝐶off ≪ 𝐼𝐼0 ≈ 𝐷𝐷diag 𝐶𝐶off 𝑡𝑡<<1𝜆𝜆𝑄𝑄 ≪ 𝑆𝑆̂𝒓𝒓 𝑛𝑛� 𝐷𝐷diag 𝑆𝑆 𝐹𝐹 𝑆𝑆 𝑆𝑆 ̂ 𝒙𝒙 𝐹𝐹 𝑆𝑆 2 ̂ 2 𝒓𝒓 𝑆𝑆 , 𝑆𝑆 𝑛𝑛 � | = | |0 N=40 𝑐𝑐 𝛹𝛹0 � → ⋯ →⟩⨂ ⟩  Growth of entanglement entropy

B>Bc  System explore large part of Hilbert space: = Ergodicity 𝑆𝑆̂𝑥𝑥 =− 𝑙𝑙𝑙𝑙𝑙𝑙 𝐼𝐼0 + 𝑆𝑆̂𝑟𝑟 𝑛𝑛� − 𝑙𝑙𝑙𝑙𝑙𝑙 𝐼𝐼0 𝐼𝐼0 Applying only to part of the spin system A ,( 𝑊𝑊�(𝐺𝐺 )) = ( ) , ( ) + L 𝑆𝑆̂𝑟𝑟 𝐿𝐿𝐴𝐴 𝑛𝑛�𝑆𝑆̂𝑟𝑟 𝐿𝐿𝐵𝐵 𝑆𝑆̂𝑟𝑟 𝐿𝐿𝐴𝐴 𝑛𝑛� 𝑆𝑆̂𝒓𝒓 𝐿𝐿𝐵𝐵 A 𝑆𝑆𝐹𝐹 𝐼𝐼0 𝐼𝐼0

= −𝛽𝛽𝐻𝐻� ,( ( ))𝑒𝑒 • Volume law: ( ) ⟶ 𝜌𝜌� 𝑍𝑍 • Thermalization 𝑆𝑆̂𝑟𝑟(𝐿𝐿𝐴𝐴 ) 𝑛𝑛�𝑆𝑆̂𝑟𝑟 𝐿𝐿𝐵𝐵 𝑆𝑆𝟐𝟐 𝜌𝜌�𝐴𝐴 ∝ 𝐿𝐿𝐴𝐴 𝑆𝑆𝐹𝐹 𝑆𝑆𝟐𝟐 𝜌𝜌�𝐴𝐴

N=40, B/Bc=0.2 | = | |0 𝑐𝑐 𝛹𝛹0 � → ⋯ →⟩⨂ ⟩ 1. Measured FOTOCs (B=0) Garttner et al Nature Physics(2017)

2. Benchmarked the Dicke Model Safavi-Naini et al Phys. Rev. Lett. (2018)

3. Implemented EIT Cooling ( ~ ) In preparation 𝒏𝒏� 𝟎𝟎 4. FOTOCS in Dicke Molel

5. Control of Decoherence (Parametric drive) Wenchao Ge et al: arXiv:1807.00924 • Measured FOTOCs in the Collective Ising model:

N=48 Garttner et al Nature Physics (2017) 𝛿𝛿 → −𝛿𝛿 Light scattering Issue: slow measurements −𝑁𝑁Γ𝑡𝑡 𝑒𝑒 ( ) = Wanted to decoupled from phonons −𝜞𝜞𝑵𝑵𝑵𝑵 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝐼𝐼0 𝑡𝑡 𝒆𝒆 𝐼𝐼0 • Fotocs in Dicke model: πy spin echo | Prepare = = | | 𝑧𝑧 −𝒊𝒊𝒊𝒊𝑯𝑯� � 𝒊𝒊𝒕𝒕𝑯𝑯� 0 ↓ ⋯|↓⟩ t | 𝒊𝒊𝝓𝝓𝑮𝑮 t | 𝜌𝜌𝟎𝟎 𝟎𝟎 𝒆𝒆 𝑾𝑾� 𝒆𝒆 𝒆𝒆 𝑽𝑽� 𝜳𝜳 ⟩⟨𝜳𝜳 𝜳𝜳𝟎𝟎⟩ 𝒕𝒕 𝜳𝜳𝒇𝒇� 𝜳𝜳 ⟩ 𝑺𝑺�𝒙𝒙 → −𝑺𝑺�𝒙𝒙 • Need to measure | /2 /2| 𝑺𝑺�|𝒛𝒛0→0−| 𝑺𝑺�𝒛𝒛 −𝛿𝛿 • Possible: Two steps: 𝛿𝛿 Probability to be dark−𝑁𝑁 (no⟩⟨− affect𝑁𝑁 motion):⊗ ⟩⟨ DONE Probability to be in ground motional state (STIRAP): Gebert et al NJP. 18 013037(2016) • Gain: No need to decoupled from phonons (faster dynamics) • Increase by an order of magnitude (parametric drive, Wenchao 𝐶𝐶 Ge et al: arXiv:1807.00924𝐵𝐵 ) Γ

N=40 With decoherence J. Bollinger

R. Lewis-Swan A. Safavi-Naini

J. Bohnet K. Gilmore M. Gärttner M. Wall M. Foss-Feig Only the beginning: Bright vista ahead

• Bounds on scrambling. Pure states? • Quantum chaos. (away from semi- classical limit)? • Error correction / information hiding? • Design of duals of black hole. • …..

Thank You! R. Lewis-Swan et al, arXiv:1808.07134 Thanks for your attention