Scrambling Quantum Information in Cold Atoms with Lightqpt.physics.harvard.edu/physicsnext/Monika_Schleier-Smith.pdfScrambling Quantum Information in Cold Atoms with Light Monika Schleier-Smith

Scrambling Quantum Information in Cold Atoms with Light

Monika Schleier-Smith August 28, 2017Emily Davis Gregory Bentsen Tracy Li Brian Swingle Patrick Hayden

How fast can an initially localized quantum bit become entangled with all degrees of freedom, i.e., scrambled?

Quantum Information Scrambling

How fast can an initially localized quantum bit become entangled with all degrees of freedom, i.e., scrambled?

Quantum Information Scrambling

Inspiration: information problem in black holesHayden, Preskill, Maldacena, Shenker, Susskind, Stanford …

Gauge/Gravity DualityQuantum many-body system

d spatial dimensionsSpacetime geometry

d+1 spatial dimensions

Figure adapted from Ramallo, arXiv:1310.4319v3[hep-th].

Gauge/Gravity DualityQuantum many-body system

d spatial dimensionsSpacetime geometry

d+1 spatial dimensions

Figure adapted from Ramallo, arXiv:1310.4319v3[hep-th].

Can we realize quantum many-body systems in table-top experimentsthat are holographically dual to black holes? How would we know?

Conjecture: black holes are the fastest scramblers in nature• Relaxation time τ = 1 / ( 2πT )• Scrambling time tS = τ log(𝕊)

Fast Scrambling Conjecture

T = Temperature 𝕊 = Entropy

Intuition: random circuit model• 𝕊 = number of qubits• τ = interaction time• Time tS ≿ τ log2(𝕊) to connect all pairs

Lashkari, Stanford, Hastings, Osborne, & Hayden, JHEP (2013).

Candidates for fast scrambling: chaotic, non-local spin models

Outline

Background

Non-local interactions mediated by light

Quantifying many-body chaos

Prospects for Cold-Atom Experiments

Kicked top: intuitions from a simple model system

Non-local hopping & many-body chaos

Outline

Background

Non-local interactions mediated by light

Quantifying many-body chaos

Prospects for Cold-Atom Experiments

Kicked top: intuitions from a simple model system

Non-local hopping & many-body chaos

Photon-Mediated Interactionsoptical cavity

cold atoms

• Non-local ⇾ entangling atoms en masse for quantum metrology ⇾ topological encoding of quantum information? ⇾ novel quantum simulations: spin glasses ; black holes?

* Sorensen & Molmer (2002); MSS, Leroux & Vuletic (2010); Hosten … & Kasevich (2016). * Jiang et al., N. Phys. (2008). * Gopalakrishnan, Lev; Sachdev; Diehl, …

cold atoms

• Non-local ⇾ entangling atoms en masse for quantum metrology ⇾ topological encoding of quantum information? ⇾ novel quantum simulations: spin glasses ; black holes?

• Easy to switch on/off and control sign• Quantitative understanding of interaction-to-dissipation ratio

Photon-Mediated Spin Interactions

lattice

controllaser

strong-couplingcavity

Two-level atomas pseudo-spin

Pairwise correlated spin flips:

⎟↓↓〉⊗⎟0〉c

⎟↑↓〉⊗⎟1〉c

⎟↑↑〉⊗⎟0〉c

(si+ + si)(sj

+ + sj) /X

Sørensen & Mølmer,PRA (2002).

latticestrong-coupling

cavitycontrollaser

• Spatial addressing enables controlled interactions between arbitrary pairs

⎟↓↓〉⊗⎟0〉c

⎟↑↓〉⊗⎟1〉c

⎟↑↑〉⊗⎟0〉c

(si+ + si)(sj

+ + sj) /X

cavitycontrollaser

• Spatial addressing enables controlled interactions between arbitrary pairs• Sign of interaction controlled by sign of detuning δ

⎟↓↓〉⊗⎟0〉c

⎟↑↓〉⊗⎟1〉c

⎟↑↑〉⊗⎟0〉c

(si+ + si)(sj

+ + sj) /X

cavitycontrollaser

• Spatial addressing enables controlled interactions between arbitrary pairs• Sign of interaction controlled by sign of detuning δ• Coherent interactions for δ≫κ and strong coupling η ≡ 4g2/(κΓ) ≫ 1

⎟↓↓〉⊗⎟0〉c

⎟↑↓〉⊗⎟1〉c

⎟↑↑〉⊗⎟0〉c

(si+ + si)(sj

+ + sj) /X

cavitycontrollaser

Experiment Design

• Strong coupling:

• Optical access for imaging & addressing

• Confinement in 3D lattice

~ 101 - 103 atoms

cavity gκ

Experiment Design

• Strong coupling:

• Optical access for imaging & addressing

• Confinement in 3D lattice

⇒ Near-concentric resonator Length L ~ 5 cm Waist w ~ 12 μm Finesse F ~ 105

~ 101 - 103 atoms

aligned

cavity gκ

Strong Coupling with Optical Access

cavity

viewportF=10 5

F=10 4F=10 6

Single-atom cooperativity η ~ 50η

Finesse 6×104

Atoms in the CavityNo atoms

Probe Frequency

cavity

viewport

200 μm

Shift of the cavity resonancedue to refractive index of a cloud of hundreds of atoms

Image of atoms

Atoms in the CavityNo atoms

Probe Frequency

cavity

viewport

200 μm

Shift of the cavity resonancedue to refractive index of a cloud of hundreds of atoms

Image of atoms

Position x

Image of spin texture

(si+ + si)(sj

+ + sj) /X

cavitycontrollaser

Simple limit:all-to-all interaction

sicollective spintwist

Spin Squeezing ID Leroux, MS-S & V Vuletic,PRL 104, 073602 (2010).

0.01 0.1 1

Twisting strength Q = N𝜒t = ( )# of photons scatteredinto cavity per atom

Q = 31

Q = 7.7

Q = 1.2Q = 0

N = 4×104 atomsη = 0.1, δ=κ/2

Global Spin Interactions

Bohnet, … & Bollinger, Science (2016). Also: Monz, … & Blatt PRL (2011).

Ion traps

Cavity QED

0.01 0.1 1

Leroux, MS-S & Vuletic, PRL (2010). Hosten, … & Kasevich, Science (2016).

Riedl, . . . & Treutlein,Nature (2010).

Hamley, . . . & Chapman,Nature Physics (2011).

Gross, . . . & Oberthaler, Nature (2010).

Vision: Non-Local Interactions• NP-hard optimization problems

• Qubit-ensemble interface ⇒ Schrödinger cat states

• Non-local + chaotic ⇒ fast scrambling?

partition problem

cavity

control light

|"i+ |#ip2

|0i+ |1ip2

Quantifying Scrambling

How to define chaos in a quantum many-body system?

Quantifying Scrambling

Quantum many-body butterfly effect: growth of commutator [V,Wt] between initially commuting operators vs. their separation in time t

How fast does Wt = e-iHt W eiHt fail to commute with V due to interactions H?

Shenker & Stanford, JHEP 2014:067. Maldacena, Shenker, & Stanford, JHEP 2016:106. Hosur, Qi, Roberts, & Yoshida, JHEP 2016:4.

How to define chaos in a quantum many-body system?

Measuring Fast Scrambling

Decay of out-of-time-order correlation function

indicates growth of commutator:

V, W: simple operators,e.g., spin rotations

[V,W]=0 at t=0

| iWtV | i

VWt| iV

| iWtV | i

VWt| iV

Measure ⟨😕|🙂⟩

Tools for measuring F • Time reversal (H → ‒H) • Many-body interferometry

| iWtV | i

VWt| iV

Measure ⟨😕|🙂⟩

Also see: Yao et al, arXiv:1607.01801.Zhu, Hafezi & Grover, arXiv:1607.00079.

B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.

Scrambling in a Cavity?

Ω1Ω2

Cavity

ControlQubit

Ensemble

C SPhoton-mediated interactions can enable…• qubit-controlled operation • switchable-sign interactions within ensemble

Ω1Ω2

Cavity

ControlQubit

Ensemble

Non-local spin models:candidates for fast scrambling

Ω1Ω2

Cavity

ControlQubit

Ensemble

Non-local spin models:candidates for fast scrambling

Globally interacting models:• ease of visualization • intuition: semiclassical limit • numerical simulations

Chaotic “Kicked Top”

Chaotic Kicked Top

• Expect nchaos ~ log N kicks for initial state of solid angle ~1/N to spreadover the entire N-atom Bloch sphere

• Probe with rotations by small angle

N = 2S = 30, k=3, p=π/2

Haake, Z. Phys. B (1987).

kick precession

C.f. kicked rotor: Rozenbaum, Ganeshan & Galitski, PRL 118, 086801 (2017).

Scrambling of a Kicked Top

Reference: time-ordered correlation function

Scrambling: out-of-time-order correlation

0 1 2 3 4 5 6 7 8Number of Kicks

1.00.80.60.40.20.0

|F|, |

Atom number N=2S

0 250 500

G = hV †t V i

1.00.80.60.40.20.0

|F|, |

1.00.80.60.40.20.0

|F||G|

|F|, |

Atom number N=2S

0 250 500

G = hV †t V i

• Scrambling time grows as tS ~ log(N) ↔ butterfly effect on Bloch sphere

1.00.80.60.40.20.0

|F|, |

1.00.80.60.40.20.0

|F||G|

|F|, |

Atom number N=2S

0 250 500

G = hV †t V i

• Scrambling time grows as tS ~ log(N) ↔ butterfly effect on Bloch sphere

• Accessible for up to atoms at cavity cooperativity η=50

1.00.80.60.40.20.0

|F|, |

1.00.80.60.40.20.0

|F||G|

|F|, |

Atom number N=2S

0 250 500

G = hV †t V i

Scrambling Experiments

Kicked top with pseudo-spin J = 5: “spin” states = momentum states of BEC

Meyer, …, & Gadway, arXiv (2017).

Twisting Hamiltonian of ~100 ions:Multiple quantum coherence method

Gärttner, Bohnet, Safavi-Naini, Wall,Bollinger, & Rey, Nat. Phys. (2017).

NMR experiments:Li, Fan, Wang, Ye, Zeng, Zhai, Peng& Du, arXiv:1609.01246.

Wei, Ramanathan & Cappellaro,arXiv:1612.05249.

cf. Davis, Bentsen, & MS-S PRL (2016).

Engineering Fast Scrambling?

All-to-all interactions restrict usto “single-particle” physics…

…but more complex non-local interactionsshould allow information to spread fastover exponentially large Hilbert space…

Can a single mode of light mediatemore complex interactions?

Entropy:𝕊 ≤ ln(N)

𝕊 ~ N

Exotic XY Models

Photon-mediated interactions for versatile control of of long-range “hopping”:

= ; = hard-core bosons = spin excitations:

Exotic XY Models

Hung, Gonzales-Tudela, Cirac & Kimble, PNAS (2016).

Approach:• Suppress hopping with magnetic field gradient• Restore hopping at arbitrary distances i-j

with modulated control field

Exotic XY Models

Hung, Gonzales-Tudela, Cirac & Kimble, PNAS (2016).

Approach:• Suppress hopping with magnetic field gradient• Restore hopping at arbitrary distances i-j

with modulated control field• Magnon dispersion relation = modulation waveform

Efficiently spread information over long distances

by coupling ith spin to i±1, i±2, i±4, i±8,…, i±2l

Dispersion Engineering

control field spectrum

-π π-

lmax=0

lmax=1lmax=2lmax=6

-π π-

lmax=0

lmax=1lmax=2lmax=6

-π π-

lmax=0

lmax=1lmax=2lmax=6

Dispersionrelation is a fractal!

-π π-

lmax=0

lmax=1lmax=2lmax=6

“Chaotic” dispersion?

-π π-

lmax=0

lmax=1lmax=2lmax=6

Looks crazy but must be integrable, since quasimomentum is conserved

-π π-

lmax=0

lmax=1lmax=2lmax=6

random phases l

“Chaotic” dispersion?

-π π-

lmax=0

lmax=1lmax=2lmax=6

Looks crazy but must be integrable, since quasimomentum is conserved

-π π-

lmax=0

lmax=1lmax=2lmax=6

random phases l

PoissonRandom-Matrix (GOE)

Energy Level Statistics

PoissonRandom-Matrix (GOE)

Poisson distributionof level spacings⇒

Engineered Chaos

Break integrability with disorder potential:

Signature of chaos: level repulsionsingle particle

single hole

n=2 n=3strongly interacting

n=1 n=4

Single-Particle vs. Many-Body Chaos?

Speed and depth of scrambling vs. boson number n?

Diagnostic: for two representative operators

0 1 2 3 4 5-0.20.00.20.40.60.81.0

Time × Energy Density

Re[F W

N=8, nearest-neighbor swap W

n = 1n = 2n = 3n = 4n = 5n = 6n = 7

W = nearest-neighbor swap

scrambling of coherences

scrambling of populations0 1 2 3 4 5

-0.20.00.20.40.60.81.0

Re[F U

N=8, phase shift U on site 4

n = 1n = 2n = 3n = 4n = 5n = 6n = 7

U = local phase shift

1,72,63,54

N = 8 sites

0 1 2 3 4 5-0.20.00.20.40.60.81.0

Re[F W

N=8, nearest-neighbor swap W

n = 1n = 2n = 3n = 4n = 5n = 6n = 7

W = nearest-neighbor swap

scrambling of coherences

scrambling of populations0 1 2 3 4 5

-0.20.00.20.40.60.81.0

Re[F U

n = 1n = 2n = 3n = 4n = 5n = 6n = 7

U = local phase shift

1,72,63,54

N = 8 sites

Deepest scrambling at half filling (“many”-body limit)… How deep?

U (phase shift) W (swap)

10 50 100

10-510-410-310-210-1100

Dimension CN,n

⟨ℛ[F]2 ⟩

Depth of ScramblingHow fully is the system scrambled at late times?

W = nearest-neighbor swapU = local phase shift

N = 10 sites

F ~ 1/(Hilbert space dimension) ?

1/dim2

n=1 2 3 4 5

10 50 100

10-510-410-310-210-1100

Dimension CN,n

⟨ℛ[F]2 ⟩

N = 10 sites

1/dim2

n=1 2 3 4 5

10 50 100

10-510-410-310-210-1100

Dimension CN,n

⟨ℛ[F]2 ⟩

N = 10 sites

1/dim2

n=1 2 3 4 5

10 50 100

10-510-410-310-210-1100

Dimension CN,n

⟨ℛ[F]2 ⟩

N = 10 sites

1/dim2⇒ interactions (hard-core) promote full scrambling

n=1 2 3 4 5

Fast Scrambling?

0 1 2 3 4 5

-0.20.00.20.40.60.81.0

Re[F U

n = 1n = 2n = 3n = 4n = 5n = 6n = 7

Fmin ~ 2-N at half filling

Scrambling time: tS ~ τ log(N); τ = 1 / ( 2πT )?

Extending numerics will help a little… Quantum simulations will help more!

1,72,63,54

n U = local phase shiftN = 10 sites

Scrambling is the ultimate form of thermalization,predicted to be subject to a fundamental speed limit.

Which systems scramble fully and how fast? Many open questions……ready to be tackled in cold-atom quantum simulations.

Photons can enable versatile engineering of interaction graphs: • single-particle chaos amenable to semiclassical intuition • interacting many-body chaos ⇒ towards black-hole analogs?

Scrambling Summary & Prospects

Acknowledgements

Research GroupEmily Davis Gregory Bentsen Tracy Li Tori Borish Ognjen Markovic Jacob Hines

CollaboratorsBrian Swingle Patrick Hayden Norman Yao Dragos Potirniche

Past visitorsAnna WangThomas Reimann Sebastian Scherg

Extras

Numerical Simulation: Chaotic Kicked Top

1.00.80.60.40.20.0

|F|, |

1.00.80.60.40.20.0

|F||G|

|F|, |

Atom number N=2S

0 250 500

G = hV †t V i

0.00.20.40.60.81.0

0 1 2 3 4 5 6 7 8

0 4 8 120 2 4 6Photons Lost

1.00.80.60.40.20.0

|F|, |

|F||G|

UnitaryN = 100

More dissipation? Hard to calculate!

Scrambling Quantum Information in Cold Atoms with Lightqpt.physics.harvard.edu/physicsnext/Monika_Schleier-Smith.pdfScrambling Quantum Information in Cold Atoms with Light Monika Schleier-Smith

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