scrambling quantum information in cold atoms with...
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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?
UV
Quantum Information Scrambling
How fast can an initially localized quantum bit become entangled with all degrees of freedom, i.e., scrambled?
UV
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
reno
rmali
zatio
n
UV
IR
u
Figure adapted from Ramallo, arXiv:1310.4319v3[hep-th].
z
∂
Gauge/Gravity DualityQuantum many-body system
d spatial dimensionsSpacetime geometry
d+1 spatial dimensions
reno
rmali
zatio
n
UV
IR
u
Figure adapted from Ramallo, arXiv:1310.4319v3[hep-th].
z
∂
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
Conjecture: black holes are the fastest scramblers in nature• Relaxation time τ = 1 / ( 2πT )• Scrambling time tS = τ log(𝕊)
Intuition: random circuit model• 𝕊 = number of qubits• τ = interaction time• Time tS ≿ τ log2(𝕊) to connect all pairs
Fast Scrambling Conjecture
T = Temperature 𝕊 = Entropy
Lashkari, Stanford, Hastings, Osborne, & Hayden, JHEP (2013).
Conjecture: black holes are the fastest scramblers in nature• Relaxation time τ = 1 / ( 2πT )• Scrambling time tS = τ log(𝕊)
Intuition: random circuit model• 𝕊 = number of qubits• τ = interaction time• Time tS ≿ τ log2(𝕊) to connect all pairs
Fast Scrambling Conjecture
T = Temperature 𝕊 = Entropy
Lashkari, Stanford, Hastings, Osborne, & Hayden, JHEP (2013).
Conjecture: black holes are the fastest scramblers in nature• Relaxation time τ = 1 / ( 2πT )• Scrambling time tS = τ log(𝕊)
Intuition: random circuit model• 𝕊 = number of qubits• τ = interaction time• Time tS ≿ τ log2(𝕊) to connect all pairs
Fast Scrambling Conjecture
T = Temperature 𝕊 = Entropy
Lashkari, Stanford, Hastings, Osborne, & Hayden, JHEP (2013).
Conjecture: black holes are the fastest scramblers in nature• Relaxation time τ = 1 / ( 2πT )• Scrambling time tS = τ log(𝕊)
Intuition: random circuit model• 𝕊 = number of qubits• τ = interaction time• Time tS ≿ τ log2(𝕊) to connect all pairs
Candidates for fast scrambling: chaotic, non-local spin models
Fast Scrambling Conjecture
T = Temperature 𝕊 = Entropy
Lashkari, Stanford, Hastings, Osborne, & Hayden, JHEP (2013).
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
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, …
**
*
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?
• Easy to switch on/off and control sign• Quantitative understanding of interaction-to-dissipation ratio
Photon-Mediated Spin Interactions
lattice
controllaser
strong-couplingcavity
gκ
Γ
|#i
Two-level atomas pseudo-spin
Photon-Mediated Spin Interactions
Pairwise correlated spin flips:
g Ω2
⎟↓↓〉⊗⎟0〉c
⎟↑↓〉⊗⎟1〉c
⎟↑↑〉⊗⎟0〉c
Ω1
δ
ΔΔ
g
H /X
i,j
(si+ + si)(sj
+ + sj) /X
i,j
six
sjx
Sørensen & Mølmer,PRA (2002).
latticestrong-coupling
cavitycontrollaser
gκ
Γ
Photon-Mediated Spin Interactions
• Spatial addressing enables controlled interactions between arbitrary pairs
Pairwise correlated spin flips:
g Ω2
⎟↓↓〉⊗⎟0〉c
⎟↑↓〉⊗⎟1〉c
⎟↑↑〉⊗⎟0〉c
Ω1
δ
ΔΔ
g
H /X
i,j
(si+ + si)(sj
+ + sj) /X
i,j
six
sjx
Sørensen & Mølmer,PRA (2002).
latticestrong-coupling
cavitycontrollaser
gκ
Γ
Photon-Mediated Spin Interactions
• Spatial addressing enables controlled interactions between arbitrary pairs• Sign of interaction controlled by sign of detuning δ
Pairwise correlated spin flips:
g Ω2
⎟↓↓〉⊗⎟0〉c
⎟↑↓〉⊗⎟1〉c
⎟↑↑〉⊗⎟0〉c
Ω1
δ
ΔΔ
g
H /X
i,j
(si+ + si)(sj
+ + sj) /X
i,j
six
sjx
Sørensen & Mølmer,PRA (2002).
latticestrong-coupling
cavitycontrollaser
gκ
Γ
Photon-Mediated Spin Interactions
• 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
Pairwise correlated spin flips:
g Ω2
⎟↓↓〉⊗⎟0〉c
⎟↑↓〉⊗⎟1〉c
⎟↑↑〉⊗⎟0〉c
Ω1
δ
ΔΔ
g
H /X
i,j
(si+ + si)(sj
+ + sj) /X
i,j
six
sjx
Sørensen & Mølmer,PRA (2002).
latticestrong-coupling
cavitycontrollaser
gκ
Γ
Experiment Design
• Strong coupling:
• Optical access for imaging & addressing
• Confinement in 3D lattice
4g2
F2
w2 1
~ 101 - 103 atoms
Lens
cavity gκ
Γ
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
4g2
F2
w2 1
~ 101 - 103 atoms
ABC
D
d
aligned
Lens
cavity gκ
Γ
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
Atoms
Tran
smiss
ion
Probe Frequency
cavity
viewport
a b c
200 μm
b
Shift of the cavity resonancedue to refractive index of a cloud of hundreds of atoms
Image of atoms
Atoms in the CavityNo atoms
Atoms
Tran
smiss
ion
Probe Frequency
cavity
viewport
a b c
200 μm
b
Shift of the cavity resonancedue to refractive index of a cloud of hundreds of atoms
Image of atoms
SΦ/S
0
2π0
2π
Mea
sure
men
t Bas
is Φ
Position x
Image of spin texture
Photon-Mediated Spin Interactions
H /X
i,j
(si+ + si)(sj
+ + sj) /X
i,j
six
sjx
latticestrong-coupling
cavitycontrollaser
gκ
Γ
Simple limit:all-to-all interaction
S =NX
i=1
sicollective spintwist
Spin Squeezing ID Leroux, MS-S & V Vuletic,PRL 104, 073602 (2010).
0.1
1
10
100
1000
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
0.1
1
10
100
1000
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).
BECs
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?
V W
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:
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
V W
W
eiHt
eiHt
eiHt
eiHt
| iWtV | i
VWt| iV
Measuring Fast Scrambling
Decay of out-of-time-order correlation function
indicates growth of commutator:
V W
W
eiHt
eiHt
eiHt
eiHt
| iWtV | i
VWt| iV
Measure ⟨😕|🙂⟩
🙂
😕
Measuring Fast Scrambling
Decay of out-of-time-order correlation function
indicates growth of commutator:
Tools for measuring F • Time reversal (H → ‒H) • Many-body interferometry
V W
W
eiHt
eiHt
eiHt
eiHt
| 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
ΩC
C SPhoton-mediated interactions can enable…• qubit-controlled operation • switchable-sign interactions within ensemble
B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.
Scrambling in a Cavity?
Ω1Ω2
Cavity
ControlQubit
Ensemble
ΩC
C SPhoton-mediated interactions can enable…• qubit-controlled operation • switchable-sign interactions within ensemble
Non-local spin models:candidates for fast scrambling
B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.
Scrambling in a Cavity?
Ω1Ω2
Cavity
ControlQubit
Ensemble
ΩC
C SPhoton-mediated interactions can enable…• qubit-controlled operation • switchable-sign interactions within ensemble
Non-local spin models:candidates for fast scrambling
Globally interacting models:• ease of visualization • intuition: semiclassical limit • numerical simulations
B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.
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
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|G|
|F|, |
G|
Atom number N=2S
0 250 500
G = hV †t V i
B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.
Scrambling of a Kicked Top
Reference: time-ordered correlation function
Scrambling: out-of-time-order correlation
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|G|
|F|, |
G|
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|F||G|
|F|, |
G|
Atom number N=2S
0 250 500
G = hV †t V i
B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.
• Scrambling time grows as tS ~ log(N) ↔ butterfly effect on Bloch sphere
Scrambling of a Kicked Top
Reference: time-ordered correlation function
Scrambling: out-of-time-order correlation
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|G|
|F|, |
G|
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|F||G|
|F|, |
G|
Atom number N=2S
0 250 500
G = hV †t V i
B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.
• Scrambling time grows as tS ~ log(N) ↔ butterfly effect on Bloch sphere
• Accessible for up to atoms at cavity cooperativity η=50
Scrambling of a Kicked Top
Reference: time-ordered correlation function
Scrambling: out-of-time-order correlation
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|G|
|F|, |
G|
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|F||G|
|F|, |
G|
Atom number N=2S
0 250 500
G = hV †t V i
B. Swingle, G. Bentsen, MS-S, & P. Hayden,PRA 040302(R) 2016.
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
Photon-mediated interactions for versatile control of of long-range “hopping”:
= ; = hard-core bosons = spin excitations:
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
B
Exotic XY Models
Photon-mediated interactions for versatile control of of long-range “hopping”:
= ; = hard-core bosons = spin excitations:
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
B
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
Efficiently spread information over long distances
by coupling ith spin to i±1, i±2, i±4, i±8,…, i±2l
Dispersion Engineering
-π π-
-
-
-
lmax=0
lmax=1lmax=2lmax=6
control field spectrum
Efficiently spread information over long distances
by coupling ith spin to i±1, i±2, i±4, i±8,…, i±2l
Dispersion Engineering
-π π-
-
-
-
lmax=0
lmax=1lmax=2lmax=6
control field spectrum
Efficiently spread information over long distances
by coupling ith spin to i±1, i±2, i±4, i±8,…, i±2l
Dispersion Engineering
-π π-
-
-
-
lmax=0
lmax=1lmax=2lmax=6
control field spectrum
Efficiently spread information over long distances
by coupling ith spin to i±1, i±2, i±4, i±8,…, i±2l
Dispersion Engineering
Dispersionrelation is a fractal!
-π π-
-
-
-
lmax=0
lmax=1lmax=2lmax=6
control field spectrum
“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
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
Time × Energy Density
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
N = 8 sites
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
Time × Energy Density
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
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
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
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
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⇒ 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
Time × Energy Density
Re[F U
]
N=8, phase shift U on site 4
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
Reference: time-ordered correlation function
Scrambling: out-of-time-order correlation
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|G|
|F|, |
G|
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8Number of Kicks
1.00.80.60.40.20.0
|F||G|
|F|, |
G|
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|, |
G|
|F||G|
UnitaryN = 100
More dissipation? Hard to calculate!