![Page 1: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/1.jpg)
Scrambling Quantum Information in Cold Atoms with Light
Monika Schleier-Smith August 28, 2017Emily Davis Gregory Bentsen Tracy Li Brian Swingle Patrick Hayden
![Page 2: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/2.jpg)
How fast can an initially localized quantum bit become entangled with all degrees of freedom, i.e., scrambled?
UV
Quantum Information Scrambling
![Page 3: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/3.jpg)
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 …
![Page 4: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/4.jpg)
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
∂
![Page 5: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/5.jpg)
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?
![Page 6: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/6.jpg)
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
![Page 7: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/7.jpg)
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).
![Page 8: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/8.jpg)
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).
![Page 9: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/9.jpg)
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).
![Page 10: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/10.jpg)
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).
![Page 11: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/11.jpg)
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
![Page 12: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/13.jpg)
Photon-Mediated Interactionsoptical cavity
cold atoms
![Page 14: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/14.jpg)
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, …
**
*
![Page 15: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/16.jpg)
Photon-Mediated Spin Interactions
lattice
controllaser
strong-couplingcavity
gκ
Γ
|#i
Two-level atomas pseudo-spin
![Page 17: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/20.jpg)
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
![Page 21: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/21.jpg)
Experiment Design
• Strong coupling:
• Optical access for imaging & addressing
• Confinement in 3D lattice
4g2
F2
w2 1
~ 101 - 103 atoms
Lens
cavity gκ
Γ
![Page 22: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/23.jpg)
Strong Coupling with Optical Access
cavity
viewportF=10 5
F=10 4F=10 6
Single-atom cooperativity η ~ 50η
Finesse 6×104
![Page 24: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/24.jpg)
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
![Page 25: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/26.jpg)
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
![Page 27: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/27.jpg)
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
![Page 28: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/28.jpg)
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
![Page 29: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/29.jpg)
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
![Page 30: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/30.jpg)
Quantifying Scrambling
How to define chaos in a quantum many-body system?
![Page 31: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/31.jpg)
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?
![Page 32: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/32.jpg)
Measuring Fast Scrambling
Decay of out-of-time-order correlation function
indicates growth of commutator:
![Page 33: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/33.jpg)
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
![Page 34: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/34.jpg)
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 ⟨😕|🙂⟩
🙂
😕
![Page 35: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/35.jpg)
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.
![Page 36: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/36.jpg)
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.
![Page 37: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/37.jpg)
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.
![Page 38: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/38.jpg)
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”
![Page 39: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/39.jpg)
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).
![Page 40: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/40.jpg)
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.
![Page 41: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/41.jpg)
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.
![Page 42: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/42.jpg)
• 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.
![Page 43: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/43.jpg)
• 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.
![Page 44: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/44.jpg)
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).
![Page 45: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/45.jpg)
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
![Page 46: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/46.jpg)
Exotic XY Models
Photon-mediated interactions for versatile control of of long-range “hopping”:
= ; = hard-core bosons = spin excitations:
![Page 47: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/47.jpg)
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
![Page 48: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/48.jpg)
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
![Page 49: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/49.jpg)
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
![Page 50: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/50.jpg)
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
![Page 51: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/51.jpg)
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
![Page 52: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/52.jpg)
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
![Page 53: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/53.jpg)
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
![Page 54: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/54.jpg)
“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
![Page 55: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/55.jpg)
“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⇒
![Page 56: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/56.jpg)
Engineered Chaos
Break integrability with disorder potential:
Signature of chaos: level repulsionsingle particle
single hole
n=2 n=3strongly interacting
n=1 n=4
![Page 57: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/57.jpg)
Single-Particle vs. Many-Body Chaos?
Speed and depth of scrambling vs. boson number n?
Diagnostic: for two representative operators
![Page 58: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/58.jpg)
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
![Page 59: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/59.jpg)
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?
![Page 60: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/60.jpg)
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
![Page 61: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/61.jpg)
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
![Page 62: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/62.jpg)
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
![Page 63: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/63.jpg)
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
![Page 64: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/64.jpg)
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
![Page 65: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/65.jpg)
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
![Page 66: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/66.jpg)
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
![Page 67: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/67.jpg)
Extras
![Page 68: 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](https://reader030.vdocuments.mx/reader030/viewer/2022011800/5ab0a1a27f8b9a284c8b80c7/html5/thumbnails/68.jpg)
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!