introduction to quantum optics [theory]2.pdf · 2018-04-19 · p zoller, cirm preschool:...

P Zoller, CIRM Preschool: Measurement & Control of Quantum Systems, 2018-04-17/18

Introduction to Quantum Optics [Theory]

University of Innsbruck and IQOQI, Austrian Academy of SciencesPeter Zoller

quantum control

perspective

Lectures 1 + 2

uibkOutline

Lectures 1+2:

Lectures 3+4:

• Basic systems & concepts of quantum optics - an overview

• Illustrations / Applications [in quantum information]

Isolated / Driven Hamiltonian quantum optical systems

Quantum Optical Systems & Control

Open quantum optical systems [a modern perspective]

• Example / Application: Ion Trap Quantum Computer

• Continuous measurement theory, Quantum Stochastic Schrödinger Equation, master equation & quantum trajectories

uibkLiteratureThe Quantum World of Ultra-Cold Atoms and Light: Book I: Foundations of Quantum OpticsBook II: The Physics of Quantum-Optical DevicesBook III: Ultra-cold Atoms

by Crispin W Gardiner and Peter Zoller

Quantum Noise

A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics

by Crispin W Gardiner and Peter Zoller

uibkA Short Preview

What is Quantum Optics?… a Short Tour & Overview

[Experiment + Theory]

• Quantum Properties of Light

• [Quantum] Interaction of Light & Mattercontrol

… a Few Examples

uibkEx 1: Quantum Computing with Trapped Ions

• general purpose quantum computing

R. Blatt

Exp.: Innsbruck, NIST, JQI, MIT, Mainz, MPQ ...

laser control of atoms + phonons = quantum gates

qubits quantum gates read out

time|√Ut |√

coherent Hamiltonian evolution - quantum gates - deterministic

quantum logic network model

• atomic physics: trapped ionsJ. I. Cirac

internal states of ions

as qubits

uibkEx 1: Quantum Computing with Trapped Ions

• general purpose quantum computing

qubits quantum gates read out

time|√Ut |√

coherent Hamiltonian evolution - quantum gates - deterministic

quantum logic network model

• atomic physics: trapped ions

R. Blatt

phonon bus

… in Lecture 1+2 we will go in detail over the underlying theory.

coherent control

uibkEx 2: Atomic Hubbard Models - QSimulators

Quantum Many-Body Atomic Physics

• Hubbard models etc.

Bosons, Fermions - strongly correlated system - quantum phase transitions

H = ∑ =

J a†a +

1

2U∑

a† a

† a a

˙ = i

h[H, ]+L

L ∑2cc† c†cc†c

HBEC = †b(r)

h

2∇2

2mb

b(r)d

3r+gbb

†b(r)

†b(r) b(r) b(r)d

3r

E0+ ∑k =0

(k) b†kbk

HBEC = E0+ ∑k =0

(k) b†kbk (1)

where

b†k

and

bk

hk

(k)

Hint = gab

†a(r) a(r)

†b(r) b(r)d

3r (2)

gab = 4 h2aab/(2mr)

mr = (mamb)/(ma+mb)

b =0+ b

b =1V∑k

ukbke

ik.r+ vkb†keik.r

, (3)

uk =Lk1L2k

, vk =11L2k

(4)

Lk =k (hk)2/(2m)mc2

mc2. (5)

k = [c2(hk)2+(hk)4/(2mb)2]1/2 (6)

c=gbb0/mb

gbb = 4 habb/mb

= 0V∑k

(uk+ vk)bkeik.r+(uk+ vk)b†ke

ik.r

.

S(k,) = (uk+ vk)2

S(k,)1/2

Hint gab∑k

S(k,)1/2w1|eik.r|w0bk|10|+h.c.

spontaneous emission rate

as = 100a0

0 = 51014cm3

= 2100 KHz

Γ= 21.1 KHz

Γ 0a2s

• atoms in optical lattices

Hubbard Hamiltonian

• We “build” a quantum system with desired Hamiltonian & controllable parameters,e.g. Hubbard models of atoms in optical lattices

Analog quantum simulation: “always on”

tunneling

onsite interactionU

optical potential

~ light intensity

AMO Hubbard toolbox

From Artificial Quantum Matter to Real Materials

Universality of Quantum Mechanics!

Quantum Regime

dl/d & 1

de Broglie Wavepackets

l

Ultracold Quantum Matter Real Materials

Same !l/d(Neuchatel)

Densities: 1014/cm3

Temperatures: few nK

Densities: 1024-1025/cm3

Temperatures: mK – several hundred K

uibkI QOIEngineered Atomic Many-Body Systems

Rydberg Spin-Models [as Quantum Simulator]

Hamiltonian Engineering

Rydbergstates

Hamiltonian

H =X

i

12≠iæ

i

x°

X

i

¢i ni +X

i< j

Vi j ni n j

æix = |gi ihri |+ |ri ihgi |

|gi ⟩

|ri ⟩

Ωi

∆i

Vi j =C6/r 6i j

Exp:Lukin-Greiner-Vuleticgroups(Harvard-MIT),Browaeys(Palaiseau),Saffman(Wisconsin),Biedermann(Sandia)

ni = |ri ihri |¥12

(1+æiz )

uibkQuantum Optics - Theory

Example: Trapped Ion Quantum Computer

P1/2

S1/2

D5/2

40Ca+

Quantumbit

QubitDetecLon,

LaserCooling

uibkTheory 1. Single Trapped Laser-Driven Ion

atominternal

harmonic trap

motion

• driven two-level atom in a harmonic oscillator trap

two-level atom trap (phonons)

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system of interestcoherent control

[quantum gates etc.]

Lectures 1+2:

laser

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…

System Hamiltonian isolated driven system

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atom: electronic atomic motion

laser drive

Engineer interesting quantum states?

uibkTheory 2. Single Trapped Laser-Driven Ion

atominternal

harmonic trap

empty radiation modes

phonon damping

reservoirs decoherence :-( laser cooling :-)

Lectures 1+2:

Lectures 3+4:

laser

system of interestcoherent control

[quantum gates etc.]

open system

laser drive

motion

• driven two-level atom in a harmonic oscillator trap

two-level atom trap (phonons)

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…∆=ωL −ωeg

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atom: electronic atomic motion

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spontaneous emission


Ion Trap Quantum Computer

• single ion- Hamiltonians, examples for quantum state engineering

•many ions - Hamiltonians, entangling gate

… as a Quantum Optical Problem

Appendices: • quantum information: qubits, quantum gates etc.

• From real atoms to two-level atoms & Rabi oscillations


Side-Remarks

From Real Atoms to Two-Level SystemsRabi Oscillations

uibkTwo-Level Atom & Rabi Oscillations

|g i

|ei|ni

|Ei

Eg =fl!g

Ee =fl!e

En =fl!n

E

fl!L

...

energy

Real Atom

laser

~Ecl(~x = 0, t ) = E~≤e°i!t +E §~≤§e+i!t

¥ ~Ecl(+)

(0, t )+~E (°)cl (0, t )

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Electric field of laser

Problem: We drive an atom near-resonant with a laser (no motion)

Schrödinger Equation

ifl d

d t|√(t )i=

°H0A °~µ ·~Ecl(~x = 0, t )

¢|√(t )i

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Two-Level System (TLS)

|√(t )i= ag (t )|g i+ae (t )|ei<latexit sha1_base64="7o0JmFRW/Q4yrZ8B5pxEyu7ChvQ=">AAACJXicfZDLSsNAFIZP6q3GW9Slm2IRKkpIKmgriAU3LivYCzSlTMbTdGhuzEykJe27uPNN3IkuiuDKV/ARTC+CRfRfnfOd/5xhfjt0mZCG8a6kFhaXllfSq+ra+sbmlra9UxVBxClWaOAGvG4TgS7zsSKZdLEeciSe7WLN7l6N57V75IIF/q3sh9j0iOOzNqNEJqilnVsNdWCFguXkocWJ77h4QVqxM0z6gTMjRwnBCcEZUa2m2tKyhm4UC8ZJMfO7MHVjouzlJ0xUbmkj6y6gkYe+pC4RooHJKSY6x6YRymbMMQy4HKoH/7oE5Ylx7FOtSGBIaJc4GEdUDH+CRiTbhV4zZn4YSfTp3DAmnvCI7PyCou/Z8xAF85O3LI90kSTpSolctSZXMx1dYm8cw/dfM38X1bxuGrp5k8+WTqd5QBr2YB9yYMIZlOAaylABCg/wBC/wqjwqz8pIeZtaU8psZxfmpHx8AZcLp10=</latexit><latexit sha1_base64="7o0JmFRW/Q4yrZ8B5pxEyu7ChvQ=">AAACJXicfZDLSsNAFIZP6q3GW9Slm2IRKkpIKmgriAU3LivYCzSlTMbTdGhuzEykJe27uPNN3IkuiuDKV/ARTC+CRfRfnfOd/5xhfjt0mZCG8a6kFhaXllfSq+ra+sbmlra9UxVBxClWaOAGvG4TgS7zsSKZdLEeciSe7WLN7l6N57V75IIF/q3sh9j0iOOzNqNEJqilnVsNdWCFguXkocWJ77h4QVqxM0z6gTMjRwnBCcEZUa2m2tKyhm4UC8ZJMfO7MHVjouzlJ0xUbmkj6y6gkYe+pC4RooHJKSY6x6YRymbMMQy4HKoH/7oE5Ylx7FOtSGBIaJc4GEdUDH+CRiTbhV4zZn4YSfTp3DAmnvCI7PyCou/Z8xAF85O3LI90kSTpSolctSZXMx1dYm8cw/dfM38X1bxuGrp5k8+WTqd5QBr2YB9yYMIZlOAaylABCg/wBC/wqjwqz8pIeZtaU8psZxfmpHx8AZcLp10=</latexit><latexit sha1_base64="7o0JmFRW/Q4yrZ8B5pxEyu7ChvQ=">AAACJXicfZDLSsNAFIZP6q3GW9Slm2IRKkpIKmgriAU3LivYCzSlTMbTdGhuzEykJe27uPNN3IkuiuDKV/ARTC+CRfRfnfOd/5xhfjt0mZCG8a6kFhaXllfSq+ra+sbmlra9UxVBxClWaOAGvG4TgS7zsSKZdLEeciSe7WLN7l6N57V75IIF/q3sh9j0iOOzNqNEJqilnVsNdWCFguXkocWJ77h4QVqxM0z6gTMjRwnBCcEZUa2m2tKyhm4UC8ZJMfO7MHVjouzlJ0xUbmkj6y6gkYe+pC4RooHJKSY6x6YRymbMMQy4HKoH/7oE5Ylx7FOtSGBIaJc4GEdUDH+CRiTbhV4zZn4YSfTp3DAmnvCI7PyCou/Z8xAF85O3LI90kSTpSolctSZXMx1dYm8cw/dfM38X1bxuGrp5k8+WTqd5QBr2YB9yYMIZlOAaylABCg/wBC/wqjwqz8pIeZtaU8psZxfmpHx8AZcLp10=</latexit><latexit sha1_base64="7o0JmFRW/Q4yrZ8B5pxEyu7ChvQ=">AAACJXicfZDLSsNAFIZP6q3GW9Slm2IRKkpIKmgriAU3LivYCzSlTMbTdGhuzEykJe27uPNN3IkuiuDKV/ARTC+CRfRfnfOd/5xhfjt0mZCG8a6kFhaXllfSq+ra+sbmlra9UxVBxClWaOAGvG4TgS7zsSKZdLEeciSe7WLN7l6N57V75IIF/q3sh9j0iOOzNqNEJqilnVsNdWCFguXkocWJ77h4QVqxM0z6gTMjRwnBCcEZUa2m2tKyhm4UC8ZJMfO7MHVjouzlJ0xUbmkj6y6gkYe+pC4RooHJKSY6x6YRymbMMQy4HKoH/7oE5Ylx7FOtSGBIaJc4GEdUDH+CRiTbhV4zZn4YSfTp3DAmnvCI7PyCou/Z8xAF85O3LI90kSTpSolctSZXMx1dYm8cw/dfM38X1bxuGrp5k8+WTqd5QBr2YB9yYMIZlOAaylABCg/wBC/wqjwqz8pIeZtaU8psZxfmpHx8AZcLp10=</latexit><latexit sha1_base64="nsJC/YBaN4924F4Bpr1Y0sW8jqM=">AAACJXicfZDLSsNAFIYn9VbjLerSTbEIFSUkFbQVhIIblxXsBZoQJuNpOjSZhJmJtKR9F3e+iTvRRRFc+SqmNYKl6L865zv/OcP8buRTIQ3jQ8ktLa+sruXX1Y3Nre0dbXevKcKYE2iQ0A9528UCfMqgIan0oR1xwIHrQ8vtX0/nrQfggobsTg4jsAPsMdqlBMsUOdql1VFHViRoSR5bHDPPhyvsJN447UdeRk5SAjMCGVEtW3W0oqEb1YpxVi0sFqZuzFREmeqONrHuQxIHwCTxsRAdSE9R0Ts1jUjaCYco5HKsHv3rEoSnxqlPtWIBESZ97EESEzH+DTqx7FYGdkJZFEtgZG6Y4EAEWPYWoBgG7jwEQVn6lhXgPuA0XSmBq9bsaqGnSxhMY/j5a+HvolnWTUM3b8vF2nkWSB4doENUQia6QDV0g+qogQh6RM/oFb0pT8qLMlHev605JdvZR3NSPr8A+ZumLg==</latexit>

keep resonant

levels

Rotating Wave Approximation

~µ ~Ecl(0, t ) °!~µeg~≤ ~Ecl(+)

(0, t )|eihg |+~µg e~≤~E(°)cl (0, t )|g ihe|

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emissionabsorption


|g i

|ei|ni

|Ei

Eg =fl!g

Ee =fl!e

En =fl!n

E

fl!L

...

energy

Real Atom

laser

Hamiltonian TLS + RWA

Problem: We drive an atom near-resonant with a laser (no motion)

H =fl!eg |eihe|°~µeg~≤E e°i!t |eihg |°~µg e~≤E

§e

i!t |g ihe|<latexit sha1_base64="KYALN75pdS+J7qiY7bHeCq8tiuQ=">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</latexit><latexit sha1_base64="KYALN75pdS+J7qiY7bHeCq8tiuQ=">AAACsXicfZFLTsMwEIad8A6vAks2ERUSixIlIAFdIJAQUpcFUR6qS3HMkFp1nMh2KpDJ9TgBG27BEUhTkKgqmI3H//z+7BmHKWdK+/6HZU9Nz8zOzS84i0vLK6uVtfVrlWSSQosmPJG3IVHAmYCWZprDbSqBxCGHm7B/NqzfDEAqlogr/ZJCJyaRYE+MEl1I3cobbjuNY9wLicRJDBHpGojyV8CSiIgD5uXiwusuHgA1OM7y0jHaQaoYT0SOY6J7lHBznuMa3JtdNoK5ehIVjaEi+Bt1bzBROi94v3DRxMsc3HG6larv+fUjf7/uTiaB55dRPflEZTS7lXf8mNAsBqEpJ0q1oaAx1asFfqo7RkKaSJ072/+6FJWFcehzcKYgJbRPIjAZVflvoZ3pp6PnjmEizTQIOlY0JFbDlidE9RKH4yIoJoq7ign1gRQ/rTVIB5dUt+dpeB6O4adX9+/kes8LfC+42KueHozmgebRJtpCOyhAh+gUNVATtRC1atal1bawvW/f2Q92OLLa1veZDTQWdv8L4GPc/w==</latexit><latexit sha1_base64="KYALN75pdS+J7qiY7bHeCq8tiuQ=">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</latexit><latexit sha1_base64="KYALN75pdS+J7qiY7bHeCq8tiuQ=">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</latexit><latexit sha1_base64="/CH18+kI2iCiIjaG14i0DF4PjWI=">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</latexit>

id

d t

µaeag

∂=

µ°¢ °~µeg~≤E

°~µg e~≤§E § 0

∂µaeag

∂

<latexit sha1_base64="FD/TsqW6dtjO4nWfI8IXasSn8og=">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</latexit><latexit sha1_base64="FD/TsqW6dtjO4nWfI8IXasSn8og=">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</latexit><latexit sha1_base64="FD/TsqW6dtjO4nWfI8IXasSn8og=">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</latexit><latexit sha1_base64="FD/TsqW6dtjO4nWfI8IXasSn8og=">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</latexit><latexit sha1_base64="TDNG2w1wkarRNuOL6yGHPjiWAJQ=">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</latexit>

ae (t ) = ae (t )e°i!t<latexit sha1_base64="vfDaR9CvuErqFx8LxD7Q6d6+34Y=">AAACFXicfZDLSsNQEIYn9VbjrerSTbEVqmhIKmi7EAtuXFawVTC1nByn7aG5kTORlpB3cOebuBNdiFs3bnwLH8H0IlhE/9XwzT8zzG/5tpCk6x9Kamp6ZnYuPa8uLC4tr2RW1+rSCwOONe7ZXnBpMYm2cLFGgmy89ANkjmXjhdU9GfQvbjGQwnPPqe9jw2FtV7QEZ5SgZmYnz5oRxgXaPjJJ2DcYsXgM8DraE6bnYJtlKc6rzUxO1/RySd8vZ38XhqYPlTv+hKGqzcy7eePx0EGXuM2kvEK3nbzU2TV0nxpRgL4XUKxu/euSPEiMA59qhhJ9xrusjVHIZfwTXIXUKvUakXD9kNDlE82IOdJh1PkFZd+xJiFK4Sa3TId1kSVpEmGgmsOt2Y5G2BvE8P1r9u+iXtQMXTPOirnKwSgPSMMGbEIBDDiECpxCFWrA4Q4e4AmelXvlUXlRXkfWlDKeWYcJKW9fJQCg2A==</latexit><latexit sha1_base64="vfDaR9CvuErqFx8LxD7Q6d6+34Y=">AAACFXicfZDLSsNQEIYn9VbjrerSTbEVqmhIKmi7EAtuXFawVTC1nByn7aG5kTORlpB3cOebuBNdiFs3bnwLH8H0IlhE/9XwzT8zzG/5tpCk6x9Kamp6ZnYuPa8uLC4tr2RW1+rSCwOONe7ZXnBpMYm2cLFGgmy89ANkjmXjhdU9GfQvbjGQwnPPqe9jw2FtV7QEZ5SgZmYnz5oRxgXaPjJJ2DcYsXgM8DraE6bnYJtlKc6rzUxO1/RySd8vZ38XhqYPlTv+hKGqzcy7eePx0EGXuM2kvEK3nbzU2TV0nxpRgL4XUKxu/euSPEiMA59qhhJ9xrusjVHIZfwTXIXUKvUakXD9kNDlE82IOdJh1PkFZd+xJiFK4Sa3TId1kSVpEmGgmsOt2Y5G2BvE8P1r9u+iXtQMXTPOirnKwSgPSMMGbEIBDDiECpxCFWrA4Q4e4AmelXvlUXlRXkfWlDKeWYcJKW9fJQCg2A==</latexit><latexit sha1_base64="vfDaR9CvuErqFx8LxD7Q6d6+34Y=">AAACFXicfZDLSsNQEIYn9VbjrerSTbEVqmhIKmi7EAtuXFawVTC1nByn7aG5kTORlpB3cOebuBNdiFs3bnwLH8H0IlhE/9XwzT8zzG/5tpCk6x9Kamp6ZnYuPa8uLC4tr2RW1+rSCwOONe7ZXnBpMYm2cLFGgmy89ANkjmXjhdU9GfQvbjGQwnPPqe9jw2FtV7QEZ5SgZmYnz5oRxgXaPjJJ2DcYsXgM8DraE6bnYJtlKc6rzUxO1/RySd8vZ38XhqYPlTv+hKGqzcy7eePx0EGXuM2kvEK3nbzU2TV0nxpRgL4XUKxu/euSPEiMA59qhhJ9xrusjVHIZfwTXIXUKvUakXD9kNDlE82IOdJh1PkFZd+xJiFK4Sa3TId1kSVpEmGgmsOt2Y5G2BvE8P1r9u+iXtQMXTPOirnKwSgPSMMGbEIBDDiECpxCFWrA4Q4e4AmelXvlUXlRXkfWlDKeWYcJKW9fJQCg2A==</latexit><latexit sha1_base64="vfDaR9CvuErqFx8LxD7Q6d6+34Y=">AAACFXicfZDLSsNQEIYn9VbjrerSTbEVqmhIKmi7EAtuXFawVTC1nByn7aG5kTORlpB3cOebuBNdiFs3bnwLH8H0IlhE/9XwzT8zzG/5tpCk6x9Kamp6ZnYuPa8uLC4tr2RW1+rSCwOONe7ZXnBpMYm2cLFGgmy89ANkjmXjhdU9GfQvbjGQwnPPqe9jw2FtV7QEZ5SgZmYnz5oRxgXaPjJJ2DcYsXgM8DraE6bnYJtlKc6rzUxO1/RySd8vZ38XhqYPlTv+hKGqzcy7eePx0EGXuM2kvEK3nbzU2TV0nxpRgL4XUKxu/euSPEiMA59qhhJ9xrusjVHIZfwTXIXUKvUakXD9kNDlE82IOdJh1PkFZd+xJiFK4Sa3TId1kSVpEmGgmsOt2Y5G2BvE8P1r9u+iXtQMXTPOirnKwSgPSMMGbEIBDDiECpxCFWrA4Q4e4AmelXvlUXlRXkfWlDKeWYcJKW9fJQCg2A==</latexit><latexit sha1_base64="lYpxvKkXiKRgIqiBvu7p6agqY1E=">AAACFXicfZDLSsNQEIZP6q3GW9Slm2ArVNGQVNB2IRTcuKxgL9DUcnI6bQ/NjZyJtIS8gzvfxJ3oQty6cePbmNYIlqL/avjmnxnmt3ybC9T1TymzsLi0vJJdldfWNza3lO2duvDCgEGNebYXNC0qwOYu1JCjDU0/AOpYNjSs4eWk37iDQHDPvcGxD22H9l3e44xigjrKUZ52IogLeHhhIre7ENE4BXAbnXDTc6BPVYzzckfJ6ZpeLumnZXW+MDR9qhxJVe0oH2bXY6EDLjKbCtECt5+8NDg2dB/bUQC+F2AsH/zrEixIjBOfbIYCfMqGtA9RyET8G7RC7JVG7Yi7fojgsplmRB3hUBzMQTF2rFkIgrvJLdOhQ6BJmogQyOZ0qzrQEEaTGH5+Vf8u6kXN0DXjupirnKWBZMke2ScFYpBzUiFXpEpqhJF78kieyYv0ID1Jr9LbtzUjpTO7ZEbS+xeHkJ+p</latexit>

Parameters: Rabi frequency ≠c ¥ 2~µeg~≤E /fl¥≠e°i' and detuning ¢=!°!eg<latexit sha1_base64="IzEoayQPS/h+RImUxyrQ6KZLzes=">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</latexit><latexit sha1_base64="IzEoayQPS/h+RImUxyrQ6KZLzes=">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</latexit><latexit 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sha1_base64="aXpr44JMNcbXf77zn7gNFZlHxnA=">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</latexit><latexit sha1_base64="hPhGQJfkPf2zzBZbdCiK8jAW9pQ=">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</latexit><latexit sha1_base64="wjckFxTURzHyFBoEHpeomUHhDfM=">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</latexit><latexit 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sha1_base64="IzEoayQPS/h+RImUxyrQ6KZLzes=">AAACinicfVHBbtNAEF0bCsUUSOHIxSJBQoIaOxRoQIhKFIkbAZG2Uje1xpuJvYq9XnbXUSPL38JPceEv+ATWMUhEFcxlZ968mad9k8icaxOGPxz3ytWta9e3b3g3d27dvtPbvXusy0oxnLAyL9VpAhpzLnBiuMnxVCqEIsnxJFm8a/snS1Sal+KLWUmcFpAKPucMjIXi3rcxKCjQWMor/zMk3J8r/FqhYCt/QD8WmEJcs4ZajC+HdImspkXVxDWmTVeh1DwvRUMLMBmDvH7fPKVZAqqb6Xb4eF7vcboEJTPeDDwQM3+GphJcpFbnCHMDb2jZUve6Z60w8OJePwzC0UH4bORfTqIgXEf/7U+yjnHc+05nJasKFIbloPUZitT6mD2JQmmmtUJZKtN4D//L0kxZYsvzaKVRAltAinXFdPM3cFaZ+cHFtOZCVsZattGsodCtJZdAvSqSTRA1F1bLOrhAsCc09hweXW/1s8DgRWvDn7/6/06Oh0EUBtGnYf/wRecH2Sb3yQPyiETkJTkkH8iYTAhztpzHzr7z3N1xh+7Ifd1RXef3zD2yEe7RL/scyc8=</latexit><latexit 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sha1_base64="IzEoayQPS/h+RImUxyrQ6KZLzes=">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</latexit><latexit sha1_base64="kcAvkcINpWhTRaOw3SrWcV8gCKM=">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</latexit>

emissionabsorption

Transformation to “rotating frame”:

H =°fl¢|eihe|° 12fl≠e

°i'|eihg |° 12fl≠e

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Validity ≠, |¢|ø!º!eg<latexit sha1_base64="oswoDySoY9zCTJaMbaBcaBKiUD4=">AAACGHicfZDPSsNAEMYn/q3xX9Wjl2IreNCQVND2ZEEP3qxgVXBL2axjGrpJlt2NtKR9Cm++iTfRg3gVL76Fj2DaKliKfqeZ33yzy3yu4L7Stv1hTExOTc/MZubM+YXFpeXsyuq5imLJsMYiHslLlyrkfog17WuOl0IiDVyOF27rsD+/uEWp/Cg80x2B9YB6oX/jM6pT1MjuFMhJgB7d7pIj5Jp2Ceck6hNChZBRe9g0EvR6BbORzduWXS7Zu+XceOFY9kD5g08YqNrIvpPriMUBhppxqtQVhl56VXPbsYWuJxJFJHXP3PzXpZhMjX2fSWKFgrIW9TCJmer9Blexvim164kfilhjyEaGCQ1UQHVzDKpO4I5CVH6Y/kUC2kKaBqo1SpMMXs01LY3tfgw/t+b+Ls6LlmNbzmkxX9kb5gEZWIcN2AIH9qECx1CFGjC4gwd4gmfj3ng0XozXoXXC+N5ZgxEZb1/8fqML</latexit><latexit sha1_base64="oswoDySoY9zCTJaMbaBcaBKiUD4=">AAACGHicfZDPSsNAEMYn/q3xX9Wjl2IreNCQVND2ZEEP3qxgVXBL2axjGrpJlt2NtKR9Cm++iTfRg3gVL76Fj2DaKliKfqeZ33yzy3yu4L7Stv1hTExOTc/MZubM+YXFpeXsyuq5imLJsMYiHslLlyrkfog17WuOl0IiDVyOF27rsD+/uEWp/Cg80x2B9YB6oX/jM6pT1MjuFMhJgB7d7pIj5Jp2Ceck6hNChZBRe9g0EvR6BbORzduWXS7Zu+XceOFY9kD5g08YqNrIvpPriMUBhppxqtQVhl56VXPbsYWuJxJFJHXP3PzXpZhMjX2fSWKFgrIW9TCJmer9Blexvim164kfilhjyEaGCQ1UQHVzDKpO4I5CVH6Y/kUC2kKaBqo1SpMMXs01LY3tfgw/t+b+Ls6LlmNbzmkxX9kb5gEZWIcN2AIH9qECx1CFGjC4gwd4gmfj3ng0XozXoXXC+N5ZgxEZb1/8fqML</latexit><latexit sha1_base64="oswoDySoY9zCTJaMbaBcaBKiUD4=">AAACGHicfZDPSsNAEMYn/q3xX9Wjl2IreNCQVND2ZEEP3qxgVXBL2axjGrpJlt2NtKR9Cm++iTfRg3gVL76Fj2DaKliKfqeZ33yzy3yu4L7Stv1hTExOTc/MZubM+YXFpeXsyuq5imLJsMYiHslLlyrkfog17WuOl0IiDVyOF27rsD+/uEWp/Cg80x2B9YB6oX/jM6pT1MjuFMhJgB7d7pIj5Jp2Ceck6hNChZBRe9g0EvR6BbORzduWXS7Zu+XceOFY9kD5g08YqNrIvpPriMUBhppxqtQVhl56VXPbsYWuJxJFJHXP3PzXpZhMjX2fSWKFgrIW9TCJmer9Blexvim164kfilhjyEaGCQ1UQHVzDKpO4I5CVH6Y/kUC2kKaBqo1SpMMXs01LY3tfgw/t+b+Ls6LlmNbzmkxX9kb5gEZWIcN2AIH9qECx1CFGjC4gwd4gmfj3ng0XozXoXXC+N5ZgxEZb1/8fqML</latexit><latexit sha1_base64="oswoDySoY9zCTJaMbaBcaBKiUD4=">AAACGHicfZDPSsNAEMYn/q3xX9Wjl2IreNCQVND2ZEEP3qxgVXBL2axjGrpJlt2NtKR9Cm++iTfRg3gVL76Fj2DaKliKfqeZ33yzy3yu4L7Stv1hTExOTc/MZubM+YXFpeXsyuq5imLJsMYiHslLlyrkfog17WuOl0IiDVyOF27rsD+/uEWp/Cg80x2B9YB6oX/jM6pT1MjuFMhJgB7d7pIj5Jp2Ceck6hNChZBRe9g0EvR6BbORzduWXS7Zu+XceOFY9kD5g08YqNrIvpPriMUBhppxqtQVhl56VXPbsYWuJxJFJHXP3PzXpZhMjX2fSWKFgrIW9TCJmer9Blexvim164kfilhjyEaGCQ1UQHVzDKpO4I5CVH6Y/kUC2kKaBqo1SpMMXs01LY3tfgw/t+b+Ls6LlmNbzmkxX9kb5gEZWIcN2AIH9qECx1CFGjC4gwd4gmfj3ng0XozXoXXC+N5ZgxEZb1/8fqML</latexit><latexit sha1_base64="6dERv0MKoNywelMkCDvtE2LufoM=">AAACGHicfZDLSsNAFIYn9VbjLerSTbEVXNSQVNB2V9CFOyvYC3RKmYynaegkGWYm0pL2Kdz5Ju5EF+JW3Pg2phfBUvRfnfOd/8xwfoczTyrL+tJSS8srq2vpdX1jc2t7x9jdq8kwEhSqNGShaDhEAvMCqCpPMWhwAcR3GNSd3sV4Xr8HIb0wuFUDDi2fuIHX8ShRCWobJzl87YNL8kN8CUyRIWYMh2OCCeci7E+bdgzuKKe3jaxlWqWidVrKLBa2aU2URTNV2sYnvgtp5EOgKCNSNiFwk6u6edviqhUL4KFQI/3oX5ekIjGOfTqOJHBCe8SFOKJy9Bs0I9Up9luxF/BIQUDnhjHxpU9UdwHKge/MQ5BekPyFfdIDkgSqFAgdT17NdE0F/XEMP7dm/i5qBdO2TPumkC2fzQJJowN0iI6Rjc5RGV2hCqoiih7QE3pBr9qj9qy9ae9Ta0qb7eyjOWkf318dodw=</latexit>

Hamiltonian in “rotating frame”


Two-Level Atom

H =°fl¢|eihe|° 12fl≠e

°i'|eihg |° 12fl≠e

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Hamiltonian in “rotating frame”

|g i

|eilaser

Discussion: on-resonance Rabi oscillations (¢= 0)<latexit sha1_base64="8OlGPRtPVvhURLzV/Go48ZaEW3U=">AAAB9nicfVBLSwJRGP3GXja9rJZtJA0MYrhjULqIhFq0NMgHOIPcuX3qxXl1751QBn9Hu6hFtO3HtOlf9BMatSCROquPc873Ok7ocqkI+dBSC4tLyyvpVX1tfWNzK7O9U5dBJBjWWOAGoulQiS73saa4crEZCqSe42LD6V+M9cY9CskD/0YNQ7Q92vV5hzOqEsou5K1LdBU9I/lDvZ3JEYOUS+S4nJ0vTINMkDv/hAmq7cy7dRuwyENfMZdK2UK/m9zcOzJJqOxYYBgINdIP/nVJJhLj2KdbkcSQsj7tYhwxOfpNtCLVKQ3smPthpNBnM2JMPelR1Zsj5dBzZkmU3E92WR7tI03iUgqFbk2mZnuGwsE4hp9fs38X9aJhEsO8LuYqJ9M8IA17sA8FMOEUKnAFVagBgzt4gCd41gbao/aivU6tKe27ZxdmoL19AYZkk9g=</latexit><latexit sha1_base64="8OlGPRtPVvhURLzV/Go48ZaEW3U=">AAAB9nicfVBLSwJRGP3GXja9rJZtJA0MYrhjULqIhFq0NMgHOIPcuX3qxXl1751QBn9Hu6hFtO3HtOlf9BMatSCROquPc873Ok7ocqkI+dBSC4tLyyvpVX1tfWNzK7O9U5dBJBjWWOAGoulQiS73saa4crEZCqSe42LD6V+M9cY9CskD/0YNQ7Q92vV5hzOqEsou5K1LdBU9I/lDvZ3JEYOUS+S4nJ0vTINMkDv/hAmq7cy7dRuwyENfMZdK2UK/m9zcOzJJqOxYYBgINdIP/nVJJhLj2KdbkcSQsj7tYhwxOfpNtCLVKQ3smPthpNBnM2JMPelR1Zsj5dBzZkmU3E92WR7tI03iUgqFbk2mZnuGwsE4hp9fs38X9aJhEsO8LuYqJ9M8IA17sA8FMOEUKnAFVagBgzt4gCd41gbao/aivU6tKe27ZxdmoL19AYZkk9g=</latexit><latexit sha1_base64="8OlGPRtPVvhURLzV/Go48ZaEW3U=">AAAB9nicfVBLSwJRGP3GXja9rJZtJA0MYrhjULqIhFq0NMgHOIPcuX3qxXl1751QBn9Hu6hFtO3HtOlf9BMatSCROquPc873Ok7ocqkI+dBSC4tLyyvpVX1tfWNzK7O9U5dBJBjWWOAGoulQiS73saa4crEZCqSe42LD6V+M9cY9CskD/0YNQ7Q92vV5hzOqEsou5K1LdBU9I/lDvZ3JEYOUS+S4nJ0vTINMkDv/hAmq7cy7dRuwyENfMZdK2UK/m9zcOzJJqOxYYBgINdIP/nVJJhLj2KdbkcSQsj7tYhwxOfpNtCLVKQ3smPthpNBnM2JMPelR1Zsj5dBzZkmU3E92WR7tI03iUgqFbk2mZnuGwsE4hp9fs38X9aJhEsO8LuYqJ9M8IA17sA8FMOEUKnAFVagBgzt4gCd41gbao/aivU6tKe27ZxdmoL19AYZkk9g=</latexit><latexit sha1_base64="8OlGPRtPVvhURLzV/Go48ZaEW3U=">AAAB9nicfVBLSwJRGP3GXja9rJZtJA0MYrhjULqIhFq0NMgHOIPcuX3qxXl1751QBn9Hu6hFtO3HtOlf9BMatSCROquPc873Ok7ocqkI+dBSC4tLyyvpVX1tfWNzK7O9U5dBJBjWWOAGoulQiS73saa4crEZCqSe42LD6V+M9cY9CskD/0YNQ7Q92vV5hzOqEsou5K1LdBU9I/lDvZ3JEYOUS+S4nJ0vTINMkDv/hAmq7cy7dRuwyENfMZdK2UK/m9zcOzJJqOxYYBgINdIP/nVJJhLj2KdbkcSQsj7tYhwxOfpNtCLVKQ3smPthpNBnM2JMPelR1Zsj5dBzZkmU3E92WR7tI03iUgqFbk2mZnuGwsE4hp9fs38X9aJhEsO8LuYqJ9M8IA17sA8FMOEUKnAFVagBgzt4gCd41gbao/aivU6tKe27ZxdmoL19AYZkk9g=</latexit><latexit sha1_base64="iQHdXRPFnl2Im8dgujHusDCEzmg=">AAAB9nicfVDLSsNAFL2prxpfVZduiq1QQcKkgrYLoaALlxXsA5pSJuNtOzSZxJmJtIR+hzvRhbj1Y9z4N6a1gqXoWV3OOfd13NDjShPyaaSWlldW19Lr5sbm1vZOZnevroJIMqyxwAtk06UKPS6wprn2sBlKpL7rYcMdXE70xgNKxQNxq0chtn3aE7zLGdUJ1S7knSv0NL0g+WOzk8kRi5RL5LScXSxsi0yRgxmqncyHcxewyEehmUeVaqHoJTf3T2wS6nYsMQykHptH/7oUk4lx4jOdSGFI2YD2MI6YGv8mWpHulobtmIsw0ijYnBhTX/lU9xdINfLdeRIVF8kux6cDpElcWqM0nenUbN/SOJzE8PNr9u+iXrRsYtk3xVzlbBZIGg7gEApgwzlU4BqqUAMG9/AIz/BiDI0n49V4+7amjFnPPszBeP8C6PSSqQ==</latexit>

Ut = e°i H t/fl =

µcos 1

2≠t °i e°i' sin 1

2≠t

°i e+i' sin 1

2≠t cos 12≠t

∂

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eg

Examples:

• transition probability g → e

Pe√g (t ) = 12

[1°cos≠t ]<latexit sha1_base64="oB9j21zABZLPzx7etG+s68hRS1U=">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</latexit><latexit sha1_base64="oB9j21zABZLPzx7etG+s68hRS1U=">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</latexit><latexit sha1_base64="oB9j21zABZLPzx7etG+s68hRS1U=">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</latexit><latexit sha1_base64="oB9j21zABZLPzx7etG+s68hRS1U=">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</latexit><latexit sha1_base64="6Gr2lkF7LjfyTYfdO1lLC11veQU=">AAACLXicfVDLSsNAFJ34rPFVdekmWAQFDUkFHwuh6MadFWwrdEKZjLfp0LyYudGWkP9x55+4UlQQt/6Gaa1gET2ryznn3ss5buwLhZb1qk1MTk3PzBbm9PmFxaXl4spqXUWJ5FDjkR/JK5cp8EUINRTow1UsgQWuDw23ezrQGzcglYjCS+zH4ATMC0VbcIY51Sqe0KZebaVAfWgjkzK6NbxsC7ePaVsyntpZWs6GWtPepTxS9DwAjxlIpfA66OjU0VvFkmVaR4fW3pHxe7BNa4gSGaHaKj7S64gnAYTIfaZUE0Ivj9rZsa0YnVRCHEnM9M1/XYrL3Djw6TRREDPeZR6kCVfZT6KZYPuw56QijBOEkI+JKQtUwLDzi1T9wB0nQYkw/0UD1gWWt4wIUqfDq0bHROgNavjOavw91MumbZn2RblU2R8VUiDrZINsEZsckAo5I1VSI5zckQfyTF60e+1Je9Pev6wT2mhnjYxB+/gEF66pRw==</latexit>

º-pulse ≠t =º inverts the TLS<latexit sha1_base64="iF+3j1tALUX8wBzGmClmiFWM03Q=">AAACF3icfZDNSsNAFIVv/K3xr+rSTbAKLjQkCmoXYsGNC8GKVoWmlMl4bYcmkyFzIy2lL+HON3EnuhC3ghvfwkcwbRUsRc/q8N1zZ7jHV4HQ5Dgfxsjo2PjEZGbKnJ6ZnZvPLixe6CiJOZZ4FETxlc80BkJiiQQFeKViZKEf4KXfOOzOL28x1iKS59RSWAlZTYobwRmlqJrdsFY9JVY3VRJoTP1JiDVm0X4XWkKmq6QtqqN1fnxmVrM5x3bye8523ho2ru30lDv4hJ6K1ey7dx3xJERJPGBal1HW0qPqG66jqNKOUUUxdcy1f1Oax2mwmzO9RKNivMFq2E647vwG5YRu9pqVtpAqIZR8YNhmoQ4Z1YegboX+IEQtZPqXF7IGsrRPIoxNr/eqVbcJm90afm61/jYXW7br2O7pVq6w0+8DMrAMK7AOLuxCAY6gCCXgcAcP8ATPxr3xaLwYr/3oiPG9swQDMt6+AFRGoDg=</latexit><latexit sha1_base64="iF+3j1tALUX8wBzGmClmiFWM03Q=">AAACF3icfZDNSsNAFIVv/K3xr+rSTbAKLjQkCmoXYsGNC8GKVoWmlMl4bYcmkyFzIy2lL+HON3EnuhC3ghvfwkcwbRUsRc/q8N1zZ7jHV4HQ5Dgfxsjo2PjEZGbKnJ6ZnZvPLixe6CiJOZZ4FETxlc80BkJiiQQFeKViZKEf4KXfOOzOL28x1iKS59RSWAlZTYobwRmlqJrdsFY9JVY3VRJoTP1JiDVm0X4XWkKmq6QtqqN1fnxmVrM5x3bye8523ho2ru30lDv4hJ6K1ey7dx3xJERJPGBal1HW0qPqG66jqNKOUUUxdcy1f1Oax2mwmzO9RKNivMFq2E647vwG5YRu9pqVtpAqIZR8YNhmoQ4Z1YegboX+IEQtZPqXF7IGsrRPIoxNr/eqVbcJm90afm61/jYXW7br2O7pVq6w0+8DMrAMK7AOLuxCAY6gCCXgcAcP8ATPxr3xaLwYr/3oiPG9swQDMt6+AFRGoDg=</latexit><latexit sha1_base64="iF+3j1tALUX8wBzGmClmiFWM03Q=">AAACF3icfZDNSsNAFIVv/K3xr+rSTbAKLjQkCmoXYsGNC8GKVoWmlMl4bYcmkyFzIy2lL+HON3EnuhC3ghvfwkcwbRUsRc/q8N1zZ7jHV4HQ5Dgfxsjo2PjEZGbKnJ6ZnZvPLixe6CiJOZZ4FETxlc80BkJiiQQFeKViZKEf4KXfOOzOL28x1iKS59RSWAlZTYobwRmlqJrdsFY9JVY3VRJoTP1JiDVm0X4XWkKmq6QtqqN1fnxmVrM5x3bye8523ho2ru30lDv4hJ6K1ey7dx3xJERJPGBal1HW0qPqG66jqNKOUUUxdcy1f1Oax2mwmzO9RKNivMFq2E647vwG5YRu9pqVtpAqIZR8YNhmoQ4Z1YegboX+IEQtZPqXF7IGsrRPIoxNr/eqVbcJm90afm61/jYXW7br2O7pVq6w0+8DMrAMK7AOLuxCAY6gCCXgcAcP8ATPxr3xaLwYr/3oiPG9swQDMt6+AFRGoDg=</latexit><latexit sha1_base64="iF+3j1tALUX8wBzGmClmiFWM03Q=">AAACF3icfZDNSsNAFIVv/K3xr+rSTbAKLjQkCmoXYsGNC8GKVoWmlMl4bYcmkyFzIy2lL+HON3EnuhC3ghvfwkcwbRUsRc/q8N1zZ7jHV4HQ5Dgfxsjo2PjEZGbKnJ6ZnZvPLixe6CiJOZZ4FETxlc80BkJiiQQFeKViZKEf4KXfOOzOL28x1iKS59RSWAlZTYobwRmlqJrdsFY9JVY3VRJoTP1JiDVm0X4XWkKmq6QtqqN1fnxmVrM5x3bye8523ho2ru30lDv4hJ6K1ey7dx3xJERJPGBal1HW0qPqG66jqNKOUUUxdcy1f1Oax2mwmzO9RKNivMFq2E647vwG5YRu9pqVtpAqIZR8YNhmoQ4Z1YegboX+IEQtZPqXF7IGsrRPIoxNr/eqVbcJm90afm61/jYXW7br2O7pVq6w0+8DMrAMK7AOLuxCAY6gCCXgcAcP8ATPxr3xaLwYr/3oiPG9swQDMt6+AFRGoDg=</latexit><latexit sha1_base64="6j8Vufvqw06F9tt+qvTRJvxPFQ8=">AAACF3icfZBLS8NAFIUn9VXjq+rSTbAVXNSQVNB2IRTcuBCs2Bc0pUzG23ZoMhkyN9JS+ifc+U/ciS7EreDGf2NaK1iKntXhu+fOcI8rPa7Qsj61xMLi0vJKclVfW9/Y3Ept71RVEIUMKizwgrDuUgUeF1BBjh7UZQjUdz2oub3z8bx2B6HigSjjQELTpx3B25xRjFErlTUyjuSZIxl5CmJ/5UOHGng2hgYX8SoqA7tglC9v9FYqbZlWIW8dF4x5Y5vWRGkyVamV+nBuAxb5IJB5VKkGiE58VDdrWxKbwxBkEOJIP/g3pVgYB8c53YkUSMp6tAPDiKnRb9CIsJ3vN4dcyAhBsJnhkPrKp9idg2rgu7MQFBfxX45Pe0DjPhEh1J3Jq0bXROiPa/i51fjbVHOmbZn2dS5dPJkWkiR7ZJ8cEpuckiK5ICVSIYzck0fyTF60B+1Je9XevqMJbbqzS2akvX8BttafCQ==</latexit>

2º pulse<latexit sha1_base64="mppQFQ7T35Bq6uLY5DZdi70N1G4=">AAAB9nicfVC7TgJBFL2LL1xfqKXNRjCxMJtZTBQqSWwsMZFHwm7I7HiBCfsYZ2YNhPAddkYLY+vH2PgXfoILaCIheqqbc859HV8EXGlCPozM0vLK6lp23dzY3Nreye3u1VWcSIY1FgexbPpUYcAjrGmuA2wKiTT0A2z4/cuJ3rhHqXgc3eihQC+k3Yh3OKM6pbxC0RW8YIkkUGi2c3lik3KJnJatxcKxyRT5i0+YotrOvbu3MUtCjDQLqFItjLrpzb0ThwjtjSSKWOqxefSvSzGZGic+000UCsr6tIujhKnxb6KV6E5p4I14JBKNEZsTRzRUIdW9BVINQ3+eRMWjdJcb0j7SNC6tUZrudKrVszUOJjH8/Gr9XdSLtkNs57qYr5zN8oAsHMAhHIMD51CBK6hCDRjcwQM8wbMxMB6NF+N1Zs0Y3z37MAfj7QuNWpSE</latexit><latexit sha1_base64="mppQFQ7T35Bq6uLY5DZdi70N1G4=">AAAB9nicfVC7TgJBFL2LL1xfqKXNRjCxMJtZTBQqSWwsMZFHwm7I7HiBCfsYZ2YNhPAddkYLY+vH2PgXfoILaCIheqqbc859HV8EXGlCPozM0vLK6lp23dzY3Nreye3u1VWcSIY1FgexbPpUYcAjrGmuA2wKiTT0A2z4/cuJ3rhHqXgc3eihQC+k3Yh3OKM6pbxC0RW8YIkkUGi2c3lik3KJnJatxcKxyRT5i0+YotrOvbu3MUtCjDQLqFItjLrpzb0ThwjtjSSKWOqxefSvSzGZGic+000UCsr6tIujhKnxb6KV6E5p4I14JBKNEZsTRzRUIdW9BVINQ3+eRMWjdJcb0j7SNC6tUZrudKrVszUOJjH8/Gr9XdSLtkNs57qYr5zN8oAsHMAhHIMD51CBK6hCDRjcwQM8wbMxMB6NF+N1Zs0Y3z37MAfj7QuNWpSE</latexit><latexit sha1_base64="mppQFQ7T35Bq6uLY5DZdi70N1G4=">AAAB9nicfVC7TgJBFL2LL1xfqKXNRjCxMJtZTBQqSWwsMZFHwm7I7HiBCfsYZ2YNhPAddkYLY+vH2PgXfoILaCIheqqbc859HV8EXGlCPozM0vLK6lp23dzY3Nreye3u1VWcSIY1FgexbPpUYcAjrGmuA2wKiTT0A2z4/cuJ3rhHqXgc3eihQC+k3Yh3OKM6pbxC0RW8YIkkUGi2c3lik3KJnJatxcKxyRT5i0+YotrOvbu3MUtCjDQLqFItjLrpzb0ThwjtjSSKWOqxefSvSzGZGic+000UCsr6tIujhKnxb6KV6E5p4I14JBKNEZsTRzRUIdW9BVINQ3+eRMWjdJcb0j7SNC6tUZrudKrVszUOJjH8/Gr9XdSLtkNs57qYr5zN8oAsHMAhHIMD51CBK6hCDRjcwQM8wbMxMB6NF+N1Zs0Y3z37MAfj7QuNWpSE</latexit><latexit sha1_base64="mppQFQ7T35Bq6uLY5DZdi70N1G4=">AAAB9nicfVC7TgJBFL2LL1xfqKXNRjCxMJtZTBQqSWwsMZFHwm7I7HiBCfsYZ2YNhPAddkYLY+vH2PgXfoILaCIheqqbc859HV8EXGlCPozM0vLK6lp23dzY3Nreye3u1VWcSIY1FgexbPpUYcAjrGmuA2wKiTT0A2z4/cuJ3rhHqXgc3eihQC+k3Yh3OKM6pbxC0RW8YIkkUGi2c3lik3KJnJatxcKxyRT5i0+YotrOvbu3MUtCjDQLqFItjLrpzb0ThwjtjSSKWOqxefSvSzGZGic+000UCsr6tIujhKnxb6KV6E5p4I14JBKNEZsTRzRUIdW9BVINQ3+eRMWjdJcb0j7SNC6tUZrudKrVszUOJjH8/Gr9XdSLtkNs57qYr5zN8oAsHMAhHIMD51CBK6hCDRjcwQM8wbMxMB6NF+N1Zs0Y3z37MAfj7QuNWpSE</latexit><latexit sha1_base64="2J3LgaxveNOWm0ixMrYwzobgJ8I=">AAAB9nicfVDLSsNAFL2prxpfVZdugq3gQsKkgra7ghuXFewDmlAm4207NI9xZiItod/hTnQhbv0YN/6Naa1gKXpWl3POfR1fBFxpQj6N3Mrq2vpGftPc2t7Z3SvsHzRVnEiGDRYHsWz7VGHAI2xorgNsC4k09ANs+cOrqd56QKl4HN3qsUAvpP2I9zijOqO8UtkVvGSJJFBodgtFYpNqhZxXreXCsckMRZij3i18uHcxS0KMNAuoUh2M+tnNgzOHCO2lEkUs9cQ8+delmMyMU5/pJgoFZUPaxzRhavKb6CS6Vxl5KY9EojFiC2JKQxVSPVgi1Tj0F0lUPMp2uSEdIs3i0hql6c6mWgNb42gaw8+v1t9Fs2w7xHZuysXaxTyQPBzBMZyCA5dQg2uoQwMY3MMjPMOLMTKejFfj7duaM+Y9h7AA4/0L7+qTVQ==</latexit>

Ut=º/≠ =µ

0 °i e°i'

°i e+i' 0

∂

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Ut=2º/≠ =µ°1 00 °1

∂

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t

Δ=2Ω

Δ=0

Rabi oscillations


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Quantum State Engineering & Quantum Computingwith Trapped Ions

P1/2

S1/2

D5/2

40Ca+

Quantumbit

QubitDetecLon,

LaserCooling

uibkSingle Laser-Driven Trapped Ion

traplaser

spontaneous emission

ion


• system = internal + external degrees of freedom

internal:electronic levels

external: motion

dipole-allowedtransition

dipole-forbiddentransition

• strong dissipation

ü laser cooling / state preparation

ü state measurement

• small dissipation

ü Hamiltonian: quantum state engineering

The System


laser

ion

laser

ion


optical wavelength

trap size


blue sideband

red sideband

…


…


red sideband


blue sideband


blue sideband

red sideband

…


…

uibkString of Laser-Driven Trapped Ions

Xlaser


laser


center of mass mode

stretchmode

ºfor N ions the separation remains with increasing N

º


center of mass mode

stretchmode

º for N ions the separation remains with increasing N


Background Material

Basic Quantum Computing

uibkQuantum Computing

system

input

ouput

• computing as a physical process

time

• quantum computing

- preparation

- unitary evolution

- state measurement

• no coupling to environment (decoherence)

uibkQuantum Memory

• Qubit:

superposition

• Quantum register

entangled stateN spin-1/2 systems

spin-1/2

-quantum parallelism

- interference of computational paths(+ cleverness) = quantum algorithms

uibkQuantum Gates

• Single qubit gate

with U unitary

U1

• A general unitary transformation can be decomposed into single bit rotations and a universal two-bit quantum gate

U1

targetcontrol

• Two qubit gate

uibk3. Read out

• state measurement

0 1 0 1 1 0


• Cold ions in a linear trap

Qubits: internal atomic states

1-qubit gates: addressing ions with a laser

2-qubit gates: entanglement via exchange of phonons of quantized collective mode

• State vector

quantum register databus

• QC as a time sequence of laser pulses • Read out by quantum jumps

Ion Trap Quantum Computer '95


state measurement via quantum jumps

qubit

auxiliary level

addressing with different light polarizations

Level scheme


• step 1: swap first qubit to phonon

laser

m nπ pulse

first atom: m

...

...

Two-qubit phase gate


• step 2: conditional sign change

laser

m n

second atom: n

-

2 pulse

flip sign

...

...


• step 3: swap phonon back to first qubit

laser

atom m


• summary: we have a phase gate between atom m and n

phonon mode returned to initial state

Rem.: this idea translates immediately to CQED


input

output

Quantum gates with ions: Nature March 2003

truth table CNOT

Ibk


Where we are today …

Digital and Analog Quantum Simulation with Trapped Ions

uibkDigital Quantum Simulation

idea: approximate time evolution by a stroboscopic sequence of gates

qubitsor

spins

time0

...

tt1

eiHeff t

H = Jz1z

2 + B(x1 + x

2 )

U(t) eiHt/ = eiHtn/ . . .iHt1/

eiHt/ eiH1t/ eiH2t/ e12

(t)2

2 [H1,H2]

first term second termTrotter errors for

non-commuting terms

Trotter expansion:

Seth Lloyd

when subducted lithosphere entered the shallowlower mantle and stagnated because of densityinversion and increased mantle viscosity (14, 27)(Fig. 3B). If heated to ambient mantle tempera-tures, carbonated basaltic lithologies form carbo-natedmelts, which can then be reduced to diamondduring reactions with surrounding mantle (8, 28).Our results also indicate that the diamonds weretransported by convection from the lower to theupper mantle, where the originally homogeneousinclusions unmixed. For example, phase relationsalong the NaAlSiO4-MgAl2O4 boundary (29) in-dicate that the bulk composition of inclusionJu5-20 would yield the observed assemblage ofnepheline plus spinel (Fig. 1A and fig. S1B) atdepths of ~150 km; other inclusions in diamondsfrom the Juina region (3, 4, 8) also suggest equil-ibration near the base of the Brazilian lithosphere(~150 to 200 km). Thus, the diamonds record ahistory of upward transport on the order of 500to 1000 km or more before being sampled by aCretaceous kimberlite and brought to the surface.

On the basis of seismological and petrologicalevidence, previous workers have argued for a man-tle plume beneath Brazil during the Cretaceous(30, 31). Furthermore, paleo-plate reconstruc-tions show that the Juina region of Brazil was lo-cated at the margin of the African large low shearvelocity provinces during the Cretaceous, whichmay be indicative of the presence of deep mantleplumes (32). We suggest that some portion ofstagnated subducted lithosphere in which the di-amonds grew was transported from the lowermantle to the base of the Brazilian lithosphere ina rising mantle plume (Fig. 3B). The Juina-5 di-amonds and their inclusions provide compellingevidence for deep cycling of oceanic crust and

surface carbon into the lower mantle and, ulti-mately, exhumation back to the upper mantle andEarth’s surface.

References and Notes1. T. Stachel, G. P. Brey, J. W. Harris, Elements 1, 73

(2005).2. B. Harte, Mineral. Mag. 74, 189 (2010).3. G. P. Bulanova et al., Contrib. Mineral. Petrol. 160,

489 (2010).4. B. Harte, N. Cayzer, Phys. Chem. Miner. 34, 647 (2007).5. F. V. Kaminsky et al., Contrib. Mineral. Petrol. 140,

734 (2001).6. T. Stachel, G. P. Brey, J. W. Harris, Contrib. Mineral. Petrol.

140, 1 (2000).7. R. Tappert et al., Geology 33, 565 (2005).8. M. J. Walter et al., Nature 454, 622 (2008).9. R. Tappert et al., Geology 37, 43 (2009).10. P. Cartigny, Elements 1, 79 (2005).11. B. Harte, J. W. Harris, M. T. Hutchison, G. R. Watt,

M. C. Wilding, in Mantle Petrology: Field Observationsand High Pressure Experimentation, Y. Fei, C. M. Bertka,B. O. Mysen, Eds. Geochemical Society SpecialPublications, 125 (1999).

12. P. C. Hayman, M. G. Kopylova, F. V. Kaminsky,Contrib. Mineral. Petrol. 140, 734 (2005).

13. T. Stachel, J. W. Harris, G. P. Brey, W. Joswig,Contrib. Mineral. Petrol. 140, 16 (2000).

14. Y. Fukao, M. Obayashi, T. Nakakuki, Deep Slab ProjectGroup, Annu. Rev. Earth Planet. Sci. 37, 19 (2009).

15. R. D. van der Hilst, S. Widiyantoro, E. R. Engdahl, Nature386, 578 (1997).

16. K. Hirose, N. Takafuji, N. Sata, Y. Ohishi, Earth Planet.Sci. Lett. 237, 239 (2005).

17. S. Ono, E. Ito, T. Katsura, Earth Planet. Sci. Lett. 190,57 (2001).

18. A. Ricolleau et al., J. Geophys. Res. 115, (B8), B08202(2010).

19. L. Heaman, N. A. Teixeira, L. Gobbo, J. C. Gaspar, U-Pbmantle zircon ages for kimberlites from the Juina andParanatinga Provinces, Brazil. Extended Abstracts, 7thInternational Kimberlite Conference, Cape Town,South Africa, 322 (1998).

20. See supporting material on Science Online.

21. F. E. Brenker et al., Earth Planet. Sci. Lett. 236, 579(2005).

22. F. Brenker, T. Stachel, J. W. Harris, Earth Planet. Sci. Lett.198, 1 (2002).

23. B. J. Wood, Earth Planet. Sci. Lett. 174, 341 (2000).24. M. B. Kirkley, J. J. Gurney, M. L. Otter, S. J. Hill,

L. R. M. Daniels, Appl. Geochem. 6, 477 (1991).25. S. Shilobreeva, I. Martinez, V. Busigny, P. Agrinier,

C. Laverne, Geochim. Cosmochim. Acta 75, 2237 (2011).26. S. Poli, E. Franzolin, P. Fumagalli, A. Crottini, Earth

Planet. Sci. Lett. 278, 350 (2009).27. S. Goes, F. A. Capitanio, G. Morra, Nature 451, 981

(2008).28. A. Rohrbach, M. W. Schmidt, Nature 472, 209 (2011).29. M. Akaogi, A. Tanaka, M. Kobayashi, N. Fukushima,

T. Suzuki, Phys. Earth Planet. Inter. 130, 49 (2002).30. S. A. Gibson, R. N. Thompson, O. H. Leonardos,

A. P. Dickin, G. J. Mitchell, J. Petrol. 36, 89 (1995).31. J. C. VanDecar, D. E. James, M. Assumpcao, Nature 378,

25 (1995).32. T. H. Torsvik, K. Burke, B. Steinberger, S. J. Webb,

L. D. Ashwal, Nature 466, 352 (2010).33. Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov,

I. N. Kupriyanov, N. V. Sobolev, Earth Planet. Sci. Lett.250, 269 (2006).

Acknowledgments: We thank I. Buisman and S. Kearnsfor assisting in the collection of electron microprobedata and L. Gobbo on behalf of Rio Tinto for providingsamples. Supported by UK Natural Environment ResearchCouncil grant NE/H011242/1 (M.J.W.) and NSF grantEAR-1049992 (S.B.S. and J.W.). See (20) for additionalcompositional data on inclusion phases, experimental runproducts, and Raman spectroscopy.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/science.1209300/DC1Materials and MethodsSOM TextFigs. S1 to S3Tables S1 to S5References (34–45)

3 June 2011; accepted 22 August 2011Published online 15 September 2011;10.1126/science.1209300

REPORTS

Universal Digital Quantum Simulationwith Trapped IonsB. P. Lanyon,1,2* C. Hempel,1,2 D. Nigg,2 M. Müller,1,3 R. Gerritsma,1,2 F. Zähringer,1,2

P. Schindler,2 J. T. Barreiro,2 M. Rambach,1,2 G. Kirchmair,1,2 M. Hennrich,2 P. Zoller,1,3

R. Blatt,1,2 C. F. Roos1,2

A digital quantum simulator is an envisioned quantum device that can be programmed to efficientlysimulate any other local system. We demonstrate and investigate the digital approach to quantumsimulation in a system of trapped ions. With sequences of up to 100 gates and 6 qubits, the fulltime dynamics of a range of spin systems are digitally simulated. Interactions beyond those naturallypresent in our simulator are accurately reproduced, and quantitative bounds are provided for theoverall simulation quality. Our results demonstrate the key principles of digital quantum simulation andprovide evidence that the level of control required for a full-scale device is within reach.

Althoughmany natural phenomena are ac-curately described by the laws of quan-tum mechanics, solving the associated

equations to calculate properties of physical sys-tems, i.e., simulating quantum physics, is in gen-

eral thought to be very difficult (1). Both thenumber of parameters and differential equationsthat describe a quantum state and its dynamicsgrow exponentially with the number of particlesinvolved. One proposed solution is to build a

highly controllable quantum system that can ef-ficiently perform the simulations (2). Recently,quantum simulations have been performed inseveral different systems (3–13), largely follow-ing the analog approach (2) whereby an analo-gousmodel is built, with a directmapping betweenthe state and dynamics of the simulated systemand those of the simulator. An analog simulatoris dedicated to a particular problem, or class ofproblems.

A digital quantum simulator (2, 14–16) is aprecisely controllable many-body quantum sys-tem onwhich a universal set of quantum operations(gates) can be performed, i.e., a quantum computer(17). The simulated state is encoded in a register

1Institut für Quantenoptik und Quanteninformation, Öster-reichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1,A-6020 Innsbruck, Austria. 2Institut für Experimentalphysik, Uni-versity of Innsbruck, Technikerstr. 25, A-6020 Innsbruck,Austria. 3Institut für Theoretische Physik, University ofInnsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria.

*To whom correspondence should be addressed. E-mail:[email protected]

www.sciencemag.org SCIENCE VOL 334 7 OCTOBER 2011 57

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experiments (≈1 to 2 ms). The current leadingsources of error, which limit both the simulationcomplexity and size, are thought to be laserintensity fluctuations (23). This is not currentlya fundamental limitation and, once properly ad-dressed, should enable an increase in simulationcapabilities.

The digital approach can be combined withexisting tools and techniques for analog simu-lations to expand the range of systems that can besimulated. In light of the present work, and cur-

rent ion trap development (35), digital quantumsimulations involving many tens of qubits andhundreds of high-fidelity gates seems feasible incoming years.

References and Notes1. R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).2. I. Buluta, F. Nori, Science 326, 108 (2009).3. S. Somaroo, C. H. Tseng, T. F. Havel, R. Laflamme,

D. G. Cory, Phys. Rev. Lett. 82, 5381 (1999).4. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch,

I. Bloch, Nature 415, 39 (2002).

5. D. Leibfried et al., Phys. Rev. Lett. 89, 247901 (2002).6. A. Friedenauer, H. Schmitz, J. T. Glueckert, D. Porras,

T. Schaetz, Nat. Phys. 4, 757 (2008).7. R. Gerritsma et al., Nature 463, 68 (2010).8. K. Kim et al., Nature 465, 590 (2010).9. B. P. Lanyon et al., Nat. Chem. 2, 106 (2010).

10. K. R. Brown, R. J. Clark, I. L. Chuang, Phys. Rev. Lett. 97,050504 (2006).

11. J. T. Barreiro et al., Nature 470, 486 (2011).12. J. Simon et al., Nature 472, 307 (2011).13. R. Islam et al., Nat Commun 2, 377 (2011).14. S. Lloyd, Science 273, 1073 (1996).15. E. Jane, G. Vidal, W. Dür, P. Zoller, J. I. Cirac, Quantum

Inf. Comput. 3, 15 (2003).

Fig. 3. Digital simulations of three-spin systems.Dynamics of the initial state |↑↑↑⟩ in three cases.(A) Long-range Ising system. Spin-spin coupling be-tween all pairs with equal strength and a transversefield. C = O2(p /32), D = O4(p /16,0). (B) Inhomog-eneous distribution of spin-spin couplings, decom-posed into an equal-strength interaction andanother with twice the strength between one pair.E = O1(p /2,1). (C) Three-body interaction, whichcouples the ∑jsy

jeigenstates |←←←⟩y and |→→→⟩y.An O3(p /4,0) operation before measurementrotates the state into the logical sz basis. F =O1(q,1), 4D = O4(p/4,0). Any point in the phaseevolution is simulated by varying the phase q ofoperation F. Inequalities bound the quantum pro-cess fidelity Fp [see (23) for details].

Fig. 4. Digital simulations of four and six spin sys-tems. Dynamics of the initial state where all spinspoint up. (A) Four spin long-range Ising system.Each digital step is D.C = O4(p/16,0).O2(p/32).Error bars are smaller than point size. (B) Six spinsix-body interaction. F = O1(q,1), 4D = O4(p/4,0).The inequality at f = 0.25 p bounds the quantumprocess fidelity Fp at q = 0.25 p [see (23) for details].Lines; exact dynamics. Unfilled shapes: ideal digitized;filled shapes: data (P0 ♦P1 P2 P3 P4 P5 P6,where Pi is the total probability of finding ispinspointing down).

7 OCTOBER 2011 VOL 334 SCIENCE www.sciencemag.org60

REPORTS

B. Lanyon C. Roos

4 & 6 Spins

remarks:• scalability (?)• error correction (?)

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LETTERdoi:10.1038/nature18318

Real-time dynamics of lattice gauge theories with a few-qubit quantum computerEsteban A. Martinez1*, Christine A. Muschik2,3*, Philipp Schindler1, Daniel Nigg1, Alexander Erhard1, Markus Heyl2,4, Philipp Hauke2,3, Marcello Dalmonte2,3, Thomas Monz1, Peter Zoller2,3 & Rainer Blatt1,2

Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons1,2. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. This has recently stimulated theoretical effort, using Feynman’s idea of a quantum simulator3,4, to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented5–7. Here we report the experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realizing (1 + 1)-dimensional quantum electrodynamics (the Schwinger model8,9) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism10,11, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron–positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields12 in favour of exotic long-range interactions, which can be directly and efficiently implemented on an ion trap architecture13. We explore the Schwinger mechanism of particle–antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulation of high-energy theories using atomic physics experiments—the long-term intention is to extend this approach to real-time quantum simulations of non-Abelian lattice gauge theories.

Small-scale quantum computers exist today in the laboratory as programmable quantum devices14. In particular, trapped-ion quan-tum computers13 provide a platform allowing a few hundred coherent quantum gates to act on a few qubits, with a clear roadmap towards scaling up these devices4,15. This provides the tools for universal digital quantum simulation16, where the time evolution of a quantum system is approximated as a stroboscopic sequence of quantum gates17. Here we show how this technology can be used to simulate the real-time dynamics of a minimal model of a lattice gauge theory, realizing the Schwinger model8,9 as a one-dimensional quantum field theory with a chain of trapped ions (Fig. 1).

Our few-qubit demonstration is a first step towards simulating real-time dynamics in gauge theories: such simulations are funda-mental for the understanding of many physical phenomena, including thermalization after heavy-ion collisions and pair creation studied at high- intensity laser facilities6,18. Although existing classical numerical methods such as quantum Monte Carlo have been remarkably success-ful for describing equilibrium phenomena, no systematic techniques exist to tackle the dynamical long-time behaviour of all but very small

systems. In contrast, quantum simulations aim at the long-term goal of solving the specific yet fundamental class of problems that currently cannot be tackled by these classical techniques. The digital approach we employ here is based on the Hamiltonian formulation of gauge theories9, and enables direct access to the system wavefunction. As we show below, this allows us to investigate entanglement generation during particle–antiparticle production, emphasizing a novel perspec-tive on the dynamics of the Schwinger mechanism2.

Digital quantum simulations described in the present work are con-ceptually different from, and fundamentally more challenging than, previously reported condensed-matter-motivated simulations of spin and Hubbard-type models4,19,20. In gauge theories, local symmetries lead to the introduction of dynamical gauge fields obeying a Gauss law6. Formally, this crucial feature is described by local symmetry generators G i that commute with the Hamiltonian of the system ˆ ˆ =H G[ , ] 0i and

restrict the dynamics to a subspace of physical states | Ψphysical⟩ which satisfy ˆ Ψ Ψ| ⟩= | ⟩G qi iphysical physical , where qi are background charges. We will be interested in the case qi = 0 for all i (see Methods). Realizing such a constrained dynamics on a quantum simulator is demanding and has been the focus of theoretical research6,7,11,21–24. Instead, to opti-mally use the finite resources represented by a few qubits of existing quantum hardware, we encode the gauge degrees of freedom in a long-range interaction between the fermions (electrons and positrons), which can be implemented efficiently on our experimental platform. This allows us to explore quantum simulation of coherent real-time

1Institute for Experimental Physics, University of Innsbruck, 6020 Innsbruck, Austria. 2Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria. 3Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria. 4Physics Department, Technische Universität München, 85747 Garching, Germany.* These authors contributed equally to this work.

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Figure 1 | Quantum simulation of the Schwinger mechanism. a, The instability of the vacuum due to quantum fluctuations is one of the most fundamental effects in gauge theories. We simulate the coherent real-time dynamics of particle–antiparticle creation by realizing the Schwinger model (one-dimensional quantum electrodynamics) on a lattice, as described in the main text. b, The experimental setup for the simulation consists of a linear Paul trap, where a string of 40Ca+ ions is confined. The electronic states of each ion, depicted as horizontal lines, encode a spin | ↑ ⟩ or | ↓ ⟩ . These states can be manipulated using laser beams (see Methods for details).

© 2016 Macmillan Publishers Limited. All rights reserved

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LETTERRESEARCH

windows. By repeating the sequence multiple times, the resulting time evolution of the system U(t) closely resembles an evolution where the individual parts of the Hamiltonian act simultaneously, as can be shown using the Suzuki–Lie–Trotter expansion:

ˆ ˆ( )= =⎛⎝⎜⎜⎜ ⊗

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Our scheme is depicted in Fig. 2f–h. It allows for an efficient realization of the required dynamics and implements the coupling matrix shown in Fig. 2d, e with a minimal number of time steps, scaling only linearly in the number of sites N. The scheme is therefore scalable to larger systems. A discussion of finite size effects can be found in Methods.

We realize the simulation in a quantum information processor based on a string of 40Ca+ ions confined in a macroscopic linear Paul trap (Fig. 1b). There, each qubit is encoded in the electronic states | ↓ ⟩ = 4S1/2 (with magnetic quantum number m = − 1/2), | ↑ ⟩ = 3D5/2 (m = − 1/2) of a single ion. The energy difference between these states is in the optical domain, so the state of the qubit can be manipulated using laser light pulses. More specifically, a universal set of high-fidelity quantum operations is available, consisting of collective rotations around the equator of the Bloch sphere, addressed rotations around the z axis and

entangling Mølmer–Sørensen (MS) gates26. With a sequence of these gates, arbitrary unitary operations can be implemented27. Thus, we are able to simulate any Hamiltonian evolution, and in particular the interactions required here, by means of digital quantum simulation techniques, as shown in Fig. 2. Each of the implemented time evolu-tions consists of a sequence of over 200 quantum gates (see Extended Data Fig. 3). In order to realize the non-local interactions Hzz and H± with their specific long-range interactions, we use global MS entan-gling gates together with a spectroscopic decoupling method to tailor the range of the interaction. For the decoupling, the population of the ions that are not involved in the specific operations are shelved into additional electronic states that are not affected by the light for the entangling operations (see Methods). The local terms in Hz correspond to z rotations that are directly available in our set of operations. The strength of all terms can be tuned by changing the duration of the laser pulses corresponding to the physical operations.

Within our scheme, a wide range of fundamental properties in one-dimensional lattice gauge theories can be studied. To demonstrate our approach, we concentrate on simulating the coherent quantum real-time dynamics of the Schwinger mechanism, that is, the creation of particle–antiparticle pairs out of the bare vacuum | vacuum⟩ , where matter is entirely absent (see Methods). After initializing the system in this state, which corres ponds to the ground state for m → ∞ (Fig. 3a), we apply HS (Fig. 2d) for different masses and coupling strengths. As a first step, we measure the particle number density

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Figure 3 | Time evolution of the particle number density, ν. a, We show the ideal evolution under the Schwinger Hamiltonian HS shown in Fig. 2d, the ideal evolution considering time discretization errors (see Fig. 2), the expected evolution including an experimental (exp.) error model (see Methods) and the experimental data for electric field energy J = w and particle mass m = 0.5w (see equation (1)). After postselection of the experimental data (see Methods), the remaining populations are 86 ± 2, 79 ± 1, 73 ± 1, 69 ± 1% after 1, 2, 3, 4 time steps (averaged over all data sets). Error bars correspond to standard deviations estimated from a Monte Carlo bootstrapping procedure. The insets show the initial state of the simulation (left inset), corresponding to the bare vacuum with particle number density ν = 0, as well as one example of a state containing one pair (right inset), that is, a state with ν = 0.5, represented as filled/empty arrows as in Fig. 2. b, Experimental data and c, theoretical prediction for the evolution of the particle number density ν as a function of the dimensionless time wt and the dimensionless particle mass m/w, with J = w.

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Figure 4 | Time evolution of the vacuum persistence amplitude and entanglement. We show the square of the vacuum persistence amplitude | G(t)| 2 (the Loschmidt echo), which quantifies the decay of the unstable vacuum, and the logarithmic negativity En, a measure of the entanglement between the left and the right halves of the system. a, b, The time evolution of | G(t)| 2 (a) and En (b) for different values of the particle mass m and fixed electric field energy J = w, where w is the rate of particle–antiparticle creation and annihilation (compare equation (1)), as a function of the dimensionless time wt. c, d, The time evolution of | G(t)| 2 (c) and En (d) changes for different values of J and fixed particle mass m = 0. Circles correspond to the experimental data and squares connected by solid lines to the expected evolution assuming an experimental error model explained in Methods. Error bars correspond to standard deviations estimated from a Monte Carlo bootstrapping procedure. e, Illustration of the creation of a particle–antiparticle pair starting from the bare vacuum state.


Digital Quantum Simulation of an Exotic Spin Model

- obtained after integrating out gauge field

experiment

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220 quantum gates

uibkI QOIAnalog Quantum Simulation

• spin models • trapped ions

• We “build” a quantum system with desired Hamiltonian & controllable parameters,e.g. Hubbard models of atoms in optical lattices

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K. Kim, C. Monroe et al., Nature (2010)

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LETTERS

Quantum simulation of frustrated Ising spins withtrapped ionsK. Kim1, M.-S. Chang1, S. Korenblit1, R. Islam1, E. E. Edwards1, J. K. Freericks2, G.-D. Lin3, L.-M. Duan3 & C. Monroe1

A network is frustrated when competing interactions betweennodes prevent each bond from being satisfied. This compromiseis central to the behaviour of many complex systems, from social1

and neural2 networks to protein folding3 and magnetism4,5.Frustrated networks have highly degenerate ground states, withexcess entropy and disorder even at zero temperature. In the caseof quantum networks, frustration can lead to massively entangledground states, underpinning exotic materials such as quantum spinliquids and spin glasses6–9. Here we realize a quantum simulation offrustrated Ising spins in a system of three trapped atomic ions10–12,whose interactions are precisely controlled using optical forces13.We study the ground state of this system as it adiabatically evolvesfrom a transverse polarized state, and observe that frustrationinduces extra degeneracy. We also measure the entanglement inthe system, finding a link between frustration and ground-stateentanglement. This experimental system can be scaled to simulatelarger numbers of spins, the ground states of which (for frustratedinteractions) cannot be simulated on a classical computer.

Linus Pauling predicted in 1945 that the frustrated oxygen–hydrogenbond lengths in the pyrochloric lattice of ice would lead to a macro-scopic degeneracy of the ground state near zero temperature14. Thiszero-point entropy has been observed in spin-ice materials5,15, wherethe competing interactions are magnetic in nature. In the simple caseof a two-dimensional triangular lattice with frustrated antiferromag-netic Ising interactions, the ground-state degeneracy can easily be seen(Fig. 1a): only two of the three spins on each triangular cell can alignantiparallel, so all possible mixed configurations in each triangle(three-quarters of all cases) are ground states. Quantum super-position of these degenerate states leads to massive entanglement thatis important in our understanding of the complex phase structure ofmany frustrated materials, ranging from molecular and liquid crystalsto high-temperature superconductors5,16.

In our experiment, we implement a quantum simulation of thesmallest possible frustrated magnetic network, which consists of threespins. This work builds on earlier results for two trapped ions12, in asystem that can be scaled to much larger numbers of spins. We controlthe sign and strength of the individual magnetic interactions, directlymeasure all possible spin correlation functions and characterizeentanglement using techniques borrowed from quantum informationscience12,17. The experiment is based on a linear chain of three trappedatomic 171Yb1 ions, where the effective spin-1/2 system is representedby the 2S1/2 jF 5 1, mF 5 0æ and jF 5 0, mF 5 0æ hyperfine ‘clock’ statesin each ion, depicted by j"æz and j#æz, respectively18, and separated infrequency by nHF 5 12.642821 GHz.

The ions are confined in a three-layer linear trap13 and form acrystal along the trap’s z axis with a centre-of-mass trap frequencyof nz 5 1.49 MHz. The three normal modes of transverse motionalong the principal x axis occur at frequencies n1 5 4.334 MHz,

n2 5 4.074 MHz and n3 5 3.674 MHz. Off-resonance laser beamsuniformly illuminate the ions, driving stimulated Raman transitionsbetween the spin states and also imparting spin-dependent forces inthe x direction. As discussed in Methods, this allows quantum simu-lation of the Ising Hamiltonian with a transverse field10–12

H~X

ivj

Ji,js(i)x s (j)

x zBy

X

i

s (i)y ð1Þ

where By is an effective uniform transverse magnetic field with eachspin having unit magnetic moment, and we have set Planck’s constant,h, equal to one. For three spins, we define J1 ; J1,2 5 J2,3 as the nearest-neighbour interaction and J2 ; J1,3 as the next-nearest-neighbourinteraction (Fig. 1b), and s (i)

a denote the Pauli matrices of the ith spin.We initialize each spin parallel to a strong transverse field and thenadiabatically lower the field relative to the Ising couplings so that the

1Joint Quantum Institute, University of Maryland Department of Physics and National Institute of Standards and Technology, College Park, Maryland 20742, USA. 2Department ofPhysics, Georgetown University, Washington DC 20057, USA. 3MCTP and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA.

–3–3 –2 –1 0

0

3

a b

J1 J1

J2

J2

J1

1 2 3?

CMTiltZigzagc

IIIIIIIV

J (k

Hz)

µ~

Figure 1 | Frustrated Ising spins. a, Simplest case of spin frustration, withthree antiferromagnetic spins on a triangle. b, Image of three trapped atomic171Yb1 ions in the experiment, taken with an intensified charge-coupled-device camera, with nearest-neighbour (J1) and next-nearest-neighbour (J2)interactions. c, Expected form of the Ising interactions J1 and J2, controlledthrough the detuning, m, of an optical spin-dependent force, scaled to theaxial (nz) and transverse (n1) centre-of-mass (CM) normal-mode frequenciesof motion such that the CM, tilt and zigzag modes of transverse motionoccur at ~mm:(m2n2

1)=n2z 5 0, 21 and 22.4, respectively13.

Vol 465 | 3 June 2010 | doi:10.1038/nature09071

590Macmillan Publishers Limited. All rights reserved©2010

LETTERS








H~X

ivj

Ji,js(i)x s (j)

x zBy

X

i

s (i)y ð1Þ




–3–3 –2 –1 0

0

3

a b

J1 J1

J2

J2

J1

1 2 3?

CMTiltZigzagc

IIIIIIIV

J (k

Hz)

µ~





LETTERS








H~X

ivj

Ji,js(i)x s (j)

x zBy

X

i

s (i)y ð1Þ




–3–3 –2 –1 0

0

3

a b

J1 J1

J2

J2

J1

1 2 3?

CMTiltZigzagc

IIIIIIIV

J (k

Hz)

µ~





P Jurcevic, BP Lanyon, R Blatt, C. Roos et al. (2014)

~ 20 - 50 ions

64

magnetization of each spin

2

ated excitations act as radiating sources of entangled quasi-particles with dispersion relations determined by the under-lying system interactions. Possibly here start wtih: in thecase of short-range interactions get well defined group veloc-ity - supoerpositions of left and right travelling wavepacketsemerge and distribute entanglement. for long-range....

Our system, realised with trapped ions in a linear Paul trap,is accurately described by the transverse Ising model

HIsing = ~X

i, j

Ji j(i)x

( j)x + ~B

X

i

(i)z

where (i) ( = x, y, z, ) are the spin-1/2 Pauli operators

for the ith spin, B is the e↵ective transverse magneticfield and Ji j is the coupling matrix with approximatelypower-law decay with the lattice distance |i j| that can betuned between infinite-range (↵ = 0) and, asymptotically,short-range (↵ = 3). The chosen interaction range deter-mines the magnon quasi-particle dispersion relationship andtheir interactions in the system (fig 1). In the case whereB >> max(Ji j) the number of spin excitations is conservedand HIsing reduces to the XY model of hopping hardcorebosons HXY =

Pi< j Ji j(+ + +). (somewhere say:?

our quasi-particles are hardcore bosons, maybe figure 1caption)We can varify the system behaviour by directlymeasuring the exact spatial distribution of the spin-spininteractions in our system, i.e. the underlying Hamiltonian,and the corresponding quasi-particle dispersion relationship(see methods), which are seen to closely match theoreticalpredictions (Fig 1b-c).

Starting with the steady state of all spins down we per-form arbitrary local and global quenches of system. For localquenches this means fully flipping or making superpositions(|+i = (| #i+| "i)/

p2) of individual spins. (something about?:

with local quench probe low lying excitations of system andQP dispersion relation. With global quench probe high lyingexcitations and QP interactions become really important.) Inthis case the spread of information from the quench site(s)around system can be seen in spatially and temporally re-solved single spin observables like the magnetisation hZ(t)i.At early times wavefronts radiating away from single spin-excitations are clearly visible while at later times reflectionsresult in complex interference patterns (fig 1a-b). Creatingexcitations at either end of the spin chain creates counter-propagating wavefronts resulting in quasi-particle collisions,with properties dependant on the underlying interaction range(Fig 1c).

For global quenches rotate all spins are prepared in thesuperposition state |+i. This corresponds to a superpositionof all possible system excitations and therefore all super-positions of magnon dynamics will occur. In this case theunderlying physics can be investigated by measuring two-spincorrelation functions (fig 1d). The propagation of correlationsfollowing global quenches in spin chains is explored in a

FIG. 2: Magnon dynamics in a systems of 7 interacting quantumspins following local and global quenches. a-c. Time evolution ofthe magnetisation hZ(t)i of each spin in the system following a localquench at: a the centre spin, for ↵ ??. a.i, data, a.ii, theory: bthe left hand spin, for ↵ ??, b.i, data, b.ii, theory and: c both endsof the chain resulting, for ↵ ??, (data only). d. Time evolutionof two-spin correlation function C = hZi(t)Zj(t)i hZi(t)ihZj(t)i fol-lowing a global quench, for ↵ ?? (see methods, where we mustexplain that done with smaller B field).

complimentary paper by Monroe [? ].

To reveal the quantum correlations carried by quasi-particles we tomographically measure the full quantum stateof pairs of spins at many points during the dynamics. In thiscase we employ short-range interactions (↵ 2) for which aclear wavefront is apparent following a local quench in thecentre of the spin chain (fig 3a). The results clearly show thatmagnons emerging from either side of the initial excitation areentangled, and distribute this entanglement across the spin-chain at a finite velocity (fig 3b-c). In general it is possible tomeasure the quantum state of any subset of spins at any pointin the dynamics.

In the case of nearest-neighbour interactions, LR boundssay that information is restricted to propagate within well de-fined wavefronts, beyond which the signal amplitude decaysexponentially (both in space and time). In that case, the sys-tems maximal group velocity tightly bounds the propagatingwave front. In the presence of power-law interactions thatdecay suciently fast, one can still obtain an approximatelight cone by considering only the nearest-neighbor interac-tions [8? ], but now the signal decays algebraically outsideof this approximate light cone. When the interactions becomeof longer range, there are more magnons that propagate fasterthan the nearest-neighbor wave front, and the light-cone con-cept breaks down.

This e↵ect is exemplified in Fig. 4a-c for local quenchesat three values of ↵, chosen roughly equally spaced around↵ = 1. In the shortest-range case (Fig. 4a, ↵ 1.41), the ob-

time

[ms]

Magnon propagation

entanglement

2 642 64

0

10

5

15

20

25Ti

me

(ms)

0

0.2

1

0.4

0.6

0.8a b

0

0.2

0.1

0.3

0.4

0.6

0.5

Con

curr

ence

0 105 15 20 3025Time (ms)

spin 3&5spin 2&6

spin 1&7

Entanglement

Light-cone-like spreading of entanglement

breakdown of the quantum speed-limit due to long range interactions

[exp & theory indistinguishable]

2

ated excitations act as radiating sources of entangled quasi-particles with dispersion relations determined by the under-lying system interactions. Possibly here start wtih: in thecase of short-range interactions get well defined group veloc-ity - supoerpositions of left and right travelling wavepacketsemerge and distribute entanglement. for long-range....

Our system, realised with trapped ions in a linear Paul trap,is accurately described by the transverse Ising model

HIsing = ~X

i, j

Ji j(i)x

( j)x + ~B

X

i

(i)z

where (i) ( = x, y, z, ) are the spin-1/2 Pauli operators

for the ith spin, B is the e↵ective transverse magneticfield and Ji j is the coupling matrix with approximatelypower-law decay with the lattice distance |i j| that can betuned between infinite-range (↵ = 0) and, asymptotically,short-range (↵ = 3). The chosen interaction range deter-mines the magnon quasi-particle dispersion relationship andtheir interactions in the system (fig 1). In the case whereB >> max(Ji j) the number of spin excitations is conservedand HIsing reduces to the XY model of hopping hardcorebosons HXY =

Pi< j Ji j(+ + +). (somewhere say:?

our quasi-particles are hardcore bosons, maybe figure 1caption)We can varify the system behaviour by directlymeasuring the exact spatial distribution of the spin-spininteractions in our system, i.e. the underlying Hamiltonian,and the corresponding quasi-particle dispersion relationship(see methods), which are seen to closely match theoreticalpredictions (Fig 1b-c).

Starting with the steady state of all spins down we per-form arbitrary local and global quenches of system. For localquenches this means fully flipping or making superpositions(|+i = (| #i+| "i)/

p2) of individual spins. (something about?:

with local quench probe low lying excitations of system andQP dispersion relation. With global quench probe high lyingexcitations and QP interactions become really important.) Inthis case the spread of information from the quench site(s)around system can be seen in spatially and temporally re-solved single spin observables like the magnetisation hZ(t)i.At early times wavefronts radiating away from single spin-excitations are clearly visible while at later times reflectionsresult in complex interference patterns (fig 1a-b). Creatingexcitations at either end of the spin chain creates counter-propagating wavefronts resulting in quasi-particle collisions,with properties dependant on the underlying interaction range(Fig 1c).

For global quenches rotate all spins are prepared in thesuperposition state |+i. This corresponds to a superpositionof all possible system excitations and therefore all super-positions of magnon dynamics will occur. In this case theunderlying physics can be investigated by measuring two-spincorrelation functions (fig 1d). The propagation of correlationsfollowing global quenches in spin chains is explored in a

FIG. 2: Magnon dynamics in a systems of 7 interacting quantumspins following local and global quenches. a-c. Time evolution ofthe magnetisation hZ(t)i of each spin in the system following a localquench at: a the centre spin, for ↵ ??. a.i, data, a.ii, theory: bthe left hand spin, for ↵ ??, b.i, data, b.ii, theory and: c both endsof the chain resulting, for ↵ ??, (data only). d. Time evolutionof two-spin correlation function C = hZi(t)Zj(t)i hZi(t)ihZj(t)i fol-lowing a global quench, for ↵ ?? (see methods, where we mustexplain that done with smaller B field).

complimentary paper by Monroe [? ].

To reveal the quantum correlations carried by quasi-particles we tomographically measure the full quantum stateof pairs of spins at many points during the dynamics. In thiscase we employ short-range interactions (↵ 2) for which aclear wavefront is apparent following a local quench in thecentre of the spin chain (fig 3a). The results clearly show thatmagnons emerging from either side of the initial excitation areentangled, and distribute this entanglement across the spin-chain at a finite velocity (fig 3b-c). In general it is possible tomeasure the quantum state of any subset of spins at any pointin the dynamics.

In the case of nearest-neighbour interactions, LR boundssay that information is restricted to propagate within well de-fined wavefronts, beyond which the signal amplitude decaysexponentially (both in space and time). In that case, the sys-tems maximal group velocity tightly bounds the propagatingwave front. In the presence of power-law interactions thatdecay suciently fast, one can still obtain an approximatelight cone by considering only the nearest-neighbor interac-tions [8? ], but now the signal decays algebraically outsideof this approximate light cone. When the interactions becomeof longer range, there are more magnons that propagate fasterthan the nearest-neighbor wave front, and the light-cone con-cept breaks down.

This e↵ect is exemplified in Fig. 4a-c for local quenchesat three values of ↵, chosen roughly equally spaced around↵ = 1. In the shortest-range case (Fig. 4a, ↵ 1.41), the ob-

P Jurcevic, et al., PZ, R Blatt & CF Roos, Nature (2014); related work by C. Monroe group

deleted

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time

Ramsey interferometer

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|#i<latexit sha1_base64="Sa9QFQcXtrRajQDumJDLF+Pi+OI=">AAAB/XicfVDLSsNAFL3xWeMr1qWbYiu4kDCpoO3KghuXFewDmlIm420bkkzCzMS2hOKfuBNdiFt/xI1/4SeYtgqWomd1Oefc13Ei35WKkA9taXlldW09s6Fvbm3v7Bp72boMY8GwxkI/FE2HSvRdjjXlKh+bkUAaOD42HO9yojfuUEg35DdqFGE7oD3udl1GVUp1jGzB9lAl9m044FSIcDAudIw8MUm5RE7LucXCMskU+YtPmKLaMd7TdhYHyBXzqZQt5L309P6JRSLVTgRGoVBj/ehfl2QiNU58uh1LjCjzaA+TmMnxb6IVq25p2E5cHsUKOZsTExrIgKr+AilHgTNPonR5ussOqIc0TU0pFLo9nZrrmwqHehrDz6+5v4t60bSIaV0X85WzWR6QgQM4hGOw4BwqcAVVqAGDITzAEzxr99qj9qK9zqxL2nfPPsxBe/sCoVuYBA==</latexit><latexit sha1_base64="Sa9QFQcXtrRajQDumJDLF+Pi+OI=">AAAB/XicfVDLSsNAFL3xWeMr1qWbYiu4kDCpoO3KghuXFewDmlIm420bkkzCzMS2hOKfuBNdiFt/xI1/4SeYtgqWomd1Oefc13Ei35WKkA9taXlldW09s6Fvbm3v7Bp72boMY8GwxkI/FE2HSvRdjjXlKh+bkUAaOD42HO9yojfuUEg35DdqFGE7oD3udl1GVUp1jGzB9lAl9m044FSIcDAudIw8MUm5RE7LucXCMskU+YtPmKLaMd7TdhYHyBXzqZQt5L309P6JRSLVTgRGoVBj/ehfl2QiNU58uh1LjCjzaA+TmMnxb6IVq25p2E5cHsUKOZsTExrIgKr+AilHgTNPonR5ussOqIc0TU0pFLo9nZrrmwqHehrDz6+5v4t60bSIaV0X85WzWR6QgQM4hGOw4BwqcAVVqAGDITzAEzxr99qj9qK9zqxL2nfPPsxBe/sCoVuYBA==</latexit><latexit sha1_base64="Sa9QFQcXtrRajQDumJDLF+Pi+OI=">AAAB/XicfVDLSsNAFL3xWeMr1qWbYiu4kDCpoO3KghuXFewDmlIm420bkkzCzMS2hOKfuBNdiFt/xI1/4SeYtgqWomd1Oefc13Ei35WKkA9taXlldW09s6Fvbm3v7Bp72boMY8GwxkI/FE2HSvRdjjXlKh+bkUAaOD42HO9yojfuUEg35DdqFGE7oD3udl1GVUp1jGzB9lAl9m044FSIcDAudIw8MUm5RE7LucXCMskU+YtPmKLaMd7TdhYHyBXzqZQt5L309P6JRSLVTgRGoVBj/ehfl2QiNU58uh1LjCjzaA+TmMnxb6IVq25p2E5cHsUKOZsTExrIgKr+AilHgTNPonR5ussOqIc0TU0pFLo9nZrrmwqHehrDz6+5v4t60bSIaV0X85WzWR6QgQM4hGOw4BwqcAVVqAGDITzAEzxr99qj9qK9zqxL2nfPPsxBe/sCoVuYBA==</latexit><latexit sha1_base64="Sa9QFQcXtrRajQDumJDLF+Pi+OI=">AAAB/XicfVDLSsNAFL3xWeMr1qWbYiu4kDCpoO3KghuXFewDmlIm420bkkzCzMS2hOKfuBNdiFt/xI1/4SeYtgqWomd1Oefc13Ei35WKkA9taXlldW09s6Fvbm3v7Bp72boMY8GwxkI/FE2HSvRdjjXlKh+bkUAaOD42HO9yojfuUEg35DdqFGE7oD3udl1GVUp1jGzB9lAl9m044FSIcDAudIw8MUm5RE7LucXCMskU+YtPmKLaMd7TdhYHyBXzqZQt5L309P6JRSLVTgRGoVBj/ehfl2QiNU58uh1LjCjzaA+TmMnxb6IVq25p2E5cHsUKOZsTExrIgKr+AilHgTNPonR5ussOqIc0TU0pFLo9nZrrmwqHehrDz6+5v4t60bSIaV0X85WzWR6QgQM4hGOw4BwqcAVVqAGDITzAEzxr99qj9qK9zqxL2nfPPsxBe/sCoVuYBA==</latexit><latexit sha1_base64="ZydewvjKMX+tmXjeHhgZ74voC34=">AAAB/XicfVDLSsNAFL3xWeMr1qWbYiu4kDCpoO2u4MZlBfuAppTJeNsOTSZhZmJbQvFP3IkuxK0/4sa/Ma0VLEXP6nLOua/jRT5XmpBPY2V1bX1jM7Nlbu/s7u1bB9m6CmPJsMZCP5RNjyr0ucCa5trHZiSRBp6PDW9wNdUb9ygVD8WtHkfYDmhP8C5nVKdUx8oW3AHqxL0Lh4JKGQ4nhY6VJzYpl8h5ObdcODaZIQ9zVDvWR9rO4gCFZj5VqoWil57eP3NIpNuJxCiUemKe/OtSTKbGqc90Y4URZQPawyRmavKbaMW6Wxq1Ey6iWKNgC2JCAxVQ3V8i1TjwFklUXKS73IAOkKapaY3SdGdTc31b48hMY/j5Nfd3US/aDrGdm2K+cjEPJANHcAyn4MAlVOAaqlADBiN4hGd4MR6MJ+PVePu2rhjznkNYgPH+BQP6ltU=</latexit>

|"i<latexit sha1_base64="NQ3bR/J+EszGvkm0vwbPuC7HA2k=">AAAB+3icfVDLSsNAFL3xWeMr6tJNsRVcSEgqaLuy4MZlBfuAppTJeNsOeQ0zk9oS8ifuRBfi1j9x41/4CaatgqXoWV3OOfd1XO4zqSzrQ1taXlldW89t6Jtb2zu7xt5+Q0axoFinkR+Jlksk+izEumLKxxYXSALXx6brXU305hCFZFF4q8YcOwHph6zHKFEZ1TWMouOhSpyYEyGi+7TYNQqWaVXK1lklv1jYpjVF4fITpqh1jXfnLqJxgKGiPpGyjWE/O3xwaltcdRKBPBIq1Y//dUkqMuPEpzuxRE6oR/qYxFSmv4l2rHrlUSdhIY8VhnROTEggA6IGC6QcB+48iZKF2S4nIB6SLDOlUOjOdGp+YCoc6VkMP7/m/y4aJdO2TPumVKiez/KAHBzCEZyADRdQhWuoQR0oDOEBnuBZS7VH7UV7nVmXtO+eA5iD9vYFCIuXHQ==</latexit><latexit sha1_base64="NQ3bR/J+EszGvkm0vwbPuC7HA2k=">AAAB+3icfVDLSsNAFL3xWeMr6tJNsRVcSEgqaLuy4MZlBfuAppTJeNsOeQ0zk9oS8ifuRBfi1j9x41/4CaatgqXoWV3OOfd1XO4zqSzrQ1taXlldW89t6Jtb2zu7xt5+Q0axoFinkR+Jlksk+izEumLKxxYXSALXx6brXU305hCFZFF4q8YcOwHph6zHKFEZ1TWMouOhSpyYEyGi+7TYNQqWaVXK1lklv1jYpjVF4fITpqh1jXfnLqJxgKGiPpGyjWE/O3xwaltcdRKBPBIq1Y//dUkqMuPEpzuxRE6oR/qYxFSmv4l2rHrlUSdhIY8VhnROTEggA6IGC6QcB+48iZKF2S4nIB6SLDOlUOjOdGp+YCoc6VkMP7/m/y4aJdO2TPumVKiez/KAHBzCEZyADRdQhWuoQR0oDOEBnuBZS7VH7UV7nVmXtO+eA5iD9vYFCIuXHQ==</latexit><latexit sha1_base64="NQ3bR/J+EszGvkm0vwbPuC7HA2k=">AAAB+3icfVDLSsNAFL3xWeMr6tJNsRVcSEgqaLuy4MZlBfuAppTJeNsOeQ0zk9oS8ifuRBfi1j9x41/4CaatgqXoWV3OOfd1XO4zqSzrQ1taXlldW89t6Jtb2zu7xt5+Q0axoFinkR+Jlksk+izEumLKxxYXSALXx6brXU305hCFZFF4q8YcOwHph6zHKFEZ1TWMouOhSpyYEyGi+7TYNQqWaVXK1lklv1jYpjVF4fITpqh1jXfnLqJxgKGiPpGyjWE/O3xwaltcdRKBPBIq1Y//dUkqMuPEpzuxRE6oR/qYxFSmv4l2rHrlUSdhIY8VhnROTEggA6IGC6QcB+48iZKF2S4nIB6SLDOlUOjOdGp+YCoc6VkMP7/m/y4aJdO2TPumVKiez/KAHBzCEZyADRdQhWuoQR0oDOEBnuBZS7VH7UV7nVmXtO+eA5iD9vYFCIuXHQ==</latexit><latexit sha1_base64="NQ3bR/J+EszGvkm0vwbPuC7HA2k=">AAAB+3icfVDLSsNAFL3xWeMr6tJNsRVcSEgqaLuy4MZlBfuAppTJeNsOeQ0zk9oS8ifuRBfi1j9x41/4CaatgqXoWV3OOfd1XO4zqSzrQ1taXlldW89t6Jtb2zu7xt5+Q0axoFinkR+Jlksk+izEumLKxxYXSALXx6brXU305hCFZFF4q8YcOwHph6zHKFEZ1TWMouOhSpyYEyGi+7TYNQqWaVXK1lklv1jYpjVF4fITpqh1jXfnLqJxgKGiPpGyjWE/O3xwaltcdRKBPBIq1Y//dUkqMuPEpzuxRE6oR/qYxFSmv4l2rHrlUSdhIY8VhnROTEggA6IGC6QcB+48iZKF2S4nIB6SLDOlUOjOdGp+YCoc6VkMP7/m/y4aJdO2TPumVKiez/KAHBzCEZyADRdQhWuoQR0oDOEBnuBZS7VH7UV7nVmXtO+eA5iD9vYFCIuXHQ==</latexit><latexit sha1_base64="f/Ol7zh1ZwyAzoBnUyzI3doPPNo=">AAAB+3icfVDLSsNAFL3xWeMr6tJNsRVcSJhU0HZXcOOygn1AU8pkvG2HJpMwM6ktIX/iTnQhbv0TN/6Naa1gKXpWl3POfR0v8rnShHwaK6tr6xubuS1ze2d3b986OGyoMJYM6yz0Q9nyqEKfC6xrrn1sRRJp4PnY9IbXU705Qql4KO70JMJOQPuC9zijOqO6llV0h6gTN46olOFDWuxaBWKTSplcVPLLhWOTGQowR61rfbj3IYsDFJr5VKk2in52+ODcIZHuJBKjUOrUPP3XpZjMjFOf6cYKI8qGtI9JzFT6m2jHulcedxIuolijYAtiQgMVUD1YItUk8BZJVFxku9yADpFmmWmN0nRnU/MDW+PYzGL4+TX/d9Eo2Q6xndtSoXo5DyQHx3ACZ+DAFVThBmpQBwYjeIRneDFS48l4Nd6+rSvGvOcIFmC8fwFrG5Xu</latexit>

phase shift

N - independent atoms N - entangled atoms

product state ª |#1i . . . |#N i<latexit sha1_base64="PpamZeh7f4ChdKulohjstiHssgM=">AAACK3icfVBNS8NAEJ34bfyqevQSrIIHCYmC2pMFL55EwapgStlsx3Zpsht2J7Yl9O9485948wMUr979CcaqYCn6TsN7b2Z4L0wiYcjznq2R0bHxicmpaXtmdm5+obC4dGZUqjlWuIqUvgiZwUhIrJCgCC8SjSwOIzwPWwef+vk1aiOUPKVugtWYNaS4EpxRTtUK5USresrJMcQInbXAiNgJWkhZUFdtybRW7ZrfC6K6IjMkHPXWaoWi53qlPW+75AwPvuv1Udx/hz6Oa4WH/ABPY5TEI2bMJcpGHrS56XsJVTONidLUs9f/dRmuc+Onzw5SgwnjLdbALOWm95u4TOlqr1PNhExSQskHxIzFJmbUHCJNNw4HSTRC5r+CmLWQ5R0TobaD/lWn6RJ27LyGn6zO38PZlut7rn+yVSzvfPUBU7ACq7ABPuxCGQ7hGCrA4Qbu4BGerFvr3nqxXr+sI9b3zjIMwHr7AGEKq60=</latexit><latexit sha1_base64="PpamZeh7f4ChdKulohjstiHssgM=">AAACK3icfVBNS8NAEJ34bfyqevQSrIIHCYmC2pMFL55EwapgStlsx3Zpsht2J7Yl9O9485948wMUr979CcaqYCn6TsN7b2Z4L0wiYcjznq2R0bHxicmpaXtmdm5+obC4dGZUqjlWuIqUvgiZwUhIrJCgCC8SjSwOIzwPWwef+vk1aiOUPKVugtWYNaS4EpxRTtUK5USresrJMcQInbXAiNgJWkhZUFdtybRW7ZrfC6K6IjMkHPXWaoWi53qlPW+75AwPvuv1Udx/hz6Oa4WH/ABPY5TEI2bMJcpGHrS56XsJVTONidLUs9f/dRmuc+Onzw5SgwnjLdbALOWm95u4TOlqr1PNhExSQskHxIzFJmbUHCJNNw4HSTRC5r+CmLWQ5R0TobaD/lWn6RJ27LyGn6zO38PZlut7rn+yVSzvfPUBU7ACq7ABPuxCGQ7hGCrA4Qbu4BGerFvr3nqxXr+sI9b3zjIMwHr7AGEKq60=</latexit><latexit sha1_base64="PpamZeh7f4ChdKulohjstiHssgM=">AAACK3icfVBNS8NAEJ34bfyqevQSrIIHCYmC2pMFL55EwapgStlsx3Zpsht2J7Yl9O9485948wMUr979CcaqYCn6TsN7b2Z4L0wiYcjznq2R0bHxicmpaXtmdm5+obC4dGZUqjlWuIqUvgiZwUhIrJCgCC8SjSwOIzwPWwef+vk1aiOUPKVugtWYNaS4EpxRTtUK5USresrJMcQInbXAiNgJWkhZUFdtybRW7ZrfC6K6IjMkHPXWaoWi53qlPW+75AwPvuv1Udx/hz6Oa4WH/ABPY5TEI2bMJcpGHrS56XsJVTONidLUs9f/dRmuc+Onzw5SgwnjLdbALOWm95u4TOlqr1PNhExSQskHxIzFJmbUHCJNNw4HSTRC5r+CmLWQ5R0TobaD/lWn6RJ27LyGn6zO38PZlut7rn+yVSzvfPUBU7ACq7ABPuxCGQ7hGCrA4Qbu4BGerFvr3nqxXr+sI9b3zjIMwHr7AGEKq60=</latexit><latexit sha1_base64="PpamZeh7f4ChdKulohjstiHssgM=">AAACK3icfVBNS8NAEJ34bfyqevQSrIIHCYmC2pMFL55EwapgStlsx3Zpsht2J7Yl9O9485948wMUr979CcaqYCn6TsN7b2Z4L0wiYcjznq2R0bHxicmpaXtmdm5+obC4dGZUqjlWuIqUvgiZwUhIrJCgCC8SjSwOIzwPWwef+vk1aiOUPKVugtWYNaS4EpxRTtUK5USresrJMcQInbXAiNgJWkhZUFdtybRW7ZrfC6K6IjMkHPXWaoWi53qlPW+75AwPvuv1Udx/hz6Oa4WH/ABPY5TEI2bMJcpGHrS56XsJVTONidLUs9f/dRmuc+Onzw5SgwnjLdbALOWm95u4TOlqr1PNhExSQskHxIzFJmbUHCJNNw4HSTRC5r+CmLWQ5R0TobaD/lWn6RJ27LyGn6zO38PZlut7rn+yVSzvfPUBU7ACq7ABPuxCGQ7hGCrA4Qbu4BGerFvr3nqxXr+sI9b3zjIMwHr7AGEKq60=</latexit><latexit sha1_base64="cYuuYzUWJFzgwfl5ehlQy8gXbhQ=">AAACK3icfVDLSsNAFJ34rPEVdekm2AouJCQVtO4KblxJBauFpoTJ9LYdmsyEmRsfhPyOO//EnQ9Q3PofRq1gKXpWl3POvZdzwiTiGl33xZianpmdmy8tmItLyyur1tr6uZapYtBkMpKqFVINERfQRI4RtBIFNA4juAiHR5/6xSUozaU4w5sEOjHtC97jjGJBBVY9UbKbMrQ1UgS74mse2/4QMPO78kpQpeRV4OV+1JWoJ4STvBJYZddxD2vu3qE9OXiO+4UyGaERWI/FAZbGIJBFVOs2iH4RdLDruQl2MgWJVJib2/+6NFOF8dNn+qmGhLIh7UOWMp3/Jtop9mrXnYyLJEUQbEzMaKxjioMJUt/E4TgJmovilx/TIdCiY0RQpv911R44CNdmUcNPVvvv4bzqeK7jnVbL9f1RISWySbbIDvHIAamTY9IgTcLILbknT+TZuDMejFfj7ds6ZYx2NsgYjPcPw5qqfg==</latexit>

GHZ ª |#1 . . . #N i+|"1 . . . "N i<latexit sha1_base64="MQGyutxA1kzT/uDEeLEu5e/KilY=">AAACPXicfZBLS8NAEMcn9R1fUY9egq0gKCGp4ONkwYM9iYptRVPKZp22S5NN2N34IPR7efNLePHiTfQg4s2raavQWnROw29+u8P8vchnUtn2o5YZGR0bn5ic0qdnZufmjYXFsgxjQbFEQz8UZx6R6DOOJcWUj2eRQBJ4Pla81n5nXrlCIVnIT9VthNWANDirM0pUimrGyUHx3My5kgWm20KVuJfhNSdChNc1x/UvQyXNPnTYXu9ZcfTL+QGH7VzNyNqWvbtjb+6aw41j2d3K7n1At45qxkO6gsYBckV9IuUF8kZ6enPDsSNVTQRGoVBtffVfS1KRih1Pd2OJEaEt0sAkprLdDy5iVd+5qSaMR7FCTgeGCQlkQFRzCMrbwBuEKBlPd7kBaSFJU1cKhe52fzWblsIbPY3h51bz76actxzbco7z2cJWLw+YhGVYgTVwYBsKUIQjKAGFO3iCF3jV7rVn7U1776kZ7fvNEgyU9vkF0Rqy6g==</latexit><latexit sha1_base64="MQGyutxA1kzT/uDEeLEu5e/KilY=">AAACPXicfZBLS8NAEMcn9R1fUY9egq0gKCGp4ONkwYM9iYptRVPKZp22S5NN2N34IPR7efNLePHiTfQg4s2raavQWnROw29+u8P8vchnUtn2o5YZGR0bn5ic0qdnZufmjYXFsgxjQbFEQz8UZx6R6DOOJcWUj2eRQBJ4Pla81n5nXrlCIVnIT9VthNWANDirM0pUimrGyUHx3My5kgWm20KVuJfhNSdChNc1x/UvQyXNPnTYXu9ZcfTL+QGH7VzNyNqWvbtjb+6aw41j2d3K7n1At45qxkO6gsYBckV9IuUF8kZ6enPDsSNVTQRGoVBtffVfS1KRih1Pd2OJEaEt0sAkprLdDy5iVd+5qSaMR7FCTgeGCQlkQFRzCMrbwBuEKBlPd7kBaSFJU1cKhe52fzWblsIbPY3h51bz76actxzbco7z2cJWLw+YhGVYgTVwYBsKUIQjKAGFO3iCF3jV7rVn7U1776kZ7fvNEgyU9vkF0Rqy6g==</latexit><latexit sha1_base64="MQGyutxA1kzT/uDEeLEu5e/KilY=">AAACPXicfZBLS8NAEMcn9R1fUY9egq0gKCGp4ONkwYM9iYptRVPKZp22S5NN2N34IPR7efNLePHiTfQg4s2raavQWnROw29+u8P8vchnUtn2o5YZGR0bn5ic0qdnZufmjYXFsgxjQbFEQz8UZx6R6DOOJcWUj2eRQBJ4Pla81n5nXrlCIVnIT9VthNWANDirM0pUimrGyUHx3My5kgWm20KVuJfhNSdChNc1x/UvQyXNPnTYXu9ZcfTL+QGH7VzNyNqWvbtjb+6aw41j2d3K7n1At45qxkO6gsYBckV9IuUF8kZ6enPDsSNVTQRGoVBtffVfS1KRih1Pd2OJEaEt0sAkprLdDy5iVd+5qSaMR7FCTgeGCQlkQFRzCMrbwBuEKBlPd7kBaSFJU1cKhe52fzWblsIbPY3h51bz76actxzbco7z2cJWLw+YhGVYgTVwYBsKUIQjKAGFO3iCF3jV7rVn7U1776kZ7fvNEgyU9vkF0Rqy6g==</latexit><latexit sha1_base64="MQGyutxA1kzT/uDEeLEu5e/KilY=">AAACPXicfZBLS8NAEMcn9R1fUY9egq0gKCGp4ONkwYM9iYptRVPKZp22S5NN2N34IPR7efNLePHiTfQg4s2raavQWnROw29+u8P8vchnUtn2o5YZGR0bn5ic0qdnZufmjYXFsgxjQbFEQz8UZx6R6DOOJcWUj2eRQBJ4Pla81n5nXrlCIVnIT9VthNWANDirM0pUimrGyUHx3My5kgWm20KVuJfhNSdChNc1x/UvQyXNPnTYXu9ZcfTL+QGH7VzNyNqWvbtjb+6aw41j2d3K7n1At45qxkO6gsYBckV9IuUF8kZ6enPDsSNVTQRGoVBtffVfS1KRih1Pd2OJEaEt0sAkprLdDy5iVd+5qSaMR7FCTgeGCQlkQFRzCMrbwBuEKBlPd7kBaSFJU1cKhe52fzWblsIbPY3h51bz76actxzbco7z2cJWLw+YhGVYgTVwYBsKUIQjKAGFO3iCF3jV7rVn7U1776kZ7fvNEgyU9vkF0Rqy6g==</latexit><latexit sha1_base64="iAC1q264ypcUN4Ypja28I9IEylc=">AAACPXicfZBLT8JAEMe3+ML6Qj16aQQTE03TYqJwI/EgJ4JGHpESsl0G2NBum92tQBq+lze/hBcv3owejFevllcCGp3T5De/3cn8bd+hQhrGsxJbWl5ZXYuvqxubW9s7id29svACTqBEPMfjVRsLcCiDkqTSgarPAbu2AxW7ezmaV+6BC+qxWznwoe7iNqMtSrCMUCNxc5W/01KWoK5mdUGGVtPrMcy512uYltP0pNDmUGF4MrEC/4czA4VhqpFIGrqRzRhnWe13Y+rGuJJoWsVG4ilaQQIXmCQOFqIGrB2d3jk1DV/WQw6+x+VQPfrXEoRH4shTrUCAj0kXtyEMiBjOg1ogW5l+PaTMDyQwsjAMsStcLDu/oBi49iIEQVm0y3JxF3CUupTAVWv8q9bRJfTVKIbZrdrfTTmtm4ZuXqeTufNpIHF0gA7RMTLRBcqhPCqiEiLoAb2gN/SuPCqvyofyOVFjyvTNPloo5esbM7mxuw==</latexit>

δϕ∼ 1"

N<latexit sha1_base64="0/GNk/51B6dyDwLkslaZ3L47GVw=">AAACEXicfVC7SgNBFL0b3+tr1dJmMSgWsuwqatIFbKxEwTwgE8Ls5CYZsi9nZkPCsn9g54+InWghtrY2dn6Ka1QwBD3V5Zxz7+UcN/K4VLb9puWmpmdm5+YX9MWl5ZVVY229IsNYMCyz0AtFzaUSPR5gWXHlYS0SSH3Xw6rbO/nUq30UkofBpRpG2PBpJ+BtzqjKqKaxQ1roKWqSPhVRl5tEct8kbUFZ4qQJkVdCJWdp2jTytmUXC/ZB0ZwcHMseIV86jG/fAeC8abySVshiHwPFPCplHYNOFqe759iRaiQCo1CoVN/+1yWZyIyfPp3EEiPKerSDScxk+puox6pdGDQSHkSxwoCNiQn1pU9Vd4KUQ98dJ1HyIPtFfNpDmjWpFAqdjK6aXUvhQM9q+Mlq/j1U9i3HtpyLrI8j+MI8bMIW7IIDx1CCUziHMjC4hjt4gEftRrvXnrTnL2tO+97ZgDFoLx/X9qHM</latexit><latexit sha1_base64="/SkDlk/Zms8hu2uEd3ZwS/7nVho=">AAACEXicfVC7SgNBFJ2Nr7i+Vi1tFoNiIcuuoiZdwMZKIpgHZEKYndxkh+zLmdmQsGxjbeePiJ1oIba2NhZ+iuAmUTAEPdXlnHPv5Rw7dJmQpvmuZGZm5+YXsovq0vLK6pq2vlERQcQplGngBrxmEwEu86EsmXShFnIgnu1C1e6eDvVqD7hggX8pByE0PNLxWZtRIlOqqe3iFriS6LhHeOgwHQvm6bjNCY2tJMbiisv4PEmaWs40zELePCzo04NlmCPkikfR3cf1Z6/U1N5wK6CRB76kLhGiDn4njePsW2YoGzGHMOAyUXf+dQnKU+PQp+JIQEhol3QgjqhIfhP1SLbz/UbM/DCS4NMJMSae8Ih0pkgx8OxJEgTz01/YI10gaZNSAlfx6KruGBL6alrDT1b976FyYFimYV2kfRyjMbJoC22jPWShE1REZ6iEyoiiG3SPHtGTcqs8KM/Ky9iaUb53NtEElNcvdVmjvg==</latexit><latexit sha1_base64="/SkDlk/Zms8hu2uEd3ZwS/7nVho=">AAACEXicfVC7SgNBFJ2Nr7i+Vi1tFoNiIcuuoiZdwMZKIpgHZEKYndxkh+zLmdmQsGxjbeePiJ1oIba2NhZ+iuAmUTAEPdXlnHPv5Rw7dJmQpvmuZGZm5+YXsovq0vLK6pq2vlERQcQplGngBrxmEwEu86EsmXShFnIgnu1C1e6eDvVqD7hggX8pByE0PNLxWZtRIlOqqe3iFriS6LhHeOgwHQvm6bjNCY2tJMbiisv4PEmaWs40zELePCzo04NlmCPkikfR3cf1Z6/U1N5wK6CRB76kLhGiDn4njePsW2YoGzGHMOAyUXf+dQnKU+PQp+JIQEhol3QgjqhIfhP1SLbz/UbM/DCS4NMJMSae8Ih0pkgx8OxJEgTz01/YI10gaZNSAlfx6KruGBL6alrDT1b976FyYFimYV2kfRyjMbJoC22jPWShE1REZ6iEyoiiG3SPHtGTcqs8KM/Ky9iaUb53NtEElNcvdVmjvg==</latexit><latexit sha1_base64="/SkDlk/Zms8hu2uEd3ZwS/7nVho=">AAACEXicfVC7SgNBFJ2Nr7i+Vi1tFoNiIcuuoiZdwMZKIpgHZEKYndxkh+zLmdmQsGxjbeePiJ1oIba2NhZ+iuAmUTAEPdXlnHPv5Rw7dJmQpvmuZGZm5+YXsovq0vLK6pq2vlERQcQplGngBrxmEwEu86EsmXShFnIgnu1C1e6eDvVqD7hggX8pByE0PNLxWZtRIlOqqe3iFriS6LhHeOgwHQvm6bjNCY2tJMbiisv4PEmaWs40zELePCzo04NlmCPkikfR3cf1Z6/U1N5wK6CRB76kLhGiDn4njePsW2YoGzGHMOAyUXf+dQnKU+PQp+JIQEhol3QgjqhIfhP1SLbz/UbM/DCS4NMJMSae8Ih0pkgx8OxJEgTz01/YI10gaZNSAlfx6KruGBL6alrDT1b976FyYFimYV2kfRyjMbJoC22jPWShE1REZ6iEyoiiG3SPHtGTcqs8KM/Ky9iaUb53NtEElNcvdVmjvg==</latexit><latexit sha1_base64="QY+xF4f/ixXdlRktx4lAO8txwe4=">AAACEXicfVDLSsNAFJ34rPEVdekmWBQXEiYK2u4KblxJBfuATimT6W07NJPEmUlpCfkDd/6JO9GFuHXrxr8xrRUsRc/qcs6593KOF/lcaYw/jYXFpeWV1dyaub6xubVt7exWVRhLBhUW+qGse1SBzwOoaK59qEcSqPB8qHn9y7FeG4BUPAxu9SiCpqDdgHc4ozqjWtYRaYOvqU0GVEY9bhPFhU06krLETROi7qROrtO0ZeWxg4sFfFa05wfXwRPk0RTllvVB2iGLBQSa+VSpBgTdLE7vxMWRbiYSolDq1Dz816WYzIxjn0liBRFlfdqFJGYq/U00Yt0pDJsJD6JYQ8BmxIQKJajuzZFqJLxZEhQPsl9E0D7QrEmtQZpkctXuORqGZlbDT1b776F66rjYcW9wvnQ+LSSH9tEBOkYuukAldIXKqIIYukeP6Bm9GA/Gk/FqvH1bF4zpzh6agfH+BXusn1I=</latexit>

δϕ∼ 1N

<latexit sha1_base64="O2ZEfxF0nSvjzKGQpXh0E6H0jrk=">AAACCnicfVC7SgNBFL0bX3F9rVpYWLgYBAtZZhU06QIiWEkE84BsCLOTm2TIvpiZDQlLSjv/xE60EFs/wca/cZMoGIKe6nLOufdyjht5XCpCPrXMwuLS8kp2VV9b39jcMrZ3KjKMBcMyC71Q1Fwq0eMBlhVXHtYigdR3Pay6vcuxXu2jkDwM7tQwwoZPOwFvc0ZVSjWNA6eFnqKm06ci6nLTkdw3nbagLLFHyc2oaeSIRQp5clYw5wfbIhPkimTvqggApabx4bRCFvsYKOZRKesYdNIc3RObRKqRCIxCoUb60b8uyURqHPt0J5YYUdajHUxiJke/iXqs2vlBI+FBFCsM2IyYUF/6VHXnSDn03VkSJQ/SX45Pe0jTCpVCoTuTq2bXUjjQ0xp+spp/D5VTyyaWfZv2cQ5TZGEfDuEYbLiAIlxDCcrA4B4e4RletAftSXvV3qbWjPa9swsz0N6/AMeLnPo=</latexit><latexit sha1_base64="fUjFwE9zSKjcokSTkoInc8sALAA=">AAACCnicfVDLSsNAFJ3UV42vqAsXLgwWwYWEiYq2u4IIrqSCbYWmlMn0th2aTMLMpLSELN35J+5EF+LWTxDEr/AXTFsFS9Gzupxz7r2c44YekwrjDy0zMzs3v5Bd1JeWV1bXjPWNigwiQaFMAy8QNy6R4DEOZcWUBzehAOK7HlTd7tlQr/ZASBbwazUIoe6TNmctRolKqYax4zTBU8R0ekSEHWY6kvmm0xKExnYSXyYNI4ctXMjjo4I5PdgWHiFXxFvn+c/j91LDeHOaAY184Ip6RMoa8Haao3Ng41DVYwFhIFSi7/3rklSkxqFPdyIJIaFd0oY4ojL5TdQi1cr36zHjYaSA0wkxJr70iepMkXLgu5MkSMbTX45PukDSCpUCoTujq2bHUtDX0xp+spp/D5VDy8aWfZX2cYLGyKJttIv2kY1OURFdoBIqI4pu0T16RE/anfagPWsvY2tG+97ZRBPQXr8AZPCe7A==</latexit><latexit sha1_base64="fUjFwE9zSKjcokSTkoInc8sALAA=">AAACCnicfVDLSsNAFJ3UV42vqAsXLgwWwYWEiYq2u4IIrqSCbYWmlMn0th2aTMLMpLSELN35J+5EF+LWTxDEr/AXTFsFS9Gzupxz7r2c44YekwrjDy0zMzs3v5Bd1JeWV1bXjPWNigwiQaFMAy8QNy6R4DEOZcWUBzehAOK7HlTd7tlQr/ZASBbwazUIoe6TNmctRolKqYax4zTBU8R0ekSEHWY6kvmm0xKExnYSXyYNI4ctXMjjo4I5PdgWHiFXxFvn+c/j91LDeHOaAY184Ip6RMoa8Haao3Ng41DVYwFhIFSi7/3rklSkxqFPdyIJIaFd0oY4ojL5TdQi1cr36zHjYaSA0wkxJr70iepMkXLgu5MkSMbTX45PukDSCpUCoTujq2bHUtDX0xp+spp/D5VDy8aWfZX2cYLGyKJttIv2kY1OURFdoBIqI4pu0T16RE/anfagPWsvY2tG+97ZRBPQXr8AZPCe7A==</latexit><latexit sha1_base64="fUjFwE9zSKjcokSTkoInc8sALAA=">AAACCnicfVDLSsNAFJ3UV42vqAsXLgwWwYWEiYq2u4IIrqSCbYWmlMn0th2aTMLMpLSELN35J+5EF+LWTxDEr/AXTFsFS9Gzupxz7r2c44YekwrjDy0zMzs3v5Bd1JeWV1bXjPWNigwiQaFMAy8QNy6R4DEOZcWUBzehAOK7HlTd7tlQr/ZASBbwazUIoe6TNmctRolKqYax4zTBU8R0ekSEHWY6kvmm0xKExnYSXyYNI4ctXMjjo4I5PdgWHiFXxFvn+c/j91LDeHOaAY184Ip6RMoa8Haao3Ng41DVYwFhIFSi7/3rklSkxqFPdyIJIaFd0oY4ojL5TdQi1cr36zHjYaSA0wkxJr70iepMkXLgu5MkSMbTX45PukDSCpUCoTujq2bHUtDX0xp+spp/D5VDy8aWfZX2cYLGyKJttIv2kY1OURFdoBIqI4pu0T16RE/anfagPWsvY2tG+97ZRBPQXr8AZPCe7A==</latexit><latexit sha1_base64="Yd4d94ldHcDVM1WihovGpI7hOIk=">AAACCnicfVC7SgNBFJ2Nr7i+Vi0tXAyChSyzCpp0ARsriWAekAlhdnKTHbKPYWY2JCwp7fwTO9FCbP0EG//GTYxgCHqqyznn3ss5ngi40hh/Grml5ZXVtfy6ubG5tb1j7e7VVJxIBlUWB7FseFRBwCOoaq4DaAgJNPQCqHv9q4leH4BUPI7u9EhAK6S9iHc5ozqj2tYh6UCgqU0GVAqf20Tx0CZdSVnqjtObcdsqYAeXivi8ZC8OroOnKKAZKm3rg3RiloQQaRZQpZoQ9bIc/qmLhW6lEkQs9dg8/telmMyME59JEgWCsj7tQZowNf5NNBPdLQ5bKY9EoiFic2JKQxVS7S+QahR68yQoHmW/SEj7QLMKtQZpkulV23c0DM2shp+s9t9D7cxxsePe4kL5YlZIHh2gI3SCXHSJyugaVVAVMXSPHtEzejEejCfj1Xj7tuaM2c4+moPx/gVbQZvu</latexit>

standard quantum noise limit Heisenberg limit

input states

Tutorial: J Huang et al., arxiv1308.6092

parameter estimation of phase shift[Fisher information, Cramer-Rao bound etc.]

prepare initial state

or: spin squeezed state

quantum measurement

quantum optical systems allow measurements beyond SQL

coherent control

uibkEx 5: Cavity QED & Quantum Networks

waveguide

• CQED

atom

cavity

photon(flying qubit)

For a review see: P. Lodahl, A Rauschenbeutel, P.Z. et al., Nature 2017

& light-matter interface

(stationary qubit)

chiral = unidirectional

input

output

input

output

open quantum system


waveguide

• CQED

qubit 1


qubit 2

• Quantum State Transfer

qubit 1 wavepacket in waveguide qubit 2

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¢|0ip |g i1 ! |g i1

°Æ |0ip +Ø |1ip

¢|g i1 ! |g i1 |0ip

°Æ |g i2 +Ø |ei2

¢

fidelity of transfer ~ quantum control problem… see Lecture 4


waveguide

• CQED

qubit 1


qubit 2

• Quantum Networks

node

channel

~ quantum computer:'stationary qubits’

~ photonic waveguide:‘flying qubits’

introduction to quantum optics [theory]2.pdf · 2018-04-19 · p zoller, cirm preschool:...

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