this week: introductory review articles: d. leibfried, r. blatt, c. … · 2018. 9. 3. ·...

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Introductory Review Articles:

• D. Leibfried, R. Blatt, C. Monroe and D. Wineland, Quantum dynamics of single trapped ions, Review of Modern Physics 75, 281 (2003)

• R. Blatt, and D. Wineland, Entangled states of trapped atomic ions, Nature 453, 1008 (2008)

3-May-18Andreas Wallraff 1

Lecture 10, May 3, 2018

This week:

Atomic Ions for QIP Ion Traps Vibrational modes Preparation of initial states

Read-Out Single-Ion Gates Two-Ion Gates

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row of qubits in a linear Paul trap forms a quantum register

Laser pulses manipulate individual ions

A CCD camera reads out the ion`s quantum state Effective ion-ion

interaction induced by laser pulses that excite the ion`s motion

Ion Trap Quantum Processor

Slide material courtesy of H. Haeffner(Innsbruck/Berkeley) and J. Home (ETHZ) with notes by A. Wallraff (ETHZ) 3-May-18Andreas Wallraff 2

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Trapping Individual Ions in a Linear Paul Trap


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Mechanical Analog

Harvard Natural Sciences Lecture Demonstrations https://www.youtube.com/watch?v=XTJznUkAmIY 3-May-18Andreas Wallraff 5

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Photo-multiplier

CCDcamera

Fluorescence detection by CCD cameraphotomultiplier

Vacuumpum

p

oven

atomic

beam

Laser beams for:• photoionization• cooling• quantum state manipulation• fluorescence excitation

Experimental Setup


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Ions with optical transition to metastable level: 40Ca+,88Sr+,172Yb+

P1/2 D5/2

τ =1s

S1/2

40Ca+

P1/2

S1/2

D5/2

Dopplercooling Sideband

cooling

metastable

opticaltransition

stable|g>

|e>

detectionQuadrupole transition

Qubit levels:

P1/2

S1/2

D5/2S1/2 , D5/2

τ ≈ 1 s

Qubit transition:S1/2 – D5/2

Ions as Quantum Bits


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The Calcium Ion as a Two State System

Quantum numbers:• principal quantum number• orbital angular momentum• electron/nuclear spin• Symbol: n2S+1LJ

• n: principal• S: total spin• L: total orbital angular

momentum• J: total angular

momentum

Simplified Level Scheme of Calcium ion:

• group 2• with one electron removed

(Alkali-like)

1 s lifetime

We are looking for two long-lived states for qubit

7 ns lifetime

year lifetime


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absorption and emissioncause fluorescence steps(digital quantum jump signal)

D

S

P

S

D

monitorweak transition

metastablelevel

Quantum jumps: spectroscopy with quantized fluorescence

• Quantum jump technique• Electron shelving technique

long lifetimeshort lifetime

no photons

lots of photons

time in excited state (average is lifetime)

Detection of Ion Quantum State

Observation of quantum jumps:Nagourney et al., PRL 56,2797 (1986),Sauter et al., PRL 57,1696 (1986),Bergquist et al., PRL 57,1699 (1986) 3-May-18Andreas Wallraff 16

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Electron Shelving for Quantum State Detection1. Initialization in a pure quantum state

3. Quantum state measurementby fluorescence detection

2. Quantum state manipulation onS1/2 – D5/2 transition

One ion: Fluorescence histogram

counts per 2 ms0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8S1/2 stateD5/2 state

P1/2 D5/2

τ =1s

S1/2

40Ca+

P1/2

S1/2

D5/2P1/2

S1/2

D5/2

Quantum statemanipulation

P1/2

S1/2

D5/2

Fluorescencedetection

50 experiments / s

Repeat experiments100-200 times


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1. Initialization in a pure quantum state

3. Quantum state measurementby fluorescence detection

2. Quantum state manipulation onS1/2 – D5/2 transition

5µm

50 experiments / s

Repeat experiments100-200 times

• Spatially resolved detection withCCD camera:

Two ions:

P1/2

S1/2

D5/2


Electron Shelving for Quantum State Detection


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Mechanical quantum harmonic oscillator

• Extension of the ground state:

• Size of the wave packet << wavelength of visible light (e.g. for Ca)

• Energy scale of interest:

harmonic trap

ions need to be very cold to be in their vibrational ground state

Mechanical Motion of Ions in their Trapping Potential


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Laser – ion interactions

g

e

⊗

2-level-atom harmonic trap

0,g

0,e1,e

2,e

2,g1,g

joint energy levels

…

Electronic structure of the ion approximated by two-level system(laser is (near-) resonant and couples only two levels)

Trap: Only a single harmonic oscillator taken into account

Ion:

tensor product dressed states diagram

An Ion Coupled to a Harmonic Oscillator


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harmonic trap

…

2-level-atom joint energy levels

External Degree of Freedom: Ion Motion

ion transition frequency 400 THz (carrier)

trap frequency 1 MHz (sideband, red/blue SB)


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carrier transitionmotional sidebands

Laser detuning ∆ at 729 nm (MHz)

motional sidebands(red / lower) (blue / upper)

• many different vibrational modes of ions in the trap

• red and blue side bands can be observed because vibrational motion of ions is not cooled (in this example)

A Closer Look at the Excitation Spectrum (3 Ions)


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4.54 4.52 4.5 4.48 0

0.2

0.4

0.6

0.8 P D

Detuning δω (MHz) 4.48 4.5 4.52 4.54 0

0.2

0.4

0.6

0.8

P D

Detuning δω (MHz)

99.9 % ground state population

after sideband cooling

after Doppler cooling

7.1=zn

Sideband absorption spectra:

red sideband blue sideband

-4.54 -4.52 -4.5 -4.48

1, −ng

1, −nene,

ng,

……

red side band transition

Cooling of the Vibrational Modes


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1, −nS

1, −nDnD,

1, +nD

1, +nSnS ,

coupled system„Blue sideband“ pulses:D

sta

te p

opul

atio

n

Entanglement between internaland motional state!

Not only cooling but also controlled excitation of the vibrational modes:

Coherent Excitation on the Sideband Transition


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• Laser slightly detuned from carrier resonance

or:

• pulses with rotation axis in equatorial plane

Arbitrary qubit rotations:

(z-rotations by off-resonant laser beamcreating ac-Stark shifts)

Gate time: 1-10 µs Coherence time: 2-3 ms

limited by

• laser frequency fluctuations• magnetic field fluctuations

(laser linewidth δν<100 Hz)

D state population

x,y-rotations

Single Qubit Operations


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Addressing Qubits

CCD

Paul trap


electro-optic deflector

coherentmanipulation of qubits

dichroicbeamsplitter

inter ion distance: ~ 4 µm

addressing waist: ~ 2 µm

< 0.1% intensity on neighbouring ions

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Exci

tatio

n

Deflector Voltage (V)


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Manipulating Single Qubits

Raman transition,hyperfine

Resonant microwaves/laser

Microwaves F > 0.999999 Harty et al. PRL 113, 220501 (2014)Lasers F > 0.9999 (Masters thesis Oxford)


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Problem: noise! – mainly from classical fields

Storing Qubits in an Atom - Phase Coherence


||Time (seconds!)

F = 2

F = 1

1207 MHz

1 GHz

119.645 Gauss

Storing Qubits in an Atom: Field-Independent Transitions

Langer et al. PRL 95, 060502 (2005)

• magnetic field independent transition

• long coherence

Hyperfine Structure in Be+


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Pulse sequence:

…

… …

…

generation of entanglement between two ions:

Generation of Bell States: Entanglement of Two Ions


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Ion 1: π/2 , blue sideband

Pulse sequence:

…

… …

…

creates entangled state between qubit 1 and oscillator



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Ion 2: π , carrier

Pulse sequence:

…

… …

…excites qubit 2



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Ion 2: π , carrier

Ion 2: π , blue sideband

Pulse sequence:

…

… …

…takes qubit 2 (with one oscillator excitation) back to ground state and removes excitation from oscillator

|SD0> is non-resonant and remains unaffected



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Fluorescencedetection withCCD camera:

• Coherent superposition or incoherent mixture ?

• What is the relative phase of the superposition ?

SSSD

DSDD SSSDDSDD

Ψ+

Measurement of the density matrix:

tomography of qubit states (= full measurement of x,y,z components of both qubits and its correlations)

Bell State Analysis


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SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

F=0.91

Bell State Reconstruction


0.5

- 0.5

0.5

- 0.5

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Quantum Gate Proposals with Trapped Ions

Some other gate proposals by:• Cirac & Zoller• Mølmer & Sørensen, Milburn• Jonathan, Plenio & Knight• Geometric phases• Leibfried & Wineland

...allows the realization of a universal quantum computer !


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Both, the phase gate as well the CNOT gate can be converted into each other with single qubit operations.

Together with single qubit gates any unitary operation can be implemented!

controlled phase gateimplementation of a CNOT for universal ion trap quantum computing:

Realizing a Controlled NOT with a Controlled Phase Gate

Cirac and Zoller, Phys. Rev. Lett. 74, 4091-4094 (1995) 3-May-18Andreas Wallraff 43

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1ε 2ε

Cirac-Zoller Two-Ion Phase Gate

Cirac and Zoller, Phys. Rev. Lett. 74, 4091-4094 (1995)

ion 1

motion

ion 2

,S D

,S D

0 0SWAP

,0S,1S

,0D,1D


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Phase gate usingthe motion andthe target bit.

ion 1

motion

ion 2

,S D

,S D

0 0SWAP

Z

1ε 2ε


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,0S,1S

,0D,1D

ion 1

motion

ion 2

,S DSWAP

,S D

0 0SWAP

Z

1ε 2ε


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Phase gate usingthe motion andthe target bit.

ion 1

motion

ion 2

,S D

,S D

0 0

Z

1ε 2ε


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How do you do this with just a two-level system?


?


• D0 blue sideband is forbidden• Complete 2π blue sideband rotations induce

phase factor -1 (c.f. geometric phase)• How to implement that for the S1-D2

sideband transition which rotates sqrt(2) faster at same drive field strength?

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Phase Gate


Composite 2π-rotation:


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A Phase Gate with 4 Pulses (2π Rotation)( ) ( ) ( ) ( )1 1 1 1( , ) , 2 2,0 , 2 2,0R R R R Rθ φ π π π π π π+ + + +=

1

2

3

4

,0 ,1S D↔on2π


S0:

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0=ϕ 0=ϕ 2πϕ =2πφ =

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (µs)

D5/

2-e

xcita

tion

2π 2π ππ2πφ =0=φ 0=φ

state preparation ,0S , then application of phase gate pulse sequence

Continuous tomography of the phase gate

A Single Ion Composite Phase Gate: Experiment

Schmidt-Kaler et al., Nature 422, 408-411 (2003) 3-May-18Andreas Wallraff 53

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Population of |S,1> - |D,2> remains unaffected

( ) ( ) ( ) ( )1 1 1 1( , ) 2, 2 ,0 2, 2 ,0R R R R Rθ φ π π π π π π+ + + +=4

3

2

1

all transition rates are a factor of sqrt(2) faster at the same Laser power


( ) ( ) ( ) ( )1 1 1 1( , ) , 2 2,0 , 2 2,0R R R R Rθ φ π π π π π π+ + + +=S0:

S1:

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Testing the Phase of the Phase Gate for |0,S>

Schmidt-Kaler et al., Nature 422, 408-411 (2003)

Time (µs)

D5/

2-e

xcita

tion

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Phase gate 2π

−2π

97.8 (5) %


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ion 1

motion

ion 2

,S DSWAP

,S D

0 0

control qubit

target qubit

SWAP

ion 1

ion 2

pulse sequence:

Cirac-Zoller Two-Ion Controlled-NOT Operation

Cirac and Zoller, Phys. Rev. Lett. 74, 4091-4094 (1995) 3-May-18Andreas Wallraff 56

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SS - SS DS - DD

SD - SD DD - DS

Cirac – Zoller CNOT Gate Operation


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input

output

Measured Truth Table of Cirac-Zoller CNOT Operation


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Approaches to Algorithms/Scaling

Pictures: T. Monz, R. BlattScience 345, 6194 (2014)

Best system thus far for algorithms – single ion string (Blatt, Roos, Innsbruck)

Universal operations:- Multi-qubit gate (all ions)- Spin rotation (all ions)- Phase rotation (individual addressing)

Quantum error correction:State of the art: Transversal operations on a topological 7-qubit Steane code


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GHZ Entanglement of up to 14 ions

Monz et al., PRL 106, 130506 (2011)

High contrast – 3 ions

Reduced contrast – 14 ions


(2, 3, 4, 5, 6, 8, 10, 12, 14) ions

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Collective Rotations - Challenges

Data: C. Hempel, C. Roos, R. Blatt (Innsbruck)

51 ion chain:

Feature:• Efficient collective Rabi oscillations

on all ions with a single laser

Challenge:• size of ion chain becomes big

compared to laser beam• variation of Rabi rotation rate with

position dependent laser amplitude


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Scaling Ion-Trap QIP Architectures – Integrated Chip-Based Traps

Wineland et al., J. Res. N.I.S.T. (1998), Kielpinski et al. Nature 417, 709 (2002)

Transport of ions is a critical ingredientHow do we scale up the optical delivery?

Logic

Cooling

"Gate" "Gate""Move, separate"


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Integrated Components - Microwaves

C. Ospelkaus et al. Nature 182, 476 (2011)

eg. Quantum control using microwaves – removes the need for high-power lasers

Gradients – produce state-dependent potentials through Zeeman shifts

2-qubit gate

Single-qubit gate


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Integrated Components - Optics

Vandevender et al. PRL 105, 023001 (2010)

Integration of excitation and detection optics:


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Integrated Components - Quantum Logic Gates by Ion Transport

Proposal: D. Leibfried et al. PRA 76, 032324 (2007)

Advantages: reduces switching opticswaveforms required anyway

parallel use of laser beams in different zones simultaneously


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Rabi Oscillations and Qubit Rotations

Home Lab at ETHZ: L. de Clercq, H-Y. Lo, M. Marinelli

Pulse sequences (multiple transports)– Ramsey separated pulse experiment

Be+ Raman transition – hyperfine qubit, ~1s coherence time

Qubit rotation ReadoutLaser Sequence

Transport

t

ttoff

Control parameters• laser power• Laser timing• Position of atom

Characterization• shelving readout


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Integrated Components - Parallel Transport Quantum Logic Gates

L. de Clercq, H-Y. Lo, M. Marinelli

Retro-reflect laser beams to different zones

zA B C

Ion in zone C

Ion in zone A

Ion in zone C

Ion in zone A

Operation chosen using the transport speed of each ion


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Scaling Ion-Trap QIP Architectures – Connecting Traps

Monroe et al. Phys. Rev. A 89 022317 (2014)

Multiple linked small processors (trapping zones)• probabilistic entanglement

generation through Hong-Ou-Mandel effect

• teleportation of states between zones

• teleported gates between zones



What’s to like (and dislike) about ions?High gate rate (us) vs coherence time (10 s)

High fidelity, single shot readout

Identical systems (Decoherence-Free-Subspace encoding)

High fidelity laser and microwave gates

Why not?Slow (compared to solid state systems). Microseconds vs nanoseconds

Cannot “pick our frequency” – wavelengths etc. set by nature

Identical systems – need to work to achieve individual addressing

Challenges:

Immature integration with scalable electronics/optics eg. CMOS, waveguides etc.

Long-distance links (coupling to single-mode photons)

Ions don’t like surfaces – charging, noise

this week: introductory review articles: d. leibfried, r. blatt, c. … · 2018. 9. 3. ·...

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