1

Quantum information processingwith trapped atoms

Introduction

Fundamentals: ion traps, quantum bits, quantum gates

Implementations: 2-qubit gates, teleportation, …

More recent, more advanced, …

Introduction

Fundamentals: ion traps, quantum bits, quantum gates

Implementations: 2-qubit gates, teleportation, …

More recent, more advanced, …

Jürgen Eschner

GDR-IQFA 1st colloquium, Nice, 23 March, 2011

Pentium 4 (2002)

1 atom

1 atomper bit

How many atoms per bit of information?How many atoms per bit of information?

ENIAC (1947)

Towards quantum technology (Moore's Law)

Quantum effects will play a role –and open up new avenuesQuantum effects will play a role –and open up new avenues

~ 2020

W. R. Keyes, IBM J. Res. Dev. 32, 24 (1988)& C. Schuck, ICFO

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Quantum information with single quantum systems

Applications in informatics and physics

P. Shor, 1994: factorization of large numbers is polynomial on a quantum computer, exponential on a classical computer

L. Grover, 1997: data base search N1/2 quantum queries, N classical

simulation of Schrödinger equations or any unitary evolution

quantum cryptography / repeaters / quantum links

improved atomic clocks (P. Schmidt et al., 2005)

(Gedanken-)experimentsfundamentals of QM, decoherence, entanglement

Fundamental phenomena Quantum information Technological applications

Quantum Information Technology

Quantum Computing & Simulation

Factorization, data base search

Quantum Cryptography (QKD)

Quantum Random Number Generation

Quantum Communication

Entanglement over large distance

Quantum Metrology (clocks, sensors)

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Quantum processors - various approaches

Ion traps

Neutral atoms in traps (opt. traps, opt. lattices, microtraps)

Neutral atoms and cavity QED

NMR (in liquids)

Superconducting qubits (charge-, flux-qubits)

Solid state concepts (spin systems, quantum dots, etc.)

Optical qubits and LOQC (linear optics quantum computation)

Electrons on L-He surfaces or in Penning trap

Ion-doped crystals or colour centers

Rydberg atoms

Superconducting qubits and circuit QED

…and more

200 nm

COLD SINGLE QUANTUM SYSTEMS WITH CONTROLLED INTERACTION

... allow full control, manipulation and measurement of quantum state(”quantum state engineering”)

Application for quantum information processing and studies of entanglement and decoherence

(work by NIST, Hamburg , Oxford, MPQ, Ibk, ...)

... are cold (< 1 mK) and well-localized (< 20 nm)

Application for precision measurements and frequency standards(work by NIST, PTB, MPQ, NRC, NPL, NML, Mainz, MIT, ...)

... allow full control, manipulation and measurement of quantum state(”quantum state engineering”)

Application for studies of entanglement and decoherence and for quantum information processing(work by NIST, Innsbruck, Hamburg , Oxford, MPQ, Michigan, ICFO)

... are individual quantum systems

Textbook experiments on fundamental quantum mechanics(work by NIST, Innsbruck, MPQ, Seattle, IBM, Hamburg, ...)

Single trapped and laser-cooled ions ...

4

~100 µm

QIP workhorse : linear Paul trap

R. Blatt

Ions = "qubits" = q.m. coherent2-level systems

Laser-controlled Interacting through

vibrational modes Individually measured

Cirac & Zoller, PRL 1995

Ion trapsIon traps

5

fluorescencedetection

ring electrode

endcapelectrode

endcapelectrode

coolingbeam

z

lens

yx

Classic Paul trap

(Intuitive picture works when )

Ion confinement requires a focusing force in 3 dimensions

rr binding force rF rr−~ 2~ reEeF rrr

Φ⇒Φ∇−==, that is

)2( 22220

0 zyxr

−+Φ

=ΦQuadrupole potential

Paul trap: tVU Ω+=Φ cos000 (Penning trap: +=Φ 00 U axial magn. field)

Equation of motion in a Paul trap: 0)cos(2 0020

=Ω++ xtVUrexm &&

This is a special case of the MATHIEU EQUATION (stability diagram…)

Full motion = secular motion at with superimposed micromotion atzyx ωωω ,, Ω

Ion storage

Ω<<ω

(saddle potential)

Intuitive picture: time-averaged kinetic energy of driven motion = effective potentialin which the ion oscillates freely with frequency ω

6

4.7 µm

Legacy : circular trap for single ions

Miniature Paul trap Single atoms (Ba+) in in trap

Ring Ø ~ 1 mmRF ~25 MHz, ~ 500V

secular frequency~ 1 MHz

RF

Paul mass filter

InnsbruckLos Alamos

Boulder, Mainz, AarhusBoulder, Mainz, Innsbruck, …

München

Linear ion trap evolution

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Trapping ion strings for QIP

Paul trap mechanismTrapped charged particles

Linear trap ~1 mm sizeIon strings

Laser coolingLocalization << λLaser

Coulomb repulsionInteraction by commonmodes of motion

Stationary / single-atom qubitsStationary / single-atom qubits

Examplestaken from:

8

0 0.5 1 1.5 20

0.5

1

(a)

Time of excitation (µs)

Quantum bits: superpositions

% p

opul

atio

n

100

50

Laser excitation switches continuously between on and off Measurement shows either on (bright) or off (dark)

TLS

TLS

Encoding of quantum information requires longlived atomic states:

microwave transitions(Hyperfine transitions, Zeeman transitions)earth alkalis: 9Be+, 25Mg+, 43Ca+, 87Sr+, 137Ba+, 111Cd+, 171Yb+

optical transitions(forbidden transitions,intercombination lines)S – D transitions in earth alkalis: Ca+, Sr+, Ba+, Ra+, (Yb+, Hg+) etc.

S1/2

P1/2D5/2

S1/2

P3/2

Qubits in trapped ions (two-level systems, TLS)

InnsbruckNIST

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Qubits in 40Ca+ ion

Ground state + long-lived excited state Zeeman substates of level

-1/2 1/2

Zeeman structure in non-zero magnetic field::

S1/2

D5/2

∆

5/23/2

-3/2-5/2- /21

- /21/21

/21

2-level-system

1/2 ⇔ 5/2

Zeeman structure of the S1/2 – D5/2 transition

levels transitions

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P3/2

S1/2

P1/2

D3/2

|D5/2(m=5/2)>

729 nm

Manipulation by laser pulseson 729 nm transition(~ 1 ms coherence time)

qubit

Optical qubit transition

|S1/2(m=1/2)>

Superpositions of S1/2(m=1/2) and D5/2(m=5/2) forms qubit

s1≈τ

|D>

Qubit dynamics

|S>

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Rabi oscillations

0 0.5 1 1.5 20

0.5

1

(a)

Time of excitation (µs)

Quantum bits: superpositions

% p

opul

atio

n

100

50

Laser excitation switches continuously between on and off Measurement shows either on (bright) or off (dark)

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Rabi oscillations cont'd.

Motion and motional qubitMotion and motional qubit

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Quantized motion

.. .motional qubit|0>

|1>|1>|n=0>

|2>

N ions → 3N oscillators

S1/2

D5/2

|n> = |0> |1> |2>

ω

coupled system & transitions

g

e

Ω Γ ⊗

2-level-atom harmonic trap

...

spectroscopy: carrier and sidebands

Laser detuning

∆n = 1∆n = -1

∆n = 0

ω 0=n12

Motional sidebands

Rabi frequencies

Carrier:

Red SB:

Blue SB:

(*) Lamb-Dicke regime : ion localised to << laser wavelength :

(*)

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Excitation spectrum of the S1/2 – D5/2 transition

ωax = 1.0 MHzωrad = 5.0 MHz

(only oneZeeman

component)

P3/2

854 nm

S1/2

P1/2

D3/2

D5/2

729 nm

|D>

|S>

qubit

Sideband cooling

S1/2

Motional state preparation bysideband cooling on 729 nm transition

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1−n

P3/2

D5/2

S1/2 n1+n

Cooling cascade

τD = 1 s : no spontaneous decay

Quenching on D5/2 - P3/2 transition :

Effective two-level system

Sideband cooling on S1/2 – D5/2 transition

> 99.9% in |n=0>

D5/2

729 nm

|1>

|0>

internal qubit

Qubits in a single 40Ca+ ion

S1/2

.. .

motional qubit

|0>|1>|1>

|n=0>

|2>

|S,n> |D,n> : carrier transition (∆ = 0)

|S,n> |D,n±1> : sideband transition (∆ = ±ω)

ω

"computationalsubspace"

|S,0>

|D,0>|D,1>

|S,1>

COHERENT LASER MANIPULATION (Rabi oscillations)

First single-ion quantum gate: Monroe et al. (Wineland), PRL 75, 4714 (1995).

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computationalsubspace

(levels with n>1ignored)

Laser-driven transitions are described by unitaryoperators (if ) :

carrier:

red sideband:

blue sideband:

Qubit rotations in computational subspace

|S,0>

|D,0>|D,1>

|S,1>

where

Example: excitation on blue sideband with

0,S

0,D1,D

1,S

carrier and sidebandRabi oscillations

with Rabi frequencies

carrier

sideband

Lamb-Dicke parameter

Coherent state manipulation, qubit rotations

Each point : average of 100-200 individual measurements, preparation – coherent rotation – state detection

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π 2π 3π 4π

Blue sideband

Blue sidebandπ/2 −π/2

D-s

tate

popu

latio

n

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

Pulse length (µs)

Rabi-flops on blue sideband

Ramsey Interference

0 500

0.2

0.4

0.6

0.8

1

D-s

tate

popu

latio

n

100 150 200 250 300Pulse length (µs)

Qubit rotations : phase of the wavefunction

|S,0>

|D,0>|D,1>

|S,1>

|S,0>

|D,0>|D,1>

|S,1>

Quantum gatesQuantum gates

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Coulomb coupling : common modes of motion

Real oscillation frequencies are 200 – 300 kHz

With several ions, the motional qubits are shared

vibrational modes

axial (c.o.m.) mode of 7 ions

breathing mode of 7 ions

2 ions + motion = 3 qubits

Motional qubit acts as the "bus" between the ions

vibrational modes computational subspace: 2 ions, 1 mode

|S,S,0>

|D,S,0>|D,S,1>

|S,S,1>

|S,D,0>|S,D,1>

|D,D,0>|D,D,1>

laser on ion 2

laser on ion 2 laser on ion 1

laser on ion 1

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Quantum gate proposal(s)

21121 εεεεε ⊕→

Further gate proposals: • Cirac & Zoller• Mølmer & Sørensen, Milburn• Jonathan & Plenio & Knight• Geometric phases

0111110110100000

→

→

→

→

control bitcontrol bit target bittarget bit

controlled NOTcontrolled NOT

1111010110100000

−→

→

→

→

phase gatephase gate

Details of C-Z gate operation (Phase gate)

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Desired result, schematic

→

→

→

→

S S S SS D S DD D DSD D D Scontrolcontrol targettarget

Experimental techniquesExperimental techniques

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Ion trap setup

RF linear trap:0.7-2Mhz5MHz

ω ≈ω ≈

Radial confinement: oscillating saddle potential

Longitudinalconfinement

Double trap apparatus

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Laser pulses for coherent manipulation etc

τcoh >> τgate

AOM = acousto-optical modulator, based on Bragg diffraction

"Ampl" = Amplitude, includes switching on/off

AOM = acousto-optical modulator, based on Bragg diffraction

"Ampl" = Amplitude, includes switching on/off

∆ , Φ , Ampl

ν

ν+∆ , Φ , Ampl

cw laser

RF

AOM I

t

I

t

Φ1

Φ2 Φ3

to trap

(That's the difficult bit!)

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Spatial addressing of ions in a string

Well-focussed laser beam

beam steering with electro-optical deflector

addressing waist ~ 2.5 - 3.0 mm

< 1/400 intensity on neighbouring ion

first demonstration: H.C. Nägerl et al., Phys. Rev. A 60, 145 (1999)

Spectral addressing: excitation spectrum of two ions

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P3/2

S1/2

P1/2

D3/2397 nm

866 nm D5/2 |D>

|S>

qubit

Discrimination of qubit states

S1/2

State detection by photon scattering on S1/2 to P1/2 transition at 397 nm

Detector

sn8≈τ

s1≈τ

Photons observed : S1/2 = |S>No photons : D5/2 = |D>

Quantum jumps & "Quantum amplification"

High power LED for photo ionisation (80 mW @ 385 nm, HWHM 15 nm) excites 393 nm transition

Ions are pumped into D5/2 state (life time ~ 1s)No fluorescence

Detection of absorption of < 1 photon/sec

40Ca+

S1/2

P1/2

P3/2

D3/2

D5/2

397

nm

393

nm

854 nm

729 nm

850 nm

866 nm

1 sec

25

P

S

D

monitor

0 20 40 60 80 100 1200

1

2

3

4

5

6

7

8

Zählrate pro 9 ms

D-Zustand besetzt S-Zustand besetzt

Anza

hl d

er M

essu

ngen

counts in 9 ms#

of m

easu

rem

ents

D state occupied S state occupied

State detection: shelving

Histogram of counts in 9 msPoisson distribution N ± N½

discrimination efficiency 99.85%

H. Dehmelt 1975

(Shelving level can, but need notbe the same as qubit state)

Individual ion detectionon CCD camera

5µm

quantum state populations pSS,pSD,pDS,pDD

|SS>|DS>

|DD> |SD>

Two-ion histogram (1000 experiments)

region 1 region 2

|SS>

|SD>

|DS>

|DD>

Quantum state discrimination with 2 ions

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computationalsubspace

,0S,1S

,0D,1D out of CS !

2~ηRabiΩ

1~ηRabiΩ

π 2π

naive idea : π-pulse on blue SB composite SWAP (from NMR)

computationalsubspace

,0S,1S

,0D,1D

π π π4

Gate pulses (I) : SWAP

(works if initial state is not |S,1>)

Swap information from internal into motional qubit and back

A.M. Childs et al., Phys. Rev. A 63, 012306 (2001)

1

2

3

π , 0 ,1D S↔on

4π ,1 , 2D S↔on

1

3

I. Chuang et al., Innsbruck (2002)

3-step composite SWAP operation

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Detect quantum state of one ion only (needed for teleportation)Protect neighbours from addressing errors

D5/2

S1/2

D5/2

S1/2

D5/2

S1/2

D5/2

S1/2

ion #1 ion #2

π

Hiding qubits

D5/2

S1/2

D5/2

S1/2

D5/2

S1/2

D5/2

S1/2

ion #1 ion #2

DD‘

superposition stateof ion #2 protected

Cirac-Zoller quantum CNOT Gatewith two trapped ions

Cirac-Zoller quantum CNOT Gatewith two trapped ions

"Realization of the Cirac–Zoller controlled-NOT quantum gate", F. Schmidt-Kaler et al., Nature 422, 408-411 (2003).

"Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate", D. Leibfried et al., Nature 422, 412 (2003).

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Quantum gate proposal

control bitcontrol bit target bittarget bit

1111010110100000

−→

→

→

→

phase gatephase gate

Detection

ion 1

motion

ion 2

,S D

,S D0 0

control qubit

target qubit

SWAP

1ε 2ε

Cirac-Zoller two-ion controlled-NOT gate

SWAP-1

Preparation |S> = bright|D> = dark

CNOT

"bus" qubit

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ion 1

motion

ion 2

,S DSWAP-1

,S D0 0SWAP

Ion 1Ion 1

Ion 2Ion 2

pulse sequence

control bitcontrol bit

target bittarget bit

Cirac-Zoller two-ion controlled-NOT operation

blueπ0

blueππ

cπ/20

blueπ

π/2

blueπ/2½

0

blueπ/2½

0

blueπ

π/2

cπ/2π

CNOTPhase

CNOT gate = Phase gate enclosed by π/2- pulsesPhase gate implemented by composite pulses

SS SS DS DD

SD SD DD DS

Result : full time evolution

Prep

arat

ion

Det

ectio

n

every point = 100 single measurements, line = calculation (no fit)

SWAP

SWAP

-1

CNOT betweenmotion and ion 2

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Measured fidelity (truth table)

D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001)

Qubits store superposition information, scalable physical system

Ability to initialize the state of the qubits

Long coherence times,much longer than gate operation time

Universal set of quantum gates:Single bit and two bit gates

Qubit-specific measurement capability

Requirements for QC

ion string, but scalability?

ground state cooling

hard workbut good progress

Coherent pulses oncarrier and sidebands,addressing

Shelving, imaging

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Quantum teleportationQuantum teleportation

"Deterministic quantum teleportation with atoms", M. Riebe et al., Nature 429, 734 (2004).

"Deterministic quantum teleportation of atomic qubits", M. D. Barrett et al., Nature 429, 737 (2004).

BOB

ALICE

Bell state

measurementin Bell basis

rotation

unknowninput state

class. communication

recoverinput state

Teleportation idea (Bennett 1993)

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Full formal procedure

Particle 2 and 3 are prepared in an entangled state

where one particle is always in the "opposite" state of the other.

When the result of the Bell measurement between 1 and 2 is

then 2 is in the "opposite" state of the unknown state of 1.

Thus particle 3, being in the opposite state to 2, will be in the samestate as 1.

If the result of the Bell measurement is a different state, theappropriate rotation has to be applied to particle 3.

Very simple description (see Bouwmeester et al.)

33

Figure fromKimble,Nature N&V

Teleportation of atomic qubit states

Teleportation protocol

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"Hide": protect ionfrom interactionwith laser pulses(manipulation ormeasurement)on the other ions

Teleportation in Innsbruck (Riebe et al.)

83 %

class.: 67 %

no cond. op.: 50 %

M. Riebe et al., Nature 429, 734 (2004)

Quantum teleportation with atoms: result

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3 Be+ ions in segmented trap, hyperfine qubits with Raman transitions, gate without addressing, instead shuttling of the ions

Teleportation protocol at NIST

F = 78%

Barrett et al., Nature 429, 737 (2004)

More recent progress: very long coherence time n>>1 qubit entanglement

high fidelity gate

More recent progress: very long coherence time n>>1 qubit entanglement

high fidelity gate

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Measured decay time

τ ≈ 1.05 s

Limit set by spontaneous emission

τ = 1.16 s0 0.1 0.2 0.3

0

1

-1

Time (s)

Parity

Superpositions of equal energy are not affected by global phase noise(see e.g. Kilpinsky et al., Science 291, 1013-1015 (2001))

Lifetime-limited Bell states (Innsbruck)

Useful to construct "decoherence-free subspaces"

Lifetime > 20 s

D5/2

S1/2S1/2

Ultra-longlived Bell states (Innsbruck)

C. Roos et al., Ibk

37

State of the art : The Quantum Byte

Blatt group, Nature 438, 643 (2005)

Density matrix

Wineland group, Nature 438, 639 (2005)

Entangled W state of 8 ions @ Innsbruck

Entangled GHZ state of 6 ions @ NIST Boulder

Fidelity 72%

High fidelity quantum gate

Blatt group, Nature Physics 4, 463 (2008)

Sta

te p

opul

atio

ns

Time (µs)

17 sequential entangling and disentangling operations

individual entangling fidelity 99.3(1)% for Ψ+ Bell state

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Ion trap QC architectureIon trap QC architecture

Manipulation of ion strings in ascalable quantum processorwith memory and processor

• segmented ion trap• coherent transport• separating and combining strings

This segmented trap:Schmidt-Kaler / Blatt, Innsbruck / Ulm

more trap architecture:A. Steane's group, OxfordD. Winelands group, NISTC. Monroe's group, U MichiganI. Chuang's group, MIT

Laser

Scalable ion trap quantum computer

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v1(t)v2(t)v3(t)v4(t)••••

“accumulator”

controlqubit

targetqubit

NIST group - Kielpinski et al, Nature 417, 709 (2002)

Quanten “CCD array”

Ion chip trap at NIST

NIST group

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Ion chip trap at NIST

NIST group

Shuttling operations

http://monroelab2.physics.lsa.umich.edu/research/trap/index.html#ttrap

2 ions are separated (also demonstrated earlier at NIST)

single ion is shuttled around corner in a T-junction linear trap

41

Single neutral atom gate: Rydberg gateSingle neutral atom gate: Rydberg gate

Dipole trap

FORT – far-off-resonance trapbased on light shift (AC Stark shift)

.. .

|1>|n=0>

|2>

focused laser beam

trapping potential

Several traps with controllable distance on the < µm scale (Grangier)

Lattice with single-site optical addressing (Kuhr)

42

Coherent Rydberg excitation

pulsed two-photon excitation

state measurement by recapture

Grangier group

Rydberg gate

Dipole – dipole couplingof 2 Rydberg atoms

Significant coupling for > 10 µm

Gate principle:

Gate pulses (UZ):

Jaksch, Grangier, Saffman

43

Upcoming system : circuit QEDUpcoming system : circuit QED

Device

2 superconducting qubits coupled by microwave resonator

44

Control

Resonant drivepopulation control

Adiabatic tuning overavoided crossing

phase control

Schoelkopf,Wallraff

Entangling gates

DiCarlo (Schoelkopf group) 2009

45

QC with atoms :

A lot has been done …

A lot remains to be done …

QC with atoms :

A lot has been done …

A lot remains to be done …