row of qubits in a linear Paul trap formsa quantum register

Ion trap quantum processor Laser pulses manipulateindividual ions

A CCD camera reads out the ion`s quantum state

Effective ion-ioninteraction induced bylaser pulses that excitethe ion`s motion

slides courtesy of Hartmut Haeffner, Innsbruck Group with some notes byAndreas Wallraff, ETH Zurich

Trapping Individual Ions

Photo-multiplier

CCDcamera

Fluorescencedetection by

CCD cameraphotomultiplier

Vacuumpum

p

oven

atomic

beamLaser beams for:

photoionizationcoolingquantum state manipulationfluorescence excitation

Experimental setup

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

Ions as Quantum Bits

opticaltransition

stable|g>

|e>

detection

Quadrupole transition

Qubit levels:

P1/2

S1/2

D5/2

S1/2 , D5/2

τ ≈ 1 s

Qubit transition:

S1/2 – D5/2

absorption and emissioncause fluorescence steps(digital quantum jump signal)

D

S

P

S

D

monitorweak transition

metastablelevel

Quantum jumps: spectroscopy with quantized fluorescence

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)

„ 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

Electron shelving for quantum state detection

1. 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 80 100 120

0

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

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 resolveddetection withCCD camera:

Two ions:

P1/2

S1/2

D5/2

Fluorescencedetection

Electron shelving for quantum state detection

Mechanical Quantum harmonical oscillator

Extension of the ground state:

Size of the wave packet << wavelength of visible light

Energy scale of interest:

harmonic trap

ions need to be very cold to be in their vibrationalground state

Mechanical Motion of Ions in their Trapping Potential:

Laser – ion interactions

g

e

⊗

2-level-atom harmonic trap

.. .

0,g

0,e

1,e2,e

2,g

1,g

joint energy levels

…

Approximations:

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:

An Ion Coupled to a Harmonic Oscillator

tensor product dressed states diagram

harmonic trap

…

2-level-atom joint energy levels

. . .

External degree of freedom: ion motion

ion transition frequency 400 THz

trap frequency 1 MHz

A closer look at the excitation spectrum (3 ions)

carrier transitionmotional sidebands

Laser detuning Δ 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)

Stretch mode excitation

no differential force

no differential force

differential force

differential force

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

Cooling of the vibrational modes

1, −ng

1, −ne

ne,

ng,

……

red side band transition

Coherent excitation on the sideband

...

1, −nS

1, −nD

nD,1, +nD

1, +nS

nS,

coupled system

...

„Blue sideband“ pulses:

D state population

Entanglement between internal and motional state !

But also controlled excitation of the vibrational modes

Single qubit operations

• Laser slightly detuned from carrier resonance

or:

• Concatenation of two 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

z-rotations

x,y-rotations

CCD

Paul trap

Fluorescencedetection

electro-opticdeflector

coherentmanipulationof 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

Exc

itatio

n

Deflector Voltage (V)

Addressing the qubits

Generation of Bell states

Pulse sequence:

…

… …

…

generation of entanglement between two ions

Generation of Bell states

Ion 1: π/2 , blue sideband

Pulse sequence:

…

… …

…

creates entangled state between qubit 1 and oscillator

Generation of Bell states

Ion 1: π/2 , blue sideband

Ion 2: π , carrier

Pulse sequence:

…

… …

…

excites qubit 2

Generation of Bell states

Ion 1: π/2 , blue sideband

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

Bell state analysis

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)

A measurement yields the z-component of the Bloch vector

=> Diagonal of the density matrix

Rotation around the x- or the y-axis prior tothe measurement yields the phase informationof the qubit.

(a naïve persons point of view)

=> coherences of the density matrix

Obtaining a single qubit density matrix

as discussed before!

Bell state reconstruction

SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

F=0.91

Controlled Phase Gate CNOT

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

Together with the three single qubit gates,we can implement any unitary operation!

controlled phase gate

implementation of a CNOT for universal ion trap quantum computing

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 !

1ε 2ε

Cirac - Zoller two-ion phase gate

ion 1

motion

ion 2

,S D

,S D

0 0SWAP

, 0S,1S

, 0D,1D

1ε 2ε

Cirac - Zoller two-ion phase gate

Phase gate usingthe motion andthe target bit.

ion 1

motion

ion 2

,S D

,S D

0 0SWAP

Z

1ε 2ε

Cirac - Zoller two-ion phase gate

, 0S,1S

, 0D,1D

ion 1

motion

ion 2

,S DSWAP

,S D

0 0SWAP

Z

1ε 2ε

Cirac - Zoller two-ion phase gate

Phase gate usingthe motion andthe target bit.

ion 1

motion

ion 2

,S D

,S D

0 0

Z

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

?

Phase gate

Composite 2π-rotation:

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

A single ion composite phase gate: Experiment

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-

exci

tatio

n

2π 2π ππ2πφ =0=φ 0=φ

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

Continuous tomography of the phase gate

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

Testing the phase of the phase gate |0,S>

Time (μs)

D5/

2-

exci

tatio

n

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 2

π−

2

π

97.8 (5) %

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 – Zoller CNOT gate operation

SS - SS DS - DD

SD - SD DD - DS

input

output

Measured truth table of Cirac-Zoller CNOT operation

Superposition as input to CNOT gate

outputprepare gate detect

Error budget for Cirac-Zoller CNOT

4 %for tgate = 600 µsOff resonant excitations

3 %5 % in Rabi frequency

(at neighbouring ion)

Addressing error

(can be corrected for partially)

~ 10 % !!!~ 100 Hz (FWHM)Laser frequency noise

(Phase coherence)

Population lossMagnitudeError source

~ 20 %

~ 2 %

0.1 %

2 %

0.4 %

November 2002Total

~ 500 Hz (FWHM)Laser detuning error

1 % peak to peakLaser intensity noise

<n>bus < 0.02

<n>spec = 6Residual thermal excitation

Easy to have thousands of ions in a trap and to manipulate them individually…

but it is hard to control their interaction!

Scaling ion trap quantum computers

Scaling of the Cirac-Zoller approach?

• Coupling strength between internal and motional states of a N-ion stringdecreases as

(momentum transfer from photon to ion stringbecomes more difficult)

-> Gate operation speed slows down

• More vibrational modes increase risk of spurious excitation of unwanted modes

-> Use flexible trap potentials to split long ion string into smaller segmentsand perform operations on these smaller strings

Problems :

• Distance between neighbouring ions decreases -> addressing more difficult

Easy to have thousands of ions in a trap and to manipulate them individually…

but it is hard to control their interaction!

Scaling ion trap quantum computers

Kielpinski, Monroe, Wineland

The solution:The solutions:

Cirac, Zoller, Kimble, Mabuchi Tian, Blatt, Zoller

“accumulator”

controlqubit

targetqubit

D. Wineland, Boulder, USAKieplinski et al, Nature 417, 709 (2002)

Idea #1: move the ions around

Segmented ion traps as scalable trap architecture

(ideas pioneered by D. Wineland, NIST, and C. Monroe, Univ. Michigan)

Segmented trap electrode allow totransport ions and to split ion strings

Transport of ions

1 mm within 50 μs

no motional heating

Splitting of two-ion crystal

tseparation ≈ 200 μs

small heating Δn ≈1

State of the art:

„Architecture for a large-scale ion-trap quantum computer“, D. Kielpinski et al, Nature 417, 709 (2002)

„Transport of quantum states“, M. Rowe et al, quant-ph/0205084

D. Wineland (NIST) : Alumina / gold trap

Coherent transport of quantum information

C. Monroe (Michigan) : GaAs-GaAlAs trap