Quantum information processing with trapped ionsmsafrono/650/Lecture16.pdf · shine the light pulse at the carrier transition, the system evolves for a time with this Hamiltonian,

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Quantum information processingwith trapped ions

Courtesy of Timo Koerber Institut für Experimentalphysik

Universität Innsbruck

1. Basic experimental techniques

2. Two-particle entanglement

3. Multi-particle entanglement

4. Implementation of a CNOT gate

5. Teleportation

6. Outlook Lectures 15 - 16

The requirements for quantum information processing

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

I. Scalable physical system, well characterized qubits

II. Ability to initialize the state of the qubits

III. Long relevant coherence times, much longer than gate operation time

IV. “Universal” set of quantum gates

V. Qubit-specific measurement capability

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S1/2

P1/2

D5/2

„quantumbit“

Experimental Setup

Important energy levels

• The important energy levels are shown on the next slides; a fast transition is used to detect ion fluorescence and for Doppler cooling, while the narrow D5/2 quadrupole transition has a lifetime of 1 second and is used for coherent manipulation and represents out quantum bit. Of course a specific set of Zeeman states is used to actually implement our qubit. The presence of other sublevels give us additional possibilities for doing coherent operations.

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P1/2

S1/2

= 7 ns

397 nm

D5/2

S1/2 – D5/2 : quadrupole transition

729 nm

= 1 s

Ca+: Important energy levels

P1/2

S1/2

D5/2

„qubit“

P1/2

= 7 ns

„quoctet“ (sp?)

Ca+: Important energy levels

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qubit

qubit

Encoding of quantum information requires long-lived atomic states:

microwave transitions

9Be+, 25Mg+, 43Ca+, 87Sr+, 137Ba+, 111Cd+, 171Yb+

optical transitions

Ca+, Sr+, Ba+, Ra+, Yb+, Hg+ etc.

S1/2

P1/2

D5/2

S1/2

P3/2

Qubits with trapped ions

Linear RF Paul trap

RF electrode

High dc potential

control electrodeLow dc voltage

control electrode

Positive ion

• Drive freq ~ 100-150 MHz• RF amp ~ 200-400 V• Secular freq

– Radial ~ 15 MHz– Axial ~ 4 MHz

Slides from :R. Ozeri

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Multi-zone ion trap

control

controlrf

rf

view along axis:

rf filter board• Gold on alumina

construction• RF quadrupole

realized in two layers

• Six trapping zones• Both loading and

experimental zones

• One narrow separation zone

• Closest electrode ~140 m from ion

segmented

linear trapping

region

Ion transport

100 m

6-zone alumina/gold trap(Murray Barrett, Tobias Schaetz et al.)

200 mseparation zone• Ions can be moved

between traps.– Electrode potentials

varied with time

• Ions can be separated efficiently in sep. zone– Small electrode’s

potential raised

• Motion (relatively) fast– Shuttling: several 10 s– Separating: few 100 s

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Innsbruck segmented trap (2004)

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

String of Ca+ ions in Paul trap

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50 µm


MHz 20.7zMHz45.1, yx

String of Ca+ ions in linear Paul trap

50 µm


MHz 20.7zMHz45.1, yx

String of Ca+ ions in linear Paul trap

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Addressing of individual ions

CCD

Paul trap

Fluorescencedetection

electrooptic deflector

coherentmanipulation of qubits

dichroicbeamsplitter

inter ion distance: ~ 4 µm

addressing waist: ~ 2.5 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)

Ion addressing

The ions can be addressed individually on the qubit transition with an EO deflector which can quickly move the focus of the 729 light from one ion to another, using the same optical path as the fluorescence detection via the CCD camera.

How well the addressing works is shown on the previous slide: The graph shows the excitation of the indiviual ions as the deflector is scanned across the crystal.

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External degree of freedom: ion motion

Notes for next slides:

Now let's have a look at the qubit transition in the presence of the motional degrees of freedom. If we focus on just one motional mode , we just get a ladder of harmonic oscillator levels.

The joint (motion + electronic energy level) system shows a double ladder structure. With the narrow laser we can selectively excite the carrier transition, where the motional state remains unchanged...

Or use the blue sideband and red sideband transitions, where we can change the motional state.

We can walk down the double ladder by exciting the red sideband and returning the ion dissipatively to the grounsstate. With this we can prepare the ions in the motional ground state with high probability, thereby initializing our quantum register.

harmonic trap

…


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harmonic trap

…

2-level-atom joint energy levels


harmonic trap

…

2-level-atom joint energy levels

Laser cooling to the motional ground state:

Cooling time: 5-10 ms

> 99% in motional ground state


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Interaction with a resonant laser beam :

: Rabi frequency

: phase of laser field

: rotation angle

Laser beam switched on for duration :

Coherent manipulation

harmonic trap2-level-atom joint energy levels

…

If we resonantly shine in light pulse at the carrier transition, the system evolves for a time tau with this Hamiltonian,

where the coupling strength Omega depends on the sqroot of the intensity, and phi is the phase of the laser

field with respect to the atomic polarization.

Coherent manipulation

Let's now begin to look at the coherent state manipulation. If we resonantly shine the light pulse at the carrier transition, the system evolves for a time with this Hamiltonian, where the coupling strength depends on the square root of the intensity, and is the phase of the laser field with respect to the atomic polarization.

The effect of such a pulse is a rotation of the state vector on the Bloch sphere, where the poles represent the two states and the equator represents superposition states with different relative phases. The roation axis is determined by the laser frequency and phase. The important message is here that we can position the state vector anywhere on the Bloch sphere, which is a way of saying that we can create arbitrary superposition states.

The same game works for sideband pulses. With a /2 pulse, for example, we entangle the internal and the motional state! Since the motional state is shared by all ions, we can use the motional state as a kind of bus to mediate entanglement between different qubits in the ion chain.

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D state population

„Carrier“ pulses: Bloch sphererepresentation

Coherent excitation: Rabi oscillations

coupled system„Blue sideband“ pulses:

D state population

Entanglement between internal and motional state !

Coherent excitation on the sideband

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P1/2 D5/2

=1s

S1/2

40Ca+

Experimental procedure

1. Initialization in a pure quantum state:laser cooling,optical pumping

3. Quantum state measurementby fluorescence detection

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

P1/2

S1/2

D5/2

Dopplercooling Sideband

cooling

P1/2

S1/2

D5/2

Quantum statemanipulation

P1/2

S1/2

D5/2


50 experiments / s

Repeat experiments100-200 times

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

=1s

S1/2

40Ca+

Experimental procedure

1. Initialization in a pure quantum state:Laser sideband cooling

3. Quantum state measurementby fluorescence detection

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

P1/2

S1/2

D5/2

Dopplercooling Sideband

cooling

P1/2

S1/2

D5/2

Quantum statemanipulation

P1/2

S1/2

D5/2


50 experiments / s

Repeat experiments100-200 times

Spatially resolveddetection withCCD camera:

Multiple ions:

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5. Teleportation

6. Outlook

Pulse sequence:

…

… …

…

Creation of Bell state

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Generation of Bell states

Ion 1: /2 , blue sideband

Pulse sequence:

…

… …

…

Creation of Bell states


Ion 2: , carrier

Pulse sequence:

…

… …

…


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Ion 2: , carrier

Ion 2: , blue sideband

Pulse sequence:

…

… …

…


Fluorescencedetection withCCD camera:

Coherent superposition or incoherent mixture ?

What is the relative phase of the superposition ?

SSSDDS

DD SSSDDSDD

Measurement of the density matrix:

Analysis of Bell states

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Reconstruction of a density matrix

Representation of as a sum of orthogonal observables Ai :

is completely detemined by the expectation values <Ai> :

For a two-ion system :

Joint measurements of all spin components

Finally: maximum likelihood estimation (Hradil ’97, Banaszek ’99)

Preparation and tomography of Bell states

SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

SSSD

DSDD SSSDDSDD

Fidelity:

Entanglementof formation:

Violation of Bell inequality:

F = 0.91F = 0.91

E(exp) = 0.79

S(exp) = 2.52(6)> 2

SSSD

DSDD SSSDDSDD

C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004)

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sensitive to:

laser frequency magnetic field exc. state lifetime

Different decoherence porperties





5. Teleportation

6. Outlook

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Ion 1: 1 , blue sideband

Ion 2,3: , carrier

Ion 2: 2 , blue sideband

Pulse sequence:

…… …

…Ion 3: 3 , blue sideband

1. 2.3.

Generation of W-states

Density matrix of W – state

experimental result theoretical expectation

Fidelity: 85 %

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DDDDDDDS

SSSSDDDD

SSSS

14.4.2005

Four-ion W-states

DDDDDDDDDS

SSSSS

SSSSS

DDDDD15.4.2005

Five-ion W-states

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Ion detectionon a CCD camera

(detection time:4ms)

5µm

5 10 15 20 25

1

2

3

5 10 15 20 25

1

2

3

5 10 15 20 25

1

2

3

5 10 15 20 25

1

2

3

5 10 15 20 25

1

2

3

5 10 15 20 25

1

2

3

5 10 15 20 25

1

2

3

5 10 15 20 25

1

2

3

all ions in |S>

ion 1 in |S>

ion 6 in |S>

ion 4 in |S>

ion 5 in |S>

ions 1 and 5 in |S>

ions 1,2,3, and 5 in |S>

ions 1,3 and 4 in |S>

123456

Detection of six individual ions

22.4.2005729 settings, measurement time >30 min.

F=73%preliminary

result

Is there 6-particle entanglement present?

• 6-particle W-statecan be distilled from thestate (O. Gühne)

• 6-particle entanglementpresent, unresolved issues with error bars

Six-ion W-state

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control

target





5. Teleportation

6. Outlook

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

controlcontrol targettarget

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

control

target

Cirac-Zoller two-ion controlled-NOT operation

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ion 1

motion

ion 2

control qubit

target qubit

SWAP


1 2

First we swap the quantum states of the control qubit and the motional qubit…

ion 1

motion

ion 2

control qubit

target qubit


1 2

Next we perform a CNOT gate between the motional qubit and ion 2 and …

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ion 1

motion

ion 2

SWAP-1control qubit

target qubit

Cirac - Zoller two-ion controlled-NOT operation

1 2

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

Finally we reverse the SWAP operation,

we have used the motional qubit as a

quantum messenger between the two ions.

ion 1

motion

ion 2

SWAP-1SWAP

Ion 1Ion 1

Ion 2Ion 2

pulse sequence:pulse sequence:

Cirac - Zoller two-ion controlled-NOT operation

control qubitcontrol qubit

target qubittarget qubit

laser frequencypulse lengthoptical phase

Phase gate

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Experimental fidelity of Cirac-Zoller CNOT operation

input

output

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

Protecting qubits (from being measured)

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Threshold

ion #1 in |D>

Tomography after the measurement result is available!

ion #1 in |S>

Selective read-out of an atom in a W-state





5. Teleportation

6. Outlook

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No : • Infinite amount of information needed to specify • Measurement on yields just one bit of information

Phys. Rev. Lett. 70, 1895 (1993)

„Alice“ „Bob“

A Bclassical communication

Is it possible to transfer an unknown quantum state

from „Alice“ to „Bob“ by classical communication ?

unknown state

Two bits

Yes, if Alice and Bob share a pair of entangled particles !

EPR pair

Quantum state teleportation

Ion 3

Ion 2

Ion 1

Bell

state

initialize #1, #2, #3

classical communication

conditional rotations

CNOT --Bell basis

Alice

Bob

Selectiveread out

recovered on ion #3

Teleportation protocol

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Ion 3

Ion 2

Ion 1

conditional rotations using electronic logic, triggered by PM signal

conditional rotations using electronic logic, triggered by PM signalP

U

U

P

C C C

B

B B B B C

CU P

B

C

C P

spin echo sequencespin echo sequence

full sequence:26 pulses + 2 measurements

B

C

blue sideband pulsesblue sideband pulses

carrier pulsescarrier pulses

P

C

B

B

Teleportation protocol, details

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Teleportation procedure, analysis

Input state Output state

TP

Initial Final

U U-1

Fidelities

Quantum teleportation with atoms: results

83 %

class.: 67 %

no cond. op. 50 %

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How to build a large-scale quantum computer with trapped ions

Linear crystal of 20 confined atomic 171Yb+ ions laser cooled to be nearly at rest

C. Monroe, and J. Kim Science 2013;339:1164-1169

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Harnessing ion-trap qubits

David P. DiVincenzo Science 2011;334:50-51

Jonathan P. Home et al. Science 2009;325:1227-1230

Schematic of the sequence of operations implemented in a single processing region for building up a computation in the architecture

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Separating memory and processor zones

Scientific American (August 2008), 299, 64-71

Ion trap structure for the shuttling of ions through a junction


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Concept of a quantum CCD trap

Image credit: National Institute of Standards and TechnologyC. Monroe, and J. Kim Science 2013;339:1164-1169

Version 2: Photonics coupling of trapped ions qubits


Energy levels of trapped ion excited with a fast laser pulse (blue upward arrow) that produces single photon whose color, represented by the state or , is entangled with the resultant qubit state.

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Version 2: Photonics coupling of trapped ions qubits

Two "communication qubit" ions, immersed in separate crystals of other ions, each produce single photons when driven by laser pulses (blue). With some probability, the photons arrive at the 50/50 beamsplitter and then interfere. If the photons are indistinguishable (in polarization and color), then they always leave the beamsplitter along the same path. The simultaneous detection of photons at the two output detectors means that the photons were different colors, this coincidence detection heralds the entanglement of the trapped ion qubits.

Scientific American (2008), 299, 64-71

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Scientific American (2008), 299, 64-71

Elementary logic unit (ELU)

Phys. Rev. A 89, 022317 (2013)

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Modular distributed quantum computer

Several elementary logic units (ELU)s are connected through a photonic network by using an optical crossconnect switch, inline fiber beamsplitters, and a photon-counting imager.


Application to quantum communication

Trapped ion quantum repeater node made up of communication qubit ions (such as Ba+) and memory qubit ions (such as Yb+), with two optical interfaces per node. Multiple communication qubits are used per optical interface to inject photons into the optical channel, while the results for successful entanglement generation at the detectors are reported back to this node. Only qubits corresponding to successful events will be transported to the memory qubit region for use in quantum repeater protocol.


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Long-distance quantum communication


A chain of quantum repeater nodes can distribute quantum entanglement over macroscopic distances. The photons generated by the ions must be converted to telecommunication wavelengths for long-distance transport, which can be achieved by nonlinear optical processes.

Documents

Quantum information processing with trapped ionsmsafrono/650/Lecture16.pdf · shine the light pulse at the carrier transition, the system evolves for a time with this Hamiltonian,