Introduction to

quantumoptics with ions

PD Dr. Kilian Singer

Universität Mainz

www.quantenbit.de/#/teach/Public%20Outreach/

start

Ion Gallery

Boulder, USA: Hg+

Aarhus, Denmark: 40Ca+ (red) and 24Mg+ (blue)

Oxford, England: 40Ca+ coherent breathing motion of a 7-ion linear crystal

Innsbruck, Austria: 40Ca+

Paul trap

fluorescence

detection

ring

electrode

endcap

electrode

endcap

electrode

cooling

beam

z

lens

y x

Paul trap

Binding in three dimensions

Electrical quadrupole potential

Binding force for charge Q

leads to a harmonic binding:

no static trapping in 3 dimensions

Laplace equation requires

Ion confinement requires a focusing force in 3 dimensions, but

such that at least one of the coefficients is negative,

e.g. binding in x- and y-direction but anti-binding in z-direction !

trap size:

Dynamical trapping: Paul‘s idea

time depending potential

with

leads to the equation of motion for a particle with charge Q and mass m

takes the standard form of the Mathieu equation

(linear differential equ. with time depending cofficients)

with substitutions

radial and axial trap radius

Regions of stability

time-periodic diff. equation leads to Floquet Ansatz

If the exponent µ is purely real, the motion is bound,

if µ has some imaginary part x is exponantially growing and the motion is unstable.

The parameters a and q determine if the motion is stable or not.

Find solution analytically (complicated) or numerically:

a=0, q =0.1 a=0, q =0.2

time time

excurs

ion

excurs

ion

a=0, q =0.3 a=0, q =0.4

time time

excurs

ion

excurs

ion

a=0, q =0.5 a=0, q =0.6

time time

excurs

ion

excurs

ion

a=0, q =0.7 a=0, q =0.8

time time

excurs

ion

excurs

ion

a=0, q =0.9 a=0, q =1.0

time time

excurs

ion

excurs

ion

6 1019

-3 1019

unstable

time

po

sitio

n in t

rap

micromotion

1D-solution of Mathieu equation single Aluminium dust particle in trap

Two oscillation frequencies

slow frequency: Harmonic secular motion, frequency w increases with increasing q

fast frequency: Micromotion with frequency W

Ion is shaken with the RF drive frequency (disappears at trap center)

Lissajous figure

3-Dim. Paul trap stability diagram

for a << q << 1 exist approximate solutions

The 3D harmonic motion with frequency wi can

be interpreted as being caused by

a pseudo-potential Y

leads to a quantized harmonic oscillator

Real 3-Dim. Paul traps

ideal 3-Dim. Paul trap with equi-potental

surfaces formed by copper electrodes

endcap electrodes at distance

ideal surfaces:

but non-ideal surfaces do trap also well:

rring ~ 1.2mm

A. Mundt, Innsbruck

ideal 3 dim. Paul trap with equi-potental

surfaces formed by copper electrodes

non-ideal surfaces

rring ~ 1.2mm

numerical calculation

of equipotental lines

similar potential

near the center

Real 3-Dim. Paul traps

determine the quadrupole part of the potential:

ideal:

real:

rring ~ 1.2mm

Paul trap: wire ring + endcaps

Paul Straubel trap: ring only

inverted Paul Straubel trap: endcaps only

Paul trap

main advantages:

good optical access

large observation angle

fully harmonic

potential

Loss factor

Real 3-Dim. Paul traps

x

y

2-Dim. Paul mass filter stability diagram

time depending potential

with

dynamical confinement in the x- y-plane

with substitutions

radial trap radius

2-Dim. Paul mass filter stability diagram

x

y

A Linear Paul trap

plug the ends of a mass filter by positive electrodes:

mass filter blade design side view

RF

RF 0V

0V Uend Uend

numerically calculate the axial electric potential,

fit parabula into the potential

and get the axial trap frequency

with k geometry factor

z0

Innsbruck linear ion trap

1.0mm

5mm

MHz5radialwMHz27.0 axialw

Blade design

eVdepthtrap F. Schmidt-Kaler, et al.,

Appl. Phys. B 77, 789 (2003).

MACOR frame

compensation

electrodes

Electrodes

Innsbruck linear ion trap

F. Schmidt-Kaler, et al.,

Appl. Phys. B 77, 789 (2003).

Aarhus, Denmark

Linear ion traps in Rood design

trap electrodes are

nearly in ideal geometry R= 4mm

r0= 3.5mm

harmonic trapping

in a large region, and

even outside the center

Rood design

Boulder, USA

Innsbruck, Austria

Leibfried, Schätz

Physik Journal 3 (2004) 23

Schulz, NJP (2008)

d ~ 250µm

Stick et al., Nature

Physics 2, 36 (2006)

d ~ 60µm

Seidelin et al.,

PRL 96, 253003 (2006)

d ~ 40µm

Hensinger et al.,

Appl. Phys. Lett.,

034101 (2006)

d ~ 200µm

Segmented micro traps overview

Labaziewicz et al,

PRL 100, 013001 (2008)

Kielpinski et al,

Nature 417, 709 (2002)

RF

DC

RF

DC

• Micro structuring - high number and density of ions

- high trap frequency and gate speed

• Segmentation - Processor and memory section

- Transport of Ion crystals

- Separation of ions

- Cavity QED region

500µm

250µm 250µm 100µm

Schulz et al., Fortschr. Phys. 54, 648 (2006)

Ulm segmented micro trap

ceramics,

fs-Laser stuctured,

Gold-coated

base

spacer

top

Ulm segmented micro trap: Fabrication

Schulz et al.,

New Journal of Physics 2008

Loading zone / Transfer zone Experiment zone Sandwich construction

• Ti/Au on Al2O3-Wafer

(10nm/400nm)

• fs-Laser cut in Au/Ti and Al2O3

• adjusting and

• gluing

into Chip Carrier

• bonding

Ulm segmented micro trap: Fabrication

Schulz et al.,

New Journal of Physics 2008

… assembled and connected

in the UHV recipient

Schulz et al.,

New Journal of Physics 2008

Planar micro traps

RF

DC

DC

RF

DC

RF DC DC RF

DC RF RF DC DC

Trapping potential

Pros:

Easy fabrication

High precision

Small sizes possible (few µm)

high surface quality

Cons:

shallow trap potential

Planar micro traps S. Seidelin, et. al.,

PRL 96 253003, (2006).

J. Labaziewicz, et. al.,

PRL 100, 013001 (2008).

Equilibrium positions in the axial potential

z-axis

mutual ion repulsion trap potential

find equilibrium positions x0: ions oscillate with q(t) arround

condition for equilibrium:

dimensionless positions with length scale

D. James, Appl. Phys.

B 66, 181 (1998)

Equilibrium positions in the axial potential

numerical solution (Mathematica),

e.g. N=5 ions

equilibrium positions

set of N equations for um

-1.74 -0.82 0 +0.82 +1.74

force of the

trap potential Coulomb force

of all ions from left side Coulomb force

of all ions from left side

-40 -30 -20 -10 0 10 20 30 400

1

2

3

4

5

6

7

8

9

10

Nu

mb

er

of

Ions

z-position (µm)

Experiment: equilibrium positions

equilibrium positions

are not equally spaced

H. C. Nägerl et al.,

Appl. Phys. B 66, 603 (1998)

theory

experiment

minimum inter-ion distance:

Eigenmodes and Eigenfrequencies

Lagrangian of the axial ion motion:

m,n=1 m=1

N N

describes small excursions

arround equilibrium positions

with

and

N

m,n=1 m=1

N N D. James, Appl. Phys.

B 66, 181 (1998)

linearized Coulomb interaction leads to Eigenmodes, but the

next term in Tailor expansion leads to mode coupling, which is

however very small.

C. Marquet, et al.,

Appl. Phys. B 76, 199

(2003)

Eigenmodes and Eigenfrequencies

Matrix, to diagonize

numerical solution (Mathematica),

e.g. N=4 ions

Eigenvectors

Eigenvalues

for the radial modes:

Market et al., Appl. Phys.

B76, (2003) 199 depends on N

pictorial

does not

Common mode excitation: Experiment

H. C. Nägerl, Optics

Express / Vol. 3, No. 2 /

89 (1998).

time

po

sitio

n

Center of

mass mode

breathing

mode

Common mode excitations

H. C. Nägerl, Optics

Express / Vol. 3, No. 2 /

89 (1998).

Breathing mode excitation

H. C. Nägerl, Optics

Express / Vol. 3, No. 2 /

89 (1998).

Basics: Harmonic oscillator

Why? The trap confinement is leads to three

independend harmonic oscillators !

here only for the linear direction

of the linear trap no micro-motion

treat the oscillator quantum mechanically and

introduce a+ and a

and get Hamiltonian

Eigenstates |n> with:

Harmonic oscillator wavefunctions

Eigen functions

with orthonormal Hermite polynoms

and energies:

Two – level atom Why? Is an idealization which is a good approximation to real

physical system in many cases

two level system is connected

with spin ½ algebra using the

Pauli matrices

D. Leibfried, C. Monroe,

R. Blatt, D. Wineland,

Rev. Mod. Phys. 75, 281 (2003)

Two – level atom Why? Is an idealization which is a good approximation to real

pyhsical system in many cases

gn ,1

en ,1en,

en ,1

gn ,1gn,

together with the harmonic oscillator leading to the ladder of

eigenstates |g,n>, |e,n>:

levels not coupled

Laser coupling dipole interaction, Laser radiation with frequency wl, and intensity |E|2

Rabi frequency:

the laser interaction (running laser wave) has a spatial dependence:

Laser

with

momentum kick, recoil:

Laser coupling in the rotating wave approximation

using

and defining the Lamb

Dicke parameter h:

Raman transition: projection of Dk=k1-k2

x-axis

if the laser direction is at an angle f to the vibration mode direction:

x-axis

single photon transition

Interaction picture

In the interaction picture defined by

we obtain for the Hamiltonian

with

coupling states with vibration quantum numbers

laser detuning D

2-level-atom harmonic trap

Laser coupling

dressed system

„molecular

Franck Condon“

picture dressed system

gn ,1

en ,1en,

en ,1

gn ,1gn,

„energy

ladder“

picture

2-level-atom harmonic trap

Laser coupling

dressed system

„molecular

Franck Condon“

picture

„energy

ladder“

picture

S

D

D

S

Sn ,1

Dn ,1Dn,

Dn ,1

Sn ,1Sn,

Lamb Dicke Regime

carrier:

red sideband:

blue sideband: laser is tuned to

the resonances:

kicked wave function is non-orthogonal to the other

wave functions

Wavefunctions in momentum space

kick by the laser:

0,S

0,DP

rob.

for

D

electronic

excitation

laser pulse length in µs

electronic

excitation

Coherent qubit rotation

Carrier flops

0,S

0,D1,D

1,S

internal

electronic

state

Vibrational quanta

laser pulse length in µs

Coherent qubit rotation

Carrier flops

carrier and sideband

Rabi oscillations

with Rabi frequencies

and

What is a quantum gate?

• QBit ist kleinste quantenmechanische

Informationseiheit:

-Verknüpfung von QBits mittels

-1Qbit Gatter

-Und 2 Qbit-Gatter=> Quanten-Computer

Damit kann man universellen Quantencomputer bauen!

1 Qubit Gatter – Licht

Wechselwirkung

1D-WW

Atom-Licht-WW

What is a Superposition?

two different quantum states

are simultaneous existing

at the very same place

two different quantum states

are simultaneous existing

at the very same place

What is a Superposition?

Vector on the Bloch-sphere

two different quantum states

are simultaneous existing

at the very same place

What is a Superposition?

10

in superposition of Zero and One

A single quantum-bit

two different quantum states

are simultaneous existing

at the very same place

What is a Superposition?

P

S

t = 7 ns

397 nm

D

729 nm

„qubit“

En

erg

y

|1>

|0>

A single Ion

in superposition

of Zero and One

two different quantum states

are simultaneous existing

at the very same place

What is a Superposition?

What is entanglement?

Solvay Konferenz 1927

Entangled particles show correlations and can only

propery discribed by a common quantum state,

regardless how far appart they are.

Still open

Problem:

Entanglement

measures and

characterization

Co-Carrier

Raman

|1>

|0>

P

opula

tion

P

opula

tion

Raman Pulsdauer [µs] Poschinger et al,

arXive 0902.2826

Bei Messung

50% der Fälle

2 Qubit gate

CNOT-GATE

Particle A Particle B

CNOT

Particle A Particle B

2 Qubit gate

CNOT-GATE

Particle A Particle B

CNOT

Particle A Particle B

Generation of Entanglement

Particle A Particle B Particle A Particle

Particle A Particle B

CNOT

1 Qbit gate on

each A and B

does not generate

entanglement:

Measurement would result in uncorrelated results

Measurement results in always the same outcomes at A and B !!!

Properties of the

complete system are fully determined,

but properties of sub-systems

completely undeterined

Erwin Schrödinger 1935:

"I would call entanglement not one but rather

the characteristic trait of quantum mechanics,

the one that enforces its entire departure from

classical lines of thought."

What is Entanglement?

Laser pulse

Entanglement by Laser pulses

get two ions

Laser-Puls

Entanglement by Laser pulses

two ions in entangled state

Entanglement by Laser pulses

Measurement

two ions in entangled state

detection

Laser

detection

Laser

|1> |0>

.... Repeated measurements

Fully determined

correlation

Measurement results

left

Measurement results

right

Subsystems completly random

Erwin Schrödinger 1935:

"I would call entanglement not one but

rather the characteristic trait of quantum

mechanics, the one that enforces its entire

departure from classical lines of thought."

Temporal sequence

of quantum logic

operations INPUT OUTPUT

Basics of a quantum computer

Single

qubit gate

two-qubit

gate time

applications in physics and informatics

P. Shor, 1994: factorization of large numbers, L digits, is much more

efficient on a quantum computer than with a classical computer:

classical computer: ~exp(L1/3), quantum computer: ~ L2

L. Grover, 1997: search data base - quantum computer: ~ L

simulation of Schrödinger equations or any unitary evolution

spin interactions, quantum phase transitions

quantum cryptography / repeaters / quantum links

improved atomic clocks

understanding the fundamentals of quantum mechanics / Gedanken-Experimente

Experiments with entangled matter

Why?

The requirements for experimental qc

• Qubits store superposition information, scalable physical system

• Ability to initialize the state of the qubits

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

• Long coherence times, much longer than gate operation time

• Qubit-specific measurement capability

D. P. DiVincenzo,

Quant. Inf. Comp. 1

(Special), 1 (2001)

Qubit

Transformation

10

Scalable device?

Experimental status

Quantum Information Roadmaps

http://qist.ect.it/

http://qist.lanl.gov/

Quantum gate proposal

control bit target bit

• single bit rotations and

quantum gates

• small decoherence

• unity detection efficiency

• scalable

J. I. Cirac P. Zoller W. Paul

• single bit rotations and

quantum gates

• small decoherence

• unity detection efficiency

• scalable

J. I. Cirac P. Zoller 21121: NOTControlled

0111

1101

1010

0000

0111

1101

1010

0000

control bit target bit

Quantum gate proposal

Cirac & Zoller gate

with two ions

S S S S

S D S D

D D DS

D D D S

ion 1

motion

ion 2

,S D

,S D

0 0

control qubit

target qubit

SWAP

1 2

control target

Controlled-NOT operation

S S S S

S D S D

D D DS

D D D S

ion 1

motion

ion 2

,S D

,S D

0 0

control qubit

target qubit

|0>, |1>

1 2

Controlled-NOT operation Controlled-NOT operation

S S S S

S D S D

D D DS

D D D S

ion 1

motion

ion 2

,S D

SWAP-1

,S D

0 0

control qubit

target qubit

|0>, |1>

1 2

Controlled-NOT operation Controlled-NOT operation

SWAP and SWAP-1

,0S,1S

,0D,1D

SWAP

,0S,1S

,0D,1D

SWAP-1

starting with |n=0> phonons,

write into and read from the common vibrational mode

-pulse on blue SB

control bit control bit

Conditional phase gate

Effect:

phase factor of -1

for all, except |D,0 >

target bit

22

Composite pulse phase gate

I.Chuang,

MIT Boston

Rabi frequency:

1W nhBlue SB:

Composite phase gate (2 rotation)

1 1 1 1( , ) , 2 2,0 , 2 2,0R R R R R f

1

2

3

4

,0 ,1S Don 2

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

1 1 1 1( , ) 2, 2 ,0 2, 2 ,0R R R R R f

4

3

2

1

ion 1

motion

ion 2

,S DSWAP-1

,S D

0 0SWAP

Ion 1

Ion 2

pulse sequence:

control bit

target bit

laser frequency

pulse duration

optical phase

Controlled-NOT operation

input

output

Fidelity of Cirac-Zoller CNOT

|<Yexp| Yideal >|2

F. Schmidt-Kaler et al.,

Nature 422, 408 (2003)

Fidelity : 73%

M. Riebe et al.,

PRL 97, 220407 (2006)

Fidelity : 92,6%

GHZ state:

W state:

C. Roos et al., Science

304, 1478 (2004)

Ion 3

Ion 2

Ion 1

Deterministic generation of GHZ state

Experiment Ideal

Fidelity: 72 %

Tomography of the GHZ state

C. Roos et al., Science

304, 1478 (2004)

What is a Schrödinger’s cat ?

Kilian Singer - Symposium

Hybride Quantensysteme Ulm

87

Schrödingerkatze im

Bewegungszustand

01.08.2013

D. Leibfried et al., Nature 422, 412 (2002)

P.J. Lee et al., J. Opt. B. 7, 371 (2005)

Bestimmung der Phasenraumtrajektorie eines verschränkten Wellenpackets

U. G. Poschinger, A. Walther, KS, F. Schmidt-Kaler, Physical Review Letters 105,

263602 (2010).