Quantum information processingwith trapped electrons

Irene Marzoli

Università di Camerino (Italy)

International School of Physics “Enrico Fermi”

Motivations

• Quantized external (motional) and internal (spin) degrees of freedom;

• Lighter mass brings to higher trapping frequencies;• Microwave and radiofrequency radiation;• Ground state cooling of cyclotron motion;• Electron spin states are very long lived;• Extremely good isolation from the environment

(negligible damping and reduced thermal fluctuations);• High-fidelity detection of the quantum state (spin and

cyclotron) via observation of quantum jumps.

Overview

• Basic properties of Penning traps• Qubit encoding and manipulation • Scalability • New trap geometries• Coupling qubits• Outlook and applications

STATIC electric and magneticFIELDS:

Homogeneous magnetic field →radial confinement (cyclotron motion)

Quadrupole potential → axial confinement and slow circular driftin the radial plane (magnetron motion)

How Penning traps workL. S. Brown and G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986)

F. G. Major, V. Gheorghe, and G. Werth, Charged particle traps (Springer, Heidelberg, 2005)

2

222

0 22),,(

dzyxVzyxV −+

=

B

Illustration of electron motion in a Penning trap

Fast radial oscillation: modified cyclotron motionSlow radial oscillation: magnetron motionRed: axial oscillation

Electron motion inside the trap

frequencycyclotron free thebeing ||with

motion axial 2

motionmagnetron 2

2

motioncyclotron 2

2

c

20

22cc

22cc

mBe

mdeV

z

z

z

≡

≡

−−≡

−+≡

−

+

ω

ω

ωωωω

ωωωω

For a trap size of 1 cm, a voltage V0 ~ 10 V, a magnetic field B ~ 3 T:• ω+/(2π) ~ 100 GHz modified cyclotron frequency,• ωz/(2π) ~ 100 MHz axial frequency,• ω-/(2π) ~ 10 kHz magnetron frequency.

Energy level structure

snklE sz 22

121

21 ωωωω h

hhh +⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +−= +−

with l, k,n = 0,1,2,Kand s = ±1

mBeg

s||

2≡ω n=0

n=2

n=1

s=-1

s=1

k=0k=1k=2k=3

l=2l=1l=0

The DiVincenzo’s criteria D. P. DiVincenzo, The Physical Implementation of Quantum Computation,

Fortschritte der Physik 48, 771 (2000).

• A scalable physical system with well defined qubits;

• The ability to initialize the state of the qubits;

• A universal set of quantum gates;

• Decoherence times much longer than the gate operation time;

• A qubit-specific measurement;

• Interconverting stationary and flying qubits;

• Transmitting qubits over long distances.

Qubits in the electron spin and cyclotron motion

{ }11,10,01,00

sω

00

01

0210

11

12

cs ωω −

cω

S. Stortini and I. M., Eur. Phys. J. D 32, 209 (2005)

Single qubit operations

Spin: resonant pulses at the spin transition frequencysω

)sincos(2

)(2

)(int tωttgtH yxs

B ωσσμ +Ω

=⋅=⋅−=hbσbμ

⎟⎠⎞

⎜⎝⎛ Ω−⎟

⎠⎞

⎜⎝⎛ Ω=

⎟⎠⎞

⎜⎝⎛ Ω−⎟

⎠⎞

⎜⎝⎛ Ω=

2sin0

2cos11

2sin1

2cos00

ss

ss

tit

tit

( ) ( ) ( )[ ]ytxtbt ˆsinˆcos0 ωω +=b

sω

00

11

10

01

( ) ( )φω += tItI dcos

Cyclotron qubit manipulation

cs ωω −

( ) ( ) ( )φω += ttbt dcos1ρb

( ) 2/522

2

14/

3dac

daIb+

=π

( ) ( )[ ] ( )φωσσμ++= tbtytxgH dyx

B cos2 1int

( ) ( )φφ σσω

μ ic

ic

c

BIP eaeam

bgH +−

−+ +≈ ~22

1int

h

in RWA

cωsω

00

11

10

01

22 2~zcc ωωω −≡

Unitary evolution operator

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−=

CDDC

ABBA

M

*

*

000000

00000000001

),( φθ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

h

tiHtUIP

intexp)(

tm

bgc

B ωμθ ~21

h≡where

( )

( )( )2/sine

2/cos

)2/sin(e2/cos

θ

θ

θ

θ

φ

φ

i

i

iD

C

iBA

−≡

≡

≡

≡

Composite pulse technique

A. M. Childs and I. L. Chuang, PRA 63, 012306 (2001); S. Gulde et al., Nature 421, 48 (2003)

Cyclotron: 1) Swap the cyclotron and spin qubits

2) Operate on the spin qubit

3) Swap back the cyclotron and spin qubits

⎟⎠⎞

⎜⎝⎛ 0,

2πM ⎟

⎠⎞

⎜⎝⎛ 0,

2πM⎟

⎠⎞

⎜⎝⎛

sM φπ ,2

2

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

2cotarccos 2 πφs

⎟⎠⎞

⎜⎝⎛ ππ ,

2M ⎟

⎠⎞

⎜⎝⎛ + sM φππ ,

22

⎟⎠⎞

⎜⎝⎛ ππ ,

2M

SWAPPING

( ) 1−SWAPPING

Two-qubit gate: Conditional phase shift

11100100 δγβα +++

( ) ( )

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−−

−−

=⎟⎠⎞

⎜⎝⎛××⎟

⎠⎞

⎜⎝⎛×

1000001000001000001000001

2,

20,

2,

20, ππππππ MMMM

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

==−

1000010000100001

)( πφPSC

11e100100 i δγβα φ+++C-PS

Towards electron‐electron entanglement

( )2

212

2222122

2122

21

)(with

)(~

zc

iizizii yxmzmmV

ωωωωω

δωωρω

ρ

ρ

−−=

−−+=x

L. Lamata et al., PRA 81, 022301 (2010)

Two electrons in a Penning trap with a weak rotating wall potential

Equilibrium position at x = ± x0/2 with

3/1

220

2

0 )(2 ⎥⎥⎦

⎤

⎢⎢⎣

⎡

−=

δωωπε ρ zmex

Two‐electron gate

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛−=Δ

Δ+=

ρzxB 2 zz

tzz a

ˆ2

)(

)cos(2

2

0CM

ρβ

ω

Axial drive at the anomaly frequency + magnetic bottle

Hamiltonian in interaction picture

( ) 4/12202

CM,2,1

CM,gate

242

)(

zc

B

tici

ti

iciI

mzg

eaeaH

ωωβμ

σσ

−=Ω

+Ω= Δ+−Δ−

=

+∑

h

h

For z0 ~ 100 μm, Ω/(2π) ~ 10 Hz

Applications

( ) acc

cc

ωπ at pulse- 02

11

drivecyclotron resonant weak 10

CM, CM,

CM,CM,

↓↑+↑↓⇒↓↓

↓↓⇒↓↓

[ ]∑ +−−+ ++−ΔΩ

=ji

jijiI nnH,

2off )1( σσσσh

Spin-spin entanglement

Two-qubit gates (off-resonance regime Δ>>Ω)

Entanglement of the spin and motional degrees of freedom (spin-cyclotron GHZ states useful for quantum metrology)

( )cc CM,CM,

202

1↓↓+↑↑=Ψ

Hyperbolical trap

From a cylindrical Penning trap…

• Microwave cavity [J. N. Tan &

G. Gabrielse, PRL 67, 3090

(1991)];

• Well defined cavity modes;

• Quality factor

• Cavity-induced suppression of spontaneous emission bysynchrotron radiation.

Q ≈ 5×1044105×=Q

S. Peil and G. Gabrielse, Phys. Rev. Lett. 83, 1287 (1999)

… to a planar Penning trap

S. Stahl, F. Galve, J. Alonso, S. Djekic, W. Quint, T. Valenzuela, J. Verdú, M. Vogel, and G. Werth, Eur. Phys. J. D 32, 139 (2005)

Electrodes printed on aninsulating substrate (Al2O3):• easily produced and miniaturized (thick- and thin-film technology)• open geometry for easy access with radiation• more traps on the same substrate to form a two-dimensional array

B

Silver plated Al2O3 ceramic disk with electrodes of

R0=2.5 mm, R1=5.8 mm, R2=9.1 mm and d=3 mm

Axial potential

R0=300 μm, R1=600 μm, and R2=900 μm

U0 =0 V, U1 =0.5 V, U2 =0 V (compensation voltage –0.417 V)

ωz/(2π) = 89.9 MHz (99.0 MHz solid line)

Operation of mm-sized planar trapsF. Galve and G. Werth, Proc. 2006 Non-Neutral Plasma Workshop, Åarhus (2006);

F. Galve, PhD thesis, Johannes-Gutenberg-Universität Mainz (2006)

Planar trap with D = 4.8 cm, operated at room temperature

Optimized parameters:U0=0 V, U1= -13.6 V, U2=33.6 Vωz/2π ~ 35 MHz

Cloud of electrons@ T = 100 mK

Planar trap with D = 2.0 cm, U0=0 V, U1= 0.5 V, U2= - 0.417 Vωz/2π ~ 100 MHzP. Bushev et al., Eur. Phys. J. D 50, 97 (2008)

Microscopic planar Penning trap withmultiple ring electrodes

Adjusting the control voltages:

from a large trap size withR0/1/2 = 1.5 / 3 / 4.5 mm

to a small trap withR0/1/2 = 50 / 100 / 150 μm

Change the distance of the electron to the electrode surfacefrom 6 mm to 50 μm

Microscopic planar Penning trap withmultiple ring electrodes

Adjusting the control voltages:variable size of the effectivecentral disc and ring electrodes

Change the distance of the electron to the electrode surfacefrom 6 mm to 50 μm

0 200 400 600 800 1000 12000

250

500

750

dist

ance

[µm

]

trap radius [µm]

Microscopic planar Penning trap withmultiple ring electrodes

Investigation of decoherenceeffects depending on theelectron-surface distance

Microscopic planar Penning trap withmultiple ring electrodes

Larger trap volumefor easier loading

Varying the coupling between spin state and axial motion

Place a small nickelpiece under the central

electrode

Large distance:homogeneous B field

for long coherence andsingle qubit operations

Short distance:inhomogeneous B field

for analysis and two-qubitoperations

Planar electron micro-trap arrays

10.5 mm

Interconnectedmicro traps withR0 = 300 μmR1 = 600 μmR2 = 900 μm

Flexible variationof the connectionsusing bonding padson the gold surface

Drawing courtesy of Michael Hellwig (University of Ulm/Mainz)

QUELE experimental setup

Microfabricated trap with multiple ring electrodes

The effective electrode size can vary from R0/1/2 = 1500/3000/4500 μmdown to R0/1/2 = 50/100/150 μm

Prototype planar trap (Univ. Mainz)

Onion trap (Univ. Ulm/Mainz)

Pixel trap (Univ. Ulm/Mainz)

Pixel trap

Mirror‐image planar Penning trap

r1r2

z

zc

V1

V2

V3

r2r1

r3

r3

J. Goldman and G. Gabrielse, Phys. Rev. A 81, 052335 (2010)J. Goldman and G. Gabrielse, Hyperfine Interact. 199, 279 (2011)

J. R. Castrejon-Pita and R. C. Thompson, Phys. Rev. A 32, 013405 (2005)

x

z

+

-

-+ +

+

B

y

Coplanar‐waveguide Penning trap

A planar trap at a distance x0 from the center of the substrate

Inhomogeneous magnetic field

(linear gradient)⎟⎠⎞

⎜⎝⎛ −−= jikB1 22

yxzb

F. Mintert and Ch. Wunderlich, Phys. Rev. Lett. 87, 257904 (2001)

Linear array of traps

Direct Coulomb interaction||4 0

2

,ji

Cji

eHrr −

=πε

The magnetic field gradient couples each electron internal (spin) and external (motional) degrees of freedom

( ) ( )iciicizziizizz

zisizizziciccimimm

NCi

aagaag

aaaaaaH

c

z,

)(,

)(~,,

,,,,,,

44

2++−+

+++

+−++

+++−≈

σσεωσεω

σωωωω

ωωhh

hhhh

zzz mmbe

mzbe

ωωωε

2|||| h

=Δ

=Size of the

ground state

The Coulomb interaction couples axial and cyclotron motionof different electrons

( )( )( )jcicjcic

c

zji

jzjzizizjiC

ji

aaaa

aaaaH

,,,,,

,,,,,,

~++

++

+−

++≈

ωωξ

ξ

h

h

2

,,0

2

3,0

2

, 41

8 ⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ==

jijijizji d

zd

edm

eπεωπε

ξh

Remove the coupling between internal and external

degrees of freedom with a canonical transformation

SS HeeH −→

( ) ( )∑=

++−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+−=

N

iiciici

c

z

a

ziziz

zi aaaagS

1,

)(,

)(,, ~4

σσωω

ωωσε

csa ωωω −≡ anomaly frequency

( )[ ]∑∑<=

+−+≈N

ji

yj

yi

xj

xi

xyji

zj

zi

zji

ziis

N

is JJH σσσσσσσω ,,,

12

22hh

3,

4

2

40

422

,

2

, 1622 jizji

zji d

bm

eggJωπε

εξ h⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

Effective spin-spin Hamiltonian

3,

4

2

40

426

42

,

26

, 64210

410

jicc

zji

xyji d

bm

eggJωπεω

ωεξ h⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=

Dipolardecay

G. Ciaramicoli, I. M., and P. Tombesi, PRA 75, 032348 (2007)

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

0

20,

20

,0 81

2 Bxb

mgeB i

isω

jzizji

N

jiizis

N

is JH ,,,,,0

1 22σσπσω ∑∑

>=

+≅′hh

3,

4

22

,

2

, 2 jizjiji d

bgJω

εξπ

∝=

the spin (qubit) frequencydepends on the trap position

the coupling constant dependson the magnetic gradient, the axial frequency, and the inter-

particle distance

G. Ciaramicoli, F. Galve, I.M., and P. Tombesi, Phys. Rev. A 72 , 042323 (2005)D. Mc Hugh and J. Twamley, Phys. Rev. A 71, 012315 (2005)

Array of trapped electrons as an NMR molecule

d = 100 μm d = 50 μm  d = 10 μm

b = 50 T/m 2.3 Hz 18 Hz 2300 Hz

b = 500 T/m 0.23 kHz 1.85 kHz 230 kHz

Estimate of the spin-spin coupling strength J

The axial frequency ωz /(2π) is 100 MHz

A channel for quantum communication

M. Avellino, A. J. Fisher, and S. Bose, Quantum Communication in Spin Systems with Long-Range Interactions, Phys. Rev. A 74, 012321 (2006).

( )[ ]∑∑<=

+−−=N

ji

yj

yi

xj

xi

xyji

zj

zi

zji

zis

N

is JJH σσσσσσσω ,,

12

2h

h

with Ji,jxy = Ji,j

z . Transmission fidelity, up to 20 spins, larger than 90%!The transfer time scales as the cube of the chain length.

Coupling the axial qubits with wires

Information exchange wire

The oscillating imagecharges of two electronsare coupled to each other

Coherent swappingof excitation between

the traps

1,0010,110 2121 ===→=== nnnn

J. Zurita-Sánchez and C. Henkel, PRA 73, 063825 (2006)J. Zurita-Sánchez and C. Henkel, New J. Phys. 10(08), 083021 (2008)

Coherent wire coupling

w0

3

20

2

20

20

2

12 21

2 CCRhR

mRe

z +⎟⎟⎠

⎞⎜⎜⎝

⎛+

=Ωω

where R0 is the central electrode radius and h is the height of the electron from the trap surface.

C0 ~ πε0R0 intrinsic electrode capacitanceCw capacitance of the wire

The shorter the wire and the smaller the trap, the stronger thecoherent coupling.

( )++ +Ω= 2,1,2,1,12 zzzz aaaaH h

Coupling strength

ωz/(2π)=100 MHzC0 = R0 x 100 fF mm-1 intrinsic electrode capacitanceCw = d x 66 fF mm-1 capacitance of the wire

d =100 mm 10 mm 1 mm 100 μm 10 μm

R0 =1 mm 0.12 Hz 1.0 Hz 3.2 Hz _ _

10 μm 1.3 kHz 13 kHz 130 kHz 990 kHz 3.2 MHz

• Universal quantum gates with trapped electrons;

• Scalable planar Penning traps of variable size and geometry;

• Coupling distant electrons;

• Design of an effective spin-spin interaction;

• Short distance quantum communication;

• Entanglement generation;

• Investigation of decoherence effects;

• Applications to quantum metrology;

• Integration into circuit QED.

Summary & future perspectives

• Giacomo Ciaramicoli

• Paolo Tombesi