イオントラップを用いた量子ゲート実験

The contents of this lecture

1.Introduction2. Ion trap and ion qubit3.Initialization and state detection 4.Coherence time5.Quantum gate6.Spin dependent force and its application

大阪大学大学院基礎工学研究科占部伸二

H22.8.27 量子情報処理サマースクール

Shinji Urabe, Utako.Tanaka, Kenji.Toyoda

Invited researcher Kazuhiro Hayasaka(NICT)

R. Yamazaki (H18.7-H21.3)

Graduate students

Doctor course S.Haze

Master course

K.Uchida, K.Tamura, K.Masuda, N.Wada,

T.Watanabe,

Y.Ibaragi, T.Ohno, T.Kanda, K.Kohda, Y.Tateishi, T.Fujiwara

Urabe Laboratory

1.Introduction

Ion trap: Trapping of charged particles with electromagnetic fieldsMass spectrometry (1950‘s), Spectroscopy ← Laser cooling (1975)

Wineland & Dehmelt, Hänsch & Schawlow

Quantum information processing

qubit: internal states of an ion in an ion string

Cirac & Zoller (1995)

High resolution spectroscopy:

Isolation from the environment

→ Optical frequency standard with single ions(Q～1015 )

Mg+－Al+: 8×10-18： quantum logic spectroscopy(NIST, 2010)

Linear rf traps

Typical trap operating parameters:

ultra high vacuum: less than 10-8 Pa

trap dimension (r0) : 0.6 mm

rf frequency (Ω) : 24 MHz

effective potential depth (Veff) : ～10V

collective motional frequency

z: 0.7MHz, x: 2.1MHz, y: 2.3 MHz

distance between ions in strings: about 7μ

2. Ion traps and ion qubit

z1

～

U1 U2 U3 U4 U5

U1’ U2’ U3’ U4’ U5’ Vrf

r0

z0 z1

Images of 1,2,3 ions

Hyperfine levels, ground state - metastable state

Ions: Be+, Mg+, Zn+, Cd+ (hyperfine qubits)

Ca+, Sr+, Ba+, Yb+, Hg+ (hyperfine or metastabe qubits)

Ion qubit

Doppler cooling : 397nm, 866nm

sideband cooling : 729nm, 854nm

quantum state control : 729nm etc.

40Ca+ ion

Zeeman qubits: 2S1/2 ,m=-1/2 - 2S1/2 ,m=1/2 (~10MHz)

Optical-transition qubits: 2S1/2 - 2D5/2 (729nm)

Terahertz-separated qubits : 2D3/2 - 2D5/2 (1.82THz)

Raman transitions driven by phase locked lasers bridged by a frequency comb

Terahertz-separated qubit:

internal state: optical pumping ～100%

external state: Doppler cooling : from 10000 K to mK

sideband cooling : to the motional ground state

3.Initialization and state detection

Initialization

S- / S+ = <nz > / (<nz >+1)

S-: height of red sideband spectrum

S+: height of blue sideband spectrum

01.0~zn

Red and blue sideband spectra of axial modes in S-D transitions after sideband cooling

0.020 ,017.0~

, stCOM nn

Single ions Two ions

l0>l1>

l2>l3>2S1/2

2D5/2

2P3/2 854nm

729nm (red sideband)

ω 0-ω v

Electron shelving method: nearly 100% efficiencyCycling transitions・・・fluorescence signal ～3x104photons/s/ion

2000 4000 6000 8000 10000 120000

200

400

600

800

Coun

ts

Photon counts /sec

Γ～1.4×10-8 s-1

Γ～1 s-1

lΨ> = c1lg> + c2le2>

Quantum jump signal of single ions Histogram of photon counts

State detection

Coherence time: hyperfine ground state・・・several minutes

metastable state・・・ about 1.0 s (40Ca+ )

External disturbance: magnetic field fluctuation, laser linewidth etc.

4.Coherence time

Coherence time of terahertz-separated (D3/2-D5/2) qubits

Fringe visibility of Ramsey signal

K.Toyoda, H.Shiibara, S.Haze, R.Yamazaki,

S.Urabe, Phys. Rev. A 79, 023419, 2009

Entanglement between qubits ・・・ mediated by the collective motional states.

Qubit : internal states, lｇ>, le>, Bus bit : motional states, l0>, l1>

Carrier

red

sideband

blue

sideband

Optical spectrum of single ions

(Electric quadrupole transitions)

manipulation of qubits・・・ laser pulsescarrier pulse,red sideband pulse, blue sideband pulse

hyperfine qubit: Raman transitions,metastable qubit: E2 transitions

5.Quantum gate

Phase-locked laser system with a frequency comb for D-D qubits

Beat signal of the phase

locked system

854nm 850nm1.8194･･ THz ~ 6.252 GHz × 291 lines + 267.・・ MHz

ΔΦrms=50 mradR.Yamazaki, T.Iwai, K.Toyoda, and S.Urabe, Opt. Lett. Vol.32,No.5,(2007)2085

S.Haze, Y.Senokuchi, R.Yamazaki, K.Toyoda, S.Urabe, Appl. Phys. B to be published

The frequency is locked to a ULE cavity by the Pound-Drever-Hall method.

line width : less than 1KHz,

frequency drift : less than 3kHz/h

729nm Ti : sapphire laser for S-D qubits

Carrier Rabi oscillation Blue-sideband Rabi oscillation

Rabi oscillations of S1/2-D5/2 transitions in single 40Ca+ ions

Carrier

transition

0,0, ge

Blue-sideband

transition

0,1, ge

2S1/2, nz=0

2D5/2,

nz=1

nz=0

Time [μｓ]

Popu

lation in D

5/2

0

)2/cos()2/sin(

)2/sin()2/cos(

)]sinˆcosˆ(2

exp[),(ˆ

)0(), (ˆ )(

:phase , :area pulse pulse,carrier 0

i

i

yx

ie

ie

iR

Rt

t

5.1 Single qubit rotation le>

geggegg 100 ,,1,0 ,,0,0

operation swap :pulse sideband red

5.2 Two-qubit gate

Z0̂

0̂1̂

1000

0100

0010

0001

Z-C : -control :pulse(aux) 2 sideband red z

, 11

11

2

1H gate Hadamard:pulse /2carrier

Cirac-Zoller C-Not gate

Cirac-Zoller Gate experiment using terahertz-separated qubits

CZ gate excitation scheme

Single 40Ca+,

Control bit : phonon states

l0>: n=0

l1>: n=1

Target bit: internal states

l↑>：D5/2(m=1/2)

l↓>: D3/2(m=1/2)

Sideband

cooling

Preparation:

(1)D5/2, n=1,

blue sideband π pulse

on S-D transitions

(2)D5/2, n=0,

carrier π pulse on S-D transitions

C-Z gate:

blue-sideband 2π pulse

on S-D transitions

2/)(ψ ,0n )2(

2/) (ψ ,1n )1(

2/)(ψ ψ

ψ )2(

ψ )1(

Control-ZRamsey π/2

pulse

Ramsey π/2

pulse

CZ gate result

K.Toyoda, S.Haze, R.Yamazaki, S.Urabe, Phys. Rev. A, 81, 032322 , 2010

Fidelity～0.74

(a) Carrier Rabi oscillation in S-D transition

(b) BSB Rabi oscillation in S-D transition

(c) Carrier Rabi oscillation in D-D transition

Initial phonon state: |0>

Initial phonon state: |1>Simulation results obtained by

solving the density matrix equation

5.3 Creation of two-particle entanglement

2 two-level ions are irradiated equally with a

laser pulse, whose frequency is tuned near the

first red sideband of the COM mode.

parameter Dicke-Lamb:

,N/ ,0

L

e

gνz

The total number of excitation of quanta is conserved

and the basis of one-quantum Hamiltonian

is composed of :

Interaction Hamiltonian (rotating frame)

}0eg, ,0ge, ,1gg,{:Basis

0

0

1

02/

02/

2/2/

ˆ

IH

}0 ,0 ,1gg,{:Basis

0

0

1

00

02/

02/

ˆ

IH

2/10

2/)eg,ge,(

,2/)eg,ge,(

n=0

n=1

}0eg, ,0ge, ,1gg,{

I.E.Linington & N.V. Vitanov:

Phys. Rev. A77,010302 (2008)

Adiabatic condition

0)ge,eg,(2

1 1gg,

2/322 )(2

1 ≪

(1) Rapid adiabatic passage method

(2)Red sideband π pulse method ( two-state system)

010 0)2/sin(1gg,)2/cos( tit

)/(t ,0)ge,eg,(2

1 0)(t ,1gg, D.B.Hume et al. PRA, 80,

052302,2009

I.E.Linington & N.V. Vitanov: Phys. Rev. A77,010302 (2008)RSB RAP pulse

,0W g,g,g, N

mm

k

kN

m

N

m PC

gg,e,e,e,1

W :state Dicke

N ions

}0 ,1gg,{:subspace state- twosymmetric

Energy of dressed states

Important process:

initialization to 0g,g 21 1g,e 211g,g 21

blue sideband π carrier π

Off resonant 854nm laser

(100GHz)

1g,g 21

differential AC Stark shift methods

kHz90

Carrier spectrum of two ions:

AC Stark shift, on

Individual addressing to

one ion is necessary.

Qubit: S1/2,m=-1/2 and D5/2, m=-5/2

for ,2cos )( 21z zP

Pulse sequence of generation of two-particle entangled states

Fidelity of the

generated states

eggegeegegeggegeF ,,,,2

1||

Diagonal terms Off diagonal terms

Analysis Pulse Ⅰ： Parity )()2cos(2)( geeg,egge,eegg,21 zzP

Analysis Pulse Ⅱ： Parity

Diagonal terms: probability of single-ion fluorescing events after the state creation

Off diagonal terms: parity signal after analysis pulse Ⅰor Ⅱ0 for Dicke states

RAP: rapid adiabatic passage pulse

06.062.0 F

)(03.040.0

Result of the RAP method

)(06.083.0

　

Histogram of photon counts Parity signal after analysis pulse II

g

e

r

0

Raman beams

1Lk

2Lk

)/2(z

)(

0

21

z

LLz

kn

zkkk

0z

)ˆˆ()2/ˆ)((ˆ )(†)(

,int

titvi

j

zj eaeaH

D.Leibfried et al., Nature,422,412,2003

G.J.Milburn et al, Fortschr. Phys.,48,801,2000

Spin dependent force Hamiltonian,

σz dependent force

Optical dipole force (state dependent)

Laser 1 Laser 2

),ˆˆ(ˆ :mode COM †aazk jz

Interaction picture and rotating wave approximation

tzkH jz

j

z cos ˆˆint

j

6.Spin dependent force and its applications

A.Sorensen & K.Molmer, PRA,62,022311, 2000

P.C.Haljan et al, PRA, 72,062316, 2005

C.F Roos, New J. Phys., 10, 013002, 2008

Two-color beams（Ω1,Ω2）, 2 ions

, :detuninng

. : addressing collective

0201

021

　

jj

σφ dependent force(Molmer & Sorensen gate)

Two pairs of Raman beams

Nkaazkaazk

NMkaazk

zsssszsssz

zccccjz

2/ ),ˆˆ(ˆ ),ˆˆ(ˆ :modestretch

2/ ),ˆˆ(ˆ :mode COM

†

2

†

1

†

..)(ˆ 21

2,1

0int cheeeHziktizikti

i

j

i

Amplitude-modulated beams

0

0

]2/)(cos[]2/)(cos[2

])cos[(])cos[(

21021

2010

zkktzkkt

zktzkt

Dressed states

)(2

1

)(2

1

ge

ge

Modulation signal Resonant carrier wave

0

E2 transition

S

D

and , :sidebands the toclose 0 ≪≪

(1)General field coupling

Interaction picture and rotating wave approximation

j

yjy

j

xjx

titi

yx

JJ

eaeaJJH

ˆ2

1ˆ ,ˆ2

1ˆ

)ˆˆ](sinˆcosˆ[ ˆ )()(†

0int

)0()(ˆ(t) tU

displacement operator

})sin(){((t)

),1()(

2

0

)(

0

tt

et ti

Time evolution:

) ˆexp( }]ˆ)(ˆ)({ˆexp[)(ˆ 2†

xgx JiatatJtU

σΦ dependence

z

p

xx

ie

xx

ie

Φ

20 )(2 :phase gained

integer: ,2)(

y trajectorclosed

n

nnt

g

)0,(:center ,: radius ),0,(:center ,: radius 0000 xxxxxxxx JJJJ

Spin dependent circular motion in the phase space

20 )(2 ),/(2 1 :ex

ggn

xj

ji

xigeff

xj

ji

xiggxg

H

iNiJiU

ˆˆˆ :nHamiltonia effective

),ˆˆexp()4/exp(]exp[ˆ 2

Spin-spin interaction

Disentangling condition

Φg: unconventional geometric phase

Φg=(geometric phase) +(dynamic phase),(dynamic phase)=-2x(geometric phase) S.Zhu, D.Z.Wang, PRL,91,187902,2003

Quantum simulation: Ising model

ix

i

xjziz

ji

x

ji BJH ,,,

,

,Ising2

1ˆ

Phase transition: paramagnetc order

⇔ (anti-) ferromagnetic order

Two ions:A.Friedenauer et.al, Nature phys. 4,757,2008

Generation of entangled states

]2/exp[ˆ )2

1(or ,

2

2

max0

xg JiU

Entanglement of two ions:

(stretch mode)

))(2/1(

))(2/1(

) )(2/1(

) )(2/1(

4

3

2

1

geiegge

geiegeg

ggieegg

ggieeee

High fidelity Bell state generation: F=0.993,

J.Benheim et al, Nature phys. Vol.4,463,2008

P.C.Haljan et al, PRA,

72,062316, 2005

(2)Weak-field coupling

)/()(~

,ˆˆ)4/~

(ˆ~

)ˆ~exp( )(ˆ ) ˆ)exp(ˆ( )(ˆ

1

2

0

2

22

0

xj

ji

xixeff

xxgx

JH

tJitUJiJDtU

≪≪

Quantum simulation : frustrated Ising spins

Three ions, transverse mode

i

i

yy

j

x

ji

i

xijeff BJH ˆˆˆ

matrixation transformmode normal:

,2

,

22

,,2

mi

m m

mjmixjiij

b

bb

M

kJ

23121 JJJ Nearest-neighbor interaction

132 JJ Next-nearest-neighbor interaction

K.Kim et.al., PRL,103, 120502(2009)

K.Kim et .al., nature, 465,3(2010)

A.Sorensen & K.Molmer, PRL,82,1971(1999)

今後の方向

１．大規模・集積化プレーナートラップ

2．他の量子糸との結合

３．量子シミュレーション

４．高速化