Quantum information processing with trapped ions...2S l3> 1/2 2D 5/2 2P 3/2 854nm 729nm (red...
Transcript of Quantum information processing with trapped ions...2S l3> 1/2 2D 5/2 2P 3/2 854nm 729nm (red...
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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
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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
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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)
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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
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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:
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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
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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
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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
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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
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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
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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>
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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(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)
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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