quantum information processing with trapped electrons
TRANSCRIPT
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Quantum information processingwith trapped electrons
Irene Marzoli
Università di Camerino (Italy)
International School of Physics “Enrico Fermi”
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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.
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Overview
• Basic properties of Penning traps• Qubit encoding and manipulation • Scalability • New trap geometries• Coupling qubits• Outlook and applications
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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
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Illustration of electron motion in a Penning trap
Fast radial oscillation: modified cyclotron motionSlow radial oscillation: magnetron motionRed: axial oscillation
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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.
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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
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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.
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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)
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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
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( ) ( )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 −≡
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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
−≡
≡
≡
≡
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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
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Two-qubit gate: Conditional phase shift
11100100 δγβα +++
( ) ( )
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−−
−−
=⎟⎠⎞
⎜⎝⎛××⎟
⎠⎞
⎜⎝⎛×
1000001000001000001000001
2,
20,
2,
20, ππππππ MMMM
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
==−
1000010000100001
)( πφPSC
11e100100 i δγβα φ+++C-PS
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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
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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
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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
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Hyperbolical trap
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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)
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… 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
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Silver plated Al2O3 ceramic disk with electrodes of
R0=2.5 mm, R1=5.8 mm, R2=9.1 mm and d=3 mm
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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)
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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)
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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
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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
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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
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Microscopic planar Penning trap withmultiple ring electrodes
Larger trap volumefor easier loading
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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
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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)
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QUELE experimental setup
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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
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Prototype planar trap (Univ. Mainz)
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Onion trap (Univ. Ulm/Mainz)
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Pixel trap (Univ. Ulm/Mainz)
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Pixel trap
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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)
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J. R. Castrejon-Pita and R. C. Thompson, Phys. Rev. A 32, 013405 (2005)
x
z
+
-
-+ +
+
B
y
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Coplanar‐waveguide Penning trap
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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
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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
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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
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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
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( )[ ]∑∑<=
+−+≈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)
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⎟⎟⎠
⎞⎜⎜⎝
⎛+= 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
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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
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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.
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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)
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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
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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
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• 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
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• Giacomo Ciaramicoli
• Paolo Tombesi