Download Ion trap quantum processor

Ion trap quantum processor
Laser pulses manipulate
individual ions
row of qubits in a
linear Paul trap forms
a quantum register
Effective ion-ion
interaction induced by
laser pulses that excite
the ion`s motion
A CCD camera reads out
the ion`s quantum state
slides courtesy of Hartmut Haeffner,
Innsbruck Group with some notes by
Andreas Wallraff, ETH Zurich
Trapping Individual Ions
Experimental setup
Fluorescence
detection by
CCD camera
photomultiplier
atomic
beam
Photomultiplier
CCD
camera
oven
Laser beams for:
Vacuum
pump
photoionization
cooling
quantum state manipulation
fluorescence excitation
Ions as Quantum Bits
Ions with optical transition to metastable level: 40Ca+,88Sr+,172Yb+
t≈1s
P
P1/2
1/2
metastable
D5/2
D5/2
Doppler
detection
cooling
SS1/2
1/21/2
S
stable
|e>
t =1s
optical
Sideband
transition
cooling
40
Ca+
|g>
Qubit levels:
S1/2 , D5/2
Qubit transition: Quadrupole transition
S1/2 – D5/2
Detection of Ion Quantum State
Quantum jumps: spectroscopy with quantized fluorescence
P
short lifetime
long lifetime
metastable
D
level
monitor
absorption and emission
cause fluorescence steps
(digital quantum jump signal)
weak transition
S
lots of photons
S
„ Quantum jump technique“
„ Electron shelving technique“
Observation of quantum jumps:
no photons
D
time in excited state (average is lifetime)
Nagourney et al., PRL 56,2797 (1986),
Sauter et al., PRL 57,1696 (1986),
Bergquist et al., PRL 57,1699 (1986)
Electron shelving for quantum state detection
P1/2
1. Initialization in a pure quantum state
D5/2
t =1s
Quantum state
Fluorescence
manipulation detection
2. Quantum state manipulation on
S1/2 – D5/2 transition
40Ca+
SS1/2
1/2
3. Quantum state measurement
by fluorescence detection
One ion : Fluorescence histogram
8
7
D5/2 state
S1/2 state
6
50 experiments / s
5
4
Repeat experiments
100-200 times
3
2
1
0
0
20
40
60
80
counts per 2 ms
100
120
Electron shelving for quantum state detection
P1/2
D5/2
Fluorescence
detection
S1/2
Two ions:
Spatially resolved
detection with
CCD camera:
1. Initialization in a pure quantum state
2. Quantum state manipulation on
S1/2 – D5/2 transition
3. Quantum state measurement
by fluorescence detection
5µm
50 experiments / s
Repeat experiments
100-200 times
Mechanical Motion of Ions in their Trapping Potential:
Mechanical Quantum harmonical oscillator
Extension of the ground state:
harmonic trap
Size of the wave packet << wavelength of visible light
Energy scale of interest:
ions need to be very cold
to be in their vibrational
ground state
An Ion Coupled to a Harmonic Oscillator
2-level-atom
harmonic trap
joint energy levels
e,2
e,1
e

g
…
e,0
g ,2
g ,1
tensor product
g ,0
dressed states diagram
Laser – ion interactions
Approximations:
Ion:
Trap:
Electronic structure of the ion approximated by two-level system
(laser is (near-) resonant and couples only two levels)
Only a single harmonic oscillator taken into account
External degree of freedom: ion motion
2-level-atom
harmonic trap
joint energy levels
…
ion transition
frequency 400 THz
trap frequency 1 MHz
A closer look at the excitation spectrum (3 ions)
(red / lower)
motional sidebands
carrier transition
(blue / upper)
motional sidebands
Laser detuning D at 729 nm (MHz)
many different vibrational modes of ions in the trap
red and blue side bands can be observed because
vibrational motion of ions is not cooled (in this example)
Stretch mode excitation
no differential force
differential force
differential force
no differential force
Cooling of the vibrational modes
e, n
e, n  1
Sideband absorption spectra
…
99.9 % ground state population
…
g, n
g, n 1
0.6
0.8
after
Doppler
cooling
nz 1.7
0.4
after
sideband
cooling
0.4
0.2
0
-4.54
4.54
0.6
PD
PD
red side band
transition
0.8
0.2
-4.52 4.5
-4.5 -4.48
4.52
4.48
Detuning  (MHz)
red sideband
0
4.48
4.5
4.52
4.54
Detuning  (MHz)
blue sideband
But also controlled excitation of the vibrational modes
Coherent excitation on the sideband
„Blue sideband“ pulses:
D, n  1
coupled system
D, n
D, n  1
S, n 1
S, n
S , n 1
Entanglement between internal and motional state !
D state population
Single qubit operations
Arbitrary qubit rotations:
• Laser slightly detuned from carrier resonance
z-rotations
(z-rotations by off-resonant laser beam
creating ac-Stark shifts)
or:
• Concatenation of two pulses with rotation
axis in equatorial plane
Gate time : 1-10 ms
D state population
x,y-rotations
Coherence time : 2-3 ms
limited by
• magnetic field fluctuations
• laser frequency fluctuations
(laser linewidth n<100 Hz)
Addressing the qubits
0.8
0.7
Paul trap
0.6
Excitation
coherent
manipulation
of qubits
0.5
0.4
electro-optic
deflector
0.3
0.2
0.1
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Deflector Voltage (V)
 inter ion distance: ~ 4 µm
dichroic
beamsplitter
 addressing waist: ~ 2 µm
< 0.1% intensity on neighbouring ions
Fluorescence
detection
CCD
Generation of Bell states
Pulse sequence:
…
…
…
…
generation of entanglement between two ions
…
Generation of Bell states
Pulse sequence:
…
…
…
Ion 1: p/2 , blue sideband
creates entangled state
between qubit 1 and oscillator
…
Generation of Bell states
Pulse sequence:
Ion 1: p/2 , blue sideband
,
…
…
Ion 2: p
…
excites qubit 2
carrier
…
Generation of Bell states
Pulse sequence:
…
…
…
Ion 1: p/2 , blue sideband
|SD0> is non-resonant and
remains unaffected
Ion 2: p
,
Ion 2: p
, blue sideband
carrier
takes qubit 2 (with one
oscillator excitation) back to
ground state and removes
excitation from oscillator
Bell state analysis
Fluorescence
detection with
CCD camera:
Coherent superposition or incoherent mixture ?
What is the relative phase of the superposition ?
tomography of qubit states (= full
measurement of x,y,z components of
both qubits and its correlations)
Measurement of the density matrix:
Y
SS
SD
DS
DD
DD DS
SD SS
Bell state reconstruction
F=0.91
SS
SD
DS
DD
SD SS
DS
DD
SS
SD
DS
DD
SD SS
DS
DD
Controlled Phase Gate  CNOT
implementation of a CNOT for universal ion trap quantum computing
Both, the phase gate as well the CNOT gate can be converted into each
other with single qubit operations.
Together with the three single qubit gates,
we can implement any unitary operation!
controlled phase gate
Quantum gate proposals with trapped ions
...allows the realization of a
universal quantum computer !
Some other gate proposals by:
• Cirac & Zoller
• Mølmer & Sørensen, Milburn
• Jonathan, Plenio & Knight
• Geometric phases
• Leibfried & Wineland
Cirac - Zoller two-ion phase gate
1
2
D, 0
S, 0
ion 1
motion
ion 2
D,1
S ,1
S ,D
0
S ,D
SWAP
0
Cirac - Zoller two-ion phase gate
1
2
Phase gate using
the motion and
the target bit.
ion 1
motion
ion 2
S ,D
0
S ,D
SWAP
0
Z
Cirac - Zoller two-ion phase gate
1
2
D, 0
S, 0
ion 1
motion
ion 2
D,1
S ,1
S ,D
0
S ,D
SWAP
SWAP
Z
0
Cirac - Zoller two-ion phase gate
1
2
Phase gate using
the motion and
the target bit.
ion 1
motion
ion 2
S ,D
0
0
S ,D
Z
How do you do this with just a two-level system?
?
Phase gate
Composite 2π-rotation:
A phase gate with 4 pulses (2p rotation)

R( , )  R1 p , p 2  R1 p


2,0 R1 p , p 2  R1 p
2p
1
3
2
4
on
2,0

S ,0  D,1
Continuous tomography of the phase gate
A single ion composite phase gate: Experiment
state preparation S ,0 , then application of phase gate pulse sequence
p 2
  00
p 2
  00
p
 p 2
p
  pp 22
1
0.9
0.8
D5/2 - excitation
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
Time (ms)
120
140
160
180
Population of |S,1> - |D,2> remains unaffected
all transition rates are a factor of sqrt(2) faster at the same Laser power




R( , )  R1 p 2, p 2 R1 p ,0 R1 p 2, p 2 R1 p ,0 
2
4
3
1
Testing the phase of the phase gate |0,S>
p

Phase gate
2
p
2
1
97.8 (5) %
0.9
D5/2 - excitation
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
Time (ms)
200
250
300
Cirac - Zoller two-ion controlled-NOT operation
ion 1
motion
ion 2
S ,D
0
S ,D
pulse sequence:
ion 1
ion 2
control qubit
SWAP
SWAP
0
target qubit
Cirac – Zoller CNOT gate operation
SS - SS
DS - DD
SD - SD
DD - DS
Measured truth table of Cirac-Zoller CNOT operation
input
output
Error budget for Cirac-Zoller CNOT
Error source
Magnitude
Population loss
Laser frequency noise
(Phase coherence)
~ 100 Hz (FWHM)
~ 10 % !!!
Residual thermal excitation
<n>bus < 0.02
<n>spec = 6
2%
0.4 %
Laser intensity noise
1 % peak to peak
0.1 %
Addressing error
(can be corrected for partially)
5 % in Rabi frequency
(at neighbouring ion)
3%
Off resonant excitations
for tgate  600 µs
4%
Laser detuning error
~ 500 Hz (FWHM)
~2%
Total
November 2002
~ 20 %