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• Introduction to Quantum Information Processing
• Trapped-ion QIP
- Qubits, preparation + measurement
- Coherent operations
- Entangled states: creation + detection
Quantum information processing with trapped ions
Christian RoosInstitute for Quantum Optics and Quantum Information
Innsbruck, Austria
Les Houches, January 18, 2012
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Computation and Physics
Are there physical limitations to the process of computation?
Can computation be understood in terms of quantum mechanics?
Can quantum-physical computation be more efficient than modelsof computation based on classical physics?
Landauer, Bennett, Benioff, Feynman, Deutsch, Josza, Lloyd,…
1960
1980
Limitations and foundations
Is quantum mechanics useful?
1990
2000
1970
Quantum algorithms, quantum error correction
Physical implementations of quantum computation?
Demonstration experiments + applications
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Bit: Quantum bit:
Classical vs. quantum information processing
Physical system with twodistinct states 0 or 1
Two-level quantum system with state
Quantum logic gateUnitary transformation
single qubit gate
Logic gatesBoolean logic operation
XOR truth table
two-qubit gate
CNOT truth table
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Bit: Quantum bit:
Classical vs. quantum information processing
Physical system with twodistinct states 0 or 1
Two-level quantum system with state
Quantum logic gateUnitary transformation
single qubit gate
two-qubit gate
Logic gatesBoolean logic operation
Quantum logic circuit
measurementinitialization
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Entangling quantum bits in a quantum register
Superpositions and entanglement
If applied at the start of a quantum algorithm, this operationenables parallel processing on all possible input states.
Putting a quantum register in a superposition
(a) (b) (a)
(b)
controlled-NOT gate
=
The computation creates quantum correlationsbetween the qubits.
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FFTNax mod
Example: Quantum factoring algorithm
Theoretical Algorithm (Shor, 1994)
1. Input superposition preparation2. Modular exponentiation (multi-qubit gates required)3. Quantum Fourier Transform4. Classical pre- and post-processing
Peter Shor: A quantum computer can efficiently find prime factors of large numbers.
x
Three main steps:
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Quantum information processing
Quantum computing
Quantum algorithms forefficient computing
Quantum simulation
tomorrow‘s lecture
Investigating many-bodyHamiltonians
using well-controlledquantum systems
Quantum communication
Secure communicationcertified by quantum physics Quantum metrology
Piet Schmidt‘s lecture
Entanglement-enhancedmeasurements
Quantum theory and its interpretations; exp. tests
Quantum foundations
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Trapped ions for quantum information processing
Ion string
State detection
Single qubit gates Entangling gates
5 μm
Qubit register
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Experimental setup
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Experimental setup
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d ~ 5 μm
Trap frequencies:
Length scales ionlocalisation
z0
iondistance
laserwavelength
d λ> >> >>
Bohr radius
a010 nm 50 pm5 μm 700 nm
λ ~ 700 nm
focused laser beam
individual addressing, spatially resolved fluorescence
Quantum physics with linear ion strings
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d ~ 5 μm
Trap frequencies:
Length scales ionlocalisation
z0
iondistance
laserwavelength
d λ> >> >>
Bohr radius
a010 nm 50 pm5 μm 700 nm
λ ~ 700 nm
focused laser beam
individual addressing, spatially resolved fluorescence
coupling internal and motional states by laser takes on simple form
Quantum physics with linear ion strings
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no direct state-dependent interactions between ions
d ~ 5 μm
Trap frequencies:
Length scales ionlocalisation
z0
iondistance
laserwavelength
d λ> >> >>
Bohr radius
a010 nm 50 pm5 μm 700 nm
λ ~ 700 nm
focused laser beam
individual addressing, spatially resolved fluorescence
coupling internal and motional states by laser takes on simple form
Quantum physics with linear ion strings
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breathingmode
centre-of-massmode
Vibrational modes
d ~ 5 μm
Trap frequencies:
Length scales ionlocalisation
z0
iondistance
laserwavelength
d λ> >> >>
Bohr radius
a010 nm 50 pm5 μm 700 nm
λ ~ 700 nm
focused laser beam
Quantum physics with linear ion strings
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Trapped ions as a quantum system
Harmonicoscillators
Motional degreesof freedoms:
Internal degreesof freedoms:
Quantumbits
negligible coupling
laser-ion interactions !
negligible coupling
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Trapped ions as a quantum system
Harmonicoscillators
Motional degreesof freedoms:
Internal degreesof freedoms:
Quantumbits
ground state ≈ 10 nm << optical wavelengths
Measurements of motional quantumstates via coupling to internal states
Entanglement between spin and motion
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Trapped ions as a quantum system
Harmonicoscillators
Motional degreesof freedoms:
Internal degreesof freedoms:
Quantumbits
Entanglement between qubits:
Interactions mediated by coupling tovibrational modes
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Trapped-ion quantum bits
Encoding, manipulation and measurement
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Be
Mg
Sr
Ba
Yb
Ca
Hg
Zn
Cd
Singly-charged ions appropriate for quantum information processing
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Ions with optical transition to metastable level: 40Ca+,88Sr+,172Yb+
Ions with hyperfine structure: 9Be+, 25Mg+, 43Ca+,111Cd+,171Yb+…
qubit manipulation requiresultrastable laser
P1/2 D5/2
τ =1s
S1/2
40Ca+
P1/2
S1/2
D5/2
Dopplercooling Sideband
cooling
metastable
Trapped ion quantum bits
opticaltransition
stable
„optical qubit“
hyperfineground states
detection
detection
„hyperfine qubit“
qubit manipulation withmicrowaves or lasers (Raman transitions)
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Qubit manipulation and measurement
P1/2
S1/2
D5/2
40Ca+
Quantumbit
Quantum statemanipulation
Quantum statedetection
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Experimental sequence
1. Initialization in a pure quantum stateP1/2 D5/2
t =1s
S1/2
40Ca+
P1/2
S1/2
D5/2
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Experimental sequence
1. Initialization in a pure quantum state
2. Quantum state manipulation onS1/2 – D5/2 transition
P1/2 D5/2
t =1s
S1/2
40Ca+
P1/2
S1/2
D5/2P1/2
S1/2
D5/2
Quantum statemanipulation
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Experimental sequence
1. Initialization in a pure quantum state
3. Quantum state measurementby fluorescence detection
2. Quantum state manipulation onS1/2 – D5/2 transition
P1/2 D5/2
t =1s
S1/2
40Ca+
P1/2
S1/2
D5/2P1/2
S1/2
D5/2
Quantum statemanipulation
P1/2
S1/2
D5/2
Fluorescencedetection
50 experiments / s
Repeat experiments100 - 1000 times
5µm
Spatially resolveddetection withCCD camera:
Two ions:
1-10 ms
0.01- 5 ms
0.2- 5 ms
2- 20 s
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Ion-laser interaction
Resonant coherent excitation:
Rabi oscillations
laser pulse
Time0 t
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Qubit superposition states
Schrödinger picture:
Phase evolution: for optical qubits ω0 ~ 1015 s-1
Interaction picture:
independent of time
The phase φ of the superposition compares two oscillatory phenomena:
Evolution of the Bloch vector in time
Evolution of the electromagnetic field of the laser exciting the qubit
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Ramsey spectroscopy for phase estimation
Two-pulse excitation: π/2 π/2T
τ τ
detection
Time
wait
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Resonant excitation in Bloch sphere picture
Bloch sphererepresentation
x
y
z
Example:
Laserphase
Time evolution operator:
For
~ Laserintensity
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0 10 20 30 40 50 60 700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (μs)
Upp
erst
ate
popu
latio
nphase of laser switched by π/2
Resonant qubit excitation
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Qubit manipulation
Resonant excitation
or
Off-resonant excitation
ac-Stark shifts shift qubittransition frequency
Arbitrary Bloch sphere rotations can be synthesized by a combination of laser pulses.
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Laser setup for manipulating the qubitTi:Sa laser @ 729nm
Fabry-Perot resonatorfinesse F = 400000, line width ≈ 5kHz
AO
M
radio frequencygenerator
feedback on laser frequency
AOM
radio frequencygenerator
tune laser frequencyinto resonance
with resonator frequency
switch laser on/offintensity controlfrequency control
Trapped ion
feedback to correct for resonator drift
drift rate < 0.2 Hz/s
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CCD
Paul trap
Fluorescencedetection
electro-optic deflector
coherentmanipulation of qubits
dichroicbeamsplitter
inter ion distance: ~ 4 µm
addressing waist: ~ 2 µm
< 0.1% intensity on neighbouring ions
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Exc
itatio
n
Deflector Voltage (V)
Addressing of individual ions with a focussed laser beam
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Measuring qubits
P1/2
S1/2
Quantumbit Quantum state
measurement
40Ca+
detection errors ~ 0.1%
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Measurement ofUnitary transformation +
fluorescence measurementsπ/2 -pulse
Fluorescence measurements:
Further quantum measurements
Measurement of
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Coupling internal and vibrational degrees of freedom
Quantum bit
motional states internal states
Harmonic oscillator
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qubit manipulation…
Trapped-ion laser interactionsE
xcita
tion
Laser detuning Δ (MHz)
redsideband
bluesideband
0
1
carriertransition
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qubit manipulation…
Trapped-ion laser interactions
…
qubit-motion coupling
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qubit manipulation…
Trapped-ion laser interactions
…
qubit-motion coupling
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Sideband excitation
Red sideband:
‘Jaynes-Cummings-Hamiltonian’
Blue sideband:
‘anti-Jaynes-Cummings-Hamiltonian’
• Coupling strength dependent on n
• Coupling strength dependent on n
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Coherent excitation on the sideband
„Blue sideband“ pulses:
upper state population
…
Entanglement between internal and motional state !
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Entangling a pair of trapped ions+
detecting the entanglement
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Generation of Bell states
Pulse sequence:
…
… …
…
Two-ion energy levels
Ion Pulse length Transition
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Generation of Bell states
Pulse sequence:
…
… …
…Ion Pulse length Transition1 π/2 blue sideband
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Generation of Bell states
Pulse sequence:
…
… …
…Ion Pulse length Transition1 π/2 blue sideband2 π carrier
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Generation of Bell states
Pulse sequence:
…
… …
…Ion Pulse length Transition1 π/2 blue sideband2 π carrier2 π blue sideband
Entanglement !
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Generation of Bell states
Pulse sequence:
…
… …
…Ion Pulse length Transition1 π/2 blue sideband2 π carrier2 π blue sideband2 π carrier
Entanglement !
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Measuring the entangled state
We hope to create the state
There are no pure states in experimental physics!
The state created in the experiment has to be described by a density matrix .
How can we analyze the state we created?
Fluorescencedetection withCCD camera:
Coherent superposition or incoherent mixture ?
What is the relative phase of the superposition ?
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Entanglement check : interference
π/2π/2
π/2π/2
To check whether the two amplitudes are phase‐coherent, we have to make an interference experiment:
Prior to the state measurement, we apply π/2to both ions which couple the and thestates to the intermediate states and .
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Entanglement check : interference
Incoherent mixture:
equal populationin all states
π/2π/2
π/2π/2
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equal populationin all states
Incoherent mixture:
Entanglement check: Scan laser phase φ and measure parity
Maximally entangled state:
destructive interference
_
+
+
_
π/2π/2
π/2π/2
constructive interference
Entanglement check : interference
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Parity
+ 1
- 1
0equal populationin all states
Incoherent mixture:
Entanglement check: Scan laser phase φ and measure parity
Maximally entangled state:
destructive interference
_
+
+
_
π/2π/2
π/2π/2
constructive interference
Entanglement check : interference
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Mølmer-Sørensen gate: parity oscillations
A = 0.990(1) 29,400 measurementsp↓↓+p↑↑ = 0.9965(4) 13,000 measurements
F = 99.3(1)%
Bell state fidelity
J. Benhelm et al., Nature Physics 4, 463 (2008)
different entanglementcreation in this experiment
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Creating entanglement vs. entangling quantum gates
Pulse sequence:
Ion Pulse length Transition1 π/2 blue sideband2 π carrier2 π blue sideband2 π carrier
Bell state generation:
This sequence of laser pulses transformsa product state into an entangled state.
Why not?
Check what the sequence does to the initial state !
Does the motional state return to the ground state at the end of the sequence?
But it is not a two-qubit quantum gate!!