Quantum information processing with trapped ions
• Introduction to Quantum Information Processing • Trapped-ion QIP - Qubits, preparation + measurement - Coherent operations - Entangled states: creation + detection
Les Houches, January 18, 2012 Christian Roos Institute for Quantum Optics and Quantum Information Innsbruck, Austria Computation and Physics
1960 Limitations and foundations Are there physical limitations to the process of computation? 1970 Can computation be understood in terms of quantum mechanics?
Is quantum mechanics useful? 1980 Can quantum-physical computation be more efficient than models of computation based on classical physics? 1990 Quantum algorithms, quantum error correction
2000 Physical implementations of quantum computation?
Demonstration experiments + applications
Landauer, Bennett, Benioff, Feynman, Deutsch, Josza, Lloyd,… Classical vs. quantum information processing
Bit: Quantum bit: Physical system with two Two-level quantum system with state distinct states 0 or 1
Logic gates Quantum logic gate Boolean logic operation Unitary transformation
single qubit gate
two-qubit gate
XOR truth table CNOT truth table Classical vs. quantum information processing
Bit: Quantum bit: Physical system with two Two-level quantum system with state distinct states 0 or 1
Logic gates Quantum logic gate Boolean logic operation Unitary transformation
single qubit gate
two-qubit gate
Quantum logic circuit
initialization measurement Superpositions and entanglement
Putting a quantum register in a superposition
If applied at the start of a quantum algorithm, this operation enables parallel processing on all possible input states.
Entangling quantum bits in a quantum register
(a) (b) (a)
(b)
The computation creates quantum correlations controlled-NOT gate between the qubits. = Example: Quantum factoring algorithm
Peter Shor: A quantum computer can efficiently find prime factors of large numbers.
Theoretical Algorithm (Shor, 1994)
x x mod Na FFT
Three main steps: 1. Input superposition preparation 2. Modular exponentiation (multi-qubit gates required) 3. Quantum Fourier Transform 4. Classical pre- and post-processing Quantum information processing
Quantum computing Quantum simulation Quantum algorithms for Investigating many-body efficient computing Hamiltonians using well-controlled quantum systems
tomorrow‘s lecture Quantum communication
Secure communication certified by quantum physics Quantum metrology
Entanglement-enhanced Quantum foundations measurements Quantum theory and its interpretations; exp. tests Piet Schmidt‘s lecture Trapped ions for quantum information processing
5 μm Ion string
Qubit register State detection Single qubit gates Entangling gates Experimental setup Experimental setup Quantum physics with linear ion strings
focused laser beam Trap frequencies:
d ~ 5 μm λ ~ 700 nm
Length scales ion laser ion Bohr distance wavelength localisation radius
d > λ >> z0 >> a0 5 μm 700 nm 10 nm 50 pm
individual addressing, spatially resolved fluorescence Quantum physics with linear ion strings
focused laser beam Trap frequencies:
d ~ 5 μm λ ~ 700 nm
Length scales ion laser ion Bohr distance wavelength localisation radius
d > λ >> z0 >> a0 5 μm 700 nm 10 nm 50 pm
individual addressing, spatially resolved fluorescence
coupling internal and motional states by laser takes on simple form Quantum physics with linear ion strings
focused laser beam Trap frequencies:
d ~ 5 μm λ ~ 700 nm
Length scales ion laser ion Bohr distance wavelength localisation radius
d > λ >> z0 >> a0 5 μm 700 nm 10 nm 50 pm
individual addressing, spatially resolved fluorescence
coupling internal and motional states by laser takes on simple form
no direct state-dependent interactions between ions Quantum physics with linear ion strings
focused laser beam Trap frequencies:
d ~ 5 μm λ ~ 700 nm
Length scales ion laser ion Bohr distance wavelength localisation radius
d > λ >> z0 >> a0 5 μm 700 nm 10 nm 50 pm
Vibrational modes
centre-of-mass breathing mode mode Trapped ions as a quantum system
Internal degrees of freedoms: Quantum bits negligible coupling
laser-ion interactions ! Motional degrees of freedoms: negligible coupling Harmonic oscillators Trapped ions as a quantum system
Internal degrees of freedoms: ground state ≈ 10 nm << optical wavelengths Quantum bits Measurements of motional quantum states via coupling to internal states
Entanglement between spin and motion
Motional degrees of freedoms: Harmonic oscillators Trapped ions as a quantum system
Internal degrees of freedoms: Quantum bits
Motional degrees Entanglement between qubits: of freedoms: Interactions mediated by coupling to Harmonic vibrational modes oscillators Trapped-ion quantum bits
Encoding, manipulation and measurement Be Singly-charged ions appropriate for quantum information processing Mg
Ca Zn
Sr Cd
Ba Hg
Yb Trapped ion quantum bits
Ions with optical transition to metastable level: 40Ca+,88Sr+,172Yb+
P1/2 metastableD5/2 „optical qubit“
τ =1s detectionDoppler qubit manipulation requires opticalSideband cooling transition ultrastable laser 40coolingCa+ SS1/21/2 stable
Ions with hyperfine structure: 9Be+, 25Mg+, 43Ca+,111Cd+,171Yb+…
„hyperfine qubit“
detection qubit manipulation with hyperfine microwaves or lasers (Raman transitions) ground states Qubit manipulation and measurement
40Ca+
P1/2
D5/2
Quantum Quantum state bit detection Quantum state manipulation S1/2 Experimental sequence
P1/2 1. Initialization in a pure quantum state D5/2
t =1s
40Ca+ SS1/21/2 Experimental sequence
P1/2 1. Initialization in a pure quantum state D5/2
t =1s Quantum state 2. Quantum state manipulation on manipulation S1/2 –D5/2 transition 40Ca+ SS1/21/2 Experimental sequence
1. Initialization in a pure quantum state P1/2 D 5/2 1-10 ms t =1s Quantum stateFluorescence 2. Quantum state manipulation on manipulation detection S1/2 –D5/2 transition 0.01- 5 ms 40Ca+ SS1/21/2 3. Quantum state measurement by fluorescence detection 0.2- 5 ms
5µm Two ions:
Spatially resolved 50 experiments / s detection with CCD camera: Repeat experiments 2- 20 s 100 - 1000 times Ion-laser interaction
Resonant coherent excitation: laser pulse 0 t Time
Rabi oscillations Qubit superposition states
Schrödinger picture:
15 -1 Phase evolution: for optical qubits ω0 ~ 10 s
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 Ramsey spectroscopy for phase estimation
Two-pulse excitation: π/2 T π/2 detection τ Time τ
wait Resonant excitation in Bloch sphere picture
Bloch sphere z representation
y
Laser ~ Laser x intensity phase
Example:
Time evolution operator:
For Resonant qubit excitation
phase of laser switched by π/2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2 Upper state population 0.1
0 0 10 20 30 40 50 60 70 Time (μs) Qubit manipulation
Resonant excitation
or
Off-resonant excitation
ac-Stark shifts shift qubit transition frequency
Arbitrary Bloch sphere rotations can be synthesized by a combination of laser pulses. Laser setup for manipulating the qubit
Ti:Sa laser @ 729nm Trapped ion
AOM
radio frequency generator feedback on laser frequency switch laser on/off intensity control
AOM frequency control
radio frequency generator feedback to correct for resonator drift
tune laser frequency Fabry-Perot resonator into resonance finesse F = 400000, with resonator frequency drift rate < 0.2 Hz/s line width ≈ 5kHz Addressing of individual ions with a focussed laser beam
0.8
coherent 0.7 manipulation Paul trap 0.6 of qubits 0.5
0.4 electro-optic 0.3 deflector Excitation 0.2
0.1
0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Deflector Voltage (V)
z inter ion distance: ~ 4 µm dichroic beamsplitter z addressing waist: ~ 2 µm < 0.1% intensity on neighbouring ions Fluorescence detection CCD Measuring qubits
40Ca+
P1/2
Quantum Quantum state bit measurement
S1/2
detection errors ~ 0.1% Further quantum measurements
Measurement of
Fluorescence measurements:
Measurement of
Unitary transformation + π/2 -pulse fluorescence measurements Coupling internal and vibrational degrees of freedom
Harmonic oscillator Quantum bit
motional states internal states Trapped-ion laser interactions
qubit manipulation …
1 carrier transition
red blue
Excitation sideband sideband
0 Laser detuning Δ (MHz) Trapped-ion laser interactions
qubit manipulation …
qubit-motion coupling
… Trapped-ion laser interactions
qubit manipulation …
qubit-motion coupling
… Sideband excitation
Red sideband:
‘Jaynes-Cummings-Hamiltonian’
• Coupling strength dependent on n
Blue sideband:
‘anti-Jaynes-Cummings-Hamiltonian’
• Coupling strength dependent on n Coherent excitation on the sideband
„Blue sideband“ pulses: …
Entanglement between internal and motional state !
upper state population Entangling a pair of trapped ions + detecting the entanglement Generation of Bell states
Two-ion energy levels
… Pulse sequence:
Ion Pulse length Transition … … … Generation of Bell states
… Pulse sequence:
Ion Pulse length Transition 1 π/2 blue sideband … … … Generation of Bell states
… Pulse sequence:
Ion Pulse length Transition 1 π/2 blue sideband
… … 2 π carrier … Generation of Bell states
… Pulse sequence:
Ion Pulse length Transition 1 π/2 blue sideband
… … 2 π carrier 2 π blue sideband …
Entanglement ! Generation of Bell states
… Pulse sequence:
Ion Pulse length Transition 1 π/2 blue sideband
… … 2 π carrier 2 π blue sideband 2 π carrier …
Entanglement ! 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?
Fluorescence detection with CCD camera: Coherent superposition or incoherent mixture ?
What is the relative phase of the superposition ? Entanglement check : interference
To check whether the two amplitudes are phase‐coherent, we have to make an interference experiment:
Prior to the state measurement, we apply π/2 to both ions which couple the and the π/2 π/2 states to the intermediate states and .
π/2 π/2 Entanglement check : interference
π/2 π/2
π/2 π/2 Incoherent mixture:
equal population in all states Entanglement check : interference
Maximally entangled state:
+ destructive interference π/2 π/2
constructive interference
_ _
π/2 π/2 Incoherent mixture: + equal population in all states
Entanglement check: Scan laser phase φ and measure parity Entanglement check : interference
Parity Maximally entangled state:
+ destructive interference + 1 π/2 π/2
constructive interference
-1 _ _
π/2 π/2 Incoherent mixture: +
equal population 0 in all states
Entanglement check: Scan laser phase φ and measure parity Mølmer-Sørensen gate: parity oscillations
different entanglement creation in this experiment
Bell state fidelity A = 0.990(1) 29,400 measurements F = 99.3(1)% p↓↓+p↑↑ = 0.9965(4) 13,000 measurements
J. Benhelm et al., Nature Physics 4, 463 (2008) Creating entanglement vs. entangling quantum gates
Bell state generation:
Pulse sequence: This sequence of laser pulses transforms a product state into an entangled state. Ion Pulse length Transition 1 π/2 blue sideband 2 π carrier 2 π blue sideband 2 π carrier
But it is not a two-qubit quantum gate!!
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?