Lecture 11, May 11, 2017 This week: Atomic Ions for QIP Ion Traps Vibrational modes Preparation of initial states Read-Out Single-Ion Gates Two-Ion Gates Introductory Review Articles: D. Leibfried, R. Blatt, C. Monroe and D. Wineland, Quantum dynamics of single trapped ions, Review of Modern Physics 75, 281 (2003) R. Blatt, and D. Wineland, Entangled states of trapped atomic ions, Nature 453, 1008 (2008) Andreas Wallraff 11-May-17 1
Ion Trap Quantum Processor row of qubits in a linear Paul trap forms a quantum register Laser pulses manipulate individual ions A CCD camera reads out the ion`s quantum state Slide material courtesy of H. Haeffner (Innsbruck/Berkeley) and J. Home (ETHZ) with notes by A. Wallraff (ETHZ) Effective ion-ion interaction induced by laser pulses that excite the ion`s motion Andreas Wallraff 11-May-17 2
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Trapping Individual Ions in a Linear Paul Trap Andreas Wallraff 11-May-17 4
Mechanical Analog Harvard Natural Sciences Lecture Demonstrations https://www.youtube.com/watch?v=xtjznukamiy Andreas Wallraff 11-May-17 5
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Experimental Setup Fluorescence detection by CCD camera photomultiplier CCD camera atomic beam Photomultiplier Laser beams for: photoionization cooling quantum state manipulation fluorescence excitation oven Vacuum pump Andreas Wallraff 11-May-17 9
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Ions as Quantum Bits Ions with optical transition to metastable level: 40 Ca +, 88 Sr +, 172 Yb + P 1/2 τ 1 s P 1/2 metastable D 5/2 D 5/2 e> τ =1s Qubit levels: S Doppler 1/2, D 5/2 detection optical cooling Sideband transition 40 cooling Ca + Qubit transition: Quadrupole transition S 1/2 S 1/2 g> S 1/2 D 5/2 1/2 stable Andreas Wallraff 11-May-17 12
Detection of Ion Quantum State Quantum jumps: spectroscopy with quantized fluorescence P short lifetime monitor S D long lifetime weak transition metastable level Quantum jump technique Electron shelving technique lots of photons S Observation of quantum jumps: Nagourney et al., PRL 56,2797 (1986), Sauter et al., PRL 57,1696 (1986), Bergquist et al., PRL 57,1699 (1986) no photons time in excited state (average is lifetime) D absorption and emission cause fluorescence steps (digital quantum jump signal) Andreas Wallraff 11-May-17 15
Electron Shelving for Quantum State Detection P 1/2 D 5/2 Quantum state τ =1s Fluorescence manipulation detection S 1/2 40 Ca + S 1/2 1. Initialization in a pure quantum state 2. Quantum state manipulation on S 1/2 D 5/2 transition 3. Quantum state measurement by fluorescence detection One ion: Fluorescence histogram 8 7 D 5/2 state S 1/2 state 6 5 4 3 2 50 experiments / s Repeat experiments 100-200 times 1 0 0 20 40 60 80 100 120 counts per 2 ms Andreas Wallraff 11-May-17 16
Electron Shelving for Quantum State Detection P 1/2 D 5/2 1. Initialization in a pure quantum state S 1/2 Two ions: Spatially resolved detection with CCD camera: Fluorescence detection 5µm 2. Quantum state manipulation on S 1/2 D 5/2 transition 3. Quantum state measurement by fluorescence detection 50 experiments / s Repeat experiments 100-200 times Andreas Wallraff 11-May-17 17
Mechanical Motion of Ions in their Trapping Potential Mechanical quantum harmonic oscillator harmonic trap Extension of the ground state: Size of the wave packet << wavelength of visible light (e.g. for Ca) Energy scale of interest: ions need to be very cold to be in their vibrational ground state Andreas Wallraff 11-May-17 18
An Ion Coupled to a Harmonic Oscillator e 2-level-atom harmonic trap joint energy levels e,0 e,1 e,2 g tensor product g,0 g,1 g,2 dressed states diagram Laser ion interactions Ion: Electronic structure of the ion approximated by two-level system (laser is (near-) resonant and couples only two levels) Trap: Only a single harmonic oscillator taken into account Andreas Wallraff 11-May-17 20
External Degree of Freedom: Ion Motion 2-level-atom harmonic trap joint energy levels ion transition frequency 400 THz (carrier) trap frequency 1 MHz (sideband, red/blue SB) Andreas Wallraff 11-May-17 21
A Closer Look at the Excitation Spectrum (3 Ions) (red / lower) motional sidebands carrier transition (blue / upper) motional sidebands 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) Laser detuning at 729 nm (MHz) Andreas Wallraff 11-May-17 22
Cooling of the Vibrational Modes Sideband absorption spectra: 99.9 % ground state population P D 0.8 0.6 0.4 after Doppler cooling n z =1.7 P D 0.8 0.6 0.4 red side band transition e, n 1 e, n after sideband cooling 0.2 0 4.54-4.54 4.52-4.52 4.5-4.5-4.48 Detuning δω (MHz) 0.2 0 4.48 4.5 4.52 4.54 Detuning δω (MHz) g, n 1 g, n red sideband blue sideband Andreas Wallraff 11-May-17 25
Coherent Excitation on the Sideband Transition Not only cooling but also controlled excitation of the vibrational modes: Blue sideband pulses: coupled system D, n 1 D,n D, n +1 S, n 1 S,n S, n +1 D state population Entanglement between internal and motional state! Andreas Wallraff 11-May-17 26
Single Qubit Operations Arbitrary qubit rotations: Laser slightly detuned from carrier resonance or: pulses with rotation axis in equatorial plane (z-rotations by off-resonant laser beam creating ac-stark shifts) x,y-rotations D state population limited by magnetic field fluctuations laser frequency fluctuations (laser linewidth δν<100 Hz) Gate time: 1-10 µs Coherence time: 2-3 ms Andreas Wallraff 11-May-17 27
Addressing Qubits electro-optic deflector coherent manipulation of qubits Paul trap Excitation 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0-10 -8-6 -4-2 0 2 4 6 8 10 Deflector Voltage (V) dichroic beamsplitter Fluorescence detection CCD inter ion distance: ~ 4 µm addressing waist: ~ 2 µm < 0.1% intensity on neighbouring ions Andreas Wallraff 11-May-17 28
Generation of Bell States: Entanglement of Two Ions Pulse sequence: generation of entanglement between two ions: Andreas Wallraff 11-May-17 29
Generation of Bell States: Entanglement of Two Ions Pulse sequence: Ion 1: π/2, blue sideband creates entangled state between qubit 1 and oscillator Andreas Wallraff 11-May-17 30
Generation of Bell States: Entanglement of Two Ions Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier excites qubit 2 Andreas Wallraff 11-May-17 31
Generation of Bell States: Entanglement of Two Ions Pulse sequence: Ion 1: π/2, blue sideband Ion 2: π, carrier Ion 2: π, blue sideband takes qubit 2 (with one oscillator excitation) back to ground state and removes excitation from oscillator SD0> is non-resonant and remains unaffected Andreas Wallraff 11-May-17 32
Bell State Analysis Coherent superposition or incoherent mixture? What is the relative phase of the superposition? Fluorescence detection with CCD camera: Ψ + tomography of qubit states (= full measurement of x,y,z components of both qubits and its correlations) SS SDDS DD SS SD DD DS Measurement of the density matrix: Andreas Wallraff 11-May-17 33
Bell State Reconstruction 0.5 0.5 F=0.91 SS SDDS SS SDDS DD DD DS SD SS DD DD DS SD SS - 0.5-0.5 Andreas Wallraff 11-May-17 35
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 Andreas Wallraff 11-May-17 37
Realizing a Controlled NOT with a Controlled Phase Gate implementation of a CNOT for universal ion trap quantum computing: controlled phase gate Both, the phase gate as well the CNOT gate can be converted into each other with single qubit operations. Together with single qubit gates any unitary operation can be implemented! Andreas Wallraff 11-May-17 38
Cirac-Zoller Two-Ion Phase Gate ε1 ε 2 D,0 D,1 S,0 S,1 ion 1 motion ion 2 S S, D 0 0, D SWAP Andreas Wallraff 11-May-17 39
Cirac-Zoller Two-Ion Phase Gate ε1 ε 2 Phase gate using the motion and the target bit. ion 1 motion ion 2 S S, D 0 0, D SWAP Z Andreas Wallraff 11-May-17 40
Cirac-Zoller Two-Ion Phase Gate ε1 ε 2 D,0 D,1 S,0 S,1 ion 1 motion ion 2 S S, D, D SWAP SWAP 0 0 Z Andreas Wallraff 11-May-17 41
Cirac-Zoller Two-Ion Phase Gate ε1 ε 2 Phase gate using the motion and the target bit. ion 1 motion ion 2 S S, D 0 0, D Z Andreas Wallraff 11-May-17 42
How do you do this with just a two-level system?? D0 blue sideband is forbidden Complete 2π blue sideband rotations induce phase factor -1 (c.f. geometric phase) How to implement that for the S1-D2 sideband transition which rotates sqrt(2) faster at same drive field strength? Andreas Wallraff 11-May-17 45
Phase Gate Composite 2π-rotation: Andreas Wallraff 11-May-17 46
A Phase Gate with 4 Pulses (2π Rotation) S0: ( ) ( ) ( ) ( ) R( θφ, ) = R ππ, 2 R π 2,0 R ππ, 2 R π 2,0 + + + + 1 1 1 1 1 2π on S,0 D,1 3 2 4 Andreas Wallraff 11-May-17 47
A Single Ion Composite Phase Gate: Experiment state preparation S,0, then application of phase gate pulse sequence 1 π φ = 0 φ = π 2 φ = 0 π φ = π 2 π 2 π 2 ϕ = 0 ϕ = 0 ϕ = π 2 0.9 0.8 0.7 D 5/2 - excitation 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180 Continuous tomography of the phase gate Time (µs) Andreas Wallraff 11-May-17 48
Population of S,1> - D,2> remains unaffected 2 S0: S1: ( ) ( ) ( ) ( ) R( θφ, ) = R ππ, 2 R π 2,0 R ππ, 2 R π 2,0 + + + + 1 1 1 1 all transition rates are a factor of sqrt(2) faster at the same Laser power ( ) ( ) ( ) ( ) + + + + R( θ, φ) = R1 π 2, π 2 R1 π,0 R1 π 2, π 2 R1 π,0 4 3 1 Andreas Wallraff 11-May-17 49
Testing the Phase of the Phase Gate for 0,S> 1 π 2 Phase gate π 2 97.8 (5) % 0.9 0.8 D 5/2 - excitation 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 Time (µs) Andreas Wallraff 11-May-17 50
Cirac-Zoller Two-Ion Controlled-NOT Operation ion 1 motion ion 2 S S, D, D SWAP SWAP 0 0 control qubit target qubit pulse sequence: ion 1 ion 2 Andreas Wallraff 11-May-17 51
Cirac Zoller CNOT Gate Operation SS - SS DS - DD SD - SD DD - DS Andreas Wallraff 11-May-17 52
Measured Truth Table of Cirac-Zoller CNOT Operation input output Andreas Wallraff 11-May-17 53
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