The requirements for quantum information processing Quantum information processing with trapped ions D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001) Courtesy of Timo Koerber Institut für Experimentalphysik I. Scalable physical system, well characterized qubits Universität Innsbruck II. Ability to initialize the state of the qubits III. Long relevant coherence times, much longer than gate operation time 1. Basic experimental techniques IV. “Universal” set of quantum gates 2. Two-particle entanglement V. Qubit-specific measurement capability 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation Lectures 16 - 17 6. Outlook 1
Experimental Setup Important energy levels • The important energy levels are shown on the next slides; a fast transition is used to detect ion fluorescence and for Doppler cooling, while the narrow D5/2 quadrupole transition has a lifetime of 1 second and is used for coherent manipulation and represents out quantum bit. Of course a specific set of Zeeman states is used to actually implement our qubit. The presence of P 1/2 other sublevels give us additional possibilities for doing D 5/2 coherent operations. „quantum bit“ S 1/2 2
Ca+: Important energy levels Ca+: Important energy levels S 1/2 – D 5/2 : quadrupole transition τ τ τ τ = 7 ns τ = 7 ns τ τ τ P 1/2 P 1/2 P 1/2 τ τ = 1 s τ τ D 5/2 D 5/2 397 nm 729 nm S 1/2 S 1/2 „qubit“ „quoctet“ (sp?) 3
Qubits with trapped ions String of Ca+ ions in linear Paul trap Encoding of quantum information requires long-lived atomic states: row of qubits in a � optical transitions � microwave transitions linear Paul trap forms a quantum register Ca + , Sr + , Ba + , Ra + , Yb + , Hg + etc. 9 Be + , 25 Mg + , 43 Ca + , 87 Sr + , 137 Ba + , 111 Cd + , 171 Yb + P 3/2 P 1/2 D 5/2 qubit S 1/2 ω ≈ 0.7 − 2 MHz qubit z S 1/2 ω ≈ 1 . 5 − 4 MHz x , y 50 µm 4
Addressing of individual ions Ion addressing 0.8 coherent 0.7 manipulation Paul trap 0.6 of qubits The ions can be addressed individually on the qubit Excitation 0.5 transition with an EO deflector which can quickly move the 0.4 electrooptic focus of the 729 light from one ion to another, using the 0.3 deflector same optical path as the fluorescence detection via the 0.2 CCD camera. 0.1 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 How well the addressing works is shown on the previous Deflector Voltage (V) slide: The graph shows the excitation of the indiviual ions � inter ion distance: ~ 4 µm as the deflector is scanned across the crystal. dichroic � addressing waist: ~ 2.5 µm beamsplitter < 0.1% intensity on neighbouring ions Fluorescence detection CCD 5
External degree of freedom: ion motion External degree of freedom: ion motion Notes for next slides: Now let's have a look at the qubit transition in the presence of the harmonic trap motional degrees of freedom. If we focus on just one motional ... … mode , we just get a ladder of harmonic oscillator levels. The joint (motion + electronic energy level) system shows a double ladder structure. With the narrow laser we can selectively excite the carrier transition, where the motional state remains unchanged... Or use the blue sideband and red sideband transitions, where we can change the motional state. We can walk down the double ladder by exciting the red sideband and returning the ion dissipatively to the grounsstate. With this we can prepare the ions in the motional ground state with high probability, thereby initializing our quantum register. 6
External degree of freedom: ion motion External degree of freedom: ion motion 2-level-atom harmonic trap joint energy levels 2-level-atom harmonic trap joint energy levels ... ... … … Laser cooling to the motional ground state: Cooling time: 5-10 ms > 99% in motional ground state 7
Coherent manipulation Coherent manipulation Let's now begin to look at the coherent state manipulation. If we resonantly shine the light pulse at the carrier transition, the system evolves for a time τ 2-level-atom harmonic trap joint energy levels with this Hamiltonian, where the coupling strength Ω depends on the ... … square root of the intensity, and φ is the phase of the laser field with respect to the atomic polarization. The effect of such a pulse is a rotation of the state vector on the Bloch sphere, where the poles represent the two states and the equator represents superposition states with different relative phases. The roation axis is determined by the laser frequency and phase. The important Interaction with a resonant laser beam : message is here that we can position the state vector anywhere on the Ω : Rabi frequency Bloch sphere, which is a way of saying that we can create arbitrary φ : phase of laser field superposition states. Laser beam switched on for duration τ : The same game works for sideband pulses. With a π /2 pulse, for example, we entangle the internal and the motional state! Since the motional state is θ : rotation angle shared by all ions, we can use the motional state as a kind of bus to mediate entanglement between different qubits in the ion chain. If we resonantly shine in light pulse at the carrier transition, the system evolves for a time tau with this Hamiltonian, where the coupling strength Omega depends on the sqroot of the intensity, and phi is the phase of the laser field with respect to the atomic polarization. 8
Coherent excitation: Rabi oscillations Coherent excitation on the sideband „Carrier“ pulses: Bloch sphere „Blue sideband“ pulses: coupled system representation ... Entanglement between internal and motional state ! D state population D state population 9
Experimental procedure Experimental procedure 1. Initialization in a pure quantum state: 1. Initialization in a pure quantum state: P 1/2 P 1/2 P 1/2 P 1/2 P 1/2 P 1/2 P 1/2 P 1/2 D 5/2 D 5/2 D 5/2 D 5/2 laser cooling,optical pumping D 5/2 D 5/2 D 5/2 D 5/2 Laser sideband cooling τ =1s 2. Quantum state manipulation on τ =1s 2. Quantum state manipulation on Quantum state Quantum state Doppler Fluorescence Doppler Fluorescence S 1/2 – D 5/2 qubit transition S 1/2 – D 5/2 transition Sideband Sideband manipulation manipulation cooling detection cooling detection cooling cooling 40 Ca + 40 Ca + 3. Quantum state measurement 3. Quantum state measurement S 1/2 S 1/2 S 1/2 S 1/2 S 1/2 S 1/2 S 1/2 S 1/2 by fluorescence detection by fluorescence detection One ion : Fluorescence histogram Multiple ions: 8 D 5/2 state S 1/2 state 7 6 Spatially resolved 50 experiments / s 50 experiments / s 5 detection with Repeat experiments 4 CCD camera: Repeat experiments 100-200 times 3 100-200 times 2 1 0 0 20 40 60 80 100 120 counts per 2 ms 10
Creation of Bell state … Pulse sequence: 1. Basic experimental techniques 2. Two-particle entanglement … … 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation … 6. Outlook 11
Creation of Bell states Creation of Bell states Generation of Bell states … … Pulse sequence: Pulse sequence: Ion 1: π /2 , blue sideband Ion 1: π /2 , blue sideband Ion 2: π , carrier … … … … … … 12
Creation of Bell states Analysis of Bell states … Pulse sequence: Ion 1: π /2 , blue sideband Fluorescence detection with Ion 2: π , carrier CCD camera: … … Ion 2: π , blue sideband Coherent superposition or incoherent mixture ? What is the relative phase of the superposition ? … Ψ + Ψ Ψ Ψ + + + Measurement of the density matrix: SS SD DS DD SS SD DS DD 13
Reconstruction of a density matrix Preparation and tomography of Bell states Fidelity: Representation of ρ as a sum of orthogonal observables A i : F = 0.91 F = 0.91 SS SS SD SD DS DS DD SS DD SS SD Entanglement SD DS DD DS DD ρ is completely detemined by the expectation values <A i > : of formation: E( ρ ρ exp ) = 0.79 ρ ρ Finally: maximum likelihood estimation (Hradil ’97, Banaszek ’99) Violation of SS Bell inequality: SS SD SD DS For a two-ion system : DS DD SS SD SS DS DD DD SD DS DD S( ρ ρ exp ) = 2.52(6) ρ ρ > 2 Joint measurements of all spin components C. Roos et al., Phys. Rev. Lett. 92 , 220402 (2004) 14
Different decoherence porperties 1. Basic experimental techniques sensitive to: 2. Two-particle entanglement laser frequency magnetic field exc. state lifetime 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook 15
Density matrix of W – state Four-ion W-states Fidelity: 85 % DDD DDD DDS DDS DSD DSD DDDD DSS DSS SDD SDD SDS DDDS SDS SSD SSD SSS SSS experimental result theoretical expectation SSSS SSSS 14.4.2005 DDDD 16
Five-ion W-states Detection of six individual ions 5µm 1 all ions in |S> 2 3 Ion detection 5 10 15 20 25 1 on a CCD camera ion 1 in |S> 2 3 (detection time:4ms) 5 10 15 20 25 1 ion 6 in |S> 2 3 5 10 15 20 25 1 ion 4 in |S> 2 3 5 10 15 20 25 1 ion 5 in |S> 2 3 5 10 15 20 25 1 DDDDD ions 1 and 5 in |S> 2 3 DDDDS 5 10 15 20 25 1 ions 1,2,3, and 5 in |S> 2 3 5 10 15 20 25 1 ions 1,3 and 4 in |S> 2 3 SSSSS 5 10 15 20 25 15.4.2005 6 5 4 3 2 1 SSSSS DDDDD 17
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