Quantum information processing with trapped ions Courtesy of Timo Koerber Institut für Experimentalphysik Universität Innsbruck 1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook Lectures 15-16 The requirements for quantum information processing D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001) I. Scalable physical system, well characterized qubits II. III. IV. Ability to initialize the state of the qubits Long relevant coherence times, much longer than gate operation time Universal set of quantum gates V. Qubit-specific measurement capability 1
Experimental Setup P 1/2 D 5/2 quantum bit S 1/2 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 other sublevels give us additional possibilities for doing coherent operations. 2
Ca+: Important energy levels = 7 ns S 1/2 D 5/2 : quadrupole transition P 1/2 = 1 s D 5/2 397 nm 729 nm S 1/2 Ca+: Important energy levels = 7 ns P 1/2 D 5/2 S 1/2 qubit quoctet (sp?) 3
Qubits with trapped ions Encoding of quantum information requires long-lived atomic states: optical transitions Ca +, Sr +, Ba +, Ra +, Yb +, Hg + etc. microwave transitions 9 Be +, 25 Mg +, 43 Ca +, 87 Sr +, 137 Ba +, 111 Cd +, 171 Yb + P 1/2 D 5/2 P 3/2 S 1/2 qubit S 1/2 qubit Linear RF Paul trap Positive ion RF electrode High dc potential control electrode Low dc voltage control electrode Slides from :R. Ozeri Drive freq ~ 100-150 MHz RF amp ~ 200-400 V Secular freq Radial ~ 15 MHz Axial ~ 4 MHz 4
Multi-zone ion trap rf filter board view along axis: rf control segmented linear trapping region control rf Gold on alumina construction RF quadrupole realized in two layers Six trapping zones Both loading and experimental zones One narrow separation zone Closest electrode ~140 m from ion Ion transport 100 m separation zone 6-zone alumina/gold trap (Murray Barrett, Tobias Schaetz et al.) 200 m Ions can be moved between traps. Electrode potentials varied with time Ions can be separated efficiently in sep. zone Small electrode s potential raised Motion (relatively) fast Shuttling: several 10 s Separating: few 100 s 5
Innsbruck segmented trap (2004) String of Ca+ ions in Paul trap row of qubits in a linear Paul trap forms a quantum register 6
String of Ca+ ions in linear Paul trap row of qubits in a linear Paul trap forms a quantum register z x, y 0.7 2 MHz 1.5 4 MHz 50 µm String of Ca+ ions in linear Paul trap row of qubits in a linear Paul trap forms a quantum register z x, y 0.7 2 MHz 1.5 4 MHz 50 µm 7
Addressing of individual ions 0.8 electrooptic deflector coherent manipulation of qubits Paul trap Excitation 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: ~ 2.5 µm < 0.1% intensity on neighbouring ions Ion addressing The ions can be addressed individually on the qubit transition with an EO deflector which can quickly move the focus of the 729 light from one ion to another, using the same optical path as the fluorescence detection via the CCD camera. How well the addressing works is shown on the previous slide: The graph shows the excitation of the indiviual ions as the deflector is scanned across the crystal. 8
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 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. External degree of freedom: ion motion harmonic trap 9
External degree of freedom: ion motion 2-level-atom harmonic trap joint energy levels External degree of freedom: ion motion 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 10
Coherent manipulation 2-level-atom harmonic trap joint energy levels Interaction with a resonant laser beam : : Rabi frequency : phase of laser field Laser beam switched on for duration : : rotation angle 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. 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 with this Hamiltonian, where the coupling strength depends on the square root of the intensity, and 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 message is here that we can position the state vector anywhere on the Bloch sphere, which is a way of saying that we can create arbitrary superposition states. The same game works for sideband pulses. With a /2 pulse, for example, we entangle the internal and the motional state! Since the motional state is 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. 11
Coherent excitation: Rabi oscillations Carrier pulses: Bloch sphere representation D state population Coherent excitation on the sideband Blue sideband pulses: coupled system Entanglement between internal and motional state! D state population 12
Experimental procedure P 1/2 D 5/2 =1s Doppler Quantum state Fluorescence cooling manipulation detection Sideband cooling S 1/2 40 Ca + S 1/2 One ion : Fluorescence histogram 8 D 7 5/2 state 6 5 4 3 2 1 S 1/2 state 0 0 20 40 60 80 100 120 counts per 2 ms 1. Initialization in a pure quantum state: laser cooling,optical pumping 2. Quantum state manipulation on S 1/2 D 5/2 qubit transition 3. Quantum state measurement by fluorescence detection 50 experiments / s Repeat experiments 100-200 times Experimental procedure P 1/2 D 5/2 =1s Doppler Quantum state Fluorescence cooling manipulation detection Sideband cooling S 1/2 40 Ca + S 1/2 1. Initialization in a pure quantum state: Laser sideband cooling 2. Quantum state manipulation on S 1/2 D 5/2 transition 3. Quantum state measurement by fluorescence detection Multiple ions: Spatially resolved detection with CCD camera: 50 experiments / s Repeat experiments 100-200 times 13
1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook Creation of Bell state Pulse sequence: 14
Generation Creation of of Bell Bell states Pulse sequence: Ion 1: /2, blue sideband Creation of Bell states Pulse sequence: Ion 1: /2, blue sideband Ion 2:, carrier 15
Creation of Bell states Pulse sequence: Ion 1: /2, blue sideband Ion 2:, carrier Ion 2:, blue sideband Analysis of Bell states Fluorescence detection with CCD camera: Coherent superposition or incoherent mixture? What is the relative phase of the superposition? Measurement of the density matrix: SS SDDS DD SS DDDS SD 16
Reconstruction of a density matrix Representation of as a sum of orthogonal observables A i : is completely detemined by the expectation values <A i > : Finally: maximum likelihood estimation (Hradil 97, Banaszek 99) For a two-ion system : Joint measurements of all spin components Preparation and tomography of Bell states Fidelity: F = 0.91 SS SDDS SS SDDS DD SS SD DD DS DD SS SD DD DS Entanglement of formation: E( exp ) = 0.79 SS SDDS SS SDDS Violation of Bell inequality: DD SS SD DD DS DD SS SD DD DS S( exp ) = 2.52(6) > 2 C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004) 17
Different decoherence porperties sensitive to: laser frequency magnetic field exc. state lifetime 1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook 18
Generation of W-states Pulse sequence: Ion 2,3:, carrier Ion 1: 1, blue sideband 1. 2. 3. Ion 2: 2, blue sideband Ion 3: 3, blue sideband Density matrix of W state Fidelity: 85 % experimental result theoretical expectation 19
Four-ion W-states DDDD DDDS 14.4.2005 SSSS DDDD SSSS Five-ion W-states DDDDD DDDDS 15.4.2005 SSSSS DDDDD SSSSS 20
Detection of six individual ions 5µm 1 2 3 1 2 3 1 2 3 5 10 15 20 25 5 10 15 20 25 all ions in S> ion 1 in S> ion 6 in S> Ion detection on a CCD camera (detection time:4ms) 5 10 15 20 25 1 2 3 ion 4 in S> 5 10 15 20 25 1 2 3 ion 5 in S> 5 10 15 20 25 1 2 3 ions 1 and 5 in S> 5 10 15 20 25 1 2 3 1 2 3 5 10 15 20 25 5 10 15 20 25 6 5 4 3 2 1 ions 1,2,3, and 5 in S> ions 1,3 and 4 in S> Six-ion W-state F=73% preliminary result Is there 6-particle entanglement present? 6-particle W-state can be distilled from the state (O. Gühne) 6-particle entanglement present, unresolved issues with error bars 22.4.2005 729 settings, measurement time >30 min. 21
1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook control target Cirac-Zoller two-ion controlled-not operation...allows the realization of a universal quantum computer! control target other gate proposals include: Cirac & Zoller Mølmer & Sørensen, Milburn Jonathan & Plenio & Knight Geometric phases Leibfried & Wineland control target 22
Cirac-Zoller two-ion controlled-not operation 1 2 ion 1 motion ion 2 SWAP control qubit target qubit First we swap the quantum states of the control qubit and the motional qubit Cirac-Zoller two-ion controlled-not operation 1 2 ion 1 control qubit motion ion 2 target qubit Next we perform a CNOT gate between the motional qubit and ion 2 and 23
Cirac - Zoller two-ion controlled-not operation 1 2 Finally we reverse the SWAP operation, we have used the motional qubit as a quantum messenger between the two ions. ion 1 motion ion 2 SWAP -1 control qubit target qubit F. Schmidt-Kaler et al., Nature 422, 408 (2003) Cirac - Zoller two-ion controlled-not operation ion 1 motion SWAP SWAP -1 control qubit ion 2 target qubit pulse sequence: Ion 1 laser frequency pulse length optical phase Ion 2 Phase gate 24
Experimental fidelity of Cirac-Zoller CNOT operation F. Schmidt-Kaler et al., Nature 422, 408 (2003) input output Protecting qubits (from being measured) 25
Selective read-out of an atom in a W-state Threshold ion #1 in D> ion #1 in S> Tomography after the measurement result is available! 1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook 26
Quantum state teleportation Phys. Rev. Lett. 70, 1895 (1993) Is it possible to transfer an unknown quantum state from Alice to Bob by classical communication? Yes, No : if Alice Infinite and amount Bob share of information a pair of entangled needed to particles specify! Measurement on yields just one bit of information unknown state EPR pair Alice A classical communication Two bits Bob B Teleportation protocol Ion 1 Alice classical communication Ion 2 Ion 3 Bell state initialize #1, #2, #3 Bob CNOT -- Bell basis Selective read out conditional rotations recovered on ion #3 27
Teleportation protocol, details Ion 1 B B B B C C P conditional rotations using electronic logic, triggered by PM signal Ion 2 C B B B C P U P Ion 3 B P U C P U C C C B C blue sideband pulses carrier pulses spin echo sequence full sequence: 26 pulses + 2 measurements 28
Teleportation procedure, analysis Initial Input state Output state Final TP U U -1 Fidelities Quantum teleportation with atoms: results 83 % class.: 67 % no cond. op. 50 % 29
How to build a large-scale quantum computer with trapped ions Linear crystal of 20 confined atomic 171 Yb+ ions laser cooled to be nearly at rest C. Monroe, and J. Kim Science 2013;339:1164-1169 30
Harnessing ion-trap qubits David P. DiVincenzo Science 2011;334:50-51 Schematic of the sequence of operations implemented in a single processing region for building up a computation in the architecture Jonathan P. Home et al. Science 2009;325:1227-1230 31
Separating memory and processor zones Scientific American (August 2008), 299, 64-71 Ion trap structure for the shuttling of ions through a junction C. Monroe, and J. Kim Science 2013;339:1164-1169 32
Concept of a quantum CCD trap Image credit: National Institute of Standards and Technology C. Monroe, and J. Kim Science 2013;339:1164-1169 Version 2: Photonics coupling of trapped ions qubits Energy levels of trapped ion excited with a fast laser pulse (blue upward arrow) that produces single photon whose color, represented by the state or, is entangled with the resultant qubit state. C. Monroe, and J. Kim Science 2013;339:1164-1169 33
Version 2: Photonics coupling of trapped ions qubits Two "communication qubit" ions, immersed in separate crystals of other ions, each produce single photons when driven by laser pulses (blue). With some probability, the photons arrive at the 50/50 beamsplitter and then interfere. If the photons are indistinguishable (in polarization and color), then they always leave the beamsplitter along the same path. The simultaneous detection of photons at the two output detectors means that the photons were different colors, this coincidence detection heralds the entanglement of the trapped ion qubits. Scientific American (2008), 299, 64-71 34
Scientific American (2008), 299, 64-71 Elementary logic unit (ELU) Phys. Rev. A 89, 022317 (2013) 35
Modular distributed quantum computer Several elementary logic units (ELU)s are connected through a photonic network by using an optical crossconnect switch, inline fiber beamsplitters, and a photon-counting imager. C. Monroe, and J. Kim Science 2013;339:1164-1169 Application to quantum communication Trapped ion quantum repeater node made up of communication qubit ions (such as Ba + ) and memory qubit ions (such as Yb + ), with two optical interfaces per node. Multiple communication qubits are used per optical interface to inject photons into the optical channel, while the results for successful entanglement generation at the detectors are reported back to this node. Only qubits corresponding to successful events will be transported to the memory qubit region for use in quantum repeater protocol. C. Monroe, and J. Kim Science 2013;339:1164-1169 36
Long-distance quantum communication A chain of quantum repeater nodes can distribute quantum entanglement over macroscopic distances. The photons generated by the ions must be converted to telecommunication wavelengths for long-distance transport, which can be achieved by nonlinear optical processes. C. Monroe, and J. Kim Science 2013;339:1164-1169 37