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Quantum Computation with Neutral Atoms Marianna Safronova Department of Physics and Astronomy Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. 1 Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. Why Quantum Computers? Computer technology is making devices smaller and smaller… …reaching a point where classical physics is no longer a suitable model for the laws of physics. 2 Bits & Qubits Fundamental building blocks of classical computers: Fundamental building blocks of quantum computers: Quantum bits or QUBITS BITS STATE: Basis states: 0 and 1 Definitely Superposition: 0 or 1 ψ =α 0 +β 1 Bits & Qubits Fundamental building blocks of classical computers: BITS STATE: Fundamental building blocks of quantum computers: Quantum bits or QUBITS Basis states: 0 and 1 Definitely 0 or 1 3 Qubit: any suitable two-level quantum system Bits & Qubits: Superposition primary differences ψ =α 0 +β 1 4 Bits & Qubits: primary differences Measurement Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. Qubit: Quantum calculation: number of parallel processes due to superposition FR QO Bits & Qubits: Superposition Measurement primary differences ψ =α 0 +β 1 Classical bit: we can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. Qubit: we cannot just measure α and β and thus determine its state! We get either 0 or 1 with corresponding probabilities |α|2 and |β|2. 2 2 α + β =1 The measurement changes the state of the qubit! 5 Hilbert space is a big place! - Carlton Caves Multiple qubits Hilbert space is a big place! - Carlton Caves Multiple qubits Two bits with states 0 and 1 form four definite states 00, 01, 10, and 11. Two qubits: can be in superposition of four computational basis set states. ψ = α 00 + β 01 + γ 10 + δ 11 2 qubits 4 amplitudes 3 qubits 8 amplitudes 10 qubits 1024 amplitudes 20 qubits 1 048 576 amplitudes 30 qubits 1 073 741 824 amplitudes 500 qubits More amplitudes than our estimate of number of atoms in the Universe!!! 6 Entanglement Results of the measurement ψ = 00 + 11 2 ψ ≠α ⊗β First qubit Second qubit 0 0 1 1 Entangled states Quantum logic gates 7 Logic gates Quantum NOT gate (X gate) Classical NOT gate NOT A A A NOT A 0 1 1 0 α 0 +β 1 α 1 +β 0 X Matrix form representation 0 1 X = 1 0 α β X = β α The only non-trivial single bit gate More single qubit gates Any unitary matrix U will produce a quantum gate! 1 0 Z = 0 −1 α 0 +β 1 Hadamard gate: α 0 +β 1 H α 0 −β 1 Z H= α 1 1 1 2 1 −1 0 +1 2 +β 0 −1 2 8 Two-qubit gates Quantum CNOT gate A B A B' AB AB' 00 00 01 01 10 11 11 10 WE NEED TO BE ABLE TO MAKE ONLY ONE TWO-QUBIT GATE! Back to the real world: What do we need to build a quantum computer? Qubits which retain their properties. Scalable array of qubits. Initialization: ability to prepare one certain state repeatedly on demand. Need continuous supply of 0 . Universal set of quantum gates. A system in which qubits can be made to evolve as desired. Long relevant decoherence times. Ability to efficiently read out the result. 9 Real world strategy “…If X is very hard it can be substituted with more of Y. Of course, in many cases both X and Y are beyond the present experimental state of the art …” David P. DiVincenzo The physical implementation of quantum computation. Experimental proposals Liquid state NMR Trapped ions Cavity QED Trapped atoms Solid state schemes And other ones … 10 1. A scalable physical system with well characterized qubits: memory (a) Internal atomic state qubits: ground hyperfine states of neutral trapped atoms well characterized Very long lived! MF=-2,-1,0,1,2 F=2 5s1/2 1 87Rb: 6.8 GHz F=1 0 Nuclear spin I=3/2 MF=-1,0,1 1. A scalable physical system with well characterized qubits: memory (b) Motional qubits : quantized levels in the trapping potential also well characterized http://www.colorado.edu/physics/2000/index.pl 11 1. A scalable physical system with well characterized qubits: memory (a)Internal atomic state qubits (b) Motional qubits Advantages: very long decoherence times! Internal states are well understood: atomic spectroscopy & atomic clocks. 1. A scalable physical system with well characterized qubits Optical lattices: loading of one atom per site may be achieved using Mott insulator transition. Scalability: the properties of optical lattice system do not change in the principal way when the size of the system is increased. Designer lattices may be created (for example with every third site loaded). Advantages: inherent scalability and parallelism. Potential problems: individual addressing. 12 2: Initialization Internal state preparation: putting atoms in the ground hyperfine state Very well understood (optical pumping technique is in use since 1950) Very reliable (>0.9999 population may be achieved) Motional states may be cooled to motional ground states (>95%) Loading with one atom per site: Mott insulator transition and other schemes. Zero’s may be supplied during the computation (providing individual or array addressing). 3: A universal set of quantum gates Hadamard gate: CNOT A A B B' AB AB' 00 00 01 01 10 11 11 10 A A' H A 0 1 A' 0 +1 2 0 −1 2 H = (X + Z)/ 2 π/8 gate: T 0 1 T = iπ / 4 0 e Phase gate S: 1 0 S= 0 i S =T2 13 3: A universal set of quantum gates 1. Single-qubit rotations: well understood and had been carried out in atomic spectroscopy since 1940’s. 2. Two-qubit gates: none currently implemented (conditional logic was demonstrated) Proposed interactions for two-qubit gates: (a) Electric-dipole interactions between atoms (b) Ground-state elastic collisions (c) Magnetic dipole interactions Only one gate proposal does not involve moving atoms (Rydberg gate). Advantages: possible parallel operations Disadvantages: decoherence issues during gate operations Two-qubit quantum gates (a) Electric-dipole interactions between atoms Brennen et al. PRL 82, 1060 (1999), PRA 61, 062309 (2000), Pairs of atoms are brought to occupy the same site in far-off-resonance optical lattice by varying polarization of the trapping laser. Two “types” of atoms: trapped in σ+ and σ- polarized wells. Near-resonant electric-dipole is induced by auxiliary laser (depending on the atomic state). Brennen, Deutch, and Willaims PRA 65, 022313 (2002) Deterministic entanglement of pairs of atoms trapped in optical lattice is achieved by coupling to excited state molecular hyperfine potentials. 14 Two-qubit quantum gates (a) Electric-dipole interactions between atoms … cont. Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Gate operations are mediated by excitation of Rydberg states (b) Ground-state elastic collisions Calarco et al. Phys. Rev. A 61, 022304 (2000) Cold collisions between atoms conditional on internal states. Cold collisions between atoms conditional on motional-state tunneling. (c) Magnetic-dipole interactions between pairs of atoms Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Why Rydberg gate? 15 Rydberg gate scheme Gate operations are mediated by excitation of Rydberg states Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Do not need to move atoms! FAST! Local blockade of Rydberg excitations Excitations to Rydberg states are suppressed due to a dipole-dipole interaction or van der Waals interaction http://www.physics.uconn.edu/~rcote/ 16 Rydberg gate scheme FAST! Rb 40p R Apply a series of laser pulses to realize the following logic gate: ∆ 1 0 00 10 01 11 5s 1 2 → 00 → 01 → 10 → − 11 Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000) Rb Rydberg gate scheme 5s 40p R π [1] ∆ 00 01 10 11 1 0 1 → → → → 00 01 R0 R1 2π [ 2] → 00 → − 01 → R0 → R1 π [1] → 00 → − 01 → − 10 → − 11 2 17 Decoherence One of the decoherence sources: motional heating. Results from atom “seeing” different lattice in ground and Rydberg states. Solution: choose the lattice photon frequency ω to match frequency-dependent polarizability α(ω) of the ground and Rydberg states. Error correction: possible but error rate has to be really small (< 10-4 ). Other decoherence sources Photoionization Spontaneous emission Transitions induced by black-body radiation Laser beam intensity stability Pulse timing stability Individual addressing accuracy 18 4. Long relevant decoherence times Memory: long-lived states. 5s1/2 F=2 1 F=1 0 6.8 GHz Fundamental decoherence mechanism for optically trapped qubits: photon scattering. Decoherence during gate operations: a serious issue. 5: Reading out a result “Quantum jump” method via cycling transitions. Advantages: standard atomic physics technique, well understood and reliable. Quantum computation with NEUTRAL ATOMS: ADVANTAGES Scalability Possible massive parallelism due to lattice geometry Long decoherence times (weak coupling to the environment) Availability of the controlled interactions Well-developed experimental techniques for initialization, state manipulation, and readout Accurate theoretical description of the system is possible. 19 Quantum computation with NEUTRAL ATOMS: PROBLEMS Decoherence during the gate operations (various sources) Reliable lattice loading and individual addressing QC architecture for lattice geometry: Error-correcting codes and fault-tolerant computation, how to run algorithms on neutral atom quantum computer. 20 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 The requirements for quantum information processing D. P. DiVincenzo, Quant. Inf. Comp. 1 (Special), 1 (2001) I. Scalable physical system, well characterized qubits II. Ability to initialize the state of the qubits III. Long relevant coherence times, much longer than gate operation time IV. “Universal” set of quantum gates V. Qubit-specific measurement capability 1 Experimental Setup P1/2 D5/2 „quantum bit“ S1/2 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 S1/2 – D5/2 : quadrupole transition τ = 7 ns P1/2 τ=1s D5/2 397 nm 729 nm S1/2 Ca+: Important energy levels τ = 7 ns P1/2 D5/2 S1/2 „qubit“ „quoctet“ (sp?) 3 Qubits with trapped ions Encoding of quantum information requires long-lived atomic states: optical transitions microwave transitions Ca+, Sr+, Ba+, Ra+, Yb+, Hg+ etc. 9Be+, 25Mg+, 43Ca+, 87Sr+, 137Ba+, 111Cd+, 171Yb+ P3/2 P1/2 D5/2 qubit S1/2 qubit S1/2 String of Ca+ ions in Paul trap row of qubits in a linear Paul trap forms a quantum register 4 String of Ca+ ions in linear Paul trap row of qubits in a linear Paul trap forms a quantum register ω z ≈ 0.7 − 2 MHz ω x , y ≈ 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 ω z ≈ 0.7 − 2 MHz ω x , y ≈ 1.5 − 4 MHz 50 µm 5 Addressing of individual ions 0.8 0.7 Paul trap 0.6 Excitation coherent manipulation of qubits 0.5 0.4 electrooptic deflector 0.3 0.2 0.1 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Deflector Voltage (V) dichroic beamsplitter inter ion distance: ~ 4 µm addressing waist: ~ 2.5 µm < 0.1% intensity on neighbouring ions Fluorescence detection CCD 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. 6 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 … 7 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 8 Coherent manipulation 2-level-atom harmonic trap joint energy levels … Interaction with a resonant laser beam : Ω : Rabi frequency φ : phase of laser field Laser beam switched on for duration τ : θ : 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 τ 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 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 π/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. 9 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 10 Experimental procedure P1/2 1. Initialization in a pure quantum state: laser cooling,optical pumping D5/2 τ =1s Quantum state Doppler Fluorescence Sideband manipulation cooling detection cooling 40 Ca+ SS1/2 1/2 2. Quantum state manipulation on S1/2 – D5/2 qubit transition 3. Quantum state measurement by fluorescence detection One ion : Fluorescence histogram 8 7 D5/2 state S1/2 state 6 50 experiments / s 5 4 Repeat experiments 100-200 times 3 2 1 0 0 20 40 60 80 100 120 counts per 2 ms Experimental procedure P1/2 D5/2 τ =1s Quantum state Doppler Fluorescence Sideband manipulation cooling detection cooling 40 Ca+ SS1/2 1/2 1. Initialization in a pure quantum state: Laser sideband cooling 2. Quantum state manipulation on S1/2 – D5/2 transition 3. Quantum state measurement by fluorescence detection Multiple ions: Spatially resolved detection with CCD camera: 50 experiments / s Repeat experiments 100-200 times 11 1. Basic experimental techniques 2. Two-particle entanglement 3. Multi-particle entanglement 4. Implementation of a CNOT gate 5. Teleportation 6. Outlook … Pulse sequence: … … … Creation of Bell state 12 … Creation ofofBell Generation Bellstates states Pulse sequence: … … … Ion 1: π/2 , blue sideband … Creation of Bell states Pulse sequence: Ion 1: π/2 , blue sideband carrier … , … … Ion 2: π 13 … Creation of Bell states Pulse sequence: Ion 2: π , Ion 2: π , blue sideband … carrier … … Ion 1: π/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 ? Ψ+ Measurement of the density matrix: SS SD DS DD SDSS DDDS 14 Reconstruction of a density matrix Representation of ρ as a sum of orthogonal observables Ai : ρ is completely detemined by the expectation values <Ai> : 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 SD DS DD DD DS SD SS SS SD DS DD DD DS SD SS Entanglement of formation: E(ρexp) = 0.79 SS SD DS DD DD DS SD SS SS SD DS DD Violation of Bell inequality: DD DS SD SS S(ρexp) = 2.52(6) > 2 C. Roos et al., Phys. Rev. Lett. 92, 220402 (2004) 15 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 16 Generation of W-states … Pulse sequence: Ion 2,3: π , carrier 1. 2. 3. Ion 2: θ2 , blue sideband … … Ion 1: θ1 , blue sideband … Ion 3: θ3 , blue sideband Density matrix of W – state Fidelity: 85 % experimental result theoretical expectation 17 Four-ion W-states DDDD DDDS SSSS SSSS 14.4.2005 DDDD Five-ion W-states DDDDD DDDDS SSSSS 15.4.2005 SSSSS DDDDD 18 Detection of six individual ions 5µm 1 all ions in |S> 2 3 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 1 ion 1 in |S> 2 3 Ion detection on a CCD camera (detection time:4ms) 1 ion 6 in |S> 2 3 1 ion 4 in |S> 2 3 1 ion 5 in |S> 2 3 1 ions 1 and 5 in |S> 2 3 1 ions 1,2,3, and 5 in |S> 2 3 1 ions 1,3 and 4 in |S> 2 3 6 5 4 3 2 1 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. 19