P 3/2 P 1/2 F = -1.5 F S 1/2. n=3. n=3. n=0. optical dipole force is state dependent. n=0

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1 (two-qubit gate): tools: optical dipole force P 3/2 P 1/2 F = -1.5 F n=3 n=3 n=0 S 1/2 n=0 optical dipole force is state dependent

2 tools: optical dipole force (e.g two qubits) ω 2 k1 d ω 1 optical dipole force beams k2 Δk trap axis walking standing wave ω 2 ω 1 =δ F F = 1.5 F Δk d = 2 πm ω = ω 2 1

3 universal geometric phase gate Gate (round trip) time, via detuning δ τ g = 2π/δ exp(i π / 2 ) Phase (area), φ = π/2 via laser intensity G = e iπ / e iπ / exp(i π / 2 ) Gives CNOT or π phase gate with add. single bit operations

4 Berry s / geometrical phase dynamic phase τ closed trip = 2π/δ p geometric phase displacement operator (drive at ω stretch δ) coherent state x stretch mode ( stretch ) in corotating frame n=0 n=1 n=2 coherent state n=3

5 - motivation for quantum-approach - ion trap approach: - idea -trap - qubit - tools - universal quantum computer - analog quantum computer/simulator - simulating a phase transition of a quantum magnet - what else? - summary and schedule

realized for trits with photons by Mattle, Weinfurter, Kwiat and Zeilinger (PRL 76, 4656 (1996)) only two Bell

6 Dense Coding Alice General scheme: entangled state BOB one of four local operations on one qubit sending one qubit receiving two bits of information Theoretically proposed by Bennett and Wiesner (PRL 69, 2881 (1992)) Experimentally realized for trits with photons by Mattle, Weinfurter, Kwiat and Zeilinger (PRL 76, 4656 (1996)) only two Bell states identifiable, other two are indistinguishable ( trit instead of bit) non deterministic (30 photon pairs for one trit)

7 Dense Coding: protocol with atomic qubits produce Alice s entangled pair π/2-pulse and phase gate on both qubits rotate Alice s qubit only σ x, σ y, σ z or no-rotation (identity) on Alice s qubit identity on Bob s qubit Bob s Bell measurement phase gate and π/2-pulse on both qubits Bob s detection separate and read out qubits individually

8 Dense Coding: results Experimental result: I σ x σ y σ z Average fidelity: 85% (perfect photon expt. 75%)

9 I shouldn t do that 2 ions in rf-trap Cavity-QED quantized oscillator = mode of motion / phonon quantized oscillator = mode of electromagnetic field / photon future? simultaneous quantum control of (1) internal states, (2) field, (3) motion

10 Where we (NIST) are (DiVincenzo requirements) I. A scalable physical system with well characterized qubits multiplexed trap architecture, hyperfine ground states II. The ability to initialize the state of the qubits to a simple fiducial state optical pumping, ground-state cooling (99.9%) 0 III. Long relevant decoherence times, much longer than the gate time T dec =1 ms (>hours), T gate =10 μs (500 ns), heating irrelevant IV. A universal set of quantum gates (single qubit rot., two qubit gate) co-carrier rotations, geometric-phase gate, heating tolerable V. A qubit-specific read out capability electron shelving method, 99% readout efficiency (100%) Individual requirements met experimentally!

11 some things take a while

12 Zuse s Z1 of replica

13 quantum computer / simulator universal QC: e.g. testing reliability of encoding implementing Shor s algorithm to beat classical computers: 1000 logical qubits necessary 100 ancillae per logical qubit 10 5 qubits for fault tollerance (error correction) can be used for universal Quantum Simulations analog QS: (one purpose QC) choose system, that can be controlled and manipulated that its evolution is described by the same Hamiltonian as the system to be simulated. less restrictive demands, e.g qubits no fault tolerance PRO LEAGUE AMATEURS

14 - motivation for quantum-approach - ion trap approach: - idea -trap - qubit - tools - universal quantum computer - analog quantum computer/simulator - simulating a phase transition of a quantum magnet - what else? - summary and schedule

15 simulations and results: Munich 1972

18 - motivation for quantum-approach - ion trap approach: - idea -trap - qubit - tools - universal quantum computer - analog quantum computer/simulator - simulating a phase transition of a quantum magnet - what else? - summary and schedule

19 simulating: quantum-spin- models quantum-spin Hamiltonians describe many solid state systems: magnets, high-tc superconductors, quantum-hall ferromagnets, ferroelectrics... H= J σ σ + B ij, σ z z x x i j i i quantum Ising model H= J σ σ + J σ σ XY model x x x y y y i j i j ij, ij, H= J σ σ + J σ σ + J σ σ x x x y y y z z z i j i j i j ij, ij, ij, Heisenberg model

20 e.g. quantum magnetism / Porras and Cirac 2004 (do not forget C.Wunderlich) e.g. quantum-ising model: H= J σ σ + B ij, σ z z x x i j i i eff. magnetic field (global qubit-rotation) effective coupling B x

21 e.g. quantum magnetism / Porras and Cirac 2004 e.g. quantum-ising model: H= J σ σ + B ij, σ z z x x i j i i eff. spin-spin Interaction J (conditional optical dipole force) eff. magnetic field (global qubit-rotation) P 3/2 P 1/2 F = -1.5 F S 1/2 B x

22 e.g. quantum magnetism / Porras and Cirac 2004 e.g. quantum-ising model: H= J σ σ + B ij, σ z z x x i j i i eff. spin-spin Interaction J (conditional optical dipole force) eff. magnetic field (global qubit-rotation) all parameters to be chosen individually: B x (e.g. amplitude, range, anti- or ferromagnetic phase )

23 quantum-antiferromagnetism control over sign and range of J i,j RADIAL MODES: ANTIFERROMAGNETIC INTERACTION short range ( ~ 1 / r 3 ) y x z pushing force in RADIAL direction reduced Coulomb energy -new ground state- pushing force in RADIAL direction

24 robust effect: quantum phase transition ground state adiabatic evolution ground state J z z σiσ j ij, J = B x B x x σi i not thermal fluctuations responsible (only for T>0) but quantum-fluctuations (also for T=0) degenerate ground state: see also: D.Bruss et al. PRA (2005) ( ) entanglement

25 summary quantum computation for wimps 1. no stroboscopic quantum gates: gates act on all ion(s) at the same time 2. study robust effects (quantum phase transitions) fault tolerance not necessary 3. read out of global fluorescence

26 is there a future? towards (huge) quantum simulations in (small) ion traps

27 simulating 2D-quantum Systems solid physicists dream of 20 x 20 arrays of individually addressable qubits to attack severe problems

28 planar, but 1D with 5-wires inspired by NIST (J.Chiaverini LANL) Field lines:

29 NIST 1D planar trap Gold on alumina Gold on glass Array is 1 mm long, consists of 5 wires, with a total width of 200 um.

30 simulating 2D -quantum systems starting with small steps again: 2 x 2 cross section top view towards 20 x x 10

31 dreaming in the right direction (dimension)?

32 Max-Planck Institute for Quantum Optics Garching Deutsche Forschungsgemeinschaft started in October 2004 (after 2a DJW) Hector Schmitz (PhD) Axel Friedenauer (PhD) collaborating with D.Porras and I. Cirac Lutz Petersen (GS)

Quantum Computation with Neutral Atoms Lectures 14-15

Quantum Computation with Neutral Atoms Lectures 14-15 15 Marianna Safronova Department of Physics and Astronomy Back to the real world: What do we need to build a quantum computer? Qubits which retain