Towards Quantum Computation with Trapped Ions

Transcription

1 Towards Quantum Computation with Trapped Ions Ion traps for quantum computation Ion motion in linear traps Nonclassical states of motion, decoherence times Addressing individual ions Sideband cooling of the common motion Heating and cooling of an ion string Entanglement with trapped ions Cavity QED with a single ion R. Blatt, Universität Innsbruck, Institut für Experimentalphysik AG Quantenoptik und Spektroskopie

2 Why Quantum Computers? applications in physics and mathematics factorization of large numbers (P. Shor, 1994) can be achieved much faster on a quantum computer than with a classical computer factorization of number with L digits: classical computer: ~exp(l 1/3 ), quantum computer: ~ L 2 fast database search (L. Grover, 1997) search data base with N entries: classical computer: O(N), quantum computer: O(N 1/2 ) simulation of Schrödinger equations spectroscopy: quantum computer as atomic state synthesizer D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996) quantum physics with information guided eye

3 Computation is is a physical process input physical object set computation physical process shift output physical object read Classical Computer - bits, registers - gates - classical processes (switches), dissipative Quantum Computer - qubits, quantum registers - quantum gates - coherent processes - preparation and manipulation of entangled states

4 Quantum bits and quantum registers classical bit: physical object in state 0 or 1 register: bit rows quantum bit (qubit): quantum register: superposition of two orthogonal quantum states ψ = c c1 L 2-level atoms, 2 L quantum states 1 2 L states correspond to numbers 0,..., 2 L most general state of the register is the superposition ψ = c c c c (binary) = c c c 7 7 (decimal)

5 A universal quantum gate: CNOT gate C-NOT: Controlled-NOT gate (XOR) CN : ε 1 ε 2 ε 1 ε 1 ε 2 addition modulo together with single qubit rotations is UNIVERSAL control bit target bit problem: physical implementation with - coherence during computation - state measurement with 100% efficiency - realization of n-qubit gates

6 General scheme of of a quantum computation quantum F (x) processor F(000) F(001) F(010) F(011) F(100) F(101) F(110) F(111) INPUT quantum logic gates OUTPUT

7 The requirements for quantum computation 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 VI. VII. Ability to interconvert stationary and flying qubits Ability to faithfully transmit flying qubits between specified locations The seven commandments for QC!!

8 Which technology? Quantum Processor Cavity QED NMR superconductors quantum dots trapped ions

9 Quantum Computer: Implementation with Trapped Ions HARDWARE/OPERATION REQUIREMENTS TRAPPED IONS Q-bits long coherence times isolated in free space (frequency standards) Q-register row of Q-bits linear ion traps, ion strings Q-gate interaction between Q-bits, operations on individual Q- bits Coulomb repulsion spatial separation allows one to address individual ions Input state preparation Optical cooling, state preparation with laser pulses Computation coherent state manipulation internal, external excitations, entanglement Output state measurement (100% efficiency) Quantum jump technique

10 Ion storage generics ion confinement requires a focusing force in 3 dimensions: r binding force F ~ 2 r F = ee = e Φ Φ ~ r quadrupole potential Φ = Φ r ( x + y 2z 2 0 ) Paul trap: Φ0 = U 0 + V0 cosωt Penning trap: Φ0 = U 0 + axial magn. field equation of motion in a Paul trap: + ( a 2q cosωt) x = 0 a ~ U V 0, q ~ 0 ẋ Ω 4 MATHIEU EQUATION 2 frequencies of secular motion: superimposed is micromotion with: ω, ω, ω x y Ω z ω 2 ( a + 1 q ) Ω 2

11 Paul trap endcap electrode z lens fluorescence detection ring electrode x y endcap electrode cooling beam

12 Linear Ion Traps Paul mass filter Innsbruck Los Alamos München Boulder, Mainz, Aarhus Boulder

13 Innsbruck linear ion trap 0.5 mm 2 mm 5 mm 10 mm ωz 700 khz ω x, 1.2 y 2 MHz

14 Innsbruck linear ion trap (2000) 1.0 mm 6 mm ω z MHz ω x, y MHz

15 L ions in linear trap Quantum Computer with Trapped Ions J. I. Cirac, P. Zoller; Phys. Rev. Lett. 74, 4091 (1995) h quantum bits, quantum register - narrow optical transitions - groundstate Zeeman coherences h 2-qubit quantum gate h state vector of quantum computer Ψ = x x xl 1 x0 c 0 CM h state measurement with 100% efficiency, quantum jump technique h decoherence small (!?), heating small (!?) h quantum computation as a series of quantum gate operations (series of laser pulses)

16 String of of Ca + ions in in a linear Paul trap 70 µm IBK'97

17 Ion strings

18 Linear Ion Trap Linear ion trap I. Waki et al., Phys. Rev. Lett. 68, 2007 (1992) M.G. Raizen et al., Phys. Rev. A 45, 6493 (1992) - up to about 30 ions (string) - ions separated by about µm collective quantized motion - anisotropic oscillator: ν z << ν x, ν y - ion motion coupled by Coulomb repulsion - eigenmodes (nearly) independent of the number of ions center of mass mode breathing mode ν 1 =ν z ν = 2 3ν z

19 Common mode excitation ν 3ν 29 / 5ν

20 Common mode excitations Center of mass mode ν position breathing mode 3ν time

21 Level scheme of of Ca+ + P 3/2 854 nm qubit on narrow S - D quadrupole transition τ 1s P 1/2 393 nm 397 nm 866 nm D 5/2 D 3/2 729 nm S 1/2

22 Three required steps D 5/2 Absorption spectrum: Resolve secular motion 200 khz Additional cooling stage: Sideband cooling S 1/2 "red" "blue" sideband n-1 n n+1 "carrier" Addressing individual ions 729 nm electrooptic deflector

23 Spectroscopy with quantized fluorescence (quantum jumps) P monitor S D spectroscopy absorption and emission cause fluorescence steps (digital quantum jump signal) S histogram of absorption events absorptions D laser detuning

24 State detection by by quantized fluorescence 8 7 D state occupied S state occupied # of measurements e g detection efficiency: 99.85% counts per 9 ms

25 Spectroscopy of of the S 1/2 1/2 D 5/2 5/2 transition Zeeman structure in non-zero magnetic field: : 5/2 D 5/2-5/2-3/2-1/2 1/2 3/2 2-level-system S 1/2-1/2 1/2 + vibrational degrees of freedom:

26 Quantized Ion Motion e 2-level-atom harmonic trap... coupled system n 1,e n,e n +1,e... g Ω Γ ν {... n 1, g n,g n +1,g excitation: various resonances spectroscopy: carrier and sidebands n = 0 D 5/2 n = -1 n = 1 S 1/2 ω n = 0 1 2

27 Excitation spectrum of of a single ion

28 Narrow Carrier Resonance D-state excitation probability (90) Hz Spectral resolution: Laser Detuning at 729 nm (Hz)

29 Ramsey spectroscopy on on quadrupole transition Excitation probability Excitation probability Laser Detuning at 729 nm (khz) Zoom-in: Data and Fit π/2 π/2 9µs T=100µs µs t Contrast (%) Vary T: measure phase coherence on superposition states S + 1/ 2 D5/ 2 1/e coherence time: 2ms Laser linewidth: 75(10)Hz Delay T in ms

30 Laser Cooling of of Trapped Atoms e ATOM TRAP... Regimes: g Ω Γ ν { ν < Γ E D = ħγ weak confinement, Doppler cooling 2, n >> 1... n 1,e n 1, g n,e n, g n +1,e n +1, g... ν > Γ E strong confinement, sideband cooling ( 2 2 Γ 4ν 1 2) S = ħν + cooling heating n << 1

31 Absorption on on quadrupole transition (with motional sidebands) ν e g P 0 A A A B B R

32 Sideband cooling on on S 1/2 1/2 D 5/2 transition 5/2 Cooling cascade: P 3/2 D 5/2 S 1/2 n 1 n n +1 Ω 2 PD Γeff 2 2 ΓSP + 4 PD Γ SP Effective two-level system: I. Marzoli et al., Phys. Rev. A 49, 2771 (1994)

33 Sideband absorption spectrum 99.9 % ground state population after Doppler cooling n z = P D 0.4 P D 0.4 after sideband cooling Detuning δω (MHz) Detuning δω (MHz) Ch. Roos et al., Phys. Rev. Lett. 83, 4713 (1999)

34 Generation and manipulation of of Fock states p,1> d,0> d,1> d,2> d,0> d,1> d,2> d,3> s,0> s,1> s,2> π-pulse s,0> s,1> s,2> s,3>

35 Preparation of of Fock states 1 Ch. Roos et al., Phys. Rev. Lett. 83, 4713 (1999) Population of D state Time (µs) n r =0, n z =0 n r =0, n z =1

36 Cooling and heating cooling Ch. Roos et al., Phys. Rev. Lett. 83, 4713 (1999) <n z > cooling: 0.2 ms phonon Cooling Time (ms) <n> heating <n y > <n z > Delay Time (ms) heating: radial: axial: ms phonon ms phonon

37 Addressing of of individual ions in in a linear Paul trap Experimental setup: CCD images: laser beam steering with an electrooptic deflector Fiber output 729nm viewport Paul trap lens 20µm P 3/2 P 1/2 40Ca + 854nm 866nm telescope 397nm 729nm D 5/2 D 3/2 dichroic beamsplitter S 1/2 detection at 397nm CCD H.C. Nägerl et al., Phys. Rev. A 60, 145 (1999)

38 Intensity of of addressing beam at at ion position H. Rohde et al., J. Opt. B: Quant. Semiclass. Opt. 3, 34 (2001) Normalized Intensity waist: 3.7 (0.3) µm Intensity determined by measuring the light shift of the addressing beam Position (µm)

39 Light power at 729nm (arb. units) Addressing of of ions in in linear trap Measurement of light shift by addressing beam Beam diameter: 2 w 0 = 2.7 µm Focus position (µm)

40 Excitation spectrum of of two ions

41 Sideband cooling of of two ions RED sidebands BLUE sidebands ν z ν ν ν y 3 ν y ν z 3 y y 0.6 D-state population P 0 > 98% P 0 > 96% P 0 > 95% Detuning at 729 nm (MHz)

42 Two ion cooling and heating cooling of the rocking mode ω R = ω 2 axial ω 2 radial

43 Fluorescence of of two ions # of measurements ee ge or eg gg Count rate in 9 ms

44 Rabi oscillations of oftwo ions, one illuminated Ground state probability e,0 e, Time (µs) g,0 g,1 Carrier of CM motion

45 Rabi oscillations of of two ions, one illuminated 1 qf0354, Ground State Probability e,0 e, Time (µs) g,0 g,1 blue sideband of CM motion

46 D,0 S,0 Problem: D,1 S,1 AC Stark shifts off-resonant (carrier) excitation (spectator modes) Off-resonant carrier excitation S - D excitation probability Ω ω = 2π = 2π carrier z interaction time (µs) 1.8 MHz 1.09 MHz Solution: cooling of all spectator modes

47 Cooling and heating transitions (tailoring absorption) n 1 strong n n +1 weak n +1 n n n n 1 n avoiding carrier excitation reduces heating cooling efficient for a wider range of frequencies

48 n n Ground state cooling with quantum interference Detuning g g> n 1 n G. Morigi, J. Eschner, C. Keitel, Phys. Rev. Lett. 85, 4458 (2000) Absorption g,ω g,k g γ e> r,ω r,k r r> Absorption of cooling laser (a.u.) n n 1 1 transitions are enhanced by bright resonance transitions are suppressed by quantum interference -2 n + +1 n 1 n n n ( g r ) /ν

49 Simultaneous ground state cooling C.F. Roos et al., Phys. Rev. Lett. 85, 5547 (2000) S 1/2 to D 1/2 excitation probability Simultaneous ground state cooling of axial and radial motion axial: P(0)=73% radial: P(0)=58% 0 axial -3.2 MHz radial -1.6 MHz lower sidebands radial +1.6 MHz upper sidebands axial +3.2 MHz

50 Ground state cooling with quantum interference measured cooling limit vs. light shift 4 Best radial (1.6 MHz) cooling: 3.5 P(0) = 80% Cooling limit <n> ν radial = 1.6 MHz Best axial (3.2 MHz) cooling: P(0) = 90% from Doppler limit: Light shift δ (MHz) <n> = 17 (radial), <n> = 8 (axial) C.F. Roos et al., Phys. Rev. Lett. 85, 5547 (2000)

51 EIT cooling: Ideal scheme for ion strings Calculated EIT cooling for a string of 10 ions mean phonon number absorption (a.u.) (MHz) EIT ω /2π = 0.7MHz z vibr. mode frequency (MHz)

52 Relevant time scales τ trap τ π-pulse τ coherence τ heating τ D 1 µs 10 µs 100 µs 1 ms 10 ms 100 ms 1 s heating is NOT the dominant problem in the near future decoherence will allow for CNOT equivalent operations with fidelity above 0.5

53 Preparation of of Bell states in in a linear ion trap State preparation and excitation of common vibrational mode D,0> D,1> Excitation of 1st ion blue sideband, π/2-pulse: qm Superposition: S1, 0> + D1, 1> S2> ( ) S,0> S,1> Read-out of vibrational mode to state of 2nd ion Excitation of 2nd ion red sideband, π-pulse: Bell state: S1, S2> + D1, D2> 0> ( ) detection efficiency close to 100%, generalization to N ions possible D,0> S,0> D,1> S,1>

54 Cavity QED with a single ion optical cavity Paul trap qubits photons P 3/2 P 1/2 S 1/2 D 5/2 D 3/2 729 nm g g = 1s -1 = 110 k = 610 s 3-1 s 5-1 N n S C = 1.4(0.2) = 510-7

55 High finesse cavity: interface from atomic to to photonic qubit Paul trap optical cavity Experimental parameters: cavity decay time: κ ~ s -1 (opt. finesse) (κ ~ s -1 ) cavity ion coupling: g ~ s -1 atom spont. decay: γ ~ 1 s -1 qubits P 1/2 D 5/2 photons Experimental sequence: i) Ion in superposition state ii) Cavity tuned to qubit transition, vacuum cavity field induces spontaneous decay iii) Photonic qubit leaves cavity S 1/2 729 cav Applications: - single photon source - Interface: static qubits flying qubits - Cavity QED with continuously trapped atoms - Quantum feedback

56 Ion trap in in a cavity

57 Ion trap in in a cavity (detail) 2 cm 0.8 mm mirror cap compensation electrode

58 Single ion interacting with a standing wave Excitation probability Visibility: 96% Piezo offset voltage (V)

59 Carrier and sideband excitation in inthe standing wave

60 Towards Quantum Computation with Trapped Ca + Ions ion strings as qubits and quantum registers in linear traps Innsbruck Ca + experiments - spherical trap (ν z = 4.5 MHz, ν x,y = 2 MHz) - linear trap (ν z = 0.7 MHz, ν x,y = 2 MHz, ν z = 1.2 MHz, ν x,y = 4 MHz) spectroscopy of the S D transition: resolution 7 x sideband cooling - using coupled transitions, Raman cooling, EIT cooling, sympathetíc cooling relevant time scales - coherence time: several ms - heating times: > 100 ms addressing of individual ions Next: preparation of Bell states, Bell measurements realization of the Cirac-Zoller gate CNOT gate operations currently possible CQED with trapped ions, interface to photonic qubits

61 Institut für Experimentalphysik F. Schmidt-Kaler S. Gulde J. Eschner G. Lancaster D. Leibfried A. Mundt C. Becher A. Kreuter C. Roos H. Häffner P. Barton P. Spitzer H. Rohde D. Rotter M. Riebe R. B. AG Quantenoptik und Spektroskopie FWF SFB F015: Control and Measurement of Coherent Quantum Systems TMR-networks: Quantum Structures, Quantum Information IHP-network: QUEST IST-network: QUBITS Austrian Industry: Institute for Quantum Information Ges.m.b.H.